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On application of optimal control to intelligent manufacturing

Sviluppo di un controllore al fine di incrementare il livello di automazione dei sistemi manifatturieri, attraverso il paradigma della produzione intelligente. Il processo di riferimento è la fresatura delle leghe di alluminio mediante macchine utensili a controllo numerico. Il sistema sviluppato è chiamato Evaluation and Perception Controller, ed è costituito da tre livelli: sensori, percezione e cognitivo. La validazione delle prestazioni del sistema qui sviluppato è svolta confrontando gli osservabili di processo con quelli ottenuti in un caso nominale, nel quale i parametri sono scelti secondo indicazioni fornite dal produttore degli utensili da taglio.

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Articoli tecnico scientifici o articoli contenenti case history
Dottorato, Università degli Studi di Trento, 2013

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da Alessia De Giosa
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Acknowledgements In ogni progetto la chiave per il successo `e data dalla coesistenza di tre condizioni: im- pegno, risorse e opportunit`a. L''impegno `e determinato dall''individuo, mentre le risorse disponibili e le opportunit`a dipendono dalle condizioni al contorno. Questo lavoro di tesi non sarebbe stato possibile senza la fiducia ed il supporto del mio supervisore prof. Paolo Bosetti, ma anche dei prof. Enrico Bertolazzi e prof. Francesco Biral. Li ringrazio per avermi continuamente accompagnato ed ispirato durante il mio dottorato. Ringrazio inoltre la prof.ssa Stefania Bruschi con cui ho lavorato in una prima fase di questo percorso. Grazie al mio compagno di avventure Antonio Di Dino con il quale ci siamo contin- uamente confrontati, spingendoci ad a'rontare gli studi da punti di vista originali e trasversali. Innumerevoli sono stati i momenti di ilarit`a che abbiamo condiviso assieme a Frego e Galva, e che mi hanno aiutato ad essere positivo anche nelle situazioni pi` u difficili. Essenziale `e stato l''a'etto ed il supporto di mio fratello Dino e dei miei amici di vec- chia data '' Marco, Daniele, Chiara, Michela ed Alice '', ma anche dei nuovi che ho conosciuto in questi ultimi anni. Ho trascorso insieme a voi momenti indimenticabili, di cui custodir`o gelosamente il ricordo. Infine, voglio sottolineare che tutto ci`o che c''`e di buono nella persona che sono diventato `e dovuto ai sacrifici ed all''immenso amore di mia madre Diana e mio padre Carlos. L''impegno che ho messo, e metter`o nell''a'rontare le sfide della vita `e continuamente guidato dal loro amore. A loro va tutto il merito del mio successo personale. Los amo. i Dedication A mi madre Diana, padre Carlos y hermano Dino que constantemente sostienen mi coraz´on con su amor. (To my mother, father, and brother who constantly sustain my heart with their love.) ii ''Once you have the vision and the clear thought together, what''s missing is just really good execution. And execution to me is all about the way you would climb a mountain you''ve never climbed before. If you waver along the way, if you debate, if you become uncertain about the objective, then you''re not going to make it. It''s important that you keep climbing. And it''s important that you acknowledge that you don''t have all the answers. So you will make mistakes, and you will have to back up, learn, and improve. That is a normal component of the innovative process. But you should not change your goal.'' S. Thrun (2013). 1 CONTENTS Contents Ringraziamenti i 1 Introduction 14 1.1 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.1 Sensing layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.1.2 Perception layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.1.3 Cognitive layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.2 Overview of Milling Process . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3 Adaptive Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.3.1 Adaptive Control with Constraints . . . . . . . . . . . . . . . . 28 1.3.2 Geometry Adaptive Control . . . . . . . . . . . . . . . . . . . . 29 1.3.3 Adaptive Control with Optimisation . . . . . . . . . . . . . . . 33 1.4 Statement of Originality . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2 Optimal Control Problem 38 2.1 Theoretical primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4 OCP for milling processes . . . . . . . . . . . . . . . . . . . . . . . . . 45 2 CONTENTS 2.4.1 Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.4.2 Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4.3 Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.4.4 DAE constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3 Simulacrum 61 3.1 Milling machine Alesamonti MB63 . . . . . . . . . . . . . . . . . . . . 62 3.2 Numerical Control FIDIA C20 . . . . . . . . . . . . . . . . . . . . . . . 65 3.3 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.1 Dynamometric table . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3.2 Accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4 Evaluation and Perception Controller 70 4.1 Process simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.1.1 Ruby Numerical Control . . . . . . . . . . . . . . . . . . . . . . 74 4.1.2 Cut simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 OCP solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.3 Learning layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3.1 Sensor Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5 Communication Layers 90 5.1 Common Object Request Broker Architecture . . . . . . . . . . . . . . 91 5.2 Interface Ruby - CORBA . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3 Characterisation and validation . . . . . . . . . . . . . . . . . . . . . . 95 5.3.1 Task Execution Time . . . . . . . . . . . . . . . . . . . . . . . . 96 3 CONTENTS 5.3.2 Validation of the acquisitions . . . . . . . . . . . . . . . . . . . 98 5.3.3 Latencies and sampling frequencies . . . . . . . . . . . . . . . . 98 6 Machine Identification 103 6.1 Dynamics of axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.1.1 Maximum feed rates and accelerations . . . . . . . . . . . . . . 104 6.1.2 Transients of feed rates and accelerations . . . . . . . . . . . . . 107 6.2 Dynamics of the spindle . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2.1 Spindle acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2.2 Mechanical torque . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.3 Dynamic compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.3.1 System identification . . . . . . . . . . . . . . . . . . . . . . . . 122 6.3.2 Regression of the dynamic compliance . . . . . . . . . . . . . . 126 6.3.3 Validation of the regression . . . . . . . . . . . . . . . . . . . . 127 6.3.4 Validation of the stability lobes diagrams . . . . . . . . . . . . . 131 7 Applications of Augmented Reality for Safe Manufacturing 134 7.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.2 3D scene reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.3 Exploitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 8 Results 144 8.1 O'ine optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 9 Conclusion 150 9.1 Summary of Thesis Achievements . . . . . . . . . . . . . . . . . . . . . 150 9.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4 CONTENTS Appendices 153 A Gaussian Processes 154 B Calculation of Stability Lobes diagram 157 C Normal Distribution Transform 164 D Identification of the cutting constants of the workpiece 167 E Technical drawings of the Alesamonti MB63 169 F Information logged by the EPC 171 Bibliography 171 5 CONTENTS List of Symbols cs [mm2] = Cross section
d [mm] = Depth of cut
Ec [W] = Cutting energy
f [m/s] = Feed rate
F [mm/min] = Feed rate ' f (') [m/s2] = Feed acceleration
Fc [N] = Cutting force
fz [mm/tooth] = Feed per tooth
Kt [N/mm2] = Tangential cutting constant
Kr [N/mm2] = Radial cutting constant
l [m] = Length of the ram
M RR [mm 3/s] = Material Removal Rate nt [ \] = Number of flutes in the mill Pc [W] = Cutting power
rf [mm] = Fillet radius of the tool
rt [mm] = Radius of the tool
S [rpm] = Spindle speed
t [s] = Time
Tc [Nm] = Cutting torque
us [J/mm3] = Specific energy of the workpiece
vc [m/min] = Cutting speed !(') [rad/s] = Spindle speed '!(') [rad/s2] = Spindle acceleration ' [m2] = Curvilinear abscissa Remaining quantities will be defined in the text. 6 CONTENTS List of Acronyms ACC = Adaptive Control with Constraints
ACO = Adaptive Control with Optimisation
AN N = Artificial Neural Network
AR = Augmented Reality
BV P = Boundary Value Problem
CN C = Computer Numerical Control
CORBA = Common Object Request Broker Architecture
DAE = Di'erential Algebraic Equations
EP C = Evaluation and Perception Controller
GAC = Geometry Adaptive Control
GP = Gaussian Process
ICP = Iterated Closest Point
M P C = Model Predictive Control
M RR = Material Removal Rate
N C = Numerical Control N LP = Non Linear Programming
N DT = Normal Distribution Transform
OCP = Optimal Control Problem
P SD = Power Spectrum Density
RN C = Ruby Numerical Control
SLAM = Simultaneous Localisation And Mapping
SLD = Stability Lobes Diagram
SSV = Spindle Speed Variation
U DP = User Datagram Protocol
V R = Virtual Reality 7 LIST OF FIGURES List of Figures 1.1 Architecture of the EPC. . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2 Examples of traditional machining processes (1). . . . . . . . . . . . . . 25 1.3 Damped tool. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.4 Examples of weights of three terms in a generic target function contex-
tualised on di'erent applications. . . . . . . . . . . . . . . . . . . . . . 34 2.1 Model of the cantilever beam with the cutting force applied at the free
end. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2 Engagement arcs definition w.r.t. feed velocity. . . . . . . . . . . . . . . 51 2.3 Discretisation of the tool path (right) and computation of the target
function at each quantised position increment (left). . . . . . . . . . . . 52 3.1 Architecture of the simulacrum. . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Configuration of the machine MB63. . . . . . . . . . . . . . . . . . . . 63 3.3 Kinematic chain of the cutting head. . . . . . . . . . . . . . . . . . . . 64 3.4 Scheme of the measurement system. . . . . . . . . . . . . . . . . . . . . 66 3.5 The dynamometric table. . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.1 Architecture of the EPC. . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 State machine of the EPC. . . . . . . . . . . . . . . . . . . . . . . . . . 72 8 LIST OF FIGURES 4.3 Trapezoidal (acceleration-limited) feed velocity profile. Short blocks can
result in triangular profiles. Ticks on time axis represent the quantisa-
tion time tq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.4 Function (t) evaluated with a sampled time of 0.01 s, t = 1.2 s, and maximum accelerations A = D = 5 m/s2; vertical dashed lines limit the constant feed rate zone . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.5 Representation of circular interpolation. . . . . . . . . . . . . . . . . . 79 4.6 Architecture of the OCP code generator and solver. . . . . . . . . . . . 81 4.7 Validation of the learning layer for the full immersion face milling. . . . 87 4.8 Validation of the learning layer for the face milling with increasing cut-
ting width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.9 Validation of the learning layer for the face milling with complicate tool
path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.1 Architecture of the software developed for the communication between
EPC and NC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.2 Communication interface and layer between EPC and NC. . . . . . . . 93 5.3 Classes architecture of the communication layer between EPC and NC. 94 5.4 Histograms of the probability density distributions of TETs measured
when reading the status of the machine (left), writing two parameters
on the NC (centre), and performing both operations (right). . . . . . . 97 5.5 Comparison of the measurements of positions and velocities for the three
axes, made with the EPC and with the oscilloscope internal to the user
panel of the machine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.6 Feed rate profiles used to estimate the latencies measured when acquiring
from the NC at '10 Hz (left), and '100 Hz (right). . . . . . . . . . . . 101 6.1 Feed rates of the axes for the test performed giving as input: feed steps
from 0 mm/min to 10000 mm/min (red), and rapid movements (blue). . . . . 105 6.2 Accelerations of the axes for the test performed giving as input: feed
steps from 0 mm/min to 10000 mm/min (red), and rapid movements (blue). 106 9 LIST OF FIGURES 6.3 Tests performed to identify the duration of the acceleration transient. . 108 6.4 Set and measured sweep profiles on the feed rate. In the test performed
along Y going from the top to the bottom the feed rate is negative
because positive Y are upwards. . . . . . . . . . . . . . . . . . . . . . . 110 6.5 Example of a deceleration test performed starting from di'erent initial
feed rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.6 Results obtained from the deceleration tests for the X (left), Y (centre),
and Z (right) axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.7 Measured spindle speed (top), and calculated accelerations (bottom) for
one of the three tests performed. . . . . . . . . . . . . . . . . . . . . . . 114 6.8 Measured free run mechanical torque at di'erent positions of the ram. . 115 6.9 Measured and fitted free run torque for the test with the ram fully
extended (top), and residuals of the regression (bottom). . . . . . . . . 116 6.10 Scaling factor K estimated for four acceleration/deceleration tests. . . . 118 6.11 Measured mechanical torque (top), and cut cross section (bottom). . . 119 6.12 Direction of impacts on the tool. . . . . . . . . . . . . . . . . . . . . . . 121 6.13 Direction of impacts on the workpiece. . . . . . . . . . . . . . . . . . . 121 6.14 Real part of the FRF matrix for the impacts on the tool, at di'erent positions of the axes (see Tab. 6.2). . . . . . . . . . . . . . . . . . . . . 123 6.15 Real and imaginary part of direct FRF along X (left) and Y (right), at di'erent positions of the ram. . . . . . . . . . . . . . . . . . . . . . . . 124 6.16 Real part of the measured (left) and regressed (right) F RFy,y. . . . . . 128 6.17 Stability lobes diagram calculated for the ram at the position -220 mm (left) and chatter frequencies (right). . . . . . . . . . . . . . . . . . . . 129 6.18 Stability lobes diagrams calculated from measured FRFs (left), and from regressed FRFs (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.19 Position of the accelerometer. . . . . . . . . . . . . . . . . . . . . . . . 131 6.20 Stability lobes diagrams validated at two positions of the ram: -300 mm (top) and -270 mm (bottom). The size of the circles is proportional to
the calculated CIER. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10 LIST OF FIGURES 7.1 Normal Distribution Transform algorithm applied to the alignment of
two 3D geometries not explicitly correspondent. . . . . . . . . . . . . . 139 7.2 Conceptual architecture of the system for the collision avoidance. . . . 141 7.3 Scene reconstructed by using the Kinfu library (left) and RGB image of
the working volume (right). . . . . . . . . . . . . . . . . . . . . . . . . 142 7.4 Conceptual design of an augmented interface for milling machines. . . . 143 8.1 Tool paths tested for the o'ine optimisation. . . . . . . . . . . . . . . 145 8.2 Nominal and optimised controls for the face milling. . . . . . . . . . . . 147 8.3 Nominal and optimised controls for the peripheral milling. . . . . . . . 148 8.4 Roughness maps for the face milling. . . . . . . . . . . . . . . . . . . . 149 8.5 Roughness maps for the peripheral milling. . . . . . . . . . . . . . . . . 149 B.1 Two-dimensional dynamical model of milling process. . . . . . . . . . . 158 D.1 Cutting forces at di'erent feed rates. The components of the force are parallel (left) to the feed, perpendicular (centre), and along the axis of
the tool (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 E.1 Lateral view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 E.2 Frontal view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 11 LIST OF TABLES List of Tables 1.1 Yardstick for automation according to Amber and Amber (2). . . . . . 16 3.1 Technical specifications of the MB63. . . . . . . . . . . . . . . . . . . . 63 3.2 Technical specifications of the spindle. . . . . . . . . . . . . . . . . . . 63 3.3 Technical specifications of the shafts in the kinematic chain, Z is the
number of teeth of the gear connected to the shaft, and D is the diameter
of the pulley. The missing information can be extracted from the CAD
drawings in Appendix E. . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4 Technical specifications of the working table. . . . . . . . . . . . . . . . 68 3.5 Technical specifications of the piezoelectric force links Kistler 9347C. . 68 3.6 Technical specifications of the charge amplifier Kistler 5073A311. . . . . 68 3.7 Technical specifications of the accelerometer Dytran 3213M6. . . . . . . 69 5.1 Measured TETs for di'erent operations, with 95% of confidence. . . . . 97 5.2 TETs required to write one parameter on the NC according to the NC
manufacturer FIDIA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.3 Measured latencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.1 Maximum velocities acceleration and deceleration of the axes. . . . . . 106 6.2 Positions at which the FRF of the tool are tested, expressed in the
absolute frame of reference of the machine. . . . . . . . . . . . . . . . . 122 12 LIST OF TABLES 6.3 Trained hyperparameters. knlmlk is the module of the negative log marginal likelihood, the higher it is, the more accurate the prediction is. 127 7.1 Examples of 3D sensors available to the market. . . . . . . . . . . . . . 140 8.1 Data of the utilised tool. . . . . . . . . . . . . . . . . . . . . . . . . . . 144 8.2 Cutting times achieved with and without the EPC. . . . . . . . . . . . 146 D.1 Process parameters used for the cutting force constants identification. . 167 D.2 Cutting force constants for the AA 6082-T6. . . . . . . . . . . . . . . . 168 F.1 Codes of the machine status parameter. . . . . . . . . . . . . . . . . . . 171 F.2 Structure of the Status C-struct. . . . . . . . . . . . . . . . . . . . . . . 172 F.3 List of parameters and sub parameters listened by the Listener instance. 173 13 CHAPTER 1. INTRODUCTION Chapter 1 Introduction This study is focused on applying the optimal control to machining process, and it has been carried out within the Italian project Michelangelo. The main objective of this work is to increment the level of automation of manufacturing machines towards the concept of intelligence manufacturing. The project has lasted three years and it has been completed on November 2013. It has involved several academic and industrial partners: University of Trento, University of Brescia, University of Bergamo, CSMT, CNR ITIA, Alesamonti S.r.l., Fidia S.p.A., Imatecno, Mandelli S.p.a., Pomini Tenova S.p.a., Scuola Superiore Sant''Anna of Pisa and Stylmeccanica. 1.1 Motivation and Objectives The 20th Century has been characterised by the growth of economies of scale where the production costs tended to reduce with the increasing size of enterprises. As a conse- 14 CHAPTER 1. INTRODUCTION quence, a rising number of manufacturing companies started to move to big production rates, where large manufacturing systems were exploited to fulfill the product demand of global market. In the last 20 years, the rapid evolution of products, changing of user needs, and global competition, have forced companies to redesign manufacture chains thus taking into ac- count not only the mere productivity, but also the quality, economy, and the flexibility of the production. These reasons have motivated an increasing number of enterprises to research and evolve their production systems towards paradigm of intelligence man- ufacturing. Increasing the level of automation in manufacturing systems generates a technological and economical evolution on a wide sectors of production chains. Op- timal control can be exploited to optimise multi-objective target functions defined in order to reduce production times, costs, and increase the quality. Small companies with custom productions need high flexibility and might benefit mainly from reduced setup costs. Large production instead can achieve high productivity from automated, integrated, short-loop process control. The cooperation of multiple intelligent systems can moreover ease the definition and calibration of reliable models of the process, thus improving the goodness of the calculated optimal controls. In this scenario, the Computer Numerical Controlled (CNC) machine tools have be- come popular and have played a key role in workshops, leading to a vast area of research on modelling and optimisation of machining processes. However, despite the contin- uous performance enhancements, since the first CNC machine tool was introduced in 1952 (3) the automation level has not been evolving as much as in other mechatronic fields, like: robotic vehicles, planes, anthropomorphic manipulators, and humanoids. In 1962 Amber and Amber defined the yardstick for automation, Tab. 1.1. Advanced robots have perception layers to sense, reconstruct, and learn their state; they can plan online the future actions to take in order to achieve a specified goal, and can cooperate with others systems to enhance their knowledge. Such systems are typi- 15 CHAPTER 1. INTRODUCTION Table 1.1: Yardstick for automation according to Amber and Amber (2). Order Human Attribute Replaced Examples A(0) None: lever, screw, Hand tools, manual machine A(1) Energy: muscles replaced Powered machines and tools, with power Whitney''s milling machine. A(2) Dexterity: self-feeding Single-cycle automatics. A(3) Diligence: no feedback but Repeats cycle; open-loop numerical repeats cycle automatically control or automatic screw; transfer lines. A(4) Judgment: positional feedback Closed loop; numerical control; self-measuring and adjusting; CAD, CAM. A(5) Evaluation: adaptive control Computer control; model of process deductive required for analysis; feedback from the
process analysis and optimisation with
data from sensors. A(6) Learning: from experience Limited self-programming; some artificial
intelligence (Al); expert systems. A(7) Reasoning: exhibits intuition; Inductive reasoning; relates causes and e'ects advanced Al in control software. A(8) Creativeness: performs Originality creates new process programs, design work unaided neural networks; fuzzy logic. A(9) Dominance: supermachine, Machine is master commands others (Hal in 2001, A Space Odyssey). cally called multi agents systems (4; 5), and in the Amber and Amber yardstick have an order of A(6). The most advanced marketed CNC machines have only the judgement attribute, corresponding to an order of A(4), with controllers that can compensate errors in positioning the axes, thermal distortions of the structure, and regulate the spindle speed at a nominal value. On the market adaptive controls are also available to mitigate vibrations and improve the quality of the process. Nevertheless, these systems rely only on the torque measured at the drives, and are simple feedback controls rather than proper model-driven intelligent systems. Intelligent systems are based on at least two factors: the ability to learn the process, and the use of a model of the process to 16 CHAPTER 1. INTRODUCTION do predictions. When an agent is a machine tool, these features ensure the operation at optimal cutting conditions, which are constantly identified based on a theoretical knowledge of the process and on the current state of the machine. The purpose of this study is to increase the level of automation of CNC machines to A(6), by designing a controller whose general framework architecture could be exported to other processes than machining. The proposed architecture is called Evaluation and Perception Controller (EPC). It is shown in Fig. 1.1 and it closes the feedback loop on the machine through three layers: sensor, perception, and cognitive. Figure 1.1: Architecture of the EPC. 17 CHAPTER 1. INTRODUCTION 1.1.1 Sensing layer In the sensing layer the status of the machine is collected directly from the CNC, and from sensors displayed in the working volume (6). The former has the advantage that is a cheap solution, since the information on the cutting power and tool condition is obtained by measuring the motor power and current at the electric drives and spindle. These measurements however are given by the feedback controllers of the drives, there- fore they tend to be a'ected by high noise and have usually a low sampling frequency. Additional sensors can be utilised to obtain reliable and accurate data, but due to their high costs they are usually used only in high-end and research applications. As instance, cutting force and torque can be measured through piezoelectric sensors and strain gauges placed on the tool holder. These devices can be exploited also for mea- suring the vibrations of the tool during the cut, however, if only such information needs to be monitored, a more suitable and cheap solution would be to use acoustic emission measuring systems (i.e. microphones), gap sensors, or accelerometers. Nevertheless, microphones should be used only when a single machine is cutting, since they might record not only the sound generated by the monitored process, but also the sound generated by others machines working in the same workshop. Recently, thanks to the improvements of the performances of personal computers, a new class of sensors have been utilised that are based on image processing techniques. Images and videos of the working volume can be acquired through cameras, which can be classified as: video, stereo, Time of Flight (ToF), and 3D depth sensors. These de- vices are used to augment the set of information provided to CNC users, thus o'ering tools to monitor, supervise, and maintain remote processes, as well as to test complex collision-free tool paths and to ease the training of novice workers (7). 18 CHAPTER 1. INTRODUCTION 1.1.2 Perception layer The perception layer fuses the sensed data to reconstruct the state of the system, which is defined by the instantaneous process status, cut cross section, and tool condition. Force, acceleration, gap, and acoustic signals can be analysed in the time as well as fre- quency domains (8). Measurements are usually studied through the Fourier transform method power spectra and Hilbert-Huang Transform (9), but the best state estimation is achieved when a model-based filter is used. The most famous state estimators are Kalman Filters (KF) (10) and Particle Filters (PF) (11). These have been utilised both individually and together in numerous appli- cations, such as: navigation of unmanned aerial and ground vehicles (12; 13; 14; 15), simultaneously localisation and mapping of robots (16; 17; 18), parameters estima- tion (19; 20; 21; 22), economic forecasting (23), wireless LAN tracking (24; 25), and vision-based tracking (26; 27; 28). The state space of KF and PF is continuos, but usually these filters are considered complementary. KF is an efficient algorithm since it scales quadratically with the di- mension of the state variables, and it is confined to estimate unimodal distributions. The tuning of KF however might be difficult since it requires to estimate both the mea- surement and model uncertainty. PF on contrary describes multimodal distributions, is easier to implement, but its efficiency might be highly a'ected when the dimension of the state is greater than four. Images acquired from cameras can provide numerous information about the ongoing machining process, with a modest economic investment. High resolution cameras and efficient Graphics Processor Units (GPU) ''which have been developed for gaming consoles and PCs'' are nowadays available on the market together with open-source 19 CHAPTER 1. INTRODUCTION state-of-the-art image processing libraries. OpenCV (Open Source Computer Vision) 1 is a computer vision and machine learning software library with more than 2500 op- timised algorithms that can be used to a wide variety of applications, such as: detect and recognise faces, identify objects, classify human actions in videos, image stitch- ing, follow eye movements, track camera movements, track moving objects, extract 3D models of objects, produce 3D point clouds from stereo cameras, recognise scenery and establish markers to overlay it with augmented reality. To augment reality it means to extend the set of information provided to the CNC user, by merging the data recorded by the NC, sensors, and video. Collision avoidance One of the main challenges in designing a milling or turning manufacturing process is to obtain a safe tool path, free from unexpected collisions between the tool and the workpiece. Safe tool paths are crucial, not only to avoid expenses damages of the milling machines, but also when expensive materials have to be cut. It is possible to classify the safety systems as passive or active, likewise it is done with vehicle safety systems. Passive systems act once the collision has occurred, and they reduce the entity of the damages. These systems tend to be used in unstructured environments, but they might be not e'ective when collisions between expensive tools and workpieces must be avoided a priori. Active systems instead can predict and avoid dangerous situations, therefore they are particularly suitable for machining processes, where the high inertia of the drive is involved during the cut. The prediction will be as much good as much precise is the knowledge of the supervised system. It is not trivial however to have a robust model of the working volume due to the complexity of the kinematics of the 5-axis machines, and 1http://opencv.org/ 20 CHAPTER 1. INTRODUCTION shape of the workpiece. Nowadays, two strategies are followed to address this problem: the g-code can be tested in free air directly on the machine, or the tool path can be designed and tested through computer aided manufacturing (CAM) softwares (29; 30). The former approach tends to be time consuming (thus reducing the productivity), and relies on the ability of the worker man to correlate the in-air process with the real one. The robustness of CAM softwares used for collision check depends on the skills of designers to take into account several factors, such as machine kinematics, geometry of raw workpiece and tool, and accuracy of theirs CAD models. Usually, the CAD model of the milling machine is provided by the machine manufacturer, but it not so for the tool and tool holder. Raw workpieces can have complex patterns, especially if they have been generated with forging or casting processes. Several issues must be addressed when applying image processing techniques to aug- ment the reality of milling processes: the presence of high frequency vibrations, debris, chips, coolant liquids inside the working volume, and reflecting surfaces of the tool and workpiece. A description of the feasibility study carried out on this topic is given in Chapter 7. 1.1.3 Cognitive layer In the cognitive layer the past history, present state and updated models of the process are used to close the loop and calculate the optimal process parameters. For small mobile robots the loop closure is done through the optimisation of simple linear mod- els, with techniques called Model predictive Control (MPC) (31; 32; 33; 34), in which a reference signal is followed. As a main advantage the MPC requires little computer resources to be run online, therefore it is particularly suitable to be implemented in small controllers. However, the use of linear models represent a strict simplification for 21 CHAPTER 1. INTRODUCTION some applications, such as machining processes. When compactness and lightness are not mandatory requirements, a personal computer can be utilised to perform non linear model-based optimal control. Example of these systems are large moving robots such as vehicles or long range drones, as well as manufacturing systems where the model of the process is formulated through di'erential equations. The optimal control tech- nique allows to maximise a set of performances, hence contextualising the controller on a specified application. In robotic vehicles the target function can be defined in order to reach a goal by optimising the path and controls, while minimising the travelling time or the energy consumption (35; 36; 37). When improving machining systems, multiple objectives can be pursued. Mechanical workshops are interested mainly in: quality, productivity, and cost reduction. Con- straints on robotic systems can be also taken into account to ensure the feasibility of the optimal solution. The equations that define the target function, model of the sys- tem, and constraints form a system of Di'erential Algebraic Equations (DAE), which are numerically integrated and minimised. These operations have a high computational cost that is addressed, even in real-time, by modern personal computers. In-line opti- mal control can be carried out to enhance the behaviour of robots while compensating external disturbances. For milling processes, online control allows to adapt the system when the workpiece material has non homogeneous properties, or when the tool wears. The cognitive layers aims thus at collecting the knowledge of skilled workers, easing the calibration of process parameters, and eliminating time-consuming manual design of optimised part programs. The e'ectiveness and reliability of intelligent systems depend on the level of knowl- edge programmed into the cognitive layer. Artificial intelligence techniques are used to learn and adapt models of the process from past records and, eventually, from a shared knowledge. Such approach is called machine learning, and the generated models can be 22 CHAPTER 1. INTRODUCTION used for classify and regress the sensed data. The former is performed on discrete finite models, and it is done to subdivide the observed quantity into groups. The latter on contrary is performed on continuos models, and its objective is to formulate continu- ous functions. In any case, models can be generated through supervised, unsupervised, and reinforcement learning. When the training is supervised, labelled data are used to train and validate the model. In unsupervised training the machine finds itself the most significant patterns and features of outputs of the process that must be taken into account to describe the system. Finally, in reinforcement learning the actions done by the machine are used to explore the set of features of the environment. The training process is, generally speaking, time consuming and requires a large amount of data. The information can be collected, inferred, and learned from a local set of sen- sors or from a pervasive network of systems deployed across distributed manufacturing plants. If cloud computing is used to o'er Data as a Service (DaaS), then the resources and e'orts needed to train large intelligence system can be significantly reduced. As suggested by Wang and Xu (38) as well as Wu et al. (39), information about process data, tools and machine performances, optimisation strategies, experience, training data, and generally speaking all the mutable information that each controller builds up with time, would possibly be moved to a cloud DaaS, thus expanding the set of features at the core of the learning process. 1.2 Overview of Milling Process In machining processes a rigid tool is used to cut a workpiece that can be a metal, polymer, or ceramic (1). The process is flexible, can be easily exploited in dynamic production chains, and combined with other non-net shape processes such as casting, forging, and sheet metal forming to obtain complicate parts. Machining of polymers is 23 CHAPTER 1. INTRODUCTION usually disadvantaged by low productivity, and the cut of ceramics is inefficient due to high tooling costs. Traditional machining processes can be subdivided into four classes, based on the relative movement between the tool and the workpiece: turning, milling, drilling, and grinding. In turning axial-symmetrical geometries are generated. The workpiece spins and the tool is feed both radially and along the rotational axis. In grinding an abrasive tool is used to remove small fractions of the workpiece and to achieve high-quality surfaces. In drilling a cylindrical or conical tool spins around its axis and is fed along this direc- tion to generate or enlarge holes. Finally, milling processes are those in which a tool, that can have a complex geometry, is spin and moved along a path on the workpiece thus generating complex patterns, Fig. 1.2. The norm of the linear velocity of the mill travelling across the workpiece is called feed rate. It is not trivial to design and control machining processes and predict their out- puts, due to several sources of unknowns, uncertainties, and multi-physical phenomena involved: ' The thermo-mechanical properties of workpiece and tool might be unknown. ' Workpiece and tool undergo to high stresses, strains, strain rates and tempera- tures. In such extreme conditions it is difficult to describe with a good approxi- mation the real behaviour of the materials. ' Cutting forces can change periodically but also instantaneously. ' Friction forces at the tool workpiece interface are high and might saturate, there- fore simple linear models cannot describe them with sufficient accuracy. ' Kinematic of the cut can be complicate since the tool and the workpiece have in general complex geometries. 24 CHAPTER 1. INTRODUCTION Figure 1.2: Examples of traditional machining processes (1). ' The dynamic properties of the machine, such as the sti'ness and damping factors, may be unknown or difficult to characterise. ' The lubricant fluid can form an unstable film at the tool-workpiece interface. Inputs of the process taken in into account during the setup can be distinguished into constrained and degree of freedoms. The formers are fixed by the context and specific application, and are related to: workpiece and tool materials, their geometries, the lubricant fluid to be used, the characteristics of the machine, and the tool path (i.e. desired geometry). The degree of freedoms are the rotational spindle speed S and the Material Removal Rate (MRR), which is correlated to the feed rate F , to the cutting 25 CHAPTER 1. INTRODUCTION depth d, and to the cutting width w: M RR = F d w (1.1) From the product between the MRR and the specific energy us of the cut material, it is possible to calculate the cutting power Pc and cutting torque Tc: Pc = MRR us (1.2) Tc = Tc S (1.3) F and S are the controls that must be carefully set to achieve an optimal process. These parameters must be chosen in order to be robust with respect to deviations of the cut from nominal conditions. Unpredictable inline variations can be related to: ' Varying cutting depths and widths. ' Tool wear. ' Non-homogeneous material properties. ' Irregularities of the surface of the workpiece. ' Geometrical accuracy and tolerances of the blanks. ' Chip accumulation. ' Thermal properties of the lubricant fluid can vary its behaviour. Usually, process planners choose the cutting parameters accordingly to tabulated values reported in the catalogues of tool manufacturers. Nevertheless, it must be considered that suggested parameters tend to be conservative and far from optimal. F and S are 26 CHAPTER 1. INTRODUCTION chosen such as they take into account and mitigate all the uncertainties and variability associated to the process, through a decision process based on the experience of most skilled technicians rather than on theoretical and methodological considerations. 1.3 Adaptive Controllers Adaptive controllers (AC) are devices that regulate process parameters to achieve a certain performance. In the case of milling process, the controls that AC regulate are the spindle speed and feed rate. A recent review of AC system was given by Stavropoulos et al. (40). Ulsoy and Koren (41) identified three types of AC: 1. Adaptive Control with Constraints (ACC): these systems are typically aimed at maximising the cutting force within the limits imposed by the maximum stress that the tool can tolerate without breakage. Such systems are mainly used in roughing operations in which a low cutting time is sought, without considering the quality of the cut. These conditions might a'ect sensibly the process productivity since a high Material Removal Rate (MRR) is associated to high cutting forces, and according to Kurada and Bradley (42) the amount of downtime caused by tool breakage is around 20%. 2. Geometry Adaptive Control (GAC): these controls are used in finishing cuts, where the main objective is to achieve a high-quality surface, which can be de- scribed by a high geometrical accuracy, good surface integrity, as well as low roughness. 3. Adaptive Control with Optimisation (ACO): the controller developed in this study belongs to this category. The process parameters are computed and regulated 27 CHAPTER 1. INTRODUCTION in order to optimise a certain index or performance, such as: reduction of vi- bration, increasing of productivity, improvement of surface quality and tool life. Constraints on the machine dynamics can be taken into account to ensure the controllability of the system. 1.3.1 Adaptive Control with Constraints Many examples of ACC systems are available in literature. Among the most recent studies, Budak and Kops (43) developed a system for maximise the cutting force by overriding the feed-rate. The same approach, but applied to end-milling was followed by Zuperl et al. (44). The authors maximised the feed-rate by utilising an adaptive neuro-fuzzy inference system. Tansel et al. (45) and Aykut et al. (46) utilised a Genet- ically Optimised Neural Network System (GONNS). Artificial Neural Networks (ANN) were trained to represent the relationship between surface roughness, cutting forces, and process inputs, and a Genetic Algorithm maximised the MRR while keeping the cutting forces below a pre-determined threshold. Such approaches however, required a time consuming training of supervised learning algorithms used to model the relation between feed rate and cutting force. This issue did the system poorly suitable to be applied in real workshops where several materials were cut. Sugita et al. (47) realised a optimal control for bone machining where the feed rate was maximised by using two di'erent policies along the tool path: an 'air-cutting mode' and a 'force control mode'. The latter was used only along the parts of the tool path in which the mill was in contact with the workpiece. The drawback of this method, was that it required the use of an expensive external sensor to measure the cutting force. This problem was solved by Kim and Kim (48), as well as Wang et al. (49) which estimated the three components of the cutting force from the current absorbed by feed-drive servo motors. 28 CHAPTER 1. INTRODUCTION 1.3.2 Geometry Adaptive Control Studies carried out in order to optimise the quality of the cut are aimed at reducing the vibrations and the thermal deformations of the machine during the process. Hao et al. (50) treated thermal errors as the superimposition of a series of error modes obtained from five measurements of temperatures of a turning centre. Neural network combined with genetic algorithm were used to predict and compensate the thermal deformation of the structure. Model-based prediction of structural deformations how- ever can be limited from the geometrical and physical complexity of the model. Yang and Ni (51) updated in real-time the coefficients of the thermal-error model through a Kalman Filter. A di'erent approach was followed by Bosetti and Bruschi (52) whom developed a new measurement system to directly provide real-time information about the deformation of the structure. The system was called reticular displacement mea- surement system (RDMS) and it was based on an array of strain sensors acquired in real-time. The main advantage of this system was its independency from expensive, sensitive, and error-prone models. The second source of geometrical errors, especially for what concerns the surface rough- ness, are the vibrations of the machine. These can be associated to the compliance of the workpiece, tool, and even of the axes of the machine. Vibrations are due to the impacts of the tool entering the workpiece, and can be classified into forced, or chatter. The former are due to the excitation of the natural frequencies of the tool- workpiece system by the discontinuous cutting forces. In this case the dynamics of the system is stable and tends to be damped. Chatter instead can be generated by two mechanisms: regeneration and mode coupling. Regenerative vibrations are due to a phase shift between vibration waves left on the inner and outer surfaces of the cut chip. Mode coupling happens when the tool traces an elliptic path along the workpiece, thus 29 CHAPTER 1. INTRODUCTION varying periodically the depth of cut. In both cases, as the cut proceeds the dynamics of the system becomes unstable, increases progressively the chip thickness, and con- sequently the magnitude of the cutting forces. Such unstable condition reduces the quality of the cut, as well as geometrical accuracy of the generated surfaces, but also might be dangerous for the tool, spindle, and machine structure. Skilled CNC users set the spindle speed with an heuristic method, that is by changing the spindle override until they don''t hear anymore the characteristic sound of chattering process. The value of the stable spindle speed however depends not only from the dynamical behaviour of tool, workpiece, and machine structure, but also from the entity of cutting forces which might change at every position of the tool along the tool path. One of the main contributions to the problem of self-excited vibrations was given by Altintas (53) who developed a model to determine the combination of spindle speed and depth of cut that delimited stable cutting conditions from unstable. Numerous chatter mitigation strategies were studied, which can be distinguished between passive and active (54). Passive chatter suppression systems Passive chatter suppression strategies are based on modifying certain machine tool el- ements to enhance the stability of the system. These approaches consist in changing the dynamical behaviour of the machine tool, the cutting tool, and tool holder (55). When the tool is the weakest element in the system, tuneable vibration absorbers (56; 57), impact dampers (58), or viscoelastic dampers (59) can be used. On the mar- ket there are available tool holders, in which a mass internal is used as a Frahm damper (60), Fig. 1.3. Tools manufacturers sell also non-standard mills with vari- able pitch and helix angles (61; 62), which increase the stability limit by disrupting the regenerative e'ect. Cau et al. (63) used numerical simulations (represented by FE models) to design a 30 CHAPTER 1. INTRODUCTION Figure 1.3: Damped tool. milling machine with a-priori known dynamic performances, which were calibrated in order to avoid as much as possible regenerative vibrations in a specific range of process parameters. Such approach can be utilised when a system was designed ex novo, while chatter often have to be addressed on already existing machines. Active chatter suppression systems Active chatter suppression systems can monitor the dynamic state of the machine tool, identify unstable conditions, and act on the system to improve its performances. One of the first implemented active chatter suppression systems was developed by Dohner et al. (64). The authors identified spindle speeds that ensured a stable process by calculating the lobes stability diagram of the cutting system. The spindle was sensed through strain gages at tool root, and it was damped through a piezo-electric actua- 31 CHAPTER 1. INTRODUCTION tor. Another example of active piezo-electric actuators utilised to reduce the dynamic displacement between the tool and workpiece was given by Abele et al. (65). Moradi et al. (66) proposed a H1 control algorithm to control the chatter under tool wear and parameter uncertainties. In a more recent study, Moradi et al. (67) exploited the a Linear Quadratic Regulator (LQR) to suppress self-regenerative vibrations. This tech- nique allowed the authors to use a non linear model of the process in which forces and chip thickness were related by a cubic relationship. The controls were the counterbal- ance forces exerted by external actuators, and the target function was defined in order to limit the chatter while minimising the e'orts of the controls. Model-based control techniques were exploited to identify the process parameters that allowed stable cuts. Poto'cnik and Grabec (68) used ANN to model the process and Nonlinear Model pre- dictive Control (NMPC) to control the cut with stable parameters. Active systems found in literature require an external-piezo electric actuator to reduce the vibrations. To accommodate these devices on the spindle might in general require expensive mechanical modification and re-designs. A common strategy nowadays fol- lowed to mitigate the chatter is called Spindle Speed Variation (SSV) (69; 70; 71). It consists in continuously changing the spindle speed with a sinusoidal patter around the mean speed to disturb the regenerative mechanism. The main advantage of this tech- nique is that it does not require expensive tool holders and it can be implemented in modern CNCs. However, the calibration of this methodology can be difficult. Hajiko- laei et al. (72) used a genetic algorithm to set the amplitude of the speed modulations such that the input energy to the process was minimised. In the study here presented, chatter and forced vibrations is addressed with a di'erent approach. It is known indeed that the stability of vibrations depends on the depth of the cut and on the spindle speed. The former is usually a fixed parameter of the process, the latter instead can be calibrated. The idea is ''for a given tool engagement'' 32 CHAPTER 1. INTRODUCTION to include into the multi-objective optimal control the information about the stability lobes diagram which drives the EPC in choosing stable spindle speed, as it was proposed by Bosetti et al. (73). The lobes stability diagrams are calculated by considering the dynamic compliance of the machine structure in the current axis configuration, and the cross section predicted by the geometrical engine simulator, as it will be explained in chapter 2.4.2. 1.3.3 Adaptive Control with Optimisation ACOs merge the feature of ACCs and GACs, and they fall in the category of intelligent systems. In the last 20 years there has been a massive e'ort from researches in outlining and designing the architecture of intelligent systems applied to manufacturing. Mekid et al. (74) defined the requirements ACO manufacturing systems should satisfy: 1. Integration: to perform conventional and non conventional processes. 2. Bi-directional data flow: to exchange information between the NC and drives on a single machine, but also through the workshop. 3. Process control loop: to in-line monitor and control tools, parts, and processes. 4. predictive maintenance: as a basis for self-reliant machine operation. 5. Autonomous optimisation: that is the development of a self-configuring and self- optimising control system, based on a knowledge of the process constantly up- dated. These systems must be reconfigurable and dynamically fuse the human and machine intelligence, as well as manufacturing knowledge and stat-of-the-art models. Intercon- nected subsystems and functions can be used to collect information about: geometries, 33 CHAPTER 1. INTRODUCTION status of the process, tool management, process planning, and control. These data can then be exploited to optimise a multi-objective target function, which is defined by a weighted sum of di'erent terms taking into account process productivity, quality, and cost. The weights of the target function can be tuned accordingly to the specific man- ufacturing context. As instance, in roughing operations the productivity and the cost of the process will be multiplied by higher coefficients (Fig. 1.4-b). On the contrary, when finishing the workpiece the term of quality will be dominant (Fig. 1.4-c). Figure 1.4: Examples of weights of three terms in a generic target function contextu-
alised on di'erent applications. An early ACO systems was developed by Amitay et al. (75). The authors optimised the MRR in grinding, subjected to constraints on workpiece burn and surface finish. The same targets were sought by Tuming and Park (76) who designed a controller to maintain a desired peak cutting force under uncertainty of model parameters. The con- troller was designed through conventional and decoupled estimation methods. Kruger et al. (77) proposed a generic architecture of a rational agent system, integrating het- erogeneous intelligent algorithms to self-learn and regulate a drilling process. ANN modelled the on-line measured data and genetic algorithms were used to find the op- timal trade o' between technical and economic factors. A more completed target function was formulated by Zuperl et al. (78), by including cutting costs and produc- tion rate (depending on MRR and tool life), while taking into account the maximum 34 CHAPTER 1. INTRODUCTION cutting power, cutting torque, surface quality, workpiece deflection, and chip build-up edge. The same performance indexes were considered by Ridwan and Xu (79) who implemented a STEP-NC ISO 14649 (80) enabled Machine Condition Monitoring sys- tem, which was composed by three layers. The first module performed a preliminary feed rate optimisation to determine the machining parameters which optimise the cut- ting time and quality. The former was minimised in roughing cuts, the latter was maximised in finishing operations. A process simulator was developed to estimate the cutting power, and a fuzzy logic processed imprecise data. Online, a second subsystem carried out an adaptive feed-rate optimisation in order to keep the spindle load constant and close to the simulated value. The third module was a knowledge-based evaluation system, that was responsible of generating accurate, informative and updated process data. The authors however did not control the spindle speed, a parameter whose tun- ing is critical when self-regenerative vibrations need to be avoided. 1.4 Statement of Originality Even if the devices developed in literature represent a remarkable e'ort in enhancing the automation level of CNC machines, they tend to be little flexible, and not practical to be used in real workshops. As instance, the models of the process rely on black-box algorithms (such as ANN) which are accurate only for a specific set of workpiece and tool materials. For every tool-workpiece combination, time consuming model trainings are required. Moreover, information about the geometry of the cut, which is neces- sary to calculate the cutting force, is missing. In those studies, process optimisation is typically performed by exploiting evolutionary algorithms that are suitable for non di'erentiable target functions and models, but require several iterations to achieve the optimal solution, and are not deterministic. It means that the convergence criteria 35 CHAPTER 1. INTRODUCTION must be carefully defined, otherwise even when starting from the same initial guess, the algorithm will reach di'erent optimal points. Finally, the dynamics of the system is not considered. The controls are computed without considering how the machine will react during transients, which might occur in complex tool paths. The aim of this study is to tackle limitations of the work done up to nowadays, by developing an intelligent manufacturing system, capable to simulate, optimise, sense, and learn the milling process. The description of the process is based on theoretical models which are enough accurate to capture the main phenomena involved during machining operations. The utilised models are well known in literature, therefore their coefficients are tabulated in manu- als, and can be calibrated through simple and short cutting tests. A process simulator is integrated into the developed system to pre-calculate the instantaneous cross section and cutting force at each position of the tool. The simulator is a geometrical kernel that is commonly used in Computer-Aided Manufacturing (CAM) softwares. The cut cross section is then interpolated along the curvilinear abscissa with a continuous and di'erentiable function. Thanks to the proposed approach it is possible to generate a continuous and di'er- entiable target function that is minimised through efficient optimisation algorithms. Machine performances and dynamic characteristics are taken into account through a system of Algebraic and Di'erential Equations (DAE). The set of target function and DAE constraints define the Optimal Control Problem (OCP). It should be mentioned that is not trivial to solve the OCP, especially when the optimal solution needs to be calculated in real time (i.e. at 10 Hz). Nevertheless this problem has been already addressed in robotics, and especially in autonomous vehicles navigation (37; 36; 81) where the dynamic behaviour is even more complicated and non-linear than that of machine tools. 36 CHAPTER 1. INTRODUCTION This thesis summarises the work done in implementing the OCP for machining of metals. The intelligent system here developed is called Evaluation and Perception Controller (EPC). Target function and DAE constraints are defined based on models of the process that are calibrated through dedicated experimental tests. The system includes a learning layer to autonomously update the coefficients of the models based on its experience. All the test carried out to characterise the machine tool are described as a guide for future utilisations of the system. The EPC is also tested on real processes to validate its performances. Process productivity, quality, and costs are sensibly improved, thus emphasising the benefits given by increasing the automation level of machine tools. 37 CHAPTER 2. OPTIMAL CONTROL PROBLEM Chapter 2 Optimal Control Problem In this chapter is proposed a brief introduction to the Optimal Control Problem (OCP). It is described the approach followed to minimise a non linear target function subjected to constraints, which are expressed as a set of Di'erential and Algebraic Equations (DAE). Further insights can be found in (82). It is moreover derived the formulation of the OCP applied to the milling process. Particular emphasis is given to the definition of the target function, starting from technological requirements of modern machining workshops. 2.1 Theoretical primer The OCP is aimed at calculating the sequence of inputs that allows a dynamical system to be controlled in an optimal way. The target function can be defined to optimise an instantaneous performance, but also to maximise the response of the system in a future time span: the horizon. The feasibility of the optimal controls is ensured by a series of path, punctual and integral constraints, which can be applied to the state 38 CHAPTER 2. OPTIMAL CONTROL PROBLEM variables and to the controls. Once the optimal solution has been calculated, the state of the machine is updated, the analysed horizon is shifted ahead in the future, and the OCP is solved again. By continuously updating the instantaneous value of the state variables it is possible to compensate unpredicted variations of the state of the system. This technique is called receding horizon. As a result, the OCP returns not only the history of the controls, but also the whole evolution of the system. Target function and constraints are generated from a model of the process. To a more accurate description of the system corresponds a solution that is closer to the true op- timum. However, complicate models can be computationally expensive and impossible to solve in real-time. The model should then be simple, but enough representative in order to capture the main phenomena at which the system is subjected. The OCP has been investigated in a wide number of robotic systems. The main con- tribution of OCP has been in the field of autonomous vehicles, where it has been applied to numerous problems, such as: obstacle avoidance (83), computation of opti- mal path (37), time-optimal and minimum-e'ort control (84). 2.2 Formulation Four are the elements that must be defined to formulate the OCP: the model of the system, the controls, the target function, and a set of constraints. The model of the system can be non linear, but it should be controllable across the whole control space. A non controllable condition might happen when it is desired to act on a system but the controls are not enough powerful, as instance when it is desired to push a heavy mass but there is not enough force available. Generally speaking, the controllability of a system might depend also on the initial conditions. In this case the system can be regulated by imposing soft constraints, that is by iteratively changing 39 CHAPTER 2. OPTIMAL CONTROL PROBLEM the initial conditions. The target function can be expressed in order to optimise the initial or final condition of the system, as well as its evolution during a future time window. The former is called problem of Mayer, and the latter problem of Lagrange. A linear combination of Mayer and Lagrange target function form the problem of Bolza. Assuming that x(t) is the state of the system and u(t) is the control, the target function J can be defined as: 1. Mayer: J = (x(tf )) (2.1) where (x(tf )) is the terminal cost function, that is independent from the controls since they are considered piecewise constant. Common definitions of the terminal cost are those used to minimise the error in reaching a desired final state, the time required to achieve a desired condition, or the fuel consumption: (x(tf )) = x T (t) P x(t) t=tf Error in final state (2.2) (x(tf )) = |tfinal tinitial | Minimum time problem (2.3) (x(tf )) = |minitial mfinal | Minimum fuel consumption (2.4) where P is a square matrix of real values, and m is the mass of the system. 2. Lagrange: J = tf Z t0 L (x(t), u(t)) dt (2.5) L (x(t), u(t)) is the Lagrangian function. By integrating the Lagrangian function ahead in the future, it is possible to optimise the expected evolution of the system. L (x(t), u(t)) can be formulated as instance to minimise the time (i.e. it becomes equal to Eq. 2.3), the quadratic trade between state and controls, or the fuel 40 CHAPTER 2. OPTIMAL CONTROL PROBLEM usage spent across the path: L (x(t), u(t)) = 1 Minimum time problem (2.6) L (x(t), u(t)) = xT (t) Q x(t) + uT (t) R u(t) Trade state controls (2.7) L (x(t), u(t)) = ' m(t) Minimum fuel usage (2.8) where m is the mass of the system. 3. Bolza: J = (x(tf )) + tf Z t0 L (x(t), u(t)) dt (2.9) Constraints are applied to the minimisation of the target function, thus ensuring the reachability of a desired state and controllability of the system. The maximum value of the state variables and controls are limited through algebraic equations and inequalities. Ordinary Di'erential Equations (ODE) define how the system and the controls evolve in time. Boundary conditions are specified if the initial state of the system is known, and if it is desired a particular final configuration. In this study, the independent variable is assumed to be the curvilinear abscissa ' representing the position of the mill in the toolpath. This change of coordinate allows to univocally define the position of the tool at each optimisation instant, even for di'erent feed rates. The complete formulation of the OCP is then: 41 CHAPTER 2. OPTIMAL CONTROL PROBLEM 8 > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > : minu('),x(') 'b R 'a J (x('), u(')) d' s.t. ( 'x (') , x (') , u (')) = 0 ODE constraints g (x (') , u ('))  1 One sided constraints 1  h (x (') , u ('))  1 Two sided constraints b(x (')) = 0 Initial constraints e(x (')) = 0 Final constraints (2.10) 2.3 Solution The OCP can be solved through di'erent strategies, which might di'er on how the time is discretised, which numerical scheme is adopted, and how constraints are han- dled. The most common methods are classified as direct and indirect. The former reformulates the OCP as a non linear programming problem (NLP) that is solved with a sequential quadratic programming (SQP). In indirects methods instead the OCP is treated as a Boundary Value Problem (BVP) through the variational calculus. Usu- ally this approach is more complex to implement but it is also more robust and efficient. In this study it is utilised the optimisation library XOptima developed by the group of Mechatronics of the University of Trento. This library exploits the symbolic engine in the software Maple v16.0 to formulate the BVP and to solve in real-time the OCP through an efficient indirect method. An comprehensive description of the algorithms implemented in XOptima is given by Bertolazzi et al. (37), and here it is briefly pre- sented. 42 CHAPTER 2. OPTIMAL CONTROL PROBLEM The inequalities constraints are eliminated with penalty formulation, and the min- imisation of the target function is obtained through an affine invariant quasi-Newton method based on Broyden update formula with globalisation strategy. Barrier functions are defined to eliminate inequality constraints in Eq. 2.10: p1,1(x) = 8 > < > : 0, x  0 1, x > 0 and p2,1(x) = 8 > < > : 0, 1  x  1 1, otherwise (2.11) The OCP then becomes: 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : minu('),x(') 'b R 'a J (x('), u(')) + P k 2I1 p1,1(h(x('), u('))) + P k 2I2 p2,1(g(x('), u('))) d' s.t. ( 'x (') , x (') , u (')) = 0 ODE constraints b(x (')) = 0 Initial constraints e(x (')) = 0 Final constraints (2.12) where I1 and I2 are two di'erent sets of indexes. Lagrange''s multipliers are used to eliminate di'erential-algebraic equality constraints and search for unconstrained minima of Eq 2.12. For sake of notation, dependency of multipliers, state, and controls variable from ' is omitted. The sum of the penalty functions is condensed into a single term called augmented lagrangian term Jp(x, u; p): min u,µ,x , ,' F (u, µ, x, , '; p) = 'b Z 'a ( 'x, x, u) + J(x, u, p) + Jp(x, u, p) d' + µ b(x('a), p('a)) + ' e(x('b), p('b)) (2.13) 43 CHAPTER 2. OPTIMAL CONTROL PROBLEM Jp(x, u; p) is the most important function since it describes the whole OCP. The BVP is then given by the first variation of F (u, µ, x, , '; p) and, by setting y = (x, )T , it is possible to formulate it as: a( 'y('), y('); p(')) d d' b(y('); p(')) = 0 s 2 ( k, k+1) (2.14a) lim l ! k lim ' ! + k j(y(l), y('); p(l), p(')) = 0 k = 1, 2, ..., d (2.14b) h(y(a), u; p(a)) = 0 (2.14c) g(y(b), '; p(b)) = 0 (2.14d) (2.14e) Eq. 2.14a is given from the first variation of Eq. 2.13. Eq. 2.14b is used to regularise internal jumps given when p(') is discontinuous at s points. For ease the solution of the problem, is set 0 = 'a and d+1 = 'b. Eqs. 2.14c and 2.14d are the boundary conditions of the degrees of freedom and Lagrange''s multipliers. These variables can be collected into the vector Z. The Jacobian of the BVP is approximated through a finite di'erence scheme, and a highly non-linear system (Z; P ) is obtained. The solution of (Z; P ) is calculated by minimising k (Z; P )k 2. In XOptima the minimisation is carried out through a damped Newton algorithm with line search. This algorithm requires the computation of the Jacobian matrix Jl = @Z l with l = (Zl; P ), and its scheme can be summarised as: Compute search direction dl by solving : Jl dl = l Update Z : Zl+1 = Zl + 'l dl (2.15) The robustness of the procedure is ensured by the damping factor 'l, which satisfies the Armijo conditions (85) and makes the algorithm affine and invariant. It should be noted that the main fraction of computational resources in indirect methods is required 44 CHAPTER 2. OPTIMAL CONTROL PROBLEM to calculate the Jacobian matrix and its factorisation. To speed up the optimisation, exact factorisation and Broyden update are combined with Newton algorithm: Jl+k+1 = Jl+k + l+k Jl+k Zl+k k Zl+kk 2 ZTl+k (2.16) where l+k = l+k+1 l+k and Zl+k = Zl+k+1 Zl+k. Moreover, when the first guess Z0 is far from the the optimal solution, non-monotone iterations are used (86). 2.4 OCP for milling processes In this study the OCP is formulated as a problem of Bolza, where the Lagrangian function is constituted by a linear combination of di'erent terms taking into account the process productivity Jp (x('), u(')), quality Jq (x('), u(')), and cost Jc (x('), u(')). The dynamical system is controlled in force, therefore the control vector u(') includes the linear acceleration of the axis ' f (') and the angular acceleration of the spindle '!('). The vector of state variables x(') includes the feed rate f (') and the spindle speed !('). Since several performances of the process need to be optimised, the OCP is formulated as a multi-objective minimisation: J (u('), x(')) = Z Z '0 wp Jp (x('), u(')) + wq Jq (x('), u(')) + wc Jc (x('), u(')) d' + + wi '' f ('i) fi fmin ''2 (2.17) The Lagrange term is integrated from the current position of the mill '0 to a future po- sition Z. The Mayer term is imposed for safety reasons, thus ensuring that the optimal feed rate is not too di'erent from the nominal one programmed by the user. Excessively high feed rates must be avoided since can damage the mill, or even break the spindle. 45 CHAPTER 2. OPTIMAL CONTROL PROBLEM Each term of the target function is multiplied by a weight, thus contextualising the OCP to a specific application. In roughing operations the process productivity has the mayor contribute on the Lagrangian function, while process quality is more important during finishing cuts. The costs function is considered if hard metals are processed, since expensive tools might be used and high current consumptions might be required by long tool paths. 2.4.1 Productivity The Material Removal Rate (MRR) quantifies the process productivity. High MRR are achieved by increasing the feed rate, as well as the depth, or width of the cut. The instantaneous MRR is calculated as: M RR(') = cs(') f (') (2.18) where cs(') is the instantaneous cut cross section. All the terms of the target function are normalised in order to have comparable values when all the weights in Eq. 2.17 are unitary. Jp (x('), u(')) is divided by the maximum MRR, which is given by setting the feed rate equal to the maximum feed of the machine axes Eq. 2.18. Moreover, since J (u('), x(')) has to be minimised, the inverse of the normalised MRR is used as descriptor of the process productivity: Jp (x('), u(')) = '' M RR(') M RRmax(') '' 1 = '' cs(') f (') cs(') fmax(') '' 1 = fmax(') f (') (2.19) 46 CHAPTER 2. OPTIMAL CONTROL PROBLEM 2.4.2 Quality The quality of the process is defined by considering the roughness and the geometrical accuracy of the cut, which are a'ected by the cutting force acting and vibrations of the machine. The term associated to process quality is defined by three functions, which take into account: the deflection of the tool when a force is applied on it (Jq,p(')), as well as forced and self-excited vibrations (Jq,f (') and Jq,c(') respectively): Jq,p(') = Jq,p(') + Jq,f (') + Jq,c(') (2.20) Deflection of the tool Cutting forces acting on the tool are calculated by modelling the ram as a cantilever beam, in which the cutting force Fc is applied at the free end, Fig. 2.1. The deflection angle of the ram is: ' = Fc(') l(')2 2 E J (2.21) where E is the elastic modulus of the ram, J its inertia, and Fc(') is the cutting force. Note that the length of the ram l(') is dependent from the curvilinear abscissa since the W axis can be extended or retracted during the process. In order to compute Fc it F c ! ! p r t Ram Tool Figure 2.1: Model of the cantilever beam with the cutting force applied at the free end. 47 CHAPTER 2. OPTIMAL CONTROL PROBLEM is necessary to estimate the cutting power Pc, which depends from the specific energy of the workpiece us: Pc(') = cs(') f (') us = MRR(') us (2.22) Fc(') = Pc(') !(') rt (2.23) By knowing the angle ' and the radius of the tool rt it is calculated the penetration p of the tool into the workpiece that generates epicyclical streaks on the cut surface. The expression of p can be linearised since the tool is sti' and the deflection angle is usually small: p(') = rt sin(') = rt '(') (2.24) p(') is normalised on its maximum value, given by setting Eq. 2.21 equal to the max- imum cross section cut along the tool path, while feed rate and spindle speed are respectively set to the maximum and minimum values supported by the drives. The expression of the function associated to the tool penetration that has to be minimised is: Jq,p(') = p(') pmax(') = cs(') f (') !(') !min csmax fmax (2.25) Self-excited vibrations The vibrations of the machine deteriorate the quality of the process, by leaving periodic patterns on the cut surface. When analysing the dynamics of the cut, it is possible to distinguish between forced and self-excited vibrations. The most dangerous self-excited vibrations are those regenerative, since as the cut pro- ceeds the dynamics of the system becomes unstable, thus increasing progressively the chip thickness and consequently the magnitude of the cutting forces. The regenerative phenomena is due to a phase shift between vibration waves left on the inner and outer surfaces of the cut chip. Regenerative self-excited vibrations are called chatter, and its 48 CHAPTER 2. OPTIMAL CONTROL PROBLEM manifestation is due to many factors that can be grouped into two main categories: process factors (material, tool geometry, lubrication, etc.) and dynamic factors (tool- workpiece relative dynamic compliance). The relationship between stability occurrence and the above-mentioned factors can be analysed through a reduced set of variables by means of two main approaches: the multi-frequency and the single-frequency 0-order approach (ZOA). For non-interrupted milling processes, the single-frequency model allows a fast and accurate calculation of the stability lobes. Due to the fact that heavy-duty milling operations are usually performed with large (75-80%) tool engage- ments and high number of flutes, the created harmonics are usually weak, and the single-frequency method can provide an accurate solution (87). Adopting this latter approach, the stability analysis is traced back to the following characteristic equation of the dynamic system ''machine tool+milling process' (see Appendix B): det ' I + 'A0 (') ' Gtool wp (s |') '' = 0 (2.26) with ' = nt 4' d Kt ' 1 e ( +i!) 2' ' nt '' (2.27) where d is the depth of cut, nt is the number of tool cutters, ( + i!) is the pole of the dynamical system, and ' is the spindle speed in rad/s. A0 is a matrix that takes into account the orientation of the average cutting force w.r.t. feed and normal axes, and depends on tool engagement condition together with tangential and radial cutting pressures (hereafter Kt and Kr, respectively). ' Gtool wp is the relative dynamic com- pliance between tool and workpiece in the complex variable s = + j!, defined w.r.t. the same reference frame of A0. Without lose of generality, since in our application the workpiece is rigid, only the tool compliance is taken into consideration, namely, ' Gtool wp ' ' Gtool. Imposing = 0 and scanning the chatter frequency !, the Eq. (2.27) can be used to 49 CHAPTER 2. OPTIMAL CONTROL PROBLEM map the stability limit w.r.t. depth of cut and spindle speed. The result of this opera- tion is the well-known stability lobes diagram (SLD). On the other side, given a depth of cut and a spindle speed, Eqs. (2.26) and (2.27) can be solved w.r.t. and !. Note that expresses the exponential growth of vibration during unstable cut and can be used as a metric to measure the instability/stability level. Unfortunately, this equation system is transcendent and the solution can be found only through time-consuming nu- merical algorithms. A viable heuristic solution consists in computing the SLD and in evaluating the instability level by the di'erence between the actual depth of cut and the limiting one. In this sense, the SLD itself is used to generate a the penalty function relating instability level to spindle speed. In Eq. (2.26) the compliance matrix ' Gtool depends on curvilinear abscissa not only in terms of tool position along the tool path, but also in terms of feed velocity direc- tion. The latter can be expressed by applying a rotation to the dynamic compliance measured w.r.t. the global machine axes: ' Gtool (' |!) = R (v) G axes tool (' |!) (2.28) with R (v) = 2 4 vx vy vy vx 3 5 (2.29) The dynamic compliance at tool tip must be measured, or estimated, for a given tool and for a suitable number of position samples along the tool path, wherever SLDs have to be computed. Since the measurement of the necessary set of dynamic compliances may be quite onerous, a viable makeshift consists in regressing an adequate mesh of primitive dynamic compliances covering the available workspace (as explained in 6.3). On the other side, the response at tool tip can be derived from that at spindle nose by exploiting the well-known receptance coupling sub-structuring technique (88): in this way the measurements do not have to be repeated whenever the tool is changed. 50 CHAPTER 2. OPTIMAL CONTROL PROBLEM Engagement arcs 1 EX 1 IN v(s) tool 2 IN 2 EX Normal direction Feed direction X Y workpiece TOOL kth slice Figure 2.2: Engagement arcs definition w.r.t. feed velocity. The orientation matrix A0 relates the vibrational displacement of the tool along feed and normal direction to the average cutting force vector due to the consequent material removal. The matrix can be computed by integrating the forces exerted by each cutter engaged in the workpiece. If the cylinder bounding the cutters is considered, tool engagement can be described by the arcs of contact between this cylinder and the workpiece. Generally speaking, the arcs of contact may be more than one for each tool section, and may vary along the tool axis (see Fig.2.2). In order to deal with such inherent complexity, the list of the contact arcs for each tool slice is computed by a dedicated code that exploites commercial 3D CSG libraries (ACIS, developed by Spatial R Corporation). Once the tool engagement is properly described, the matrix 51 CHAPTER 2. OPTIMAL CONTROL PROBLEM A0 is computed as it follows (i.e., the generalisation of the corresponding quantity presented in (53)): A0 = 2 4 aff afn anf ann 3 5 with aff = 1 atot P k a P i 1
2 (cos2 2Kr + Krsin2 ) | EX ki
IN ki afn = 1 atot P k a P i 1
2 ( sin2 2 + Krcos2 ) | EX ki
IN ki anf = 1 atot P k a P i 1
2 ( sin2 + 2 + Krcos2 ) | EX ki
IN ki ann = 1 atot P k a P i 1
2 ( cos2 2Kr Krsin2 ) | EX ki
IN ki (2.30) where atot is the total tool depth engaged in the workpiece, a is the tool discretisation spacing in axial direction, IN ki and EX ki are respectively the initial and the exit angle of the ith arc of the kth tool slice. The approach presented in this study is illustrated in Fig. 2.3 where SLDs are calcu- lated ad each sampled curvilinear abscissa. At each position of the tool, the stability limit dlim, and position of the highest peak dmax in SLD are extracted. Then, the 0 500 1000 1500 0 0.5 1 1.5 2 Spindle Speed S (RPM) Cutting Depth d (mm) Lobes diagram
Set d =0.7 mm
Stable S 0 500 1000 1500 0 5 10 15 20 Spindle Speed S (RPM) Target Function (/) d max d lim Cross section !(') Curvilinear abscissa ' (tool path) Depth of cut y x z Workpiece Figure 2.3: Discretisation of the tool path (right) and computation of the target func-
tion at each quantised position increment (left). 52 CHAPTER 2. OPTIMAL CONTROL PROBLEM term Jq,c(') of the target function necessary to mitigate the chatter is obtained from the intersection of the SLD with the actual cutting depth d('). Three conditions are identified: 1. d(') < dlim : Jq,c(') is zero since all the spindle speed allow a stable cut. 2. d(') > dmax : it is not possible to achieve a stable cut, but in order to limit unavoidable chatter, Jq,c(') is formulated as a cubic spline with the minimum at the spindle speed corresponding to dmax. 3. dlim  d(')  dmax : the abscissa coordinate of the stability pockets are identified and used to generate a set of nodes that are interpolated with a Piecewise Cubic Hermite Polynomial (89). Peaks close to actual cutting depth less than 0.1 mm are neglected in order to deal with the uncertainty of the model parameters used to compute the stability lobes diagrams. Auxiliary nodes with non-null ordinate are placed between the stable spindle speeds to avoid instability pockets. In the end, the minimum and maximum point are characterised by defining each node as a stationary point. A similar approach was followed by Merdol and Altintas (90). The authors developed a process optimisation system to calculate the cutting speed, feed rate, depth and width of the cut which maximised the MRR, while considering the dynamics of the process along the tool path. This methodology however did not take into account the e'ect of forced vibrations, and it was tested only on peripheral milling operations. It is worth to be remarked that ZOA approach assumes that A0 and ' Gtool (! |') do not depend on time, while, as matter of fact, this assumption does not hold in the case of complex tool trajectories generated by a general part program, since tool engagement varies continuously as well as structural machine dynamics. However, the assumption 53 CHAPTER 2. OPTIMAL CONTROL PROBLEM can be considered still valid on the basis of the so-called frozen time approach based on the 'Adiabatic Theorem', also adopted to tackle variable time delay systems (91). It states: A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the system spectrum. (92). If, in a time window T , the variation of the product A0 ' Gtool is negligible and T is much greater than the tooth passing period, the system dynamics (whose pole is given by ( + i!) /' ) is much more faster than the perturbation action. The time-varying A0 ' Gtool can then be assumed to be frozen in the interval T , i.e.: A0 ' Gtool (' (t)) ' const. for t 2 T (2.31) Since the chatter frequency is a function of spindle speed, the variability of ' Gtool must be evaluated for a suited range of frequency, namely, in the neighbourhood of the resonance that are candidate to be source of chatter problems. Algorithmically, the following expression is proposed to verify the consistency of the frozen time assumption: ||A0 (' (tk+1)) ' Gtool (!, ' (tk+1)) A0 (' (tk)) ' Gtool (!, ' (tk)) || ||A0 (' (tk)) ' Gtool (!, ' (tk)) || < ', (2.32) 8! 2 Fc and tk+1 tk > M'TPF where ' (tk) and ' (tk+1) are two subsequent curvilinear abscissas, 'tpf is the tool passing frequency, ' and M are two parameters to be set after proper experiments, and Fc is the union of neighbourhoods of the relevant tool tip resonances. In case the (2.32) is not verified, the stability analysis is unreliable and the corresponding term in the global objective function must be neglected. The last issue to be tackled concerns the rate of the stability analysis that must be performed along the curvilinear abscissa during the process simulation. In order to increase the computational efficiency, the SLDs should be updated and stored only if necessary. Only the eigenvalue ' varies along the tool path (due to changes in 54 CHAPTER 2. OPTIMAL CONTROL PROBLEM tool engagements and dynamics), this quantity itself can be used to drive the SLDs evaluation density. The SLD is then re-calculated and stored if the eigenvalue change in two consecutive positions is more than a given threshold: 1 N X i |'k (!i) 'k+j (!i) | > ' with !i 2 {!1, !2, ..., !N} (2.33) where k refers to the last abscissa sample corresponding to a new SLD, j is the sample lapse till the current abscissa, !i are the chatter frequency samples on which ' is calculated, and ' is a parameter to be set on the basis of a trade-o' between precision and computational power. Finally, SLDs are used only in a spindle speed range which is feasible from a technological point of view. Forced vibrations In spite of process stability, even forced vibrations could be significant and detrimental, so they must be considered in the optimisation objective as well. Forced vibration level is evaluated by considering the response of the tool tip to the dynamic components of cutting force, that can be computed from tool engagement and cutting parameters (in a similar way than the average component, as explained in (93)). Starting from the ideal cutting mechanics, the dynamic cutting force components in frequency domain are harmonics of the tool passing frequency. The rth harmonic can be written as it follows: f (r) dyn (' ) = 1 4' Kt atot N (') f (') 8 < : a (r)
f f a (r)
nf 9 = ; (2.34) 55 CHAPTER 2. OPTIMAL CONTROL PROBLEM with a (r)
f f = 1 atot P k a P i j
2 c0Kre jrNcut + c1e jp1 c2ejp2 | EX ki
IN ki a (r)
nf = 1 atot P k a P i 1
2 c0e jrNcut + c1e jp1 + c2ejp2 | EX ki
IN ki (2.35) where p1 = 2 + rNcut, p2 = 2 rNcut, c0 = 2 rNcut , c1 = Kr j p1 and c2 = Kr+j p2 , with j denoting the imaginary unit (the other symbols have been defined after (2.30)). The number of relevant harmonics (r) of the tooth passing frequency (!tpf ) to be considered for an accurate reconstruction of the dynamic cutting force components depends on the immersion conditions and on the number of teeth in the cut. Finally, the vibration index can be identified with the magnitude of the resulting tool tip vibrational displacement: Fforced = || X r 2([ R,R] {0}) ' Gtool (r!TPF |') · f (r) dyn|| (2.36) where R is the number of harmonics to be considered (usually 3 ÷ 5). 2.4.3 Costs In a constant cutting-speed, rough-machining operation, the production cost Cpr per component can be calculated by (94): Cpr = Cm + M tl + M tm + (M tct + Ct) tm t (2.37) where Cm is the material cost, M is the total machine and operator rate, tl is the nonproductive time, tm is the machining time, tct is the tool changing time, and Ct is the cost of providing a sharp tool. The minimum cost of production can be calculated 56 CHAPTER 2. OPTIMAL CONTROL PROBLEM for each operation, through Taylor''s tool life equation: C = vc t n (2.38) vc = 2 ' rt s = 0.06 ! r (2.39) where C is a constant, vc is the cutting speed, s and ! is the spindle speed in rpm and rad/s respectively, rt is the radius of the tool, n depends on the cutting conditions, and t is the cutting time. The machining time tm is: tm = K vc (2.40) K is a constant, di'erent for each operation, that can be interpreted as the distance traveled by the tool cutting edge corner relative to the workpiece. By substituting Eqs. 2.38 and 2.40 into Eq. 2.37, after some calculations that are omitted for brevity, the minimum cost of production Cmin is: Cmin = M tl + M K (1 n) vr '' tc
tr ''n (2.41) Cmin depends on the type of the operation carried out, as well as on the tool wear. The former is a constant that cannot be optimised if not by redesigning the whole process. Tool wear instead can be reduced with an accurate selection of process parameters. Taylor''s wear model has been formulated mainly for turning operations, and it is too simple for milling processes since it does not take into account the cut cross section. The wear model 2.38 is therefore extended, and tool wear wt is calculated from the multiplicative inverse of the tool life ttl, that is expressed in minutes: ttl(') = nt '' C vc(') fz(')a cs(')b '' 1 n (2.42) 57 CHAPTER 2. OPTIMAL CONTROL PROBLEM fz(') = f (') nt s(') = 2' 60 f (') nt !(') (2.43) wt(') = 1 ttl(') (2.44) where nt is the number of flutes in the mill, fz(') is the instantaneous feed per tooth, while C, n, a and b are the tool''s wear coefficients. wt(') is normalised on the maximum tool wear, given when the spindle speed is minimum, and the cross section together with feed rate are maximum. The final expression of the normalised tool wear that has to be minimised is: Jc,w(') = wt(') wtmax = " !(') rt !max rt '' f (') nt!(') nt!max fmax '' '' cs(') csmax ''b# 1n (2.45) The model in Eq. 2.37 lacks in considering the energy spent during the process. Energy consumption is a factor with respect to which CNC users have grown interests in the last years, due to the raise of energy price and more strict environmental regulamentations. During a cut, electric current is mainly absorbed by the spindle, which rotates at high speeds. Electric energy absorbed by a motor is proportional to the kinetic energy developed. It means that the rotational kinetic energy of the spindle can be used as an estimator of energy consumption: Ee / Ek = 1
2 I !2 (2.46) J is the moment of inertia of all the kinematic chain from the tool to the electric drive. Once Ek it is normalised on the maximum kinetic energy developable by the spindle, the target function associated to energy consumption is defined as: Jc,e(') = Ee Emax = '' !(') !max '' (2.47) 58 CHAPTER 2. OPTIMAL CONTROL PROBLEM Finally, Eqs. 2.45 and 2.47 are summed to form Jc('): Jc(') = Jc,w(') + Jc,e(') (2.48) 2.4.4 DAE constraints Algebraic constraints are imposed in order to calculate controls whose values are within the performance of the CNC machine, and respect safety conditions. One sided con- straints take into account the maximum mechanical torque and power generated by the spindle: 2 4 Tc(x('), u(')) Pc(x('), u(')) 3 5  2 4 Tmax Pmax 3 5 (2.49) where Tmax = 57.3 Nm, and Pmax = 11 kW. A third inequality is added to Eq. 2.49 in order to limit the roughness of the cut. The maximum feed rate is calculated through an heuristic relationship taking into account the tool fillet radius rf = 0.8 mm. This equation is suggested by the tool manufacturer: f (')  0.6 2' rf !(') (') (2.50) the (') function is zero for null cut cross sections, otherwise its value is one. By doing so, it is possible to include this extra constraint without a'ecting the process productivity during in-air fast movements. Two sided constraints are set to avoid excessively high feed rates and spindle speeds, but also to void dangerous situations, such as a non rotating mill impacting against 59 CHAPTER 2. OPTIMAL CONTROL PROBLEM the workpiece: 2 4 !min fmin 3 5  2 4 !(') f (') 3 5  2 4 !max fmax 3 5 (2.51) where !min = 10.5 rad/s, !max = 157 rad/s, fmin = 0 mm/min, and fmax = 10000 mm/min. The last set of constraints included into the OCP take into account the dynamics of the machine. From preliminary tests, it has been observed that the dynamics of the state variables is fast, instead the controls are limited in acceleration. Feed rate and spindle speed velocity are modelled as a first order system: 2 6 4 f (') d d' f (' ) vf (') f (') d d' ! (' ) v ! (') 3 7 5 = 2 4 0 0 3 5 (2.52) 60 CHAPTER 3. SIMULACRUM Chapter 3 Simulacrum Electrical Cabinet Control Panel Dynamometric table and accelerometer Workpiece Cutting head Ruby CORBA EPC Network switch CN Compact RIO Milling machine Alesamonti MB-63 Bidirectional bus RJ45 bidirectional RJ45 monodirectional Measurements from the dynamometric table Measurements from the accelerometer Figure 3.1: Architecture of the simulacrum. The simulacrum developed to test and evaluate the Evaluation and Perception Con- troller (EPC) is schematised in Fig. 3.1. The system is constituted by three elements: 61 CHAPTER 3. SIMULACRUM NC milling machine, measurement system, and EPC that is installed on an industrial fan-less computer with a Dual Core Atom 1.6 GHz CPU. A Local Area Network (LAN) has been created in order to allow the communication between the three subsystems of the simulacrum. The state of the process is reconstructed through the data recorded by the NC and by the CompactRIO. The latter is used to acquire a triaxial accelerom- eter on the cutting head, and a triaxial dynamometric table on which the workpiece is clamped. EPC and NC are communicating through a bi-directional connection, which is bridged by the user panel of the machine. Each hardware and software component of the system is described in this chapter, with exception of the communication layer which will be outlined and characterised in Chapter 5. 3.1 Milling machine Alesamonti MB63 The five-axis milling machine Alesamonti MB63 is used as simulacrum to test and validate the EPC. According to the norm ISO 3070 part I, the MB63 is classified as a machine with a fixed column and movable table, Fig. 3.2. The technical drawings of the structure are reported in Appendix E. Movements along X and Z axes are associated to the working table and workpiece. The cutting head is moved vertically along the Y axis, while B and W are used to turn the working table and to extract the ram respectively. The three linear axes are moved through recirculating ball screws, and they are actuated by brushless motors. Digital drives Fidia XPower are used to control the electric motors of axes and spindle. The technical specifications of the machine are summarised in Tabs. 3.1 and 3.2. The kinematic chain of the cutting head in Fig. 3.3 is analysed to characterise the cutting torque at the spindle (as it will be explained in Chapter 6.2.2). In Tab. 3.3 are reported the geometrical information of all the shafts. 62 CHAPTER 3. SIMULACRUM MICHELANGELO
Codice Progetto: MI01_00237 '' Deliverable T1.1 File: T1-1 Deliverable - Definizione specifiche simulacro r00.doc Page 10 of 44 2. DEFINIZIONE DEL SIMULACRO: MACCHINA 2.1 Descrizione generale della macchina
Per simulacro si intende una vera e propria macchina utensile in grado di effettuare lavorazioni per asportazione di truciolo ed in particolare operazioni di fresatura, alesatura e foratura.
Nell'ambito della produzione di ALESAMONTI è stata individuato un modello che per dimensioni e
potenze può sicuramente soddisfare le esigenze richieste dal progetto di ricerca. Il modello
adottato, denominato MB63, è un'alesatrice a montante fisso con tavola girevole. Dal punto di vista
normativo la configurazione della macchina MB63 fa riferimento alla norma ISO 3070 Parte 1:
macchine con montante fisso e tavola girevole; la figura 2.1 riporta la configurazione della
macchina con l'indicazione degli assi della macchina come previsto dalla norma. Figura 2.1: configurazione della macchina
La macchina è caratterizzata dalle seguenti strutture portanti: - il basamento (fusione in ghisa); - il montante solidale al basamento (fusione in ghisa); - il carro trasversale che corre sul basamento (fusione in ghisa); - il carro porta-tavola che corre sul carro trasversale (fusione in ghisa); - la tavola che ruota sul carro porta-tavola (fusione in ghisa); e dalla seguente struttura portata: - la testa che scorre sul montante (fusione in ghisa).
La macchina MB63 ha cinque assi controllati. Gli assi sono così denominati: 1) asse X : movimento trasversale della tavola 2) asse Y : movimento verticale della testa 3) asse W : movimento longitudinale del cannotto 4) asse Z : movimento longitudinale della tavola 5) asse B : movimento rotatorio della tavola N. Italian English French German 1 banco bed banc Maschinenbett 2 montante column montant Machinenstander 3 testa spindle head chariot porte-broche Spindelstock 4 slitta trasversale table saddle trainare Bettschlitten 5 tavola table table Aufspanntisch 6a mandrino di alesatura boring spindle broche d''alesage Bohrspindel 6b mandrino di fresatura milling spindle broche de fresage Frässpindel MICHELANGELO
Codice Progetto: MI01_00237 '' Deliverable T1.1 File: T1-1 Deliverable - Definizione specifiche simulacro r00.doc Page 10 of 44 2. DEFINIZIONE DEL SIMULACRO: MACCHINA 2.1 Descrizione generale della macchina
Per simulacro si intende una vera e propria macchina utensile in grado di effettuare lavorazioni per asportazione di truciolo ed in particolare operazioni di fresatura, alesatura e foratura.
Nell'ambito della produzione di ALESAMONTI è stata individuato un modello che per dimensioni e
potenze può sicuramente soddisfare le esigenze richieste dal progetto di ricerca. Il modello
adottato, denominato MB63, è un'alesatrice a montante fisso con tavola girevole. Dal punto di vista
normativo la configurazione della macchina MB63 fa riferimento alla norma ISO 3070 Parte 1:
macchine con montante fisso e tavola girevole; la figura 2.1 riporta la configurazione della
macchina con l'indicazione degli assi della macchina come previsto dalla norma. Figura 2.1: configurazione della macchina
La macchina è caratterizzata dalle seguenti strutture portanti: - il basamento (fusione in ghisa); - il montante solidale al basamento (fusione in ghisa); - il carro trasversale che corre sul basamento (fusione in ghisa); - il carro porta-tavola che corre sul carro trasversale (fusione in ghisa); - la tavola che ruota sul carro porta-tavola (fusione in ghisa); e dalla seguente struttura portata: - la testa che scorre sul montante (fusione in ghisa).
La macchina MB63 ha cinque assi controllati. Gli assi sono così denominati: 1) asse X : movimento trasversale della tavola 2) asse Y : movimento verticale della testa 3) asse W : movimento longitudinale del cannotto 4) asse Z : movimento longitudinale della tavola 5) asse B : movimento rotatorio della tavola N. Italian English French German 1 banco bed banc Maschinenbett 2 montante column montant Machinenstander 3 testa spindle head chariot porte-broche Spindelstock 4 slitta trasversale table saddle trainare Bettschlitten 5 tavola table table Aufspanntisch 6a mandrino di alesatura boring spindle broche d''alesage Bohrspindel 6b mandrino di fresatura milling spindle broche de fresage Frässpindel Figure 3.2: Configuration of the machine MB63. Table 3.1: Technical specifications of the MB63. Data X Y Z W B Travel 729 mm 556 mm 734 mm 369 mm 360 Max. speed 10000 mm/min 10000 mm/min 10000 mm/min 5000 mm/min 720 /min Stall torque 20 Nm 13 Nm 20 Nm 13 Nm 20 Nm of the motor
Max. speed 3000 rpm 3000 rpm 3000 rpm 3000 rpm 3000 rpm of the motor Table 3.2: Technical specifications of the spindle. Data Value Brand Siemens Model 1PH7107-2NF02-0CA3 Max. speed 1500 rpm Power in S6 (60%) 11 kW Constant torque of the torque 57.3 Nm Constant torque in high range 72,77 Nm Cone type ISO 40 DIN 2080 Diameter of the boring spindle 70 mm 63 CHAPTER 3. SIMULACRUM Figure 3.3: Kinematic chain of the cutting head. Table 3.3: Technical specifications of the shafts in the kinematic chain, Z is the number
of teeth of the gear connected to the shaft, and D is the diameter of the pulley. The
missing information can be extracted from the CAD drawings in Appendix E. Number Description Material 1 spindle with gear, Z = 58 Steel 18 NiCrMo5 2 Shaft, Z1 = 38 and Z2 = 61 Steel 18 NiCrMo5 3 Shaft, Z1 = 21 and Z2 = 61 Steel 18 NiCrMo5 4 Shaft, Z1 = 21 and Z2 = 20 Steel 18 NiCrMo5 5 Shaft, Z1 = 35 and Z2 = 25 Steel 18 NiCrMo5 6 Shaft, Z1 = 30 and Z2 = 30 Steel 18 NiCrMo5 7 Shaft, Z1 = 24 and D2 = 140 mm Steel 18 NiCrMo5 and GG25 DIN 1691 8 V-belt, long 375 mm and width 16 mm polyamide 9 Pulley D = 100 mm GG25 DIN 1691 10 Electric motor 1PH7107-2NF01-0CA3 See data sheet 64 CHAPTER 3. SIMULACRUM 3.2 Numerical Control FIDIA C20 The numerical control installed on the simulacrum is the FIDIA C20. The architecture of this unit is characterised by two processors dedicated to di'erent tasks: a RISC Power PC is utilised for axis and PLC management, while the PC side is aimed at managing the user interface. Thanks to this architecture it is possible to perform even onerous operations on the user panel without interfering with the ongoing machining. 3.3 Sensors The process is sensed not only through the information available in the NC, but also through a dynamometric table and a triaxial accelerometer, Fig. 3.4. These sensors are acquired through a National Instrument CompactRIO cRIO 9012 with a reconfig- urable chassis NI 9103. The cRIO has a FPGA module which is used to sample at high frequency (5 kHz) the input/output modules to which the sensors are connected. The accelerometer is plugged to a NI 9234 module, while the dynamometric table is connected to a NI 9215 module. The specifications of the sensors are summarised in the next sections. The collected information is elaborated, and sent to the EPC at a low frequency (5 Hz), in order to have a communication that is synchronous with the optimisation loop run- ning at 5 Hz. The software loaded on the cRIO has been developed by the group of mechatronic engineering of the university of Trento. 65 CHAPTER 3. SIMULACRUM CompactRIO NI 9022 Chassis
NI 9103 Acquisition module NI 9215 Acquisition module NI 9234 Charge Amplifier Kistler 5073A311 Force links Kistler 9347C Accelerometer Dytran 3216M6 x y z x y z x y z x y z Ethernet interface with the EPC Figure 3.4: Scheme of the measurement system. 66 CHAPTER 3. SIMULACRUM 3.3.1 Dynamometric table The dynamometric table consists of three piezoelectric Kistler 9347C 3-component force links on top of which is fixed a customised working table, Fig. 3.5. The speci- fications of the working table, and of the force links are summarised in Tab. 3.4 and in Tab. 3.5 respectively. A charge amplifier Kistler 5073A311 is used to convert the charge measured by the piezoelectric sensors into a voltage signal, Tab 3.6. Note that the three components of the force measured by each dynamometer are analogically summed, thus reducing the number of acquired channels to three. By doing so it is possible to cut the cost of the system, since only one charge amplifier and one NI 9215 module are needed. Working table Force link Force link Force link Figure 3.5: The dynamometric table. 67 CHAPTER 3. SIMULACRUM Table 3.4: Technical specifications of the working table. Parameter Value Weight 300 kg Dimensions [L x W x H] 600 mm x 600 mm x 60 mm Width of T-slots 14 mm Flatness tolerance 0.01 mm Parallelism tolerance between 0.01 mm upper and lower planes 0.02 mm Table 3.5: Technical specifications of the piezoelectric force links Kistler 9347C. Parameter Value Range along X and Y ±15 kN Range along Z ±30 kN Overload 10% Sensibility along X and Y -7.8 pC/N Sensibility along Z -3.7 pC/N Linearity  ±0.5 % FSO Table 3.6: Technical specifications of the charge amplifier Kistler 5073A311. Parameter Value Range ±100 pC, ±1,000,000 pC Output voltage 0 V, ±10 V Maximum output current ±5 mA Measurement accuracy  ± 0.5% 68 CHAPTER 3. SIMULACRUM 3.3.2 Accelerometer The technical specifications of the Integrated Electronic PiezoElectric (IEPE) three- axial accelerometer Dytran 3213M6 are reported in Tab. 3.7. Table 3.7: Technical specifications of the accelerometer Dytran 3213M6. Parameter Value Range ±50 g Sensitivity ±10% 100 mV/g Frequency response ±10% 1.5 to 5000 Hz Linearity ±1% FSO Maximum trasversal sensitivity 5% 69 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER Chapter 4 Evaluation and Perception Controller The Evaluation and Perception Controller is constituted by three elements: a simulator, the optimisation library, and the perception layer (Fig. 4.1). In the process simulator a simplified Numerical Control (RNC) analyses the tool path. The G-Code can be generated either manually or through a CAM software, then it is loaded and parsed by the RNC. This software calculates also the feed profiles for every block in the G- Code (i.e. G01, G02, and G03), and interpolates the tool trajectory with the desired sample time. The parsed tool path, together with a model of the kinematic of the machine, workpiece and tool geometries, is fed into a geometric kernel in order to compute the cut cross section, cutting forces and stability lobes diagram along the curvilinear abscissa. The simulation of the cut is the most time consuming task since it requires the continuous update of the workpiece geometry at each simulated tool position. Once all the blocks of G-code are simulated, the generated models are used to perform an o'ine optimisation, in which the optimal feed rate and spindle speed profiles are computed by solving an Optimal Control Problem. When the optimisation 70 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER Figure 4.1: Architecture of the EPC. converges, the process is started and the nominal controls programmed into the G-code are overridden with the pre-optimised ones. During the cut, the data recorded by the NC and sensors in the working area are utilised by a perception layer to reconstruct and learn the state of the process. By having the current status of the system, it is possible to maintain constantly updated the models of the process and to re-optimise online the controls. Through the online loop, deviations of the process state from the predictions made during the o'ine optimisation can be compensated. As instance, it is possible to correct the controls profiles by estimating online the specific energy of the cut material, whenever a workpiece is harder (or even softer) than expected. It must be remarked that the o'ine optimisation is essential to have an online adaptive control, since the pre-calculated optimal control ease the convergence of the online optimisation. The o'ine optimal controls are used as a first guess during the online 71 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER optimisation, and are applied to the system whenever the online optimisation struggles to converge. The EPC is implemented as a state machine, whose architecture is schematised in Fig. 4.2. This code structuration allows the software to be highly flexible, expandable, and easily maintainable. Whenever a new functionality has to be added to the system, it is only necessary to implement a new state and to re-define the tree structure of the state machine by specifying the dependences between the new state and the already existing ones. The implemented states are the following: Figure 4.2: State machine of the EPC. 72 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER 1. StateParse: the G-code is loaded, parsed, and the velocity and positions profiles are generated. 2. StateSimulate: the instantaneous MRR, cutting force, and SLD are calculated by computing the cut cross section along the toolpath. 3. StateO'LineOptimise: the OCP is solved o'ine. 4. StateCRioConnect: the UDP connection with the CRio is initialised. 5. StateCncConnect: the CORBA connection with the CNC is initialised. 6. StateWait: the state machine waits until the user starts the execution of the part program on the NC user pannel. 7. StateRun: the optimal controls are communicated to the NC, the sensors dis- played in the working area are acquired, and the current status of the machine is reconstructed. 8. StateOnlineOptimise: with the updated model of the process the controls are re-optimised online. 9. StateStop: the EPC is disconnected from the CNC and CRio, and a log file with all the records of the process is stored. 10. StateError : errors risen during the process are caught and the system is shut down. This is a particularly important state since it increases the reliability of the system during critical situations. In the next sections are described the functionalities and features of the simulator, OCP solver, and perception modules of the EPC. 73 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER 4.1 Process simulator The process is simulated at two levels: at first the tool path is reconstructed by analysing the programmed part program, then the cut is simulated by calculating the cut geometry and the dynamics of the cut. 4.1.1 Ruby Numerical Control The numerical control implemented into the EPC is a simplified CNC software de- veloped by the Authors (95). The numerical control software is written in the Ruby language1 (and thus called RNC, Ruby Numerical Control ). It supports a subset of the ISO G-Code language, and implements acceleration-limited velocity profiles, as well as three-axes linear and circular interpolation. Linear (G01) and circular (G02 and G03) interpolation commands are directly imple- mented into the RNC software, while rapid positioning blocks (G00) and other non- machining instructions are not of interest for optimisation of spindle speed and feed rate, and are thus transparently passed through the parser (no-op). The reference machine tool architecture adopted is that of a 3-axes milling machine tool with the kinematic chain depicted in Fig 3.2. Generation of Feed Velocity Profiles The profiler module cited in the previous section has the role of calculating a profile (or time-sequence) of tool tangent velocity (or instant feed rate) for each G01, G02, or G03 block, given the maximum available accelerations of the axes and the nominal feed rate specified in the G-Code. Once the feed velocity profile is known, it is used 1http://www.ruby-lang.org 74 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER Figure 4.3: Trapezoidal (acceleration-limited) feed velocity profile. Short blocks can
result in triangular profiles. Ticks on time axis represent the quantisation time tq to calculate the time history of the axes positions (operation that is usually called interpolation (96)). There are numerous available approaches to the problem of feed velocity profiles gen- eration, which usually try to provide profiles that minimise jerk and path following errors. The simplest model of profiler, though, assumes that the tool feed velocity goes to zero at the beginning and at the end of each block, and that the nominal feed rate fm is reached within each machining block after a constant acceleration step and before a constant deceleration step. The resulting profile has the trapezoidal shape depicted in Fig. 4.3. In case of short blocks, the acceleration and deceleration steps could not allow to reach the nominal feed rate fm, and in these cases the resulting feed velocity profile is triangular. It is worth noting that the resulting piecewise profile has discontinuous acceleration (and thus infinite jerk) on the profile corners. This is usually an undesir- able condition, since high jerk values cause vibrations that can harm the surface finish and the tool itself. Also, the condition of null velocity at the nodes between blocks can result in an actual feed rate significantly and needlessly reduced with respect of the nominal one when a sequence of very short, almost collinear positioning blocks is 75 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER performed. Nevertheless, in order to evaluate the e'ectiveness of process optimisation by continuous variation of the feed rate (and spindle speed), neither the jerks nor the case of short collinear blocks are of concern. To compute a trapezoidal velocity profile, the time duration of the two acceleration and deceleration steps, t1 and t2, are estimated as: t1 = fm/A (4.1) t2 = fm/D (4.2) where A and D are the maximum acceleration and deceleration available, and fm is the nominal feed rate. The integral of the velocity profile is equal to the block length L, which is defined by the G-Code: t Z 0 f (' )d' = L (4.3) where t = t1 + tm + t2. Consequently, the value of tm is: tm = L fm t1 2 t2 2 (4.4) Since the axes drive controllers work on a discrete time basis, the trapezoidal velocity profiles are evaluated at discrete time steps tk = k tq k 2 N, where tq is the quantisation time, thus the block duration t is a multiple of the quantisation time. This quan- tisation is obtained by extending tm up to its nearest upper quantised value, t'm, keeping constant t1 and t2, and reducing fm to a value f 'm that satisfies Eq. 4.3. In 76 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER formulas: t' = ceil( t/tq) tq = t1 + t'm + t2 (4.5) t'm = tm + ( t' t) (4.6) L = t' Z 0 f (' )d' = f 'm( t1/2 + t'm + t2/2) (4.7) f 'm = 2L t1 + t2 + 2 t'm (4.8) 0 < a = f 'm t1 < A (4.9) 0 > d = f 'm t2 > D (4.10) where a and d are reduced acceleration and deceleration as a consequence of the re- duction of fm down to f 'm. Assuming that each profile starts at t0 = 0, the resulting velocity profile is: f (t) = 8 > > > > > < > > > > > : a t t < t1 f 'm t1 < t < t1 + t'm fm d(t t1 t'm) t > t1 + t'm (4.11) Note that for short tool movements, the accelerations A and D might not allow the tool to reach the desired feed rate fm. In this case, which occurs when t'm < 0, the velocity profile is triangular. Its calculation is performed from analogous considerations made on the trapezoidal profiles, and it is omitted here for brevity. 77 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ' @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) λ(t) Figure 4.4: Function (t) evaluated with a sampled time of 0.01 s, t = 1.2 s, and maximum accelerations A = D = 5 m/s2; vertical dashed lines limit the constant feed rate zone Interpolation of Axes Position The velocity profile evaluated by the Eq. 4.11 is analytically integrated and normalised by the block length L, thus obtaining the function (Fig. 4.4): (t) = 1 L t Z 0 f (' ) d' (4.12) where t is the time elapsed since the block start, and where it has to be noted that (0) = 0 and ( t') = 1. Let ~s0 = (x0, y0, z0) be the vector with the three components of the tool position when the current block starts, and ~s = ( x, y, z) the vector of the total travel made 78 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER Figure 4.5: Representation of circular interpolation. during the current block interpolation. The instantaneous tool position ~s(tk) for a linear interpolation is evaluated as: ~s(tk) = ~s0 + ~s (tk) (4.13) The RNC can also perform circular interpolations with rotation around the three carte- sian axes. As instance, if the rotation is around Z at the k-th time instant the z(tk) position is evaluated with the corresponding Eq. 4.13, while x(tk) and y(tk) are: x(tk) = xc + R cos('0 + ' (tk)) y(tk) = yc + R sin('0 + ' (tk)) (4.14) where '0 is the phase of the vector joining the arc centre C and the start point O of the block, ' is the angle included by the interpolated arc, and R is its radius (see Fig. 4.5). 79 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER 4.1.2 Cut simulator A cut simulator is included into the EPC to calculate the Material Removal Rate (MRR), cutting force, and Stability Lobes Diagrams (SLD) along the curvilinear ab- scissa of the tool path. The simulator has been implemented in C/C++ by the group of Mechatronic Engineering of the University of Trento, and it is based on the Spatial ACIS c geometric modelling kernel. The main advantage of the ACIS library is its capability to compute intersections between complex workpiece and tools geometries moving along complicate tool paths. Tool and the raw workpiece are modelled as 3D solids and are reinterpreted as a series of canonical geometries (circles, lines, planes), which are parameterised into polynomials. Matematically, the position of these ge- ometries in the 3D space is represented by a non linear system of algebraic equations, whose solution gives the intersections. The tool path is discretised into a finite number of steps, at which it is calculated the interference, i.e. the engagement, of the mill into their workpiece. The engagement is described by the cut cross section, as well as by the entry and exit angles of the cutters. Once the cut material has been calculated, the geometry of the workpiece is updated, the mill is placed at the next position along the curvilinear abscissa, and the computation of the intersections is repeated. The information about the engagement is used to calculate the cutting force vector, and the stability lobes diagram. Both are given by the methodology proposed by Altintas and Budak (53) (see Appendix B). 4.2 OCP solver The scheme of the OCP solver is in Fig. 4.6. The OCP is formulated in XOptima c , an optimisation library developed by the Mechatronic Engineering group of the University 80 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER Figure 4.6: Architecture of the OCP code generator and solver. of Trento. XOptima is installed as an add-on package for Maple c software, which is used to develop the equations of the model of the dynamical system. Target function, DAE constraints, and initial conditions are processed and formulated as a Boundary Value Problem (BVP); the OCP is then automatically converted into a C++ code by the Maple CodeGeneration package. Thanks to the followed approach, it is possible to define external C++ functions and include them into the BVP formulation, thus easing the integration of complex models into the OCP. The BVP is compiled and linked to the OCP solver library Mechatronix c , imple- mented in C++ by the Mechatronic Engineering group of the University of Trento. Once the OCP is solved, the controls are communicated to the NC through the CORBA infrastructure. 4.3 Learning layer The online optimisation is performed once the instantaneous status of the process has been reconstructed. Such information is given by the set of data recorded from NC, 81 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER and sensors displayed in the working volume. In order to have a robust and reliable state reconstruction, it is necessary to fuse all the collected measurements. The parameter with respect to which the model of the process is more sensitive, is the specific energy of the cut material us. It is defined as the energy required to cut 1 mm3 of material, and it is calculated from the Eq. 2.22: us = Pc M RR (4.15) Particular attention should be given to the adopted unit of measurements since the material removal rate is expressed in mm 3/min, while us is expressed in J/mm3. The cutting power can be estimated online, by measuring the spindle speed and the cutting torque directly from the NC. In this case the specific energy becomes: us = ! Tc M RR (4.16) The cutting torque is measured from the feedback loop on the spindle speed. As a consequence, it tends to be noisy and change significantly when the cutting force changes abruptly. A filter is used to improve the estimation of the instantaneous us. The first step consists in individuating the sections of the tool path where the measurements and the process are stable. The estimation of us is considered valid when three indicators reach a true value: 1. D0 is an index defined on the measurement percentage error '% of us. It avoids to learn values of the specific energy that are too di'erent with respect to those reported in literature and used for the o'ine optimisation. At the k-th position 82 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER of the mill, D0 is calculated as: D0,k = 8 > < > : 1 if '%,k  25% 0 otherwise (4.17) '%,k = |¯us,k usn | usn (4.18) ¯ us,k = N P i=0 us,k 1 N (4.19) where usn is the nominal specific energy, us,k is the measured one, and ¯ us,k is the average of the last N measurements. The latter takes into account not only the most recent measurement, but also its records. The number of samples to be considered, i.e. the amount of time to which look backward in the history, is N = 15. It corresponds to 3 s of cut, since the sampling frequency is 5 Hz. This amount of time has been chosen as a compromise between the time required to ensure that the cut reaches a stable condition, but also to avoid to include an excessive delay on the state estimation procedure. 2. D1 estimates the variation of us along the curvilinear abscissa, such that local modifications of the cutting torque due to noise of measurement or transitory events ''e.g. when the mill enters into the workpiece'' do not concur to estimate the specific energy. On contrary, variations of the material properties ''e.g. due to local thermal treatments'' modify the cutting torque with slower dynamics, and this event is captured and learnt by the EPC. D1 is true whether the specific us does not double its value in 1 mm of travelled tool path: D1,k = 8 > < > : 1 if us ' k  us 0 otherwise (4.20) 83 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER us ' k = us,k us,k 1 's,k 's,k 1 (4.21) 3. D2 estimates the curvature (i.e. the second derivative) of the specific energy with respect to the curvilinear abscissa. This quantity is used to evaluate the trend of us. When the second derivative is large, the jerks on the spindle tend to be high, therefore in such instants the measurement is discarded. D2 is defined as: D2,k = 8 > < > : 1 if 2us '2 k  us 0 otherwise (4.22) The threshold us is set by assuming that the measured cutting torque is stable if the specific energy increases less than twice its value in 1 mm2 of cut cross section. The second derivative is calculated by the backward di'erence quotient: 2us '2 k = 1 ' '' us ' k us ' k 1 '' (4.23) When D0, D1, and D2 are true (i.e. 1) it is available a measurement of us enough robust to be integrated into the sensor fusion layer. According to the considerations here made, when the process is stable it is assumed that the noise of measurement: has null mean, has a power spectral density (PSD) uniformly distributed along the whole frequency band considered, is normally probability distribution, and the samples are uncorrelated along the time. It is so assumed that the measurements are a'ected by white noise. These assumptions are needed in order to utilise the state estimators commonly used in literature. 84 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER 4.3.1 Sensor Fusion The most famous algorithms used for the sensor fusion are the Kalman Filter (KF) (10) and the Particle Filter (PF) (11). Kalman filter is used in this study because it is an efficient algorithm and it is easy to implement in the scalar case. KF requires that the model of the process is linear with respect to the state variables. Such condition is not satisfied by the model in Eq. 4.16, where the state is constituted by the cutting torque, MRR, and spindle speed. A common solution to this problem is to linearise locally the non linear model, or to find a proper transformation of the state with respect to which the model becomes linear. These approaches are called Extended Kalman Filter (EKF) and Unscented Kalman Filter respectively (97; 98). In the EPC it is used a modified version of KF, where the prediction phase is substituted by the value of specific energy known from the literature. By doing so it is avoided that the learnt us diverges excessively from its reference value. During a cut indeed, the eventuality in which us changes significantly with respect to its nominal value might be due to an improper setup of the OCP, therefore it represents an extremal condition which must be handled not by correcting the estimated us, but rather by redesigning the process optimisation. The update of the specific energy at the k-th instant is calculated as: ' us,k = usn 2m + us,k 2 n 2 m + 2 n (4.24) where 2m is the variance of the nominal specific energy and it is obtained from the lit- erature, assuming that the data reported in the manuals are a'ected by an uncertainty of 50 %. The variance of the measurement is 2n = 1.3 · 10 3 J2/mm6, and it is estimated by propagating the uncertainty associated to Tc and ! across the Eq. 4.16. The lat- ter have been estimated by cutting a fixed cross section of workpiece with a constant spindle speed and feed rate, and their values are Tc = 2.5 Nm and ! = 0.72 rad/s. 85 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER 4.3.2 Validation The learning layer has been validated for three di'erent face milling processes: with full immersion and constant cross section (Fig. 4.7), with increasing cutting width (Fig. 4.8), and with a complicate path arbitrarily defined (Fig. 4.9). The cut material is the AA6082-T6. For the three case studies are shown the plots of the nominal, learnt and measured specific energies. The former is equal to 0.7 J/mm3. The measured specific energy is rep- resented by three colours: red for the sections of the tool path in which the mill is not cutting, orange for the parts in which the mill is cutting but the torque is excessively noisy, and green for the instants when D0 is true. The same colour scheme is applied for the plots of D1 and D2. At the bottom it is shown the cut cross section calculated by the process simulator, and it is green when the three index are true, that is when the EPC is actually learning the process. From the results obtained it is observed that us is adjusted when the measurements are stable. The estimated specific energy is a linear combination of the measured and nominal ones, suitably weighted by the confidence with which those values are known. The proposed approach is robust even for highly noising measurements such as the ones in Fig. 4.8. The results here obtained allow to maintain the models of the process constantly up- dated, therefore this learning layer can be exploited to the online adaptive control of the cut. 86 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER 160 180 200 220 240 260 280 300 320 340 0 0.5 1 1.5 2 (mm) e s (J/mm 3 ) Measured e s Nominal e s Estimated e s 160 180 200 220 240 260 280 300 320 340 0 1 2 (mm) D 1 160 180 200 220 240 260 280 300 320 340 0 1 2 (mm) D 2 160 180 200 220 240 260 280 300 320 340 0 5 10 x 10 ''5 (mm) Cross section (m 2 ) Figure 4.7: Validation of the learning layer for the full immersion face milling. 87 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER 160 180 200 220 240 260 280 300 320 340 0 0.5 1 1.5 2 (mm) e s (J/mm 3 ) Measured e s Nominal e s Estimated e s 160 180 200 220 240 260 280 300 320 340 0 1 2 (mm) D 1 160 180 200 220 240 260 280 300 320 340 0 1 2 (mm) D 2 160 180 200 220 240 260 280 300 320 340 0 2 4 6 8 x 10 ''5 (mm) Cross section (m 2 ) Figure 4.8: Validation of the learning layer for the face milling with increasing cutting
width. 88 CHAPTER 4. EVALUATION AND PERCEPTION CONTROLLER 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.5 1 1.5 2 (mm) e s (J/mm 3 ) Measured e s Nominal e s Estimated e s 200 400 600 800 1000 1200 1400 1600 1800 0 1 2 (mm) D 1 200 400 600 800 1000 1200 1400 1600 1800 0 1 2 (mm) D 2 200 400 600 800 1000 1200 1400 1600 1800 0 1 2 x 10 ''5 (mm) Cross section (m 2 ) Figure 4.9: Validation of the learning layer for the face milling with complicate tool
path. 89 CHAPTER 5. COMMUNICATION LAYERS Chapter 5 Communication Layers In this chapter it is described interface developed for the communication between the EPC and the NC Fidia CN20. The architecture of the software is outlined and its performances are reported. A bi-directional protocol is used for the communication between EPC and NC, thus allowing the controller to read the status from the ma- chine and to copy the optimal feed rate and spindle speed. As it will be shown in the next sections, the most time consuming operation is the communication of the opti- mal feed rate and spindle speed from the EPC to the user panel. According to the manufacturer of the NC Fidia, the mean time required to communicate one parameter to the NC is approximately 10 ms. Since two parameters are communicated at each optimisation loop, at least 20 ms for this operation are required. Moreover, it has to be considered that the EPC should work at more than 5 Hz. During this time, several operations need to be performed: the acquisition of the NC data, the measurement of the forces and accelerometer data from the CompactRio, the reconstruction of the whole status of the system, the online calculation of the optimal process parameters, and the communication of these to the NC. It is essential therefore to have an estima- tion of the latencies and task execution times associated to the communication. 90 CHAPTER 5. COMMUNICATION LAYERS In the next sections are described the tests performed to characterise the latency associ- ated the communication between EPC and NC. The communication software is written in C++ and in Ruby, whilst the interface between EPC and NC exploits CORBA. 5.1 Common Object Request Broker Architecture CORBA has been developed by the Object Management Group, and it is an indepen- dent architecture and infrastructure that allows applications to communicate across the network (99; 100; 101; 102). Thanks to this framework applications can commu- nicate through the Internet Inter-ORB Protocol (IIOP), that is independent from the programming language and hardware in which they have been implemented. CORBA is constituted by three elements: 1. Client: requires the services; 2. Server: provides the services; 3. ORB: manages the information exchange between client and server. In this study the client is run in the EPC, while the server is run in the user panel that is interfaced with the NC. The software architecture of the optimisation system is schematised in Fig. 5.1. All the services provided by server and interface are defined in the Interface Description Language (IDL). Thanks to the flexibility of CORBA, it is possible to instantiate clients that require services even to multiple servers. The service requests raised from clients are intercepted and driven to the servers by the skeleton. When a service request reaches the servers, the IDL is used to decode the received parameters. The server utilises the same IDL to encode the outputs of the service, to send them back to the client, and to communicate to it how to decode the data when 91 CHAPTER 5. COMMUNICATION LAYERS they are received. By using the IDL the client invoke remote methods as if they were functions of a local instance of the server object, though the client is actually calling a stub that acts as a proxy. The stub is a local simulation of a remote object and it is interfaced with the IDL. Milling machine CORBA Milling machine Object Request Broker Services Read machine status Set process parameters Industrial PC AD-Link EPC Interface FFI Ruby-C Interface C-CORBA (client) Sensor fusion Compact Rio Sensors IDL stub IDL skeleton CN User panel (server) Process Drives Figure 5.1: Architecture of the software developed for the communication between
EPC and NC. 5.2 Interface Ruby - CORBA The Ruby-CORBA interface is defined by two layers: the first layer allows the EPC to instantiate clients and to require services from the server, the second layer is the interface with CORBA, Fig. 5.2. The EPC is implemented in Ruby, while CORBA services are accessible via C++ classes. With this design of the communication layer it is possible to combine the advantages of Ruby and C/C++ thus obtaining an efficient software. Ruby is an interpreted language, highly flexible, and with a simple syntax. This language is ideal for implementing the whole architecture of the EPC. On con- trary, C/C++ is a compiled language so it is particularly suitable to perform complex and time-consuming tasks like the handling of the complex CORBA architecture. It 92 CHAPTER 5. COMMUNICATION LAYERS CNC User pannel Windows XP Layer C (C - C++) Layer FFI (Ruby - C) CORBA (C++) EPC (Ruby) AD-Link Debian Figure 5.2: Communication interface and layer between EPC and NC. is not possible to wrap C++ functions in Ruby due to lack of an Application Binary Interface (ABI) standard that makes the C++ source code independent from the com- piler used. Nevertheless, the pure C code has the compilation standard ANSI X3J11 ''ANSI C'', and ISOIEC JTC1 SC22 WG14 ''ISO C''. For Ruby is available the FFI library to wrap C and Ruby code 1. In Fig. 5.3 are outlined the modules and functions implemented in the communica- tion layer between EPC and user panel. The EPC includes the Interface Ruby-CORBA (IRC) module containing the classes and the FFI functions necessary to communicate with C/C++ layer. The ConnectionParameters class allows the EPC to access to the ConnectionPar C structure, where there are stored the parameters used to configure the connection with the NC (IP-address, port number, and NC tag-name). The Ma- chineStatus class is wrapped to the Status C structure, containing the status acquired from the NC. The Connection class has all the methods necessary to manage the connection, set or override the process parameters, and to acquire the status of the machine (whose fields are listed in Appendix F.2). The most important FFI functions implemented are, and are grouped in three structures: 1https://github.com/ffi/ffi 93 CHAPTER 5. COMMUNICATION LAYERS 1. connect IRC : initialises the connection and the Status structure; 2. disconnect IRC : closes the connection and de-allocates the memory; 3. set parameters IRC : sets the feed rate and the spindle speed to a specific value. 4. get status IRC : acquires the instantaneous state of the machine. The FFI methods are wrapped to C/C++ functions that access to the CORBA client. The architecture here presented allows a single client to require services from multi- ple servers, once a ConnectionPar structure has been defined for each communication channel. Note in Fig. 5.3 that the function get status IRC has a bidirectional arrow because it can be called by the EPC and returns the current status of the machine. CORBA Ruby machine
address
port Class: ConnectionParameters t
xm, ym, zm
xv, yv, zv
xa, ya, za
S, F
I
P
machine_status Class: MachineStatus check_connection
connect
disconnect
set_override
set_parameter
get_status pConectionParameters
pStatus Class: Connection C structs _ConnectionPar _Status C functions connect_IRC disconnect_IRC set_flag_acquireStatus_IRC set_override_IRC set_parameter_IRC get_status_IRC Listner
~Listner
OnDisplay pConectionParameters
pStatus Class: Listner cnc
~cnc
ConnectParameterDisplay
ReadCncStringParameter
WriteCncStringParameter
DisconnectParameterDisplay m_hostName
m_hostAddress
m_hostPort Class: cnc Machine
~Machine pMachConPar
Status Class: Machine init_status_IRC Figure 5.3: Classes architecture of the communication layer between EPC and NC. 94 CHAPTER 5. COMMUNICATION LAYERS On the CORBA side of the architecture, three C++ classes are defined: Machine, cnc, and Listener. The former is used to store the information about the connection, the second one initialises the CORBA connection, while the latter acquires the status of the machine by requiring the OnDisplay service to the CORBA server. This is done by utilising a parallel thread that is killed by the NC when it registers a modification of one or more of the listened parameters. The parameters and sub parameters that are monitored by the instance of the Listener are summarised in Appendix F: 1. M.XAC : returns the positions of the axes in the active workpiece frame. 2. MIC BLK F INFO: retrieves the information about active block and executed file. 3. MIC INST DATA: collects the remaining information necessary to define the status of the system. The NC measures the status of the machine at a higher frequency (1000 Hz) than the one used to communicate them to the EPC ( '500 Hz). Therefore, for some outputs, the NC measures the data as fast as possible and communicates to the user panel theirs statistical descriptors at a lower frequency (100 Hz). The chosen descriptors are mean, minimum, and maximum values, that are calculated between two consecutive data transmissions. 5.3 Characterisation and validation Tests are performed to characterise and validate the communication layer between EPC and CORBA. TET and latencies are measured for both directions of the EPC-NC com- munication. 95 CHAPTER 5. COMMUNICATION LAYERS 5.3.1 Task Execution Time The communication between EPC and NC has to be fast, in order to have a controller that runs at 5 Hz. In this study fast operations are those accomplished in less than 100 ms. During the TET tests it is ensured that only the essential processes are active on the EPC and NC. The tests carried out to validate the communication interface are performed by acquir- ing the state of the machine both through the EPC and the internal oscilloscope in the NC. The data acquisition is triggered by the NC (i.e. its operation mode), thus minimising the acquisition of non significant information. It is possible to consider a null TET for the communication between EPC and Compact RIO since it is made through UDP that is a light and fast protocol. Moreover, the Compact RIO has built in a FPGA chip that ensures hard real time performances (sampling frequency >50 kHz). On contrary, the communication between EPC and NC is more complex due to CORBA and to the control panel that acts as a proxy to the NC. Analysis of TETs are carried out by running the program List. 6.1 and measuring the time required to perform three di'erent operations: 1. Writing on the user panel new values of feed rate and spindle speed. 2. Reading the machine status from the user panel. 3. Reading the status, and subsequently send the new feed rate and spindle speed to the user panel. The measurements are made on the EPC side, and the results are reported in Tab. 5.1 and Fig. 5.4. It possible to observe that the fastest operation is the reading of the 96 CHAPTER 5. COMMUNICATION LAYERS machine status, with an average TET of 0.01755 ms. To write two parameters on the user panel it takes 32.300 ms, therefore one parameter is updated in 16.150 ms that is in agreement with the TET measured by the NC manufacturer (Tab. 5.2). Outliers in Fig. 5.4 are due to stochastic delays of the non-realtime operating systems running on the EPC (Linux Debian) and on the user panel (Windows XP ). Figure 5.4: Histograms of the probability density distributions of TETs measured when
reading the status of the machine (left), writing two parameters on the NC (centre),
and performing both operations (right). Table 5.1: Measured TETs for di'erent operations, with 95% of confidence. Operation Mean TET (ms) Standard deviation (ms) 95 % confidence Operation interval (ms) Read the status 0.01755 0.00388 [0.01752, 0.01757] Write 2 parameters 32.300 4.133 [31.653, 32.947] Read - write 32.545 4.374 [31.786, 33.304] 97 CHAPTER 5. COMMUNICATION LAYERS Table 5.2: TETs required to write one parameter on the NC according to the NC
manufacturer FIDIA. Parameter Value (ms) Minimum latency 3.291 Maximum latency 10.019 Median latency 9.787 1-st percentile 4.275 99-th percentile 10.018 99.9-th percentile 10.019 Standard deviation 1.028 5.3.2 Validation of the acquisitions The data acquired with CORBA are validated by comparing them with measurements collected from the internal oscilloscope on the user panel. The programmed tool path is a spiral with a radius of 25 mm. The experiment is repeated twice, thus acquiring the status of the machine through both the EPC and oscilloscope. In Fig. 5.5 are compared measured positions and feed rates for the three cartesian axes X, Y , Z. The acquisitions made through CORBA are in agreement with the collected data from the oscilloscope. Moreover, the former are o'set in advance of approximately 170 ms with respect to the latter, meaning that the user panel manages the CORBA commu- nication process with a higher priority than the oscilloscope process. It is a desired feature, and it is necessary in order to have a reliable communication with the EPC, especially when multiple operations are executed on the user panel. 5.3.3 Latencies and sampling frequencies To have a robust optimal control, it is necessary to characterise the time associated to the EPC-user panel communication, the latencies associated to it, as well as the 98 CHAPTER 5. COMMUNICATION LAYERS Figure 5.5: Comparison of the measurements of positions and velocities for the three
axes, made with the EPC and with the oscilloscope internal to the user panel of the
machine. sampling frequencies. It means to estimate the time tEPC NC required to send a parameter from the EPC to the NC, and the time tNC ME elapsed since the parameter is updated on the NC and it is measured back by the EPC. The information exchanged is the feed rate F and the spindle speed S. tEPC NC is the sum of T ETEPC UP necessary to communicate one parameter to the user panel, and T ETUP NC associated to the communication between the user panel and NC: tEPC NC = T ETEPC UP + T ETUP NC (5.1) 99 CHAPTER 5. COMMUNICATION LAYERS tNC ME is obtained by the sum between the actuation time ta necessary to accel- erate the axis (or the spindle) to the nominal F (or S), and T ETNC EPC required to read the status of the machine that can be neglected as exposed in Sec. 5.3.1: tNC ME = ta + T ETNC EPC (5.2) The total latency tL is thus: tL = tEPC NC + tNC ME (5.3) The tests are performed by moving only the X axis, while linearly modifying the feed rate from 0 mm/min to the maximum value 10000 mm/min. The EPC sends to the user panel a periodical triangular feed profile with period 1 s. The measurements for one period are reported in Fig. 5.6. Sent is the time at which the nominal F is sent by the EPC, set is instant when F is received by user panel, and measured is the time at which the feed rate reaches physically the nominal value. This test allows to identify the minimum sampling frequency that has to be set on the NC. From preliminary tests it is observed that the set triangular feed profile (dashed line) has a quantised trend (Fig. 5.6 on the left). This is due to the fact that the communication between the EPC and the user panel is fast ( '30 Hz), while the sam- pling frequency between user panel and NC is slower ( '10 Hz). The EPC reads for several times the same data from the NC, thus resulting in a quantised acquisition. Note that the same trend is observed for the measured feed rate (dot-dashed line), but here it is not appreciable because the data are pre-filtered by eliminating the duplicate values in order allow the calculation of the acceleration for further analysis. The data quantisation problem is addressed by increasing the user panel-NC sampling frequency to 100 Hz, thus obtaining smoother data, see Fig. 5.6 on the right. The latencies are measured by time-shifting and superimposing the sent and the mea- 100 CHAPTER 5. COMMUNICATION LAYERS Figure 5.6: Feed rate profiles used to estimate the latencies measured when acquiring
from the NC at '10 Hz (left), and '100 Hz (right). Table 5.3: Measured latencies. Operation Sampling freq. 30 Hz Sampling freq. 100 Hz tEPC NC 60 ms 60 ms tNC ME 150 ms 115 ms tL 210 ms 180 ms sured curve to the set lines. The time shifts are calculated through a manual bisection method, by minimising the residual sum of squares, and the results are summarised in Tab. 5.3. The total latency is 180 ms. This results means that during the online process control the EPC will have to anticipate the sent controls in order to be in-phase along the tool path with the controls profile calculated in the OCP. The total latency is given by the contribute of di'erent sources of delay: 1. Communication of the new feed from the EPC to the user panel. It takes 16.15 ms for each sent parameter. 2. Communication of the new feed from the user panel to the NC. According to 101 CHAPTER 5. COMMUNICATION LAYERS Eq. 5.1 it is equal to T ETUP NC = 43.85 ms. 3. Actuation dynamics, that is estimated equal to ta = 115 ms from Eq. 5.2. During this period, the NC modifies the nominal feed rate in order to take into account the maximum linear acceleration, to respect the quantisation time of the NC, and to limit jerks. This regularising e'ect is applied not only when computing the jerk-limited velocity profiles of the tool path, but also when the feed rate is instantaneously changed. 4. Measurement of the machine state through the EPC. As seen in the previous chapter this operation can be considered instantaneous but, due to the non- realtime nature of the operating systems running on the EPC and on the user panel, the acquisition might be a'ected by stochastic delays. 102 CHAPTER 6. MACHINE IDENTIFICATION Chapter 6 Machine Identification The goodness and robustness of the controls calculated by the Optimal Control Problem depend on the accuracy of the models used to describe the milling process, machine kinematics, and machine dynamics. The more accurate the models of the system are, the better is the optimal solution. In this section are described the tests carried out to characterise the machine dynamics. These tests are focussed on the identification of velocity profiles, maximum feed rate, and accelerations of axes and spindle of the machine Alesamonti MB-63. The dynamics of the drives are studied by changing the feed rate instantaneously (stair input), periodically (sweep input), and continuously (ramp input). The spindle is studied with tests aimed at identifying the free run performances and looses. 103 CHAPTER 6. MACHINE IDENTIFICATION 6.1 Dynamics of axes 6.1.1 Maximum feed rates and accelerations The maximum feed rate and acceleration for each axis are studied by setting the feed rate from zero to the maximum value 10000 mm/min with step and ramp inputs. The tool path designed for the step input test is reported in List. 6.1. 1 N1 ( F i d i a S . p .A. ISOGRAPH v . 8 . 2 . 3 C1 10/12/12 1 8 : 5 8 : 4 2 ) N2 ( C o m p i l a t i o n o f F i l e : Y: \ i d e n t i f i c a t i o n r a p i d m a x ) 3 N3 G00 X0 . 0 0 0 Y0 . 0 0 0 Z0 . 0 0 0 B0 . 0 0 0 W0. 0 0 0 F10000 S1500 N4 X600 . 0 0 0 M03 5 N5 Z600 . 0 0 0 M05 N6 B90 . 0 0 0 7 N7 W 250.000 N8 Y260 . 0 0 0 9 N9 X0 . 0 0 0 Y0 . 0 0 0 Z0 . 0 0 0 B0 . 0 0 0 W0. 0 0 Listing 6.1: Part program executed during the characterisation of the axes. The axis are moved one at time with the sequence: X, Z, B, W , and Y . Each axis is moved though a path long enough to reach the nominal feed rate, in order to have trapezoidal velocity profiles. The last block brings back all the drives to the origin with a rapid movement, thus allowing to verify that the maximum feed rate and acceleration achieved during the interpolated movement are limited by the slowest drive. The movements are repeated five times, and they are done in air, in order to minimise the risk for the machine and the worker man. The accelerations are calculated from the measured velocities through a finite di'erence scheme, after filtering the data with a moving mean. The axes are moved with two approaches: with linear interpolated G01 movements where the feed rate is set at its 104 CHAPTER 6. MACHINE IDENTIFICATION 0 10 20 30 40 ''5000 0 5000 10000 Time (s) V x (mm/min) fast
rapid 0 10 20 30 40 ''5000 0 5000 10000 Time (s) V y (mm/min) 0 10 20 30 40 ''5000 0 5000 10000 Time (s) V z (mm/min) 0 10 20 30 40 ''5000 0 5000 Time (s) V w (mm/min) 0 10 20 30 40 ''1000 ''500 0 500 1000 1500 Time (s) V b (°/min) Figure 6.1: Feed rates of the axes for the test performed giving as input: feed steps
from 0 mm/min to 10000 mm/min (red), and rapid movements (blue). maximum theoretical value (10000 mm/min), and with rapid G00 movements. This is done to verify if the latter are interpolated too. The measured velocity profiles and calculated accelerations are shown in Fig. 6.1 and Fig. 6.2 respectively. Note that in the two figures, between 30 s and 41 s, velocities and accelerations are lower. This is due to the fact that in the last block the dynamic of the system is actually limited by the slowest drive ''B'' which rotates the working table. Tab. 6.1 summarises the measured maximum velocities and accelerations for the five tests. The former are in agreement with the data reported on the data sheet of 105 CHAPTER 6. MACHINE IDENTIFICATION 0 10 20 30 40 ''0.5 0 0.5 1 1.5 Time (s) A x (m/s 2 ) fast
rapid 0 10 20 30 40 ''0.5 0 0.5 1 1.5 Time (s) A y (m/s 2 ) 0 10 20 30 40 ''0.5 0 0.5 1 1.5 Time (s) A z (m/s 2 ) 0 10 20 30 40 ''0.5 0 0.5 Time (s) A w (m/s 2 ) 0 10 20 30 40 ''0.05 0 0.05 0.1 Time (s) A b (°/s 2 ) Figure 6.2: Accelerations of the axes for the test performed giving as input: feed steps
from 0 mm/min to 10000 mm/min (red), and rapid movements (blue). Table 6.1: Maximum velocities acceleration and deceleration of the axes. Axis Velocity Acceleration Deceleration X 9997.4 ± 18.1 mm/min 0.580 ± 0.033 m/s2 0.572 ± 0.032 m/s2 Y 9999.6 ± 5.8 mm/min 0.522 ± 0.015 m/s2 0.557 ± 0.025 m/s2 Z 9999.2 ± 7.6 mm/min 0.534 ± 0.021 m/s2 0.564 ± 0.024 m/s2 W 4998.1 ± 4.4 mm/min 0.298 ± 0.015 m/s2 0.330 ± 0.018 m/s2 B 719.6 ± 3.6 /min 0.063 ± 0.003 /s2 0.058 ± 0.003 m/s2 106 CHAPTER 6. MACHINE IDENTIFICATION the machine Tab. 3.1. It is observed that accelerations and decelerations of the axes have the same absolute value. Since the EPC controls only the Cartesian axes X, Y , and Z, the acceleration of remaining drives W and B are neglected. It is therefore assumed that the acceleration of the axes is 0.4 m/s2, thus being slightly conservative in the calculation of the optimal feed rate profiles. 6.1.2 Transients of feed rates and accelerations When severals feed values are sent to the NC in few seconds, the velocity profiles are dominated by transients. It is necessary to characterise how the NC handles these transients to design properly the velocity profiles calculated by the optimal control. Three might be the strategies followed by the NC to adapt online the velocity profiles when a new feed rate is set: 1. Constant acceleration: the axes are moved at the maximum acceleration. The time duration of the transient is variable and depends on the increment of speed applied. 2. Constant time: the acceleration of the axis is calculated from the increment of feed between two consecutive time instants. 3. Variable time and acceleration: the intensity and duration of the accelerations are variable and they are calculated online by the look ahead planner. To identify the adopted strategy, X, Y , and Z are moved independently, by feed increments with amplitude increasing in time, Fig. 6.3. The test starts with a low feed, that is doubled every two seconds. On the X and Z axis the last speed increment is set to a value that is higher than the maximum feed supported, therefore the feed cannot follow the reference. From the acceleration plots (on the right) it is possible 107 CHAPTER 6. MACHINE IDENTIFICATION to state that when the feed is modified, the NC applies accelerations for a constant period, but with di'erent amplitudes. The time span required to accelerate the axes is 394.74 ±57.29 ms. The uncertainty of the time durations is estimated by calculating the standard deviation among all the transients. However, under step inputs the NC has to adapt the feed profiles only during a short time interval. To analyse how the system reacts to inputs continuously changing, the feed rate is changed with a ramp input. 10 15 20 0 5000 10000 Time (s) Feed (mm/min) Feed along X Sent
Measured 10 15 20 ''0.5 0 0.5 Time (s) Acceleration (m/s 2 ) Acceleration along X 75 80 85 0 5000 10000 Time (s) Feed (mm/min) Feed along Y 75 80 85 ''0.5 0 0.5 Time (s) Acceleration (m/s 2 ) Acceleration along Y 28 30 32 34 36 38 0 5000 10000 Time (s) Feed along Z Feed (mm/min) 28 30 32 34 36 38 ''0.5 0 0.5 Time (s) Acceleration (m/s 2 ) Acceleration along Z Figure 6.3: Tests performed to identify the duration of the acceleration transient. 108 CHAPTER 6. MACHINE IDENTIFICATION As preliminary tests, this behaviour is studied by changing the feed rate with trian- gular sweep profiles. The height of the triangles is set equal to the maximum feed rate. The first period of the sweep is defined accordingly to the axis travel, and it is subsequently halved. The test along the Y axis is performed moving the cutting head from the top to the bottom and vice versa, to take into account the e'ect of inertia of the moved masses. It is observed that the deceleration of Y is higher when the cutting head is lowered, and the acceleration is higher when the cutting head is risen. This e'ect is also shown in Fig 6.4, where for the Y axis the measured feed profiles are slightly asymmetrical. Note that, due to the inertia of the axes, the phase displacement between the set and the measured feed becomes bigger as the period of the sweep is reduced. From this test is not possible to obtain useful information about the frequency response of the system because the traveled length is too short and the latency of the communication is too high. To avoid these problems another test is designed. Only the deceleration of the axes is studied with this second experiment. The drives are slowed down starting from three di'erent feed rates to zero, with di'erent decelera- tions. The starting feed are 10000 mm/min, 6666 mm/min, and 3333 mm/min. The magnitude of the maximum deceleration is calculated through a linear regression of the the feed profile during the transients. An example of the acquisition made along X for a deceleration of 0.325 m/s2 is shown in Fig. 6.5. The red, green, and blue lines are the measured feeds for tests performed starting from 3333 mm/min, 6666 mm/min, and 10000 mm/min respectively. Note that the deceleration ramps are smoothed when the drives are decelerated. This is due to the fact that the NC generates velocity profiles that are jerk limited. The magenta line represents the set feed profile, calculated from the chosen deceleration. The black solid and dashed lines are the fit of the measured feeds, estimated taking into account only the data sampled during the feed transitions. The slope of the magenta line is therefore 109 CHAPTER 6. MACHINE IDENTIFICATION 0 0.5 1 1.5 2 2.5 0 5000 10000 Time (s) Feed (mm/min) Sweep along X Sent
Measured 0 0.5 1 1.5 2 2.5 ''10000 ''5000 0 Time (s) Feed (mm/min) Sweep along Y '' from top to bottom 0 0.5 1 1.5 2 2.5 0 5000 10000 Time (s) Feed (mm/min) Sweep along Y '' from bottom to top 0 0.5 1 1.5 2 2.5 0 5000 10000 Time (s) Feed (mm/min) Sweep along Z Figure 6.4: Set and measured sweep profiles on the feed rate. In the test performed
along Y going from the top to the bottom the feed rate is negative because positive Y
are upwards. the measured deceleration. Qualitatively, as much similar are the slopes of the magenta and black lines, as better the drive can follow the feed reference, and so the imposed deceleration. 110 CHAPTER 6. MACHINE IDENTIFICATION 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 5000 10000 15000 Time (s) F (mm/min) Axis:X, deceleration set =''0.325 m/s 2 F(t=0) = 3333 mm/min, D =''0.26481 m/s 2) F(t=0) = 6666 mm/min, D =''0.29638 m/s 2) F(t=0) = 10000 mm/min, D =''0.31377 m/s 2) Regression line for F(t=0) = 3333 mm/min
Regression line for F(t=0) = 6666 mm/min
Regression line for F(t=0) = 10000 mm/min Set feed profile, D =''0.325 m/s 2 Figure 6.5: Example of a deceleration test performed starting from di'erent initial feed
rates. In Fig. 6.6 are shown the data collected for all the performed tests. On the abscissa is reported the measured deceleration, while the set deceleration is on the ordinate. At a critical value the curves start to diverge, and the measured decelerations become smaller than the set ones, until they saturate to a maximum value. For the X and Y axes the value of the critical deceleration depend on the initial value of the feed, and the deflection points are shifted to lower accelerations, as lower is the initial feed rate. This is due to the fact that the NC does not apply instantaneously the feed rate passed by the EPC, but rather it interpolates the movements between the current feed and the desired one. For each test the percentage error in following the set deceleration 111 CHAPTER 6. MACHINE IDENTIFICATION is calculated. The results obtained from the star input tests in Sec. 6.1.1 can be con- firmed, scilicet that the maximum acceleration achievable under a feed ramp input is 0.4 m/s2, with a tracking error lower than 25%. 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 Acceleration set (m/s 2) Acceleration measured (m/s 2 ) Accelerations deviations X axis F3333
F6666
F10000 0 0.2 0.4 0.6 0 20 40 60 80 100 Acceleration set (m/s 2) Error % Acceleration errors X axis F3333
F6666
F10000 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 Acceleration set (m/s 2) Acceleration measured (m/s 2 ) Accelerations deviations Y axis F3333
F6666
F10000 0 0.2 0.4 0.6 0 20 40 60 80 100 Acceleration set (m/s 2) Error % Acceleration errors Y axis F3333
F6666
F10000 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 Acceleration set (m/s 2) Acceleration measured (m/s 2 ) Accelerations deviations Z axis F3333
F6666
F10000 0 0.2 0.4 0.6 0 20 40 60 80 100 Acceleration set (m/s 2) Error % Acceleration errors Z axis F3333
F6666
F10000 Figure 6.6: Results obtained from the deceleration tests for the X (left), Y (centre),
and Z (right) axes. 112 CHAPTER 6. MACHINE IDENTIFICATION 6.2 Dynamics of the spindle 6.2.1 Spindle acceleration The experimental tests here described are aimed at identifying the maximum rotational acceleration and deceleration of the spindle. The former has a lower module than the latter because the NC can decelerate the spindled through a mechanical brake. The model of the spindle integrated into the EPC assumes that the magnitude of the max- imum rotational acceleration and deceleration of the spindle are equal. In order to ensure the feasibility of the optimal solution, the spindle acceleration and deceleration are set equal to the smallest between the two. A step pyramid on the spindle speed is imposed and the mechanical torque is measured. The pyramid goes from 100 rpm to 1500 rpm, with increments of 100 rpm. The tests are repeated three times to perform statistical analysis on the measurements. Each step lasts 20 s to ensure the attainment of the steady state conditions. In Fig. 6.7 are reported the measurements collected during one test. The spindle speed exhibits jerks because it is measured from the feedback loop controller on the spindle speed. It should be mentioned that the tool tip rotates with a more regular speed, due to the inertia and frictions of the kinematic chain bringing the movement from the electric motor to the ram. The spindle acceleration is calculated with a finite dif- ference scheme after the data have been filtered with moving mean whose period is of 5 samples. The average maximum acceleration is 123.5 rad/s2. Shapiro''s test on the acceleration data give a p-value of 0.47, therefore a Student''s t-test is used to estimate the confidence interval [110.8, 136.2]rad/s2, with a confidence level of 95%. In this test the null hypothesis is: measured mean equal to zero. When modelling the spindle, it is assumed a maximum acceleration of 104.7 rad/s2 113 CHAPTER 6. MACHINE IDENTIFICATION Figure 6.7: Measured spindle speed (top), and calculated accelerations (bottom) for
one of the three tests performed. ( '1000 rpm/s), thus being slightly conservative and taking into account the high un- certainty of measurement. 6.2.2 Mechanical torque The mechanical frictions in the kinematic chain of the cutting head are estimated by measuring the free run mechanical torque. These frictions are due to the contact between gears, bearings, and viscosity of the lubricant oil. The mean mechanical torque is measured at di'erent steady state spindle speeds (Fig. 6.7), and it is repeated for the ram (W axis) at di'erent positions. The ram is fully retracted when W =0 mm, and it is fully extended when W =300 mm. As can be seen in Fig. 6.8, the free run 114 CHAPTER 6. MACHINE IDENTIFICATION 0 500 1000 1500 15 20 25 30 35 40 45 50 55 Free''Run Mechanical Torque Spindle Speed (RPM) |Torque| (Nm) w = 0 mm
w = 100 mm
w = 200 mm
w = 300 mm Figure 6.8: Measured free run mechanical torque at di'erent positions of the ram. torque is not a'ected by the position of the ram. Moreover, the free run torque is non linearly dependent from the spindle speed. A robust model of the friction torque with respect to residuals is obtained by interpolating with a second order polynomial the free run torque in function of the spindle speed. In the adopted model, the constant term can be associated to the Coulomb friction Tc, the linear term to the viscous friction Tv, but the quadratic term T2 has not physical meaning. The latter might be wrongly associated to the aerodynamic drag by virtue of its dependency from the quadratic velocity. However, aerodynamic forces are generated only when a relative wind is impacting against a body, that is not the case for the milling process. Moreover, the drag torque should quadratically increase with the spindle speed. The quadratic term in the regression is therefore included because it allows to obtain residuals that are independent from the fitted spindle speed, Fig. 6.9. The expression of the free run 115 CHAPTER 6. MACHINE IDENTIFICATION torque Tfr is: Tfr = T2 + Tv + Tc = 1.629 · 10 5 S2 + 0.047 S + 15.670 (6.1) where S is the spindle speed in RPM. The corrected mechanical torque Tcm is obtained subtracting Tfr to the raw measurements: Tcm = Traw Tfr (6.2) Figure 6.9: Measured and fitted free run torque for the test with the ram fully extended
(top), and residuals of the regression (bottom). 116 CHAPTER 6. MACHINE IDENTIFICATION Validation From some preliminary tests it is observed that ''due to a bad scaling of the torque measured from the CN'' the measured cutting power is higher than the absorbed power current. This issue is addressed by identifying the correct scaling factor, through tests in which the spindle is continuously accelerated and decelerated with four di'erent accelerations. By knowing the acceleration and the moment of inertia of all the shafts in the kinematic chain of the cutting head, it is possible to estimate the mechanical torque generated by the motor. Given an acceleration at the spindle, the dynamics of the remaining shafts are calcu- lated by analysing the kinematic chain through the depth-first search algorithm (103). The theoretical mechanical torque at the motor Tem is calculated by summing the inertial torque of each shaft in the kinematic chain: Tem = Ns X i=1 '!i Ji (6.3) where Ns is the total number of shafts, while Jk and '!k are the moment of inertia and rotational acceleration of the i-th shaft (see Tab. 3.3). The scaling factor is given by the ratio between Tem and the badly-scaled measured torque Tcm: K = Tem
Tcm (6.4) The average scaling factor calculated among all the tests is K = 0.67, Fig. 6.10. The corrected measured mechanical torque Tm is obtained by subtracting the free run torque to Traw, and by multiplying the result for the correction factor K and for the mechanical efficiency of the transmission system ''m = 0.85 given by the machine manufacturer: Tm = (Traw Tfr)K ''m (6.5) 117 CHAPTER 6. MACHINE IDENTIFICATION Figure 6.10: Scaling factor K estimated for four acceleration/deceleration tests. The torque measurements are validated by cutting a workpiece with a known specific energy. A face milling is performed with a cutting depth of 1.3 mm. The mill has 5 flutes and a diameter of 63 mm. It is cut the AA6082-T6 with a specific energy of 0.75 J/mm3. As can be observed in Fig. 6.11, the measured and the theoretical values of the mechanical torque are in agreement. Anomalous torque peaks at the beginning and end of the cut are due to the fact that the measured signal comes from the feedback loop on the spindle speed. When the mill enters in contact with the workpiece, the spindle tends to be slowed down, therefore the feedback controller compensates this deviation from the reference speed by increasing the current, and so the torque, at the spindle. 118 CHAPTER 6. MACHINE IDENTIFICATION 180 200 220 240 260 280 300 0 2 4 6 8 10 12 14 16 18 20 Curvilinear Abscissa (mm) T (Nm) Cutting Torque Predicted
Measured 180 200 220 240 260 280 300 0 20 40 60 80 100 Curvilinear Abscissa (mm) Cross Section (mm 2 ) Figure 6.11: Measured mechanical torque (top), and cut cross section (bottom). 119 CHAPTER 6. MACHINE IDENTIFICATION 6.3 Dynamic compliance In order to analyse the stability of the process, the dynamical response of the system (machine, tool, and workpiece) is described through the Frequency Response Function. The FRF should cover a frequency range enough wide to include all the main modes of the structure. The shape of the stability lobes diagram is indeed given not only by the stability pockets generated by each mode, but also by the intersection of lobes associated to di'erent modes. If a mode is missing in the FRFs, then the lobes diagram will appear distorted or will not describe the true stability space at all. The system is modelled as Multiple Input Multiple Output (MIMO). The inputs are the forces on the tool tip, while outputs are the displacements, therefore the FRF of the system is represented by a 3 ' 3 matrix: F RF = 2 6 6 6 4 F RFx,x F RFy,x F RFz,x F RFx,y F RFy,y F RFz,y F RFx,z F RFy,z F RFz,z 3 7 7 7 5 (6.6) The notation is the following: the first subscript indicates the direction along which the force is applied, while the second subscript is used to specify the direction along which the displacement is measured. In this study impact tests are performed to measure the FRF matrix. The ram of the machine is excited through a PCB 086D20 hammer with a 084A63 hard plastic tip. A three-axial PCB T356A15 accelerometer is mounted on the tool tip and it is acquired at 5000 [Hz], Fig. 6.12. The impacts are given in a position close to the accelerometer, and are repeated seven times in order to reduce the uncertainty of measurement associated to its variable intensity and orientation. The same procedure is applied to identify the FRF of the workpiece and working table, Fig. 6.13. 120 CHAPTER 6. MACHINE IDENTIFICATION Figure 6.12: Direction of impacts on the tool. Figure 6.13: Direction of impacts on the workpiece. The software used for acquisition is CUTPRO MalTF, which calculates the Frequency Response Function (FRF) of the machine with the procedure proposed by Altintas (96). Sampled force and acceleration signals are multiplied by an exponentially decaying window, thus neglecting the data associated to a null force. The exponential window decays from unity to 0.05 in the total sample time of 10 s. The Fourier transform of the displacement is calculated by integrating twice the Fourier transform of the accelera- tions. Finally, the FRFs are computed through the cross-power spectrum of measured signals, and the obtained response is validated by verifying that the coherence function is unitary. The homogeneity of dynamic properties of the machine and the workpiece across the 121 CHAPTER 6. MACHINE IDENTIFICATION Table 6.2: Positions at which the FRF of the tool are tested, expressed in the absolute
frame of reference of the machine. Test X mm Y mm Z mm W mm B At -350.3 -250.3 +350.3 +0.3 0 Bt -350.3 +250.3 +350.3 +0.3 0 Ct -350.3 +250.3 +350.3 +300.3 0 Dt -350.3 -250.3 +350.3 +300.3 0 working volume are verified by changing and combining the positions of linear axes accordingly to a design matrix (104). The real part of the recorded FRFs for di'erent position of the axes are in Fig. 6.14. As expected, the direct FRFs (on the diagonal of the figures) are higher in magnitude with respect to the transversal ones. The workpiece can be considered sti' since all its FRFs are close to zero. The compliance of the whole system is independent from the positions of the X, Y , and Z axes. Only the position of the ram, that is the W axis, a'ects the frequency response of the ram. In Fig. 6.15 are reported, for four positions of the ram, the real and imaginary part of the measured direct FRFs. When the ram is retracted (W =0 mm) the first mode is around 850 Hz and low. In this case the system can be considered sti'. On contrary, when the ram is fully extended, (W =-300 mm) the first mode is dominant and at a lower frequency, approximately 200 Hz. 6.3.1 System identification In order to compute lobes diagrams that are accurate across the whole working volume, it must be characterised the influence of the position of the ram on the dynamical prop- erties of the machine. The FRF of the ram are measured at di'erent position of the W axis, which is changed from -220 mm to -300 mm with increments of 5 mm. A robust 122 CHAPTER 6. MACHINE IDENTIFICATION 200 400 600 800 1000 ''1 ''0.5 0 0.5 1 x 10 ''6 Freq. (Hz) Displacement along X Re(FRF x, x ) (m/N) Impulse along X 200 400 600 800 1000 ''1 ''0.5 0 0.5 1 x 10 ''6 Freq. (Hz) Re(FRF y, x ) (m/N) FRF of the TOOL Impulse along Y 200 400 600 800 1000 ''1 ''0.5 0 0.5 1 x 10 ''6 Freq. (Hz) Re(FRF z, x ) (m/N) Impulse along Z 200 400 600 800 1000 ''1 ''0.5 0 0.5 1 x 10 ''6 Freq. (Hz) Displacement along Y Re(FRF x, y ) (m/N) 200 400 600 800 1000 ''1 ''0.5 0 0.5 1 x 10 ''6 Freq. (Hz) Re(FRF y, y ) (m/N) 200 400 600 800 1000 ''1 ''0.5 0 0.5 1 x 10 ''6 Freq. (Hz) Re(FRF z, y ) (m/N) 200 400 600 800 1000 ''1 ''0.5 0 0.5 1 x 10 ''6 Freq. (Hz) Displacement along Z Re(FRF x, z ) (m/N) 200 400 600 800 1000 ''1 ''0.5 0 0.5 1 x 10 ''6 Freq. (Hz) Re(FRF z, y ) (m/N) 200 400 600 800 1000 ''1 ''0.5 0 0.5 1 x 10 ''6 Freq. (Hz) Re(FRF z, z ) (m/N) A t B t C t D t Figure 6.14: Real part of the FRF matrix for the impacts on the tool, at di'erent
positions of the axes (see Tab. 6.2). 123 CHAPTER 6. MACHINE IDENTIFICATION 200 400 600 800 1000 ''1 ''0.5 0 0.5 1 x 10 ''6 Freq. (Hz) Re(FRF x, x ) (m/N) FRF x,x 200 400 600 800 1000 ''2 ''1 0 1 2 x 10 ''6 Freq. (Hz) Im(FRF x, x ) (m/N) 200 400 600 800 1000 ''1 ''0.5 0 0.5 1 x 10 ''6 Freq. (Hz) Re(FRF y, y ) (m/N) FRF y,y 200 400 600 800 1000 ''2 ''1 0 1 2 x 10 ''6 Freq. (Hz) Im(FRF y, y ) (m/N) W = 0
W = ''150
W = ''220
W = ''300 Figure 6.15: Real and imaginary part of direct FRF along X (left) and Y (right), at
di'erent positions of the ram. 124 CHAPTER 6. MACHINE IDENTIFICATION calculation of the lobes diagram at intermediate positions is performed by generating a model of the dynamic compliance of the ram in function of W . This operation can be performed with two approaches: system identification and regression. When identifying a dynamic system, a model is defined a priori and its coefficients are estimated by fitting the model to real data. The identified coefficients describe a physical property of the system (mass, sti'ness, damping...). Di'erent approaches are available in literature for system identification, and a comprehensive review was given by Kerschen et al. (105). In this study several well-knowns strategies have been tested, such as: lumped model inverse analysis (106), Linear Parameter-Varying (LPV) mod- elling (107), pole/residue parametrisation (108) and estimation of state-space model using a subspace method (109). Nevertheless, none of them is enough accurate to catch all the modes in the FRFs that concur to form the lobes diagram. For this reason, a regression is carried out on the acquired data. The di'erence be- tween regression and system identification is that in the former the coefficients of the model don''t have necessarily a physical meaning. Autoregressive exogenous (ARX) model (110) and estimation of Output-Error polynomial model (111) have been fit to the data. However, in both cases it has been observed that a low order polynomial is not enough accurate, while a high order polynomial is a'ected by over-fitting. The problem of regression and classification is well known in the field of machine learn- ing, where large amount of data must be analysed. Classification is usually performed by means of K-mean algorithm and Support Vector Machines (SVM) (112). Regression can be still performed by using SVM, but more efficient algorithms are Artificial Neural Networks (ANN) (103), Locally Weighted Projection Regression (LWPR) (113), and Gaussian Processes (GP) (114). ANN were used in the past 60 years in a wide range of applications, however the correct 125 CHAPTER 6. MACHINE IDENTIFICATION design of their architecture is still a complex task. The main drawback of these models is their lack of explanation capability (115). ANNs are considered 'black-box' since it is difficult to interpret their output and the e'ect of the parameters of the network on its performance. LWPR can be used to fit non linear function in high dimensional spaces with redundant and irrelevant input dimensions. This method is particularly suitable for large amount of data, which can be learnt online through an incremental learning. In this study the dynamic behaviour of the machine is described by few data points, which are collected beforehand the process optimisation. For such conditions LWPR loses its edge against the generalisation capability of GP. 6.3.2 Regression of the dynamic compliance Here are summarised the results obtained with the regression of the FRFs through GP. The code is implemented in Matlab by exploiting the open source GPML Toolbox v3.2 (116). The 2D dynamical model of the machine used to compute the lobes diagram takes into account the behaviour of the system along X and Y axes. Thanks to this simplification, only the elements of the FRF matrix associated to these directions are regressed, and those associated to Z are neglected. Moreover, from preliminary tests it has been observed that the computation of the lobes diagram is a'ected mainly by the direct FRFs, meaning that only two of its components have to be regressed: F RFx,x and F RFy,y. These two elements are represented as vectors of complex numbers, which are split into real and imaginary parts. Therefore, in total four GPs are trained. The formulation of the GP is the one described in the Appendix. A. As remarked in the previous chapter, the regression given by GP is closely related to the behaviour of the data rather than to a parametric function. It means that di'erent functions can be described by the same hyperparameters as long as the di'erent sets of training data 126 CHAPTER 6. MACHINE IDENTIFICATION exhibit similar probabilistic characteristics (i.e. a null mean, and a similar covariance). In Tab. 6.3 are reported results of the trainings of the four FRFS. Note that it is possible to generalise the set of hyperparameters, thus performing the training only for one set of data. In this study are used the hyperparameters found by training the GP with the real part of F RFx,x. An example of the regression is reported in Fig. 6.16 for the real part of F RFy,y. On the left it is shown the measured FRF (top) and its contour plot (bottom), on the right it is shown the regressed FRF. The FRF plot of the measurement appears to be quantised, while the regressed FRF is more continuous since it has been evaluated for a more dense grid of ram positions and frequencies. The prediction is in agreement with the measured data. It can be observed that a main mode dominates the dynamics of the machine. As the ram is extended (W is decreased to -300 mm) the main mode moves to lower frequencies and its amplitude increases. This means that the machine becomes less sti' and vibrates more. A second lower mode is slightly visible at 380 Hz. 6.3.3 Validation of the regression The accuracy of the regression achieved through GP has a direct impact on the reli- ability of the calculated stability lobes diagrams, see Appendix B. The stability lobes Table 6.3: Trained hyperparameters. knlmlk is the module of the negative log marginal likelihood, the higher it is, the more accurate the prediction is. Hyperparameters l f d n knlmlk Re (F RFx,x) 9.0 1.6 · 10 7 m/N 2.2 · 10 8 m/N 4.2 · 10 9 mN/ 23720 Im (F RFx,x) 8.8 1.6 · 10 7 m/N 2.0 · 10 8 m/N 4.1 · 10 9 mN/ 23777 Re (F RFy,y) 10.2 1.7 · 10 7 m/N 2.1 · 10 8 m/N 4.1 · 10 9 mN/ 23866 Im (F RFy,y) 10.3 1.7 · 10 7 m/N 2.2 · 10 8 m/N 4.1 · 10 9 mN/ 23861 127 CHAPTER 6. MACHINE IDENTIFICATION Figure 6.16: Real part of the measured (left) and regressed (right) F RFy,y. diagrams are calculated by considering as reference process the full immersion face milling, since for this cut the calculation of the engagement arcs is simplified: the radial immersion is between 0 and 180 . The mill has a diameter of 63 mm, with 5 WIDIA XPHT160408-AL inserts. The cut material is the AA 6082-T6 with cutting constants of Krc = 264.9 N/mm2 and Ktc = 614.1 N/mm2. These values have been identi- fied as explained in Appendix D. The process damping is set to zero. In Fig. 6.18 are compared the stability lobes diagram calculated along the W axis with the measured and with the regressed FRFs, and it is observed that the two stability 128 CHAPTER 6. MACHINE IDENTIFICATION lobes surfaces are in agreement. When W is set to bigger values (i.e. the ram was shorter), the stability limit is at a higher cutting depth and the stability pockets move to higher spindle speeds, meaning that the machine becomes more sti'. To better interpret the stability lobes surfaces they are sliced by fixing W to a specific value, e.g. to -220 mm in Fig. 6.17. This is the condition at which the machine is more sti' and exhibits many di'erent strong eigenmodes. In such complicate situation the stability lobes diagram tends to be complicated, but it is still well predicted by the FRFs regressed through GP. 600 800 1000 1200 1400 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 S (RPM) d (mm) W = 220 mm Measured
Estimated 500 1000 1500 220 240 260 280 300 320 340 S (RPM) Chatter freq. (Hz) 600 800 1000 1200 1400 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 S (RPM) d (mm) W = 220 mm Measured
Estimated 500 1000 1500 220 240 260 280 300 320 340 S (RPM) Chatter freq. (Hz) Figure 6.17: Stability lobes diagram calculated for the ram at the position -220 mm
(left) and chatter frequencies (right). 129 CHAPTER 6. MACHINE IDENTIFICATION Figure 6.18: Stability lobes diagrams calculated from measured FRFs (left), and from
regressed FRFs (right). 130 CHAPTER 6. MACHINE IDENTIFICATION 6.3.4 Validation of the stability lobes diagrams The stability lobes diagrams are validated by recording the vibrations of the cutting head during cuts carried out at di'erent spindle speeds and cutting depths. Two posi- tions of the ram are considered during the tests: W = 300 mm and W = 270 mm. The triaxial accelerometer Dytran 3213M6 is acquired at 10 kHz, and it is screwed directly on the cutting head, as much close as possible to the tool, Fig. 6.19. The transmittance between the tool tip and the position of the accelerometer is neglected, since the aim of these tests is only to identify the chatter frequency (and not the am- plitude of vibrations). The choice of spindle speed and depth of cut to be tested is performed according to the automatic methodology proposed by Bediaga et al. (117). In this approach the boundary of the SLD is searched by iteratively adjusting the cut- ting depth and the spindle speed. The first cuts are aimed at identifying the location of an instability pocket in the SLD. The cutting depth is low in the beginning, and it is increased until chatter is detected. The spindle speed is then adjusted in a such way that the new tooth passing frequency is equal to a multiple value of the chatter frequency. Once a stable cut is achieved, the cutting depth is increased again and the iterative process is repeated. The tests are interrupted when the chatter persists after five consecutive iterations, since that condition means that the maximum stable depth has been overshot. Accelerometer Ram Tool Figure 6.19: Position of the accelerometer. 131 CHAPTER 6. MACHINE IDENTIFICATION The chatter onset is quantified by calculating the energy ratio chatter indicator (CIER) (118): CIER = Ec E = 1 Ep + En E (6.7) where CIER is given by integral quantities related to the Power Spectrum Density (PSD) (119) of: recorded accelerations (E), asynchronous component of the signal related to the chatter (Ec), synchronous components associated to the forced vibrations (Ep), and noise (En). All the PSD are calculated on signals that are windowed with a Hamming window (120). High CIER are given by unstable cuts. Chatter indeed is associated to phase shift of half-period between the vibration of the tool and the tool pass frequency. When this phase shift occurs, the integral of the PSD of Ep becomes smaller, while the integral of the PSD of Ec increases. In Fig. 6.20 are presented the SLD for the two positions of the ram tested. Each point in the SLD has a colour and a dimension that is related to the measured CIER; the bigger and more red the point is, the more severe the chatter is. It can be observed that when W = 300 mm, the amplitude of the predicted stability pockets is in agreement with the measurements, but they are slightly shifted to higher spindle speed. This is due to the fact that the SLD have been measured and validated in di'erent moments of the day, during which the temperature and lubrication of the machine has changed. For W = 270 mm the abscissa positions of the stability pockets are well predicted, but their amplitude is over estimated by the calculated SLD. With the ram at this position the machine is sensibly more sti', therefore the FRF are a'ected by a lower signal-to-noise ratio. Nonetheless, it should be remarked that in the strategy adopted for integrating the information of SLD into the OCP it is more important to estimate accurately the abscissa position of the stability pockets rather than their amplitude (see Chapter 2.4.2). It is therefore concluded that the calculated SLD are sufficiently in agreement with the measurements made. 132 CHAPTER 6. MACHINE IDENTIFICATION 900 1000 1100 1200 1300 1400 1500 0.0 1.0 2.0 W = -300 mm Spindle speed (rpm) Cutting depth (mm) 900 1000 1100 1200 1300 1400 1500 0.0 1.0 2.0 W = -270 mm Spindle speed (rpm) Cutting depth (mm) Figure 6.20: Stability lobes diagrams validated at two positions of the ram: -300 mm
(top) and -270 mm (bottom). The size of the circles is proportional to the calculated
CIER. 133 CHAPTER 7. APPLICATIONS OF AUGMENTED REALITY FOR SAFE MANUFACTURING Chapter 7 Applications of Augmented Reality for Safe Manufacturing In this chapter it is proposed an approach to enhance the sensing layer of the EPC through the integration of images and videos recorded by cameras. The aim of this study is to state the feasibility of 3D scene reconstruction of the working volume through 3D depth cameras. Several issues must be addressed when applying image processing techniques to augment the reality of milling processes, such as: the presence of high frequency vibrations, debris, chips, coolant liquids inside the working volume, as well as reflecting surfaces of the tool and workpiece. Image processing system can be exploited in numerous applications thanks to their flexibility. The information dis- played to the CNC user can be augmented by integrating data from the NC, process simulators, and auxiliary safety systems. 134 CHAPTER 7. APPLICATIONS OF AUGMENTED REALITY FOR SAFE MANUFACTURING 7.1 State of the art In the past years several authors applied the AR in design and manufacturing (121; 7). Nevertheless, up to now the e'orts made to augment the information collected from the machining processes are limited to few studies, whose main purpose has been to design efficient tool paths, and to ease the training of students and novice workmen. Weinert et al. (122) used AR to test o'-line the tool path by including a virtual tool and a virtual workpiece in the video acquisitions of the real working volume. The system exploited a monocular camera and tracked the position of the tool holder by localising a marker sticked on top of it. Zhang et al. (123; 124) predicted o'ine the cutting forces and collisions by cutting a virtual workpiece with virtual tool. Online, the virtual model was aligned to the real working table. The tracking of the real tool was made with a monocular camera, by means of template matching. The system was used also to superimpose to the video acquisition of the machining volume the information about actual cutting forces, and active block. Olwal et al. (125) developed a system called ASTOR, an auto-stereoscopic optical see-through spatial AR system, to project information about the cutting forces, process parameters and G-code on the safety glass of the machine. The main limitation of the approaches here reviewed is the same of collision avoidance performed by CAM softwares: they rely on a virtual model of the bulk workpiece that has to be generated with sufficient accuracy. In this study it is carried out a feasibil- ity analysis to try to overcome this drawback through state of the art algorithms and hardware in 3D reconstruction. The geometry of the raw workpiece and tool can be learnt o'ine, directly from the scene, thus eliminating the error-prone human factor from the tool path safeness verification. It is moreover proposed the conceptual design of a collision avoidance system for safe tool paths based 3D dense scene reconstruction. 135 CHAPTER 7. APPLICATIONS OF AUGMENTED REALITY FOR SAFE MANUFACTURING 7.2 3D scene reconstruction Complex tool paths and workpieces force CNC users to carefully design and check on air the programmed G-codes. However, the tool path verification is a time consuming task that often, for the sake of productivity, is neglected. The problem of online collision monitoring and avoidance is well known in many robotic applications where mobile robots have to travel across an unstructured environment, without to impact with objects that can be static (walls, columns, etc..) or moving (people, animals, other robots). In machining process the environment is partially structured since ''in the worst scenario'' only the raw workpiece has an unknown shape. However, more attention has to be given to the definition of collision in machining processes. The tool has to touch the workpiece to cut it, hence the contact between them is unavoidable, nevertheless three undesired conditions can be identified: ' The tool hits the workpiece without spinning or during rapid movements. ' The tool hits the working table. ' The tool holder hits the workpiece or working table. The system developed to supervise the working area should be able to classify dan- gerous situations on the basis of an extended set of features, reconstructed from the status of the process, and relative position of tool and workpiece. The former is already gathered by the simulator module of the EPC, while the latter might be obtained by aligning a virtual reality (VR) model of the simulated process with a 3D set of points acquired from the real scene. To extend the knowledge of the real world by merging 136 CHAPTER 7. APPLICATIONS OF AUGMENTED REALITY FOR SAFE MANUFACTURING measured and virtual data is the main paradigm of the Augmented Reality (AR). In order to obtain a reliable 3D model of the scene, di'erent camera poses must be acquired and fused together. The Simultaneous Localisation And Mapping (SLAM) technique can be used to obtain a map of the environment and the poses of the camera with respect to the reconstructed scene. At the core of the SLAM there is the reg- istration process that allows to combine di'erent spatial data sets in a single global model, thus generating large, complex, and coherent virtual scenes, while mitigating the e'ects associated to the measurement uncertainty. In a virtual environment, objects can be represented by a set of points, namely point clouds. The most di'used algorithms developed for the registration of 2D and 3D point clouds are the Iterated Closest Point (ICP), and the Normal Distribution Transform (NDT) (126). The ICP extracts key points from the images and then aligns two point clouds by min- imising a metric that can be defined as: the point-to-point Euclidean distance (127), the point-to-plane distance (128), or the plane-to-plane distance of Mitra et al. (129). The ICP is a fast method, suitable for real-time applications, and it has been used in many projects for real-time 3D dense reconstruction (130; 131; 132). One of the first robust SLAM algorithms was developed by Davison (133), who registered sets of key points extracted from images through the ICP. This approach resulted to be robust with respect to scaling, rotation, translation and affine transforms. Klein et al. (134) paralleled the feature tracking and mapping processes by distributing the SLAM computation on a dual core computer, thus increasing the real time capabilities of the system. As a drawback, the ICP requires an explicit correspondence between the entities in the two clouds, otherwise the convergence of the registration would be slow, or nor reached at all. This implies that slow movements should occur between two consecutive acquisitions, and the images should be highly textured. 137 CHAPTER 7. APPLICATIONS OF AUGMENTED REALITY FOR SAFE MANUFACTURING When the spatial distribution of the two point clouds are not homogeneous and theirs density are not comparable, the assumption of explicit correspondence cannot be sat- isfied and the ICP would fail. In such cases it is possible to utilise the NDT (Ap- pendix C). This algorithm exploits the normal distribution to generate a probabilistic model of the acquired scene, and it finds the transformation between two acquisitions by maximising the probability that the new point cloud is part of the statistical model of the environment. This method does not require to establish a direct correspondence between primitives (points, lines, or surfaces) therefore it is more robust to affine trans- formations than the ICP. As instance, the NDT has been applied successfully to align the CAD model and 3D measured shape of a sheet deformed through the Single Point Incremental Forming, in order to evaluate the spring back Fig. 7.1. However, the NDT requires more computational resources to perform a 3D registra- tion in realtime, therefore up to now it has been used in real time only for the map alignment of coarse 2D laser scans (135; 136; 137). The success of AR-based applications has been driven by the increasing of the per- formances of personal computers and by the di'usion of several techniques for 3D reconstruction: SLAM with monocular cameras, stereo imaging, structured light, laser range finders, time-of-flight cameras, and range cameras. Structured light is the tech- nology that allows to achieve the higher accuracy of measurement, indeed, dimensions down to 50 µm can be detected (138; 139; 140; 141). This technique is however expen- sive, the acquisitions tend to be slow, and the observed volume is usually small. Range cameras instead, are particularly cheap devices that have been developed for gaming consoles, and that have had a massive di'usion within research groups thanks to their high quality-cost ratio. The most famous 3D sensors are: the Microsoft Kinect R , the SoftKinect DS325, and the Asus Xtion R (Tab. 7.1). The accuracy of these sensors is not outstanding, nevertheless it is sufficient for numerous applications involving rela- 138 CHAPTER 7. APPLICATIONS OF AUGMENTED REALITY FOR SAFE MANUFACTURING Measurement Nominal geometry Structured light sensor Alignment - Normal Distribution Transform Input measured model Target CAD model Figure 7.1: Normal Distribution Transform algorithm applied to the alignment of two
3D geometries not explicitly correspondent. tively large objects (i.e. human tracking and robot navigation). Moreover, the availability of open source libraries for 3D scene reconstruction is reduc- ing the e'orts and the time required to develop robust applications. The most famous and robust registration libraries are the Kinfu1 developed by Newcombie et al. (142), and ReconstructMe2 from Held et al. (131). These libraries exploit Graphic Process- ing Units (GPU) to parallelise the calculations, and reconstruct in real-time (30 fps) a dense 3D map of the environment. Such approach has been exploited in many 3D 1http://pointclouds.org/documentation/tutorials/using kinfu large scale.php
2http://reconstructme.net/ 139 CHAPTER 7. APPLICATIONS OF AUGMENTED REALITY FOR SAFE MANUFACTURING Table 7.1: Examples of 3D sensors available to the market. Data Kicrosoft SoftKinect Asus Xtion Data Kinect DS325 Pro Live Depth Camera Resolution 640x480 320x240 640x480 RGB Camera Resolution 1280x960 1280x720 1280x1024 Field of view (HxVxD) 43 x 57 74 x 58 x 87 58 x 45 x 70 Frame rate (fps) 30 60 30 Operating range (m) 1.2 - 3.5 0.15 - 1.0 0.8 - 3.5 Depth resolution (mm) 50 (at 4 m) 14 (at 1 m) 50 (at 4 m) AR applications, like: 3D mapping (143), gaming (130; 144), surveillance (145), and architecture (146). 7.3 Exploitations The feasibility study for the 3D scene reconstruction of the working volume is carried out by using the 3D depth camera Asus Xtion Pro Live. The conceptual architecture of the system developed in this study for 3D reconstruction and collision avoidance is outlined in Fig. 7.2. It follows the same structure adopted for the EPC, and it is or- ganised in three layers: sensor, perception and cognitive. In the sensor layer, RGB and 3D images (i.e. point clouds) of the working volume are recorded by the depth camera. The point clouds are then registered in real time by exploiting the Graphic Processing Unit of a graphic card supporting the CUDA library3. The graphic card used is the NVidia GeForce GTx 690, which has a 3.0 Kepler architecture (147), 3072 cuda cores, and 4 Gb of dedicated ram. Two registration libraries are tested, the Kinfu and the ReconstructMe, both utilising an efficient implementation of the ICP. The Kinfu is available as module of the Point Cloud Library (PCL), and since it is open source, it is 3https://developer.nvidia.com/what-cuda 140 CHAPTER 7. APPLICATIONS OF AUGMENTED REALITY FOR SAFE MANUFACTURING easy to integrate into large applications. Moreover, the Kinfu is capable to reconstruct large spaces, such as interior of houses and offices (148). The ReconstructMe instead is a licensed software that has better registration capabilities, but it is limited to acquire small volumes. In Fig. 7.3 it is shown the result of a 10 s 3D reconstruction of the working volume through the Kinfu. The accuracy of the virtual model is enough to locate all the rigid bodies that the tool could hit when an erroneous G-code is programmed. Workpiece, vise, and cutting table are clearly identifiable. It is possible to conclude that 3D sensors are suitable for the generation of accurate maps of the working volume. Figure 7.2: Conceptual architecture of the system for the collision avoidance. 141 CHAPTER 7. APPLICATIONS OF AUGMENTED REALITY FOR SAFE MANUFACTURING In future works these maps can then be utilised, as instance, for collision detection and avoidance. The objects in the acquired scene, might be classified in three categories: 1. Static, constituted by all the non-moving parts. 2. Dynamic, comprising all the moving components. 3. Noise, due not only the measurement uncertainty, but also to the debris and chips produced during the cut. The moving objects might be then tracked through particle filter, and verified against the global model by collision detection algorithms4 such as CULLIDE (149) or RAPID (150). Whether the distance between objects falls down to a predetermined threshold, or a collision is detected, the control loop is closed by acting on the CNC to stop the axes. Exploitation of the 3D models acquired could be also the enhancement of the infor- mation provided to the CNC user. The Augmented Reality would be used to ease the 4A collection of collision detection algorithms is available at http://gamma.cs.unc.edu/research/collision/packages.html Figure 7.3: Scene reconstructed by using the Kinfu library (left) and RGB image of
the working volume (right). 142 CHAPTER 7. APPLICATIONS OF AUGMENTED REALITY FOR SAFE MANUFACTURING supervision of the process, Fig. 7.4. On the video stream of the operation it might be visualsed auxiliary data, such as: process parameters, tool trajectory, tool life, and measured forces or vibrations. It could be possible to superimpose a coloured patch on the tool, thus emphasising the regions of the cut in which the cutter are more stressed, or in which there is the risk of having a collision. The instantaneous shape of the cut, which is usually occluded by the mill and evacuated chip, could also be visualised. F: 200
S: 1500
X: 0.32
Y: 150.03
Z: 10.78
''F: 95 %
''S: 120 %
CN status: working
EPC: active
Path collisions: OK
Figure 7.4: Conceptual design of an augmented interface for milling machines. 143 CHAPTER 8. RESULTS Chapter 8 Results In this chapter are summarised the results obtained from the validation of the EPC on real processes. The controller is tested on several tool paths, with complicate geometries. The raw workpiece is made of AA6082-T6 and it has cubic shape in order to ease the definition of its initial geometry in the process simulator. This simplification is not limited by the geometric kernel used to simulate the cut, but eases the setup of the EPC. The ACIS libraries can however handle complicate geometries generated through CAD softwares. The technical specifications of the used tool are reported in Tab. 8.1. Table 8.1: Data of the utilised tool. Parameter Value Diameter of the mill 63 mm Number of flutes 5 Insert WIDIA XPHT160408-AL Coating of the inserts TiN 144 CHAPTER 8. RESULTS 8.1 O'ine optimisation In the tests here described the optimisation is performed o'ine, that is before the process starts. The target function is defined as the sum of terms considering: process productivity Jp, deflection of the tool Jq,p, and cutting costs Jc. Forced and self-excited vibrations of the machine are neglected. This simplification is suitable for roughing operations, where the quality of the cut surface is not a mandatory requirement, but the process productivity and tool life have to be maximised. The weights of all the terms of the target function are set to unitary values. 100 Start End 60 A) 100 Start End 30 B) Figure 8.1: Tool paths tested for the o'ine optimisation. Two G-codes are programmed: a face milling with a complicate toolpath and cutting depth of 1 mm (Fig. 8.1-A), and a peripheral milling with a cutting depth of 3 mm (Fig. 8.1-B). In both cases, the optimised process is compared with the nominal process. The latter is designed according to the data specified by the tool manufacturer, that is by setting the feed rate to 1000 mm/min, and the spindle speed to 1500 rpm. The nominal and optimised controls for each operation are reported in Fig. 8.2 and 8.3. In blue are plot the measured feed rate and spindle speed. The solutions calculated by the EPC are plot with a green line. The black line called set indicates the instantaneous controls passed by the EPC to the CN. Note that these values are anticipated of 180 ms with respect to the controls computed o'ine, in order to compensate the latencies 145 CHAPTER 8. RESULTS Table 8.2: Cutting times achieved with and without the EPC. Test without EPC with EPC Time saving Face milling 82.5 s 44.1 s - 46.6 % Peripheral milling 23.3 s 19.5 s - 22.9 % associated to the communication and actuation of the drives. By receiving the optimal controls in advance, the machine can follow correctly the desired profile of feed rate and spindle speed. The controls are as much in phase as much the green and the blue line are overlapped. At the end of each block the Optimized feed rate goes instantaneously to zero, thus slowing down the axes at the maximum deceleration. This condition has been imposed to speed up the convergence of the OCP, and does not a'ect the performance of the real process. For both case studies, the EPC increases the feed rate and reduces the spindle speed when the mill is not cutting (i.e. the cross section is null). As result, the process productivity is maximised and the energy consumption is minimised. On contrary, when the tool is cutting, the feed rate is kept low and the spindle speed is high, thus reducing the force that tends to deflect the tool, and improving the quality of the cut. The optimal controls allow to reduce the cutting time up to 23 % for the peripheral milling, and up to 46 % for the face milling, Tab. 8.2. However, the improvement of process productivity does not a'ect the quality of the cut. Roughness measurements are carried out in di'erent zones of the cut surface, as shown in Fig. 8.4 and 8.5. A Mitutoyo Surftest SJ-401 profilometer is used. The probe has spherical-conical tip with a radius of 2 µm. Each coloured segment is 25 mm long and represents the travel of the probe across the surface. For a easier interpretation of the results, the roughness maps of the tests performed with the optimal controls are coloured in green when the quality of the surface is improved, in blue when it remains almost unchanged, and in red when it is worsened. It is concluded that even if the process productivity is significantly improved, the quality of the cut remains overall unchanged. 146 CHAPTER 8. RESULTS Figure 8.2: Nominal and optimised controls for the face milling. 147 CHAPTER 8. RESULTS Figure 8.3: Nominal and optimised controls for the peripheral milling. 148 CHAPTER 8. RESULTS 0.310
3.793 0.218
1.688 0.253
1.919 0.379
2.134 0.611
2.693 0.282
2.112 0.460
2.257 0.319
1.891 0.486
3.917 0.404
2.333 0.570
3.187 0.633
3.561 0.395
4.592 0.359
4.087 0.531
5.280 0.303 2.099 0.425 3.305 0.417 3.334 0.312 2.120 0.455 2.853 0.371 2.985 0.605
5.171 0.380
3.500 0.227
2.991 0.393
4.357 0.314
2.853 0.330
2.238 0.336
3.168 0.280
2.903 0.259
2.687 0.584
3.800 0.366
4.525 0.368
3.917 0.324
2.988 0.239
2.369 0.426
3.448 0.455 4.333 0.376 3.423 0.342 2.929 0.480 4.344 0.370 4.276 0.369 3.450 Roughness:
Improved
Unchanged
Worsened Without EPC Legend:
Ra
Rz With EPC Frontal view Real surfaces Figure 8.4: Roughness maps for the face milling. 1.465
7.043 0.558
3.269 0.996
5.582 0.752
4.924 0.302
2.361 0.557
5.783 0.497
3.497 0.484
3.252 0.387
3.548 0.623
3.476 0.505 2.91 1 0.431 2.722 1.561
8.276 2.074 10.441 1.744 10.329 0.860
5.545 0.242
1.931 0.607
4.503 0.577
4.356 0.593
6.272 0.393
3.058 0.574
4.128 0.452 2.381 0.510 3.674 Roughness:
Improved
Unchanged
Worsened Legend:
Ra
Rz Without EPC With EPC Frontal view T op view Real surfaces Figure 8.5: Roughness maps for the peripheral milling. 149 CHAPTER 9. CONCLUSION Chapter 9 Conclusion 9.1 Summary of Thesis Achievements It has been proposed and tested a framework to increase the level of automation of manufacturing processes towards the paradigm of intelligent manufacturing. The de- veloped intelligent system was called Evaluation and Perception Controller, and it was applied to the milling process. The intelligence of the EPC was represented by its ca- pability to enhance the performances of the process, while learning its real behaviour. These features were pursued by developing a modular and flexible architecture, which was constituted by three layers: sensor, perception, and cognitive. The process was sensed through accelerometers, dynamometers, and 3D video cameras displayed in the working volume. Such information was processed by the perception layer, which recon- structed the whole state of the system. In the cognitive layer the process was simulated to pre-calculate the cut cross section, cutting forces, and stability lobes diagrams. A multi-objective optimisation was then carried out to identify the optimal feed rate and spindle speed. 150 CHAPTER 9. CONCLUSION In order to ease the adaptation of this approach to di'erent manufacturing systems, a language- and OS-independent communication interface between EPC and CNC was defined. Moreover, the experimental tests necessary to calibrate the EPC were de- scribed, thus identifying not only the dynamical model of the machine, but also the material properties of the cut workpiece. The optimisation problem was formulated as an Optimal Control Problem, where a multi objective target function was minimised, while taking into account constraints associated to the dynamic behaviour of the machine and process. The objective func- tion was formulated through terms describing productivity, quality, and costs of the process. These contributes were weighted according to the specific context in which the process was carried out. As instance, for roughing cuts the productivity was weighted more than the remaining terms. During the cut, the models of the process were kept updated by a learning module, thus ensuring the robustness of the OCP. Finally, the EPC was validated on real processes. Three-axes tool paths were tested, with complicate geometries. The performance of the EPC were validated against nom- inal processes, designed according to guidelines given by the tool manufacturer. When all the terms of the target function were weighted equally (i.e. for a generic process), the EPC outperformed the productivity achieved in the nominal cuts, without degrading their quality. 9.2 Future Works This study represents a further step in the direction of developing intelligent manufac- turing systems, in which non homogeneous production lines will cooperate as multi- agent robotic system to maximize the performances of production plants. In this vision, a distributed knowledge of the processes involved will be learnt by each single produc- tion agent and shared among cloud services to other entities in the work-plant. As 151 CHAPTER 9. CONCLUSION instance, di'erent CNC machine tools will collaborate to define reliable models of the tool wear, or to identify robustly the material properties of the cut workpiece. In future production plants workmen and process engineers will no more set up pro- cesses by identifying the parameters based on their expertise, rather they will specify a production context. Set of requirements will be defined by users, and analysed by intel- ligent production systems, thus creating a human-machine cooperation in the pursuing of efficiency. 152 Appendices 153 APPENDIX A. GAUSSIAN PROCESSES Appendix A Gaussian Processes A GP is defined as a distribution over functions, and is specified by mean µ(x) and covariance k(x, x0) functions: f ' GP (µ(x), k(x, x0)) (A.1) In GP there is not an algebraic formulation of the regressed function, but each regressed point can be thought as a sampled from a multivariate Gaussian distribution. The main advantage of this approach is that the form of the classifier is not limited by a parametric form, instead it can represent a function by letting the data 'speak' more clearly for themselves. Let''s consider a data set x 2 R n with zero mean. Two observations x and x0 are related to each other through the covariance function k(x, x0). Di'erent formulations of k(x, x0) can be utilised, to take into account complex relations between points. As instance, it is possible to include constraints in order to model the dynamics of the system (151; 152; 153; 154). The covariance function used to regress the FRFs is 154 APPENDIX A. GAUSSIAN PROCESSES formulated as a Squared Exponential function with isotropic distance measure: k(x, x0) = f e (x x0)I P 1 (x x0) 2 l + 2d x,x0 (A.2) where I is the identity matrix, 2 f is the signal variance, l is a scaling factor, 2 d is the variance of the training data, while x,x0 is the Kronecker delta function. The second term in Eq. A.2 is included to take into account the noise of measurement of the training data. In order to regress data from the GP, it is necessary to define the likelihood function, which specifies the probability of observations given a latent function. In this study it is used the Gaussian likelihood function since it is the first- choice commonly used in regression: lik(x) = 1 p 2 ' 2n e (x µ)T I (x µ) 2 2 n (A.3) µ and n are the mean value and the standard deviation of the likelihood respectively. The former is to set µ = 0 from the assumption made on the null mean. The coefficients l, f , d, n are called hyperparameters '' and their values are calculated by training the GP. Given a set of training data (x, f ) and a set of test data (x', f '), the training is carried out by finding hyperparameters that maximise the logarithmic marginal likelihood: L = log (y | x, '') = 1
2 log |''| 1
2 (y µ) T '' 1 (y µ) n 2 log (2') (A.4) where y is the output values, and µ is the mean of the inputs, '' is the covariance of the training data. The first term of L penalises the complexity of the model, the second is the one that guides the fit of the data, while the third is a normalisation term. Note that the first two terms have opposite e'ects on the regression: 1
2 log |''| tends to keep the model simple and general, while 1 2 (y µ) T '' 1 (y µ) increases the accuracy of 155 APPENDIX A. GAUSSIAN PROCESSES the regression. This definition simplifies the training of the GP since there is no need of a cross validation. L can be minimised by di'erentiating the Eq. A.4 with respect to the hyperparameters and solving the associated system of equations. Up to here it has been summarised how to compute the GP that is used as a prior for the Bayesian inference-driven regression. The prior is independent from the training data, but specifies the properties of the function described by the hyperparameters. The joint normal distribution over all the data is defined as: 2 4 f f ' 3 5 = N 0 @ 2 4 µ µ' 3 5 , 2 4 '' '' ' '''T '''' 3 5 1 A (A.5) ''' is the covariance for the training-test data set, and '''' is the covariance of test data. The values of f are known, while f ' has to be determined, therefore the conditional distribution of f ' ''given f '' is: f ' | f = N µ' + ''' T '' 1(f µ), '''' '''T '' 1''' (A.6) that is the posterior distribution for a set of test data x'. The corresponding posterior process for a set of '' hyperparameters is then defined by the mean µ'' and variance k''(x, x0), as follows: µ''(x) = µ(x) + ''(X, x) T '' 1(f µ) (A.7) k''(x, x0) = k(x, x0) ''(X, x)T '' 1''(X, x0) (A.8) with ''(X, x) being the covariance calculated between every training case and x. It can be noted that adding a data to the training process enriches the processed information, with a consequent reduction of the posterior variance k''(x, x0). Indeed, to the prior variance k(x, x0) it is subtracted a positive quantity calculated from the training tests. 156 APPENDIX B. CALCULATION OF STABILITY LOBES DIAGRAM Appendix B Calculation of Stability Lobes diagram The model here described is the one developed by Altintas and Budak (53). This appendix summarises the basic concepts that allow to compute cutting forces and stability lobes diagrams. Cutting forces The model considers two degrees of freedom, which are the directions orthogonal to the axis of the tool, Fig. B.1. It is assumed that the mill has nt teeth, and a null helix angle. The feed is along X. During the cut, the dynamic displacement of the j-th tooth along the radial direction is given by vj = x sin( j) y cos( j) (B.1) where j is the instantaneous angular immersion. For a tooth rotating with a ' angular speed, the dynamics of the immersion angle is j(t) = ' t. The total chip thickness is given by two contributes, associated to the rigid movement of the cutter, and to the 157 APPENDIX B. CALCULATION OF STABILITY LOBES DIAGRAM Figure B.1: Two-dimensional dynamical model of milling process. vibrations of the tool at the present and previous tooth periods: h( j) = [fz sin( j) + (vj,0 vj)] g( j) (B.2) where fz is the feed per tooth and (vj, vj,0) are the dynamic displacements of the cutter at the current and previous tooth period. The function g( j) is unitary in the range of immersion angles in which the mill is cutting, that is when j is between the entry ( st) and exit ( ex) angles, otherwise it is zero. The static component of the chip thickness does not contribute to the vibrations of the tool, therefore the Eq. B.3 is simplified and by substituting the Eq. B.1 it becomes: h( j) = [ x sin( j) + y cos( j)] g( j) (B.3) 158 APPENDIX B. CALCULATION OF STABILITY LOBES DIAGRAM where x = x x0 and y = y y0. The dynamic displacements of the cutter at the present and previous tooth period are (x, y) and (x0, y0) respectively. The cutting force Fc,j on the X Y plane can be decomposed into its tangential Ft,j and radial Fr,j components. The latter are proportional to the depth of cut d, chip thickness h( j), as well as the constants Kt and Kr taking into account the material properties of the workpiece, tool, the geometry of the cutter, and the lubrication conditions: Ft,j = Kt d h( j) , Fr,j = Kr Ft,j (B.4) The forces can be projected along the X and Y directions and summed among all the teeth, then: Fx = nt 1 X j=0 [ Ft,j cos( j) Fr,j sin( j)] Fy = nt 1 X j=0 [Ft,j sin( j) Fr,j cos( j)] (B.5) the immersion angle is calculated as j = p + j , where p = 2'/nt is the pitch angle. A more practical representation of the cutting forces is given by substituting the Eq. B.3 and Eq. B.4 into Eq. B.5, and rearranging the expression in matrix form it gives: {Fc(t)} = 1
2 d Kt [A(t)] { (t)} (B.6) 159 APPENDIX B. CALCULATION OF STABILITY LOBES DIAGRAM where {Fc(t)} = (Fx(t), Fy(t)) T , { (t)} = ( x, y)T , and [A(t)] is the directional dynamic milling force matrix whose components are: ax,x = nt 1 X j=0 g [sin(2 j) + Kr (1 cos(2 j))] ax,y = nt 1 X j=0 g [(1 + cos(2 j)) + Kr sin(2 j)] ay,x = nt 1 X j=0 g [(1 cos(2 j)) Kr sin(2 j)] ay,y = nt 1 X j=0 g [sin(2 j) Kr (1 + cos(2 j))] (B.7) As the cutter rotates, the milling forces vary periodically. [A(t)] is periodic at tooth passing frequency ! = nt ' (or tooth period T = 2'/!), therefore it can be expanded into Fourier series: [A(t)] = 1 X r= 1 [A(t)] ei r ! t [Ar] = 1 T T Z 0 [A(t)] e i r ! t dt (B.8) r is the number of harmonics to be considered in the approximation of [A(t)] and depends on the immersion conditions and on the number of teeth in cut. In the most simple model r = 0 and only the average component of [A(t)] is considered: [A0] = 1 T T Z 0 [A(t)] dt (B.9) 160 APPENDIX B. CALCULATION OF STABILITY LOBES DIAGRAM Since [A0] is valid only when the mill is cutting (i.e. st   ex), it is equal to the average value of [A(t)] at = p: [A0] = 1 p ex Z st [A( )] d = nt 2' 2 4 'x,x 'x,y 'y,x 'y,y 3 5 (B.10) where: 'x,x = 1
2 [cos(2 ) 2Kr + Kr sin(2 )] ex st 'x,y = 1
2 [ sin(2 ) 2 + Kr cos(2 )] ex st 'y,x = 1
2 [ sin(2 ) + 2 + Kr cos(2 )] ex st 'y,y = 1
2 [ cos(2 ) 2Kr Kr sin(2 )] ex st (B.11) and the dynamic expression of the cutting force in Eq. B.6 becomes: {Fc(t)} = 1
2 d Kt [A0] { (t)} (B.12) note that [A0] is time invariant, but it depends on the immersion angle which, in general, varies in time. Stability Lobes Diagrams The transfer function [G(i !)] experimentally identified at the cutter-workpiece contact zone is represented as a 2 ' 2 matrix: [G(i !)] = 2 4 Gx,x(i !) Gx,y(i !) Gy,x(i !) Gy,y(i !) 3 5 (B.13) 161 APPENDIX B. CALCULATION OF STABILITY LOBES DIAGRAM where Gx,x(i !) and Gy,y(i !) are the direct transfer functions in x and y directions, and Gx,y(i !) and Gy,x(i !) are the cross transfer functions. The vibrations vectors at present time and previous tooth period T are called: {r} = {x(t), y(t)} T ; {r0} = {x(t T), y(t T)}T which, at the chatter frequency !c, can be calculated as: {r(i !c)} = [G(i !)] {F } e i !c t {r0(i !c)} = {r(i !c)} e i !c T (B.14) The displacement { } = {(x x0)(y y0) } T is then given by: { (i !c)} = {r(i !c)} {r0(i !c)} = ' 1 e i !c T ' ei !c t [G(i !c)] {F } (B.15) note that !c T is the phase delay between the vibrations at successive tooth periods T . Substituting Eq. B.15 into Eq. B.12 defined to calculate the cutting force, it is obtained the equation: {F } e i !c t = 1
2 d Kt [1 e i !c T ][A0][G(i !c)] {F }e i !c t whose non trivial solution is given by the characteristic equation of the closed loop dynamical milling system: det  [I] 1
2 d Kt(1 e i !c T )[A0][G(i !c)] = 0 (B.16) The eigenvalues ' of the characteristic equation are calculated after the chatter fre- quency, material properties (Kt, Kr), radial immersion, and transfer function of the structure have been defined. Since [G(i !)] is matrix of complex number, the eigenval- 162 APPENDIX B. CALCULATION OF STABILITY LOBES DIAGRAM ues have a real and an imaginary part: ' = 'R + i 'I . After some simple algebraic calculation steps here neglected for brevity, it is possible to obtain the critical axial depth of cut at ! = !c: dlim = 2' 'R nt Kt 1 + 2 (B.17) where  = 'I /'R. Given a chatter frequency, the Eq. B.17 allows to calculate the maximum cutting depth that gives a stable process. The stable spindle speed S is calculated by estimating the tooth passing period T , given by: T = 1 !c (' + 2 k ') S = 60 nt T (B.18) where k is the integer number of the lobes considered, and ' = ' 2(tan 1(k)) is the phase shift between present and previous vibration marks. 163 APPENDIX C. NORMAL DISTRIBUTION TRANSFORM Appendix C Normal Distribution Transform The NDT is a registration algorithm that has been developed to align two di'erent sets of 2D or 3D points. Each set is called point cloud. Here it is described the NDT implementation in the open source Point Cloud Library (PCL), (155). First, the input cloud Pinput (i.e. the cloud to be aligned) is down-sampled, in order to increase the speed of the registration. This operation is performed through a voxel grid filter. A voxel is a 3D cube, that groups N samples and describes the associated space through their centroid. The reduced cloud is called ' Pinput. The NDT algorithm is then outlined in two steps: the creation of a statistical model of the 3D target cloud Ptarget, and the probability-driven registration of Pinput. Statistical model The Ptarget is subdivided into cells with constant size and containing at least three samples. The size of the cells is determined according to the dimension of the point 164 APPENDIX C. NORMAL DISTRIBUTION TRANSFORM cloud. In general a trade-o' should be sought, since alignments performed with small cells are more precise, but require more computational resources. For a k-th cell, its centroid qtarget,k and covariance matrix ''target,k is calculated as: qtarget,k = 1 np np X j=1 Ptarget,j (C.1) ''target,k = 1 np np X j=1 (Ptarget,j q) (Ptarget,j q) T (C.2) where np is the number of points in the cell. A normal probability distribution N(q, '') is associated to each cell, thus obtaining a global statistical model of the 3D cloud Ptarget. Since a derivative optimisation algorithm is used minimise the alignment error, the global statistical model is smoothed by partially overlapping the local normal dis- tributions. Given a point P , the probability that it belongs to the global probability distribution p is then calculated as: p(P ) ' exp (P q) T '' 1 (P q) 2 ! (C.3) Alignment The roto-translation matrix used to align the input cloud to the target cloud can be represented by a vector x of six components: three for the rotations, and three for the translations. These elements are identified through Newton''s optimisation algorithm. The initial transformation guess xo is set equal to the identity matrix, thus assuming that initially the two clouds are already aligned. The first guess is then corrected according to the value of a score function that is calculated by iterating across the following steps: 165 APPENDIX C. NORMAL DISTRIBUTION TRANSFORM ' project ' Pinput into the frame of Ptarget according to the calculated transformation x. The mapped input cloud is called ' Minput. ' compute the corresponding normal distribution for each ' Minput,j point, with j = 1, ..., np. ' evaluate the distribution for each ' Minput,j point, and sum the result, thus obtain- ing the score: score(x) = ni X j=1 exp 0 B @ ' ' Minput,j qj ''T '' 1 j ' ' Minput,j qj '' 2 1 C A (C.4) where ni is the number of points in the input cloud, while qi and ''j are the mean and covariance matrix of the normal distribution corresponding to ' Minput,j, looked up in the NDT of the target cloud. The convergence criteria can be established based on the maximum number of itera- tions, as well as on the minimum step of transformation. 166 APPENDIX D. IDENTIFICATION OF THE CUTTING CONSTANTS OF THE WORKPIECE Appendix D Identification of the cutting constants of the workpiece In this study the reference material of the workpiece is the AA 6082-T6. To compute the lobes diagram and cutting force it is necessary to estimate the cutting constants of the material through experimental tests. It is followed the procedure developed by Budak et al. (156). Full-immersion face milling (i.e. slotting) cuts are carried at di'erent feed rates. The used tool, inserts, and spindle speed are listed in Tab. D.1. Cutting forces along the three cartesian axes of the machine ( ¯ Fx, ¯ Fz, and ¯ Fy) are measured with a dynamometric table. The average cutting forces at each feed rate are reported in Fig. D.1. The relation between cutting forces and cutting force constants Table D.1: Process parameters used for the cutting force constants identification. Parameter Value Material of the workpiece AA 6082-T6 Diameter of the mill 63 mm Number of flutes 5 Insert WIDIA XPHT160408-AL Coating of the inserts TiN Spindle speed 1500 rpm Depth of cut 2 mm Width of cut 63 mm 167 APPENDIX D. IDENTIFICATION OF THE CUTTING CONSTANTS OF THE WORKPIECE is linear, and it is defined as: 8 > > > > > < > > > > > : ¯ Fx = nt d 4 Krc f nt d 4 Kre ¯ Fy = + nt d 4 Ktc f + nt d 4 Kte ¯ Fz = + nt d 4 Kac f + nt d 4 Kae (D.1) where f is the feed rate nt is the number of flutes in the mill, and d is the cutting depth. Krc, Ktc, and Kac are the cutting constants due to the shear stresses in radial, tangential, and axial directions respectively, while Kre, Kte, and Kae are due to the frictions between the edge of the tool and the workpiece. The values of the coefficients are reported in Tab. D.2. The axial component of the cutting force, that is Fz, is small and it is almost constant with respect to feed rate. Therefore Kac is considered to be 0 N/mm2. Such assumption does not a'ect the calculation of the lobes diagram, since the model used in this study considers only the forces perpendicular to the axis of the spindle. Figure D.1: Cutting forces at di'erent feed rates. The components of the force are
parallel (left) to the feed, perpendicular (centre), and along the axis of the tool (right). Table D.2: Cutting force constants for the AA 6082-T6. Constant Krc Kre Ktc Kte Kac Kae Value 264.9 N/mm2 53.4 N/mm 614.1 N/mm2 21.1 N/mm 0 N/mm2 3.9 N/mm 168 APPENDIX E. TECHNICAL DRAWINGS OF THE ALESAMONTI MB63 Appendix E Technical drawings of the Alesamonti MB63 Figure E.1: Lateral view. 169 APPENDIX E. TECHNICAL DRAWINGS OF THE ALESAMONTI MB63 Figure E.2: Frontal view. 170 APPENDIX F. INFORMATION LOGGED BY THE EPC Appendix F Information logged by the EPC Table F.1: Codes of the machine status parameter. Code Meaning 0 Manual mode. 1 Automatic mode. 2 Block syntax error. 3 Waiting to start. 5 Stop. 6 Executing G-code program or block. 171 APPENDIX F. INFORMATION LOGGED BY THE EPC Table F.2: Structure of the Status C-struct. Data Field Unit of meas. Description Time t ms Time in milliseconds elapsed since the
milling process has started. Curvilinear blkspace mm Travelled length since the last block abscissa has started. Axes positions xm mm Position of the five axis referred to the ym double local user-defined workpiece frame. zm double wm double bm double Axes speed xv mm/min Mean axis speed calculated starting from yv double the values transduced by the main counting zv double device. max and min data available. wv double bv double Feed rate F mm/min Programmed feed rate, upstream of the
overrides. max and min data available. F o mm/min Programmed feed rate, downstream of the
overrides. max and min data available. F instvel mm/min Instantaneous feed rate. Spindle speed S RP M Programmed spindle speed, upstream of
the overrides. max and min data available. S o RP M Programmed spindle speed, downstream of
the overrides. max and min data available. S instvel RP M Instantaneous spindle speed. Absorbed current current % Absorbed current by the spindle. max and min data available. Absorbed power power W Absorbed electrical power by the spindle. max and min data available. Torque at the spindle torque N m Torque at the spindle. max and min data available. Machine status jobstatus / Code used to identify the machine status.
See table F.1 Block number nblk / Number of the executed block. Active block blk / String with the block currently active. Executed file iso file / String with the executed part program. 172 APPENDIX F. INFORMATION LOGGED BY THE EPC Table F.3: List of parameters and sub parameters listened by the Listener instance. M.XAC Sub parameter Unit of Meas. Note XM , Y M , ZM , W M , BM mm Axis positions. MIC BLK F INFO Sub parameter Unit of Meas. Note ACT BLK / String of the executed block. ACT F ILE / Name of the executed program. MIC INST DATA Sub parameter Unit of Meas. Note M S T IM E ms Time elapsed since the process started. M XA X mm Axes position in the machine frame. M XA Y mm M XA Z mm M XA A1 mm M XA A2 deg BLKSP ACE mm Curvilinear abscissae of the active block. JOBST AT U S / Machine state, see Tab. F.1 IN ST V X AV G mm/min Average axis speed. IN ST V Y AV G mm/min IN ST V Z AV G mm/min IN ST V A1 AV G mm/min IN ST V A2 AV G mm/min IN ST V X M IN mm/min IN ST V Y M IN mm/min IN ST V Z M IN mm/min IN ST V A1 M IN mm/min IN ST V A2 M IN mm/min IN ST V X M AX mm/min IN ST V Y M AX mm/min IN ST V Z M AX mm/min IN ST V A1 M AX mm/min IN ST V A2 M AX mm/min F N OV AV G mm/min Feed rate upstream the override. F N OV M IN mm/min F N OV M AX mm/min F OV AV G mm/min Feed rate downstream the override. F OV M IN mm/min F OV M AX mm/min S N OV AV G mm/min Spindle speed upstream the override. S N OV M IN mm/min S N OV M AX mm/min S OV AV G mm/min Spindle speed downstream the override. S OV M IN mm/min S OV M AX mm/min SP D CU RREN T AV G 1000 = 100 % Absorbed current. SP D CU RREN T M IN 1000 = 100 % SP D CU RREN T M AX 1000 = 100 % SP D P OW ER AV G W Absorbed electrical power. SP D P OW ER M IN W SP D P OW ER M AX W SP D T ORQU E AV G N m Torque at the spindle. 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Document Outline

Ringraziamenti Introduction Motivation and Objectives Sensing layer Perception layer Cognitive layer Overview of Milling Process Adaptive Controllers Adaptive Control with Constraints Geometry Adaptive Control Adaptive Control with Optimisation Statement of Originality Optimal Control Problem Theoretical primer Formulation Solution OCP for milling processes Productivity Quality Costs DAE constraints Simulacrum Milling machine Alesamonti MB63 Numerical Control FIDIA C20 Sensors Dynamometric table Accelerometer Evaluation and Perception Controller Process simulator Ruby Numerical Control Cut simulator OCP solver Learning layer Sensor Fusion Validation Communication Layers Common Object Request Broker Architecture Interface Ruby - CORBA Characterisation and validation Task Execution Time Validation of the acquisitions Latencies and sampling frequencies Machine Identification Dynamics of axes Maximum feed rates and accelerations Transients of feed rates and accelerations Dynamics of the spindle Spindle acceleration Mechanical torque Dynamic compliance System identification Regression of the dynamic compliance Validation of the regression Validation of the stability lobes diagrams Applications of Augmented Reality for Safe Manufacturing State of the art 3D scene reconstruction Exploitations Results Offline optimisation Conclusion Summary of Thesis Achievements Future Works Appendices Gaussian Processes Calculation of Stability Lobes diagram Normal Distribution Transform Identification of the cutting constants of the workpiece Technical drawings of the Alesamonti MB63 Information logged by the EPC Bibliography


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