Static Accuracy Enhancement of Redundantly ...

Static Accuracy Enhancement of Redundantly Actuated Parallel Kinematic Machine Tools Thesis for the award of the academic title Doktor-Ingenieur (Dr.-Ing.) Docteur de l’INSA de Rennes approved by the Fakult¨at f¨ur Maschinenbau, Technische Universit¨at Chemnitz and by the Institut National des Sciences Appliqu´ees de Rennes submitted by Ga¨el Ecorchard, M.Sc., born January 9, 1979 in Vannes (France), on October 6, 2008. Referees: Univ.-Prof. Dr.-Ing. habil. Prof. E.h. Dr.-Ing. E.h. Dr. h.c. Reimund Neugebauer o. Prof. Dr.-Ing. Alexander Verl Dr. Fran¸cois Pierrot

URL: http://archiv.tu-chemnitz.de/pub/2009/0206

Chemnitz, March 4, 2010

Bibliographic Description Ecorchard, Ga¨el

Title Static Accuracy Enhancement of Redundantly Actuated Parallel Kinematic Machine Tools Dissertation at the Fakult¨at f¨ur Maschinenbau, Technische Universit¨at Chemnitz, Institut f¨ur Werkzeugmaschinen und Produktionsprozesse and at the Institut National des Sciences Appliqu´ees, Rennes, November 19, 2009. 164 pages 101 figures 34 tables 151 bibliographic references

Short summary Redundantly actuated parallel kinematic machines are a new type of mechanism derived from classical parallel kinematic machines by adding one or more redundant links. In this dissertation, new calibration methods have been developed to improve the static positioning of such machines. These methods are based on geometrical and elasto-geometrical modeling. The latter takes into account the elements’ elastic deformations due to the redundancy.

Keywords Redundantly actuated parallel kinematic machines, static accuracy, calibration, elasto-geometrical modeling

Abstract

Abstract Redundant parallel kinematic machines are parallel mechanisms to which one or more kinematic branch is added in order to improve their mechanical properties, in particular, their stiffness. Redundant parallel kinematic machines have then more actuators than their degree of freedom. New calibration methods are developed in this thesis in order to deal with the particularities related to the actuation redundancy. First, calibration methods using geometrical models are tested. Several measurement systems and control models are compared. A self-calibration is also carried out, where the redundant branches are switched to a passive mode. Thus, they play the role of the measurement system and the mechanism can be calibrated without the help of extra sensors. Geometrical calibration methods, however, do not take into account the internal constraints due to the redundancy. Elastic deformations are neglected although they are shown to have an influence on the positioning accuracy after the calibration. Modeling methods are then developed that take into account the geometry of the mechanism as well as the stiffness of its elements to improve the accuracy of the calibration. With such modeling methods, it is possible to determine the tool-center-point position for redundantly actuated parallel mechanisms from geometrical and stiffness parameters and given positions for all actuators. The modeling methods are first demonstrated on a simple mechanism. They are then tested on a real machine and used in calibration processes.

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R´esum´e

R´ esum´ e Am´elioration de la pr´ecision statique des machines-outils `a cin´ematique parall`ele redondante

Les m´ecanismes `a cin´ematique parall`ele redondante sont des m´ecanismes parall`eles auxquels a ´et´e ajout´ee une branche cin´ematique, ou plus, dans le but d’am´eliorer leurs propri´et´es m´ecaniques, en particulier, leur rigidit´e. Les m´ecanismes `a cin´ematique parall`ele redondante poss`edent donc plus d’actionneurs que leur degr´e de libert´e. De nouvelles m´ethodes d’´etalonnage sont d´evelopp´ees dans cette th`ese afin de prendre en compte les sp´ecificit´es li´ees `a la redondance d’actionnement. Les m´ethodes d’´etalonnage utilisant des mod`eles g´eom´etriques sont d’abord test´ees. Plusieurs syst`emes de mesure et plusieurs mod`eles de contrˆole sont compar´es. Un auto-´etalonnage est aussi r´ealis´e. Pour cette m´ethode d’´etalonnage, les actionneurs redondants sont mis en mode passif et jouent le rˆole de syst`eme de mesure. Le m´ecanisme peut ˆetre ´etalonn´e sans ajout de codeurs. Cependant, les m´ethodes d’´etalonnage g´eom´etriques ne prennent pas en compte les contraintes internes li´ees `a la redondance. Les d´eformations ´elastiques sont n´eglig´ees bien qu’il soit montr´e qu’elles ont une influence sur la pr´ecision de positionnement apr`es ´etalonnage. Des m´ethodes de mod´elisation qui prennent en compte la g´eom´etrie du m´ecanisme ainsi que la rigidit´e des ´el´ements sont donc d´evelopp´ees pour am´eliorer la pr´ecision de l’´etalonnage. Avec de telles m´ethodes, il est possible de d´eterminer la position de l’outil des m´ecanismes `a redondance d’actionnement `a partir de param`etres g´eom´etriques et ´elastiques et de la position de tous les actionneurs. Les m´ethodes de mod´elisation sont d’abord appliqu´ees sur un m´ecanisme simple. Elles sont ensuite test´ees sur une machine r´eelle et utilis´ees dans des processus d’´etalonnage.

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Zusammenfassung

Zusammenfassung Verbesserung der statischen Genauigkeit von Werkzeugmaschinen mit redundanter Parallelkinematik

Mechanismen mit redundanter Parallelkinematik sind Parallelmechanismen, denen eine oder mehrere kinematischen Ketten zugef¨ugt werden, um die mechanischen Eigenschaften, insbesondere die Steifigkeit, zu verbessern. Maschinen mit redundanter Parallelkinematik besitzen dann mehrere Antriebe als ihr Freiheitsgrad erforden w¨urde. In dieser Dissertation werden neue Kalibrierungsmethoden entwickelt, um die mit der Antriebsredundanz gebundenen Besonderheiten zu betrachten. Zuerst werden Kalibrierungsmethoden basierend auf geometrischen Modellen getestet. Verschiedene Messmethoden und Messsysteme werden verglichen. Eine Selbstkalibrierung wird durchgef¨uhrt. Bei dieser Kalibrierungsmethode werden die redundanten Antriebe freigeschaltet und als Messsystem genutzt. Die Maschine kann dadurch ohne externes Messsystem kalibriert werden. Dennoch betrachten geometrische Kalibrierungsmethode keine internen Verspannungen, die mit der Redundanz verbundenen sind. Die elastischen Verformungen werden vernachl¨assigt, obwhol gezeigt wird, dass sie einen Einfluss auf die Positioniergenauigkeit nach der Kalibrierung haben. Es werden deshalb Modellierungsmethoden entwickelt, die sowohl die Geometrie des Mechanismus als auch die Elementsteifigkeit betrachten, um die Genauigkeit der Kalibrierung zu verbessern. Mit solchen Methoden ist es m¨oglich, die Werkzeugposition redundanter Parallelkinematiken aus den Geometrie- und Steifigkeitsparametern und allen Antriebspositionen zu bestimmen. Die Modellierungsmethoden werden zuerst an einem einfachen Mechanismus angewandt. Sie werden danach an einer realen Maschine getestet und in einem Kalibrierungsprozess genutzt.

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Acknowledgements

Acknowledgements With these few words I would like to express my gratitude to all persons and institutions who helped me to fulfill my thesis. In advance, I apologize to those who I might forget or not directly mention. At first, I would like to greatly thank my two supervisors, Prof. Reimund Neugebauer for the German part and Prof. Vigen Arakelyan for the French one. They both accepted me as a student in their institution and, thus, took the risk and made the effort to support the fact that I decided to write my thesis as a cooperation between the University of Technology TU Chemnitz, and the National Institut for Applied Sciences INSA Rennes. I specially would like to thank Prof. Neugebauer for its financial and technical support at the Fraunhofer Institut for Machine Tools and Forming Technology IWU Chemnitz. Without this financial support and without the fact that the Scissors-Kinematics, on which my work is based, was made available for my experiments my thesis would not have been possible. Secondly, I would like my tutors Dr. Welf-Guntram Drossel and Dr. Patrick Maurine for their sensible advices and their availability despite their heavy workload. I would like to address Patrick a special thank for his closer technical exchanges. For their financial support, I’m grateful to the German Academic Exchange Service DAAD and the French-German University UFA-DFH. I would like to thank all my colleagues from the Frauhnofer IWU, the INSA and the IWP for their help and support and for making me good and pleasant work conditions. I also thank my friends for their support. A special thank goes particularly to Maria Bobrova who greatly helped me avoiding lots of English mistakes and unclarities, making my thesis less unreadable. I’m also grateful to all unknown friends of the Open Source Community who contributed in the softwares I used during my thesis, among others LaTeX & Friends, jEdit, Inkscape, The Gimp. I’m also really thankful to my family, especially to my parents, for helping and encouraging me. I apologize for my choosing of writing my thesis too far from them, at least during a part of PhD studies. At last, I want to thank my son Goulven for motivating me, even though he was not aware of it, during the final phase of my thesis because I wanted to finish quickly to have more time with him. I apologize to him not having submitted my thesis before his birth, as was the first plan. The last person I want to thank is my wife Petra. I really want to express her my gratitude for her great day-to-day love, support and encouragements.

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Contents

Contents 1 Introduction

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2 State of the Art 2.1 Redundancy for Parallel Kinematic Machines . . . . . . . . . . . 2.1.1 Redundancy Types for Parallel Manipulators . . . . . . . 2.1.2 Research on Actuation Redundancy . . . . . . . . . . . . 2.1.3 Existing Mechanisms with Actuation Redundancy . . . . . 2.2 Calibration and Self-Calibration of Classical and Redundant PKMs 2.2.1 Existing Methods for the Calibration of PKMs . . . . . . 2.2.2 Existing Methods for the Self-Calibration of PKMs . . . . 2.2.3 Calibration of Redundant PKM . . . . . . . . . . . . . . 2.3 Elastic Modeling Methods for PKMs . . . . . . . . . . . . . . . 2.3.1 Use of the Jacobian Matrix . . . . . . . . . . . . . . . . 2.3.2 Lumped Models . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Finite-Element Methods . . . . . . . . . . . . . . . . . . 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Calibration Principle and Optimization Parameters 3.1 Calibration Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Error Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Specificity of the Calibration of Parallel Mechanisms against Serial Ones 3.1.3 Formulation of the Least-Squares Problem . . . . . . . . . . . . . . . 3.2 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Existing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Constrained and Unconstrained Optimization . . . . . . . . . . . . . . 3.3 Calibration-Compatible Modeling . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Minimality, Completeness and Consistency of Models . . . . . . . . . . 3.3.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Equation System Conditioning . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Jacobian Normalization . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Determination of a Set of Identifiable Parameters . . . . . . . . . . . . 3.4 Calibration Lead-Through . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Elimination of Measurement Outliers . . . . . . . . . . . . . . . . . . 3.4.2 Calibration Certification . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Parameter Dependencies . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Final Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

4 External and Autonomous Geometrical Calibrations of Redundantly Actuated Parallel Kinematic Machines 4.1 Presentation of the Scissors-Kinematics . . . . . . . . . . . . . . . . . . . . . 4.1.1 Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Complete Planar Model . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Model Identification through External Calibration . . . . . . . . . . . . . . . . 4.2.1 Calibration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Jacobian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Observability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Model Identification through Self-Calibration . . . . . . . . . . . . . . . . . . 4.3.1 Self-Calibration Method . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Jacobian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Observability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Calibration with the Laser-Tracker . . . . . . . . . . . . . . . . . . . . 4.4.2 Calibration with the Grid-Encoder . . . . . . . . . . . . . . . . . . . . 4.4.3 Calibration with the Double Ball-Bar . . . . . . . . . . . . . . . . . . 4.4.4 Calibration with the Laser-Interferometer . . . . . . . . . . . . . . . . 4.4.5 Influence of the Platform Angle Measurements . . . . . . . . . . . . . 4.4.6 Self-Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Synthesis and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Static Elastic Deformation Models of Redundantly Actuated matic Mechanisms 5.1 Statistical Method . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Least-Squares Minimization Method . . . . . . . . . . . . . . . 5.3 Lumped Elastic Models . . . . . . . . . . . . . . . . . . . . . 5.4 Finite-Element Modeling . . . . . . . . . . . . . . . . . . . . . 5.4.1 Rod Modeling . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Joint Modeling . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Structure Assembly . . . . . . . . . . . . . . . . . . . 5.4.4 Modeling of the Base Links . . . . . . . . . . . . . . . 5.4.5 Resolution . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Actuator-Displacements Method . . . . . . . . . . . . . . . . . 5.6 Structure-Division Method . . . . . . . . . . . . . . . . . . . . 5.7 Joint-Internal-Forces Method . . . . . . . . . . . . . . . . . . 5.8 Thermo-Mechanical Model . . . . . . . . . . . . . . . . . . . . 5.9 Method Comparison and Conclusion . . . . . . . . . . . . . . . 5.9.1 Method Comparison . . . . . . . . . . . . . . . . . . . 5.9.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 6 Calibration havior 6.1 Elastic 6.1.1 6.1.2 6.1.3

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Parallel Kine. . . . . . . . . . . . . . . .

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of the Scissors-Kinematics with Consideration of its Elastic Be108 Modeling of the Scissors-Kinematics . . . . . . . . . . . . . . . . . . . 109 Statistical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Least-Square Minimization Method . . . . . . . . . . . . . . . . . . . 109 Lumped Elastic Models . . . . . . . . . . . . . . . . . . . . . . . . . 111 8

Contents

6.2

6.3

6.4

6.1.4 Finite-Element Methods . . . . . . . . . . . 6.1.5 Actuator-Displacements Method . . . . . . . 6.1.6 Structure-Division Method . . . . . . . . . . 6.1.7 Joint-Internal-Forces Method . . . . . . . . 6.1.8 Thermo-Mechanical Model . . . . . . . . . 6.1.9 Method Comparison . . . . . . . . . . . . . Calibration with Elasto-Geometrical Models . . . . . 6.2.1 Calibration Method . . . . . . . . . . . . . 6.2.2 Jacobian Matrix . . . . . . . . . . . . . . . 6.2.3 Sensitivity Analysis . . . . . . . . . . . . . . 6.2.4 Observability Analysis . . . . . . . . . . . . Experimental Validation . . . . . . . . . . . . . . . 6.3.1 Validation of the Elasto-Geometrical Models 6.3.2 Calibration with the Laser-Tracker . . . . . . 6.3.3 Calibration with the Grid-Encoder . . . . . . 6.3.4 Calibration with the Double Ball-Bar . . . . 6.3.5 Calibration with the Laser-Interferometer . . Synthesis and Conclusion . . . . . . . . . . . . . .

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7 Conclusion 135 7.1 General Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.3 Perspectives and future work . . . . . . . . . . . . . . . . . . . . . . . . . . 136 A Intersection of Two Circles B Calculation of the Jacobian Matrix for Calibrations B.1 Differential of the Closure-Loop Equations . . . . . . . . . . . . . B.2 Inverse Kinematic Calibrations . . . . . . . . . . . . . . . . . . . . B.3 Calibration with the Double Ball-Bar . . . . . . . . . . . . . . . . B.4 Calculation of the Jacobian Matrix for the Self-Calibration Method

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C Results of the Circular Tests 155 C.1 Calibration with the laser-tracker and model 24 . . . . . . . . . . . . . . . . . 156 C.2 Calibration with the laser-tracker and model 29 . . . . . . . . . . . . . . . . . 157 C.3 Calibration with the grid-encoder and model 24 . . . . . . . . . . . . . . . . . 158 C.4 Calibration with the grid-encoder and model 29 . . . . . . . . . . . . . . . . . 159 C.5 Calibration with the laser-tracker, platform angle measurements and model 29 160 C.6 Calibration with the grid-encoder, platform angle measurements and model 29 161 C.7 Self-calibration with model 24 . . . . . . . . . . . . . . . . . . . . . . . . . . 162 C.8 Self-calibration with model 29 . . . . . . . . . . . . . . . . . . . . . . . . . . 163

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List of Abbreviations

List of Abbreviations PKM TCP

parallel kinematic machine tool center point

LT LI KGM DBB

Laser-Tracker Laser-Interferometer grid-encoder (Kreuzgittermessger¨at in German) Double ball-bar

igm fgm rgm fkm fegm

inverse geometrical model forward geometrical model redundant geometrical model forward kinematic model forward elasto-geometrical model

dof dom

degree of freedom degree of measurement

FE

finite-element

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List of Mathematical Symbols

List of Mathematical Symbols In the following is a list of the most important mathematical symbols used in this dissertation. In a general manner, scalars are noted with a small italic letter, e.g. i. Vectors are noted with a bold italic capital letter, e.g. X. Capital upright bold letters denote matrices, e.g. J. a n n0 m r b s

number number number number number number number

of of of of of of of

actuators machine parameters identified parameters measurement values measurement directions mechanism kinematic loops nodes for finite-element models

X x y ψ

TCP position vector TCP x-position TCP y-position platform rotation around z-axis

P pi li xair ,yair xbir ,ybir xaia ,yaia xbia ,ybia di γi

vector of machine parameters i-th machine parameter length of rod i x and y relative co-ordinates of point Ai the same for point Bi co-ordinates of point Ai in the absolute reference frame the same for point Bi absolute position of guide i angle of guide i to the y-axis

Q Qnr qi

vector of all actuator positions vector of non-redundant actuator positions position of actuator i as well as the actuator itself

J 0n 0n,m

general notation for Jacobian matrices n × n null-matrix n × m null-matrix

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List of Mathematical Symbols

Wmeas Wmod Wref

vector of measurement values vector of model values vector of reference values

diag

diagonal function. diag(B) is the diagonal matrix which diagonal elements are the elements of B in the same order.

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List of Figures

List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13

Sensor redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . Kinematic redundancy . . . . . . . . . . . . . . . . . . . . . . . . Serial actuation redundancy. . . . . . . . . . . . . . . . . . . . . . Parallel actuation redundancy. . . . . . . . . . . . . . . . . . . . . Non-redundant (a) and redundant PKMs (b). . . . . . . . . . . . . Eclipse machining center (a) and its kinematic structure (b). . . . . The PA-R-Ma prototype (a) and its kinematic structure (b). . . . . The Shoulder Joint and its schematic representation. . . . . . . . . Kinematic structure of the shoulder joint. . . . . . . . . . . . . . . The ARCHI mechanism (a) with its kinematic structure (b). . . . . CAD view of the Eureka prototype. . . . . . . . . . . . . . . . . . Kinematic structure of the Eureka mechanism (source: [Krut 03b]). Lumped model of one kinematic chain of the Delta robot. . . . . .

3.1

Calibration process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18

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CAD representation of the Dynapod machine (Source: VRCP, Chemnitz). . . . The Scissors-Kinematics during the assembly. . . . . . . . . . . . . . . . . . . CAD representation of the Scissors-Kinematics. . . . . . . . . . . . . . . . . . CAD views of the Scissors-Kinematics. . . . . . . . . . . . . . . . . . . . . . Kinematic structure of the Scissors-Kinematics. . . . . . . . . . . . . . . . . . Simplified top view of the mechanism. . . . . . . . . . . . . . . . . . . . . . . Geometrical parameters for model 24. . . . . . . . . . . . . . . . . . . . . . . Geometrical parameters for model 29. . . . . . . . . . . . . . . . . . . . . . . Used measurement systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity for the calibration with the laser-tracker and model 24. . . . . . . . Sensitivity for the calibration with the grid-encoder and model 24. . . . . . . . Sensitivity for the calibration with the double ball-bar and model 24. . . . . . . Sensitivity for the forward calibration with the laser-interferometer with model 24. Sensitivity for calibration with the laser-tracker and model 29. . . . . . . . . . Sensitivity for the calibration with the grid-encoder and model 29. . . . . . . . Sensitivity for the calibration with the double ball-bar and model 29. . . . . . . Sensitivity for the calibration with the laser-interferometer and model 29. . . . Sensitivity for calibrations with the laser-tracker, platform angle measurements and model 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.19 Sensitivity for the calibration with the grid-encoder, platform angle measurements and model 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20 Self-calibration principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.21 Sensitivity analysis for the self-calibration and model 24. . . . . . . . . . . . .

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List of Figures

4.22 4.23 4.24 4.25 4.26 4.27

Sensitivity analysis for the self-calibration and model 29. . . . . . . . . . . . Calibration results with the laser-tracker and model 24. . . . . . . . . . . . . Calibration results with the laser-tracker and model 29. . . . . . . . . . . . . Calibration results with the grid-encoder and model 24. . . . . . . . . . . . Calibration results with the grid-encoder and model 29. . . . . . . . . . . . Calibration results with the laser-tracker, platform angle measurements and model 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.28 Calibration results with the grid-encoder, platform angle measurements and model 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.29 Self-calibration results with model 24. . . . . . . . . . . . . . . . . . . . . . 4.30 Self-calibration results with model 29. . . . . . . . . . . . . . . . . . . . . . 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 6.1 6.2

. . . . .

70 74 75 76 77

. 80 . 81 . 83 . 83

The Redundant Triglide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Kinematic structure of the Redundant Triglide. . . . . . . . . . . . . . . . . . 88 Statistical method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Modeling of elastic elements subject to bending or tension. . . . . . . . . . . 90 Method of the lumped elastic model - Steps 1-3 . . . . . . . . . . . . . . . . 91 Method of the lumped elastic model - Steps 4-6 . . . . . . . . . . . . . . . . 91 FE model of the Redundant Triglide. . . . . . . . . . . . . . . . . . . . . . . 92 Local wrenches and displacements on an isolated beam projected in the x-y-plane. 92 Beam weight distribution on the nodes. . . . . . . . . . . . . . . . . . . . . . 96 Deformation of flexible joints with initial coincident nodes. . . . . . . . . . . . 96 Deformation of flexible joints with non-coincident nodes. . . . . . . . . . . . . 97 Biglide with an actuated redundant branch. . . . . . . . . . . . . . . . . . . . 97 Actuator-displacements method - Steps 1-3 . . . . . . . . . . . . . . . . . . . 99 Extra nodes for rotatory actuators. . . . . . . . . . . . . . . . . . . . . . . . 100 Actuator-displacements method - Step 4 . . . . . . . . . . . . . . . . . . . . 100 Structure-division method - Steps 1-4 . . . . . . . . . . . . . . . . . . . . . . 101 Structure-division method - Steps 5-6 . . . . . . . . . . . . . . . . . . . . . . 101 Joint-internal-forces method - Steps 1-3. . . . . . . . . . . . . . . . . . . . . 103 Joint-internal-forces method - Step 4. . . . . . . . . . . . . . . . . . . . . . . 103 TCP x displacement of the Redundant Triglide. . . . . . . . . . . . . . . . . 105 TCP y displacement of the Redundant Triglide. . . . . . . . . . . . . . . . . . 105

Influence of the redundancy on the repeatability of the Scissors-Kinematics. . . Two assembly modes near the singularity for the subsystem q2 -q4 of the ScissorsKinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Statistical method for the Scissors-Kinematics. . . . . . . . . . . . . . . . . . 6.4 Method of the lumped elastic model - Steps 1-3 . . . . . . . . . . . . . . . . 6.5 Passive and compliant joints on rods 2 and 3. . . . . . . . . . . . . . . . . . . 6.6 Method of the lumped elastic model - Steps 4-6 . . . . . . . . . . . . . . . . 6.7 Beam-element model of the Scissors-Kinematics. . . . . . . . . . . . . . . . . 6.8 One rod of the Scissors-Kinematics. . . . . . . . . . . . . . . . . . . . . . . . 6.9 Error on the calculation of the TCP displacement with respect to the stiffness coefficients for passive and rigid joints. . . . . . . . . . . . . . . . . . . . . . 6.10 Condition number of the stiffness matrix with respect to the stiffness coefficients for passive and rigid joints. . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Actuator-displacements method for the Scissors-Kinematics - steps 1 to 3. . . 6.12 Actuator-displacements method for the Scissors-Kinematics - step 4. . . . . .

108 110 110 111 112 114 115 116 116 117 117 118 14

List of Figures

6.13 Representing the structure by the equivalent stiffness of substructures. . . . . . 6.14 Solving the equivalent FE Model. . . . . . . . . . . . . . . . . . . . . . . . . 6.15 Solving the problem of non-coincident joints by considering internal forces step 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.16 Solving the problem of non-coincident joints by considering internal forces step 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.17 Output of the thermo-mechanical method with ANSYS code. . . . . . . . . . 6.18 TCP y displacement (δl3 = 10 µm) . . . . . . . . . . . . . . . . . . . . . . . 6.19 TCP x displacement (δl2 = δl3 = 10 µm) . . . . . . . . . . . . . . . . . . . . 6.20 Sensitivity analysis with the laser-tracker and the elasto-geometrical model. . . 6.21 Sensitivity analysis with the grid-encoder and the elasto-geometrical model. . . 6.22 Sensitivity analysis with the double ball-bar and the elasto-geometrical model. . 6.23 Sensitivity analysis with the laser-interferometer and the elasto-geometrical model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.24 y TCP displacement for test 1. . . . . . . . . . . . . . . . . . . . . . . . . . 6.25 x TCP displacement for test 2. . . . . . . . . . . . . . . . . . . . . . . . . . 6.26 Sensitivity analysis on the TCP x position. . . . . . . . . . . . . . . . . . . . 6.27 Sensitivity analysis on the TCP y position. . . . . . . . . . . . . . . . . . . .

118 120 121 122 122 123 124 126 127 127 128 129 129 130 130

A.1 Intersection of two circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 C.1 C.2 C.3 C.4 C.5

Circular test for the calibration with the laser-tracker and model 24. . . . . . Circular test for the calibration with the laser-tracker and model 29. . . . . . Circular test for the calibration with the grid-encoder and model 24. . . . . . Circular test for the calibration with the grid-encoder and model 29. . . . . . Circular test for the calibration with the laser-tracker, platform angle measurements and model 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.6 Circular test for the calibration with the grid-encoder, platform angle measurements and model 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.7 Circular test for the self-calibration with model 24. . . . . . . . . . . . . . . C.8 Circular test for the self-calibration with model 29. . . . . . . . . . . . . . .

. . . .

156 157 158 159

. 160 . 161 . 162 . 163

15

List of Tables

List of Tables 2.1 2.2

Common measurement systems for the calibration of parallel mechanisms. . . . 27 Selection of references for common measurement systems. . . . . . . . . . . . 28

4.1 4.2 4.3 4.4

Nominal parameters for model 24. . . . . . . . . . . . . . . . . . . . . . . . . 50 Complete parameter set for geometrical models (non-redundant part). . . . . . 53 Complete parameter set for geometrical models (redundant part). . . . . . . . 54 Sorted geometrical parameters and the associated observability index for several calibration methods and model 24. . . . . . . . . . . . . . . . . . . . . . . . 66 Sorted geometrical parameters and the associated observability index for several calibration methods with model 29. . . . . . . . . . . . . . . . . . . . . . . . 67 Sorted geometrical parameters and the associated observability index for several calibration methods with model 29 and platform angle measurements. . . . . . 68 Sorted geometrical parameters and the associated observability index for the self-calibration method and model 24. . . . . . . . . . . . . . . . . . . . . . . 71 Parameters for the self-calibration with model 29 sorted in decreasing order of influence and the associated condition number. . . . . . . . . . . . . . . . . . 72 Compatibility test for the parameters of the calibration with the laser-tracker and model 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Compatibility test for the parameters of the calibration with the laser-tracker and model 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Compatibility test for the parameters of the calibration with the grid-encoder and model 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Compatibility test for the parameters of the calibration with the grid-encoder and model 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Compatibility test for the parameters of the calibration with the double ball-bar and model 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Compatibility test for the parameters of the calibration with the double ball-bar and model 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Compatibility test for the parameters of the calibration with the laser-interferometer and model 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Compatibility test for the parameters of the calibration with the laser-interferometer and model 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Compatibility test for the parameters of the calibration with the laser-tracker, platform angle measurements, and model 29. . . . . . . . . . . . . . . . . . . 80 Compatibility test for the parameters of the calibration with the grid-encoder completed by platform angle measurements with model 29. . . . . . . . . . . 81 Compatibility test for the parameters of the self-calibration with model 24. . . 82 Compatibility test for the parameters of the self-calibration with model 29. . . 84

4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20

16

List of Tables

4.21 Summary of geometrical calibration results. . . . . . . . . . . . . . . . . . . . 84 5.1 5.2

Parameters of a beam for the FE Method. . . . . . . . . . . . . . . . . . . . 93 Nominal parameter values of the Redundant Triglide (mm). . . . . . . . . . . 106

6.1 6.2

Repeatability test for the Scissors-Kinematics. . . . . . . . . . . . . . . . . . 109 Comparison of the various modeling methods for the computation of the influence of parameter l2 on the TCP position. . . . . . . . . . . . . . . . . . . . 123 Sorted geometrical parameters and the associated observability index for several calibration methods with the elasto-geometrical model. . . . . . . . . . . . . . 128 Summary for the sensitivity analysis on the real machine and the simulated one. 131 Compatibility test for the parameters of the calibration with the laser-tracker and the elasto-geometrical model. . . . . . . . . . . . . . . . . . . . . . . . . 132 Compatibility test for the parameters of the calibration with the laser-tracker, pre-identified parameters and the elasto-geometrical model. . . . . . . . . . . 132 Compatibility test for the parameters of the calibration with the grid-encoder and the elasto-geometrical model. . . . . . . . . . . . . . . . . . . . . . . . . 133 Compatibility test for the parameters of the calibration with the double ball-bar and the elasto-geometrical model. . . . . . . . . . . . . . . . . . . . . . . . . 133 Compatibility test for the parameters of the calibration with the laser-interferometer and the elasto-geometrical model. . . . . . . . . . . . . . . . . . . . . . . . . 134

6.3 6.4 6.5 6.6 6.7 6.8 6.9

17

Chapter 1. Introduction

Chapter 1 Introduction The development of parallel kinematic machines began at the end of the fifties. The mechanism that is considered to be the pioneer for this technology is the Gough platform used in the tire industry from the middle of the twentieth century on. Compared to serial mechanisms, they can offer better stiffness, accuracy, dynamics and ratio payload/mass of the mechanism, among others. However, their acceptance is still pretty low in industry, especially for the use of these mechanisms as machine tools. One reason for this may be the size of the workspace, which is limited both by the mechanics and the presence of singularities within the workspace. The passive joints of parallel kinematic machines are also a key point in the mechanism accuracy. The high requirements imposed on these elements are, however, difficult to meet. The mechanism accuracy is, for example, negatively influenced by even small backlashes in these joints. A redundantly actuated parallel kinematic mechanism is in principle a parallel mechanism to which one or more extra actuated chains are added whereas the degree of freedom of the Tool Center Point (TCP) remains constant. The idea behind actuation redundancy is to improve the properties of classical non-redundant parallel mechanisms by fighting particularly against two drawbacks, the singularities and the joint backlashes. A redundantly actuated mechanism can be seen as the association of two or more non-redundant subsystems. Since the different subsystems have different singular positions, there is, for all positions in the workspace, at least one of the subsystems which is not at a singular position. One of the advantages of redundantly actuated mechanisms is that an internal preload can be applied so that the forces in the mechanism have a direction which vary only slightly. The backlashes are thus reduced and the accuracy is improved. An important problem for redundantly actuated parallel mechanisms is their calibration. Indeed, whereas for classical non-redundant parallel mechanisms, parameter errors induce positioning error exclusively, such errors are also responsible for internal constraints in redundantly actuated parallel mechanisms because of actuators acting against each other. Besides, the calibration algorithms must be adapted to account for the actuation redundancy, synonymous of over-determined equation system of loop-closure equations. My dissertation is divided into five chapters. ˆ Chapter 1 presents the state of the art on three main topics: the redundancy in parallel kinematic machines, the calibration of non-redundant and redundant parallel mechanisms, and their elastic modeling. ˆ In Chapter 2, the mathematical and technical background for calibration is given. This Chapter is necessary to clarify the notations and to present the algorithms and other notions that are used throughout my dissertation.

18

Chapter 1. Introduction

ˆ Chapter 3 is twofold. First, it introduces the Scissors-Kinematics, the mechanism that was the practical base of my research. Second, it describes the particularities of the calibration of geometrical models for redundant parallel kinematics. ˆ Chapter 4 is the core of my work. In this Chapter, the problem of the accurate tool center point position calculation for redundantly actuated parallel kinematic machines is treated. ˆ In Chapter 5, before the conclusion, the various methods developed in Chapter 4 are adapted to the Scissors-Kinematics. They are then experimentally validated and used for the calibration of the mechanism.

19

Chapter 2. State of the Art

Chapter 2 State of the Art The bibliographic work related to my dissertation focuses on three topics. The different types of redundancy are first explained. Previous works on redundantly actuated parallel mechanisms are naturally deeply studied, and in particular existing mechanisms are given as examples. The second Section deals with the calibration of parallel kinematic machines and redundantly actuated parallel mechanisms. Quick methods for the modeling of the stiffness of mechanisms are the subject of the last part of this Chapter.

2.1 2.1.1

Redundancy for Parallel Kinematic Machines Redundancy Types for Parallel Manipulators

The kinematic structure of the mechanisms are represented with the following symbols: passive joint actuated joint joint with sensor P prismatic joint R revolute joint S spherical joint U universal joint 2.1.1.1

Sensor Redundancy

The term sensor redundancy is used when the number of sensors built on a mechanism is greater than the number of controlled axes. On non-redundant parallel mechanisms the extra sensors are built on passive links, cf. Fig. 2.1. Redundantly actuated parallel mechanisms can be regarded as non-redundant parallel mechanisms with an intrinsic sensor redundancy if the redundant actuators can be used as followers, i.e. switched to a passive mode. The sensor redundancy can be used on parallel mechanisms to reach a simplification of the forward kinematic model, [Merlet 93], [Bonev 99], [Bonev 00], [Bonev 01], [Stoughton 91]. In an ideal case, their presence even enables one to calculate the TCP position from actuator positions analytically, thus allowing, for instance, the use of dynamical models in the Cartesian space. They also provide extra boundary conditions that limit the number of solutions of this transformation. Redundant sensors can be also implemented as security features on PKMs. Their measured values can be confronted to the values obtained from actuator positions and allow the detection

20

Chapter 2. State of the Art

Figure 2.1: Sensor redundancy of mechanical problems or the need of a new calibration. Moreover, redundant sensors can be used in self-calibration processes, cf. [Daney 03]. They provide some information about the internal state of the machine, thus allowing the identification of machine parameters. 2.1.1.2

Kinematic Redundancy

A kinematically redundant mechanism is a mechanism in which the internal actuated degreeof-freedom is higher than the dimension of the task-space, cf. Fig. 2.2 and [Marquet 02a]. The mechanism is not fully parallel anymore. It is called hybrid mechanism because it has some properties of both serial and parallel mechanisms. The kinematic redundancy allows one to access a single TCP pose (position and orientation) with an infinite number of machine configurations. Kinematic redundancy should not be mistaken for actuation redundancy. When not explicitly specified, redundancy refers to actuation redundancy in my dissertation.

Figure 2.2: Kinematic redundancy

2.1.1.3

Actuation Redundancy

A redundantly actuated mechanism is a statically over-determined mechanism, where the number of actuators is greater than the mechanism degree of freedom. Such mechanisms are usually subject to internal constraints. Two types of actuation redundancy exist, a serial actuation redundancy, cf. Fig. 2.3, and a parallel actuation redundancy, cf. Fig. 2.4. However, one can hardly find them being distinguished in the literature. The only author who distinguishes between them is Kim in [Kim 97a]. He denotes serial actuation as redundancy of Type I and parallel actuation as redundancy of Type II. The serial actuation redundancy in robotic research is mostly a collaboration of two serial arms manipulating a single object, [Koeppe 05].

21

Chapter 2. State of the Art

In [Kock 98], the authors studied a 2-dof mechanism with parallel actuation redundancy. For both types of actuation redundancy, a given platform position corresponds to a unique set of actuator positions, naturally for the same assembly mode. Some slight deviations of these actuator positions can then be applied to obtain an internal preload, [Lee 05], [M¨uller 05]. On the contrary to parallel mechanisms with kinematic redundancy or serial actuation redundancy, the link considered to be redundant can be removed from parallel redundantly actuated mechanisms without altering the mechanism function. The redundant link is kinematically parallel to all other links.

P

R

P

R

R

Figure 2.3: Serial actuation redundancy.

Figure 2.4: Parallel actuation redundancy. The most typical example of the use of actuation redundancy is the tendon-driven parallel manipulator. Since cables cannot be used under compression loads, the redundant cables are used to ensure that all cables are permanently tensed, cf. [Sch¨aper 06]. The main advantage of actuator redundancy is the avoidance of singularities. One of the problems of classical PKMs is the presence of singularities which is often the main factor responsible for workspace reduction. In the case of redundant PKMs, if a non-redundant subsystem reaches a singularity within the workspace, another subsystem exists which is not in a singular position and which allows the mechanism to maintain its global stiffness and controllability, [Liao 04], [Marquet 02a]. Fig. 2.5(a) and 2.5(b) illustrate this fact. The mechanism represented on Fig. 2.5(a) is a planar 3-RPR mechanism with actuated prismatic joints. R stands for revolute joint and P stands for actuated prismatic joint. It is represented here in the singular central position. The fourth leg maintains the global stiffness of the mechanism because, e.g., the subsystem containing the actuators 1, 2 and 4 is not in a singular position.

2.1.2

Research on Actuation Redundancy

Since a few years, researchers have begun to show interest in actuation redundancy for PKMs. As already mentioned, its main advantage is a higher stiffness owing to singularity avoidance, 22

Chapter 2. State of the Art

(a)

(b)

Figure 2.5: Non-redundant (a) and redundant PKMs (b). cf. [Liao 04]. This was what also motivated the addition of two redundant actuators to the Eclipse mechanism, [Iura¸scu 99], [Bo¨er 99], [Kim 01]. Another fact that is specific to redundant PKM is that backlash can be reduced owing to an internal preload. This allows a stiffness improvement and a mechanism stiffness behavior that is closer to be linear. This was developed in [M¨uller 05]. With an appropriate control, redundant PKMs also allow an adjustable stiffness at the end-effector for example for assembly tasks, cf. [Kim 97b], [Kock 98]. In the case of breakdown of one of the actuators, a redundant PKM keeps its mobility degree, which avoids great security problem, cf. [Cheng 01].

2.1.3

Existing Mechanisms with Actuation Redundancy

Only few mechanisms with actuation redundancy can be cited as examples. The number of these mechanisms designed to be used as machine tools is even more limited. As for now, no redundantly actuated PKM found an application in industry as a machine tool. This Section contains practical implementations of the kinematics presented in Section 2.1.1.3. 2.1.3.1

Eclipse

The Eclipse machine is probably the most famous PKM with actuation redundancy, cf. Fig. 2.6(a). It was designed through a collaboration between the Seoul University and Sena Technologies as a five-axis machining center. In [Kim 01], [Iura¸scu 99], the authors propose the redundancy as a solution for singularity avoidance. The proposed mechanism would be then 2-order redundant with the hybrid kinematics presented in Fig. 2.6(b). The main goal of the redundancy on this mechanism is the expansion of the workspace, in particular of the tilting capability, through singularity avoidance. The machine head can reach an inclination of ± 90◦ and a continuous sweep of 360◦ over the lateral surfaces of the workpiece. 2.1.3.2

Scissors-Kinematics

The Scissors-Kinematics is a machine tool recently developed at the Fraunhofer Institute for Machine Tools and Forming Technology (IWU), Chemnitz, Germany for large-tool-piece milling. Entire dies can be machined for the automotive industry, [Neugebauer 06]. The whole 23

Chapter 2. State of the Art

(a)

(b)

Figure 2.6: Eclipse machining center (a) and its kinematic structure (b). machine is a hybrid mechanism. The parallel part of the machine is a mechanism with parallel actuation redundancy. A more detailed description is given in Section 4.1 of this dissertation. 2.1.3.3

Other Redundant Parallel Mechanisms

A few examples can be found in the research on actuation redundancy as test benches. A prototype of planar 2-dof robot was developed, for example, at the Technical University Carolo-Wilhelmina at Brunswick (Braunschweig), Germany for the test of control strategies, cf. Fig. 2.7(a), [Kock 98], [Kock 01].

(a)

(b)

Figure 2.7: The PA-R-Ma prototype (a) and its kinematic structure (b). The most advanced applications of actuation redundancy in robotics are tendon-driven mechanisms. For these mechanisms, the redundancy is a must as it compensates for the fact that the links cannot be compressed, cf. [Fang 04], [Krut 04], [Gouttefarde 06]. Nahvi, Hollerbach, and Hayward in [Nahvi 94], [Hayward 94] developed a hydraulic shoulder mechanism based on the parallel actuation redundancy principle, cf. Fig. 2.8(a) and Fig. 2.8(b). It is characterized by a large workspace free of singularities for spherical movements. The four actuators provide a first-order redundancy, cf. Fig. 2.9. The mechanism ARCHI is a 3-dof over-actuated parallel mechanism, designed as a substructure of a 5-axis hybrid machining robot, [Marquet 01], [Marquet 02b]. Four linear drives provide two translations and one rotation degree of freedom. The mechanism can be considered as the association of two 2-dof robots that manipulate a moving platform, cf. Fig. 2.10(a) 24

Chapter 2. State of the Art

(a)

(b)

Figure 2.8: The Shoulder Joint and its schematic representation.

Figure 2.9: Kinematic structure of the shoulder joint.

25

Chapter 2. State of the Art

and 2.10(b). According to the authors, it has been demonstrated that owing to redundancy, there is no limitation on the nacelle rotation (no overmobility-type singular position).

(a)

(b)

Figure 2.10: The ARCHI mechanism (a) with its kinematic structure (b). Eureka, presented in Fig. 2.11, is a parallel mechanism concept providing five motions: three translations plus two rotations, cf.[Krut 03b], [Krut 03a] and [Krut 06]. This device is able to reach high tilting angles (± 90◦ ). This is, once again, the result aimed with the redundancy. The rod base points are articulated by six linear actuators cf. Fig. 2.12.

Figure 2.11: CAD view of the Eureka prototype.

2.2 2.2.1

Calibration and Self-Calibration of Classical and Redundant PKMs Existing Methods for the Calibration of PKMs

Calibration methods essentially differ in the used measurement method, which, in turn, is primarily dependent upon the geometry of the mechanism. A mechanism with a greater degree

26

Chapter 2. State of the Art

Figure 2.12: Kinematic structure of the Eureka mechanism (source: [Krut 03b]). of freedom will generally require measurements with a greater sensor index, [Hollerbach 96b], i.e. the number of values informing about the state of the mechanism must be greater. The first and most intuitive method for the accuracy enhancement of serial as well as parallel mechanisms is the direct measurement of geometrical parameters, [Großmann 01], [Sato 02]. There are several reasons explaining the lack of references for this method. The first is that it is rather a technical problem which is very specific to each machine. The second reason is that the obtained accuracy is generally not sufficient for machining applications. However, the a priori knowledge of parameter values allows one to improve the optimization method by including it into the iterative method, thus reducing the risk of convergence to a false parameter set, [Sato 04]. The measurement methods classically involved in external calibration processes are based on the direct measurement of the position or orientation of the TCP relatively to the machine base. The most common measurement systems are summed up and asserted in Table 2.1. The abbreviation ”acc.” stands for accuracy; ”dom” denotes degree of measurement, i.e. the number of directions along which the measurement system can measure simultaneously in one setup. In this Table, a ”+” denotes a positive property compared to other measurement systems and ”−” a negative one. For the accuracy of the laser-tracker, ”+−” expresses the fact that laser-tracker systems have a good accuracy in the direction of the laser beam but the accuracy in the other two directions is limited by the accuracy of the rotatory sensors. The degree of measurement of the laser-tracer is also assessed with ”+−”, because, in principle, it is a one dimensional measurement system but the associated software allows one to retrieve the complete TCP position by driving the machine at least four times along the same path. The corresponding references are given in Table 2.2. Table 2.1: Common measurement Measurement system double ball-bar grid-encoder laser-tracker laser-tracer laser-interferometer theodolites dial-gauges

systems for the calibration of acc. dom range setup /use + −− −− ++ ++ − − − +− ++ ++ ++ ++ +− ++ + ++ −− ++ −− −− ++ ++ −− + −− −− −

parallel mechanisms. price + − −− − − + ++

27

Chapter 2. State of the Art

Table 2.2: Selection of references for common measurement systems. Measurement References system double ball-bar [Großmann 04], [Denkena 04], [Mart´ınez 04], [Altenburger 04], [Heisel 04], [Ibaraki 04], [Kuhfuss 06] grid-encoder [Jywe 03] laser-tracker [B¨ohler 02], [Jokiel 01], [Dong 02], [Altenburger 04], [Schoppe 02] laser-tracer [Schneider 04] laser-interferometer [Val´aˇsek 02], [Ding 05], [Chai 02] theodolites [Besnard 00], [Masory 93], [Zhuang 98] dial-gauges [Deblaise 04] Some other measuring systems have been proposed but are still not commercially available. In [Thomas 03] and [Geng 94], the authors proposed a cable-based measurement system that measures the distance to the TCP. In this case, the measurement system is a parallel structure itself. The precision is limited through the cable deformations and the precision with which the cable lengths can be measured. The measurement system for three rotations and one length proposed in [Rauf 04], and [Rauf 06] can be cited. In [Vischer 00], the author proposed a sixdimensional measurement system composed of two units, one for the position of the virtual rotation center, and the other for the rotations of the platform. In [Bleicher 04], a special device is used for measuring the three position co-ordinates and three small angles deviations from the nominal angle of the TCP of the 3-dof tripod. The device is used in an automated table-based error compensation. In [Corbel 08], the authors propose the use of a 6-dof Gough platform to perform a on-line calibration and compensation of a 3-dof linear Delta mechanism. Artifact-based calibrations were carried out in [Maurine 96b], [Maurine 02], [Ibaraki 04], and [Val´aˇsek 03]. In this method, a probe detects the contact of the TCP with a work-piece, whose dimensions are known. In [Deblaise 05], the authors developed a low-cost artifact constituted by a steel plate with borings and a single steel sphere. The measurement is done with dial-gauges to determine the sphere center at various known positions. In [Neugebauer 02], the authors propose to combine three probes and an inclinometer to reach a measurement in 6 directions. A similar method is the milling and measuring of a work-piece, cf. [Franitza 02]. The risk is then that the influence of machining forces is non-negligible. Some more exotic measurement methods can be cited. A camera is used in [Klein 02], [Andreff 03], [Renaud 02], and[Renaud 06] as an external detection of the mechanism positions. In [Olea 02], and [Vischer 98], the authors use a serial CMM to calibrate small PKMs. To raise the calibration index, which is the difference between the degree of sensing and the degree of mobility [Hollerbach 96b], several measurement systems can be combined, [Daney 03]. One can also reduce the movement possibilities of the mechanism. The usual way of doing this is to block the position of the TCP, [Rauf 01], [Meggiolaro 02] or constrain it into a plane, [Maurine 96a].

2.2.2

Existing Methods for the Self-Calibration of PKMs

A self-calibration process allows the calibration of mechanisms without the need of setting up an external measurement system. The main objective compared to classical calibration 28

Chapter 2. State of the Art

methods is to improve the machine accuracy without or with minimal human intervention. Redundantly actuated parallel mechanisms are well suited for self-calibration method, because of the intrinsic presence of extra-sensors, cf. Section 2.1.1.1. The mostly used means of carrying out a self-calibration process is the installation of rotatory sensors on one or more passive joints. This method was used, for example, in [Marquet 02c], [Daney 03], [Enev 04], [Yiu 03], [Yang 01], [Yang 02], [Ecorchard 05]. If the internal measurements used in self-calibration were compared to measurement systems in the same way as in Table 2.1, it would be assessed with ”++” for all properties except for the degree of measurement which is dependent upon the number of used redundant sensors and that would be marked ”+−”. In [Neumann 06], the author describes a sensing device that was specially designed already at the machine design phase to integrate into the Exechon Machine for a self-calibration process. As already mentioned, instead of using an external calibration method, one can reduce the degree of mobility of the mechanism so that the degree of sensing becomes greater than the degree of freedom. In the case of self-calibration processes, this is achieved by blocking the movement of one or more passive joints. In [Daney 98], [Daney 00], [Khalil 99], and [Maurine 99], the authors present some calibration methods where the rotation of a passive joint is locked by an appropriate device. Chiu and Perng in [Chiu 04] chose to block two directions of movements of a hexapod through an extra strut mounted between the base and the platform. A similar method consists in fixing the length of one or more struts, which can be done at the control level. This method was used in [Zhuang 93]. In all such calibration strategies, one or more actuators must be set as passive. In [Yu 05], the authors use the same method but with virtual constraints. The endpoint position is measured with an external portable CMM in the form of a serial measuring arm. The TCP is then driven so that it remains first in a horizontal plane, secondly along a vertical line. Another method that can be classified as self-calibration method is the use of inclinometers, as in [Besnard 99] and [Besnard 00]. It is called self-calibration because the measurement system is built-in on the machine. However, its principle is the one of classical calibration, since one measures the orientations of the platform relative to the environment rather than the internal state of the mechanism. The method has then the advantages of the both. The measurements can also be performed by mounting permanent cables and sensors between the base and the moving platform, as in [Cheok 93]. The calibration can then be carried out automatically. The PRPU measuring limb presented in [Gao 04] works similarly.

2.2.3

Calibration of Redundant PKM

The number of publications on redundant PKMs is small. The number of those treating the problem of their calibration is consequently smaller. As far back as 1994, however, the authors of [Nahvi 94] proposed a calibration method for a hydraulic redundant shoulder-joint. This method is very similar to the self-calibration method, since the inherent redundant sensing is used to provide the necessary information about the internal state of the mechanism. A similar method is used in [Iura¸scu 99]. In [Jeong 04], the authors include the measurements of constraints due to the actuation redundancy into the optimization process. The constraint measurements are combined with position measurements with a laser ball-bar to calibrate a planar redundant PKM. In [Val´aˇsek 04], the authors propose to use the redundant sensors of a redundant parallel mechanism for an on-line model-based compensation for dimension changes, due, for example, to thermal effects. 29

Chapter 2. State of the Art

2.3 2.3.1

Elastic Modeling Methods for PKMs Use of the Jacobian Matrix

In the design step of articulated mechanisms, a quick modeling of elastic deformations is a key point to efficient optimization algorithms that include stiffness study. An efficient method for the approximation of the structure stiffness is the use of the Jacobian matrix giving the linear dependencies between actuator forces or moments and the resulting forces and moments on the moving platform. The actuators’ stiffness factors and the geometry are then the only parameters influencing the resulting stiffness of the mechanism. All the actuators being usually identical on a single machine, the stiffness study through the Jacobian matrix is often equivalent to the study of how the geometry of a mechanism has an influence on its stiffness. The elements are considered to be infinitely stiff. In [Gosselin 90], the approach used by the author is to establish the stiffness map of the workspace of two manipulators by using the condition number of the Jacobian matrix. This method allows one to determine and visualize singularities of the mechanism workspace. Such a modeling enables one to estimate the influence of actuator stiffness on machine accuracy, cf. [Svinin 01] for the application on a Gough-Platform. In [El-Khasawneh 99], the Jacobian matrix is used to determine the minimum and maximum stiffness along the Cartesian directions for a single pose. In [Chakarov 04], the method is used to achieve an end-effector variable stiffness by controlling the redundant actuator forces. A special attention should be paid to the fact that the condition number of the Jacobian matrix can have no physical meaning if both translations and rotations are involved in actuator or platform movements, cf. [Majou 04a]. The rotational and translational types of stiffness have to be studied separately. This method gives good results as far as the constituting elements are sufficiently stiff. It is, however, limited to the study of the influence of forces or moments applied to the endplatform on the structure deformation. If gravitation forces due to the elements’ own weight are to be taken into account, their effects must be expressed as equivalent effects on the TCP.

2.3.2

Lumped Models

The method of the Jacobian matrix gives relatively good results for PKMs with varying strut lengths, thus with push-pull loads, such as hexapods. However, the results are inaccurate in the case when some elements are subject to bending, [Majou 04a]. To improve the quality of the results with a kinematic Jacobian matrix, some local compliance elements can be included. The deformation of some elements can be represented as local stiffness, whereas the elements themselves are considered to be infinitely stiff. Such models are called lumped models. A lumped model of one kinematic chain of the Delta robot is presented in Fig. 2.13 with real joints in normal line and virtual compliance joints in dashed line, [Ecorchard 05]. In [Lee 05], the authors represented the bending deformations of the slender elements as a rotation spring for the control of a redundantly actuated PKM. In [Company 05], the authors used a lumped model to represent the compliance of the actuators and bending, torsion and tension deformations of some elements. In [Zhang 01], [Zhang 02], and [Gosselin 02], Zhang and Gosselin integrated the stiffness of joints and links into the method for the modeling of several parallel mechanisms. In [Majou 07], a lumped model was used for the modeling of the Orthoglide. The parametric formulation allowed the authors to quickly identify the critical link parameters when isotropy is aimed. 30

Chapter 2. State of the Art z

y q 2i

x

q1i u1i

qi

q 3i q 5i

v 1i q 4i

q 6i

q 6i

q 7i

Figure 2.13: Lumped model of one kinematic chain of the Delta robot. In [Huang 01] and [Huang 02], the lumped model of the parallel structure is combined to a finite-element method of the frame holding this structure to retrieve the global stiffness owing to the principle of superposition. In [Ota 02], the authors compensate for the elastic deformations of some elements with actuator corrections calculated from the element compliance and its own weight.

2.3.3

Finite-Element Methods

Finite-Element Methods are widely used among machine tool developers. They give good results on the elastic deformations or vibration modes of any structure, [Rizk 06]. Combined with Multi-Body System simulation tools, a finite-element model can give precise information about the behavior of a moving mechanism. Due to the required refinement, such methods are very demanding for time and computer resources. For this reason, only a coarse modeling can be used during the early design phase, a fine modeling being reserved for the last design steps, usually for the optimization of single parts. One of the advantages of PKMs that can be fully exploited by FE models is the fact that some elements are used several times in the mechanism, [Kauschinger 06]. The modeling of the whole mechanism can thus be simplified by the use of super-elements because only the deformations of some characteristic points, such as the TCP, are needed, [Rizk 06]. Modeling time can be reduced. PKMs, such as Delta Robots and Hexapods, are often constituted by slender elements. Beam-elements can then be advantageously applied to in the modeling of such mechanisms, [Corradini 04], [Deblaise 06b], [Deblaise 06c]. They correspond to two-node elements with deformation capabilities along all directions. An enormous time gain can be achieved compared to a FEM modeling with a large number of tri-dimensional small elements. In [Clinton 97], the authors simplify the representation of the struts of a hexapod structure as beam elements exclusively subject to tension/compression loads. Joint compliance can be also modeled with this method. This is usually done through two-node elements with a diagonal stiffness matrix, [Deblaise 06b]. In [Yoon 04], the authors modeled the joints taking into account the two assembled bearings. In [Dong 05], the authors integrated a finite-element modeling into the control of flexure hinge-based hexapod owing to a non-linear resolution of the inverse kinematics analysis.

31

Chapter 2. State of the Art

2.4

Conclusion

The study of the state of the art revealed the following facts. ˆ Most of the research on redundantly actuated parallel mechanisms concentrates on their control and does not consider any parameter errors. ˆ A few papers on the calibration of redundantly actuated parallel mechanisms refer to the self-calibration method. They do not mention, however, the obtained internal constraints. ˆ When researchers discuss these constraints, they exclusively consider the stiffness of force driven actuators. The element stiffness is not considered and a position control on all actuators, which is necessary for machine tools, is not mentioned. ˆ The calibration of redundantly actuated parallel kinematic machine tools is a new scientific field.

In this dissertation, I propose several methods that can resolve the above-mentioned problems.

32

Chapter 3. Calibration Principle and Optimization Parameters

Chapter 3 Calibration Principle and Optimization Parameters The calibration process consists in estimating the parameters of a co-ordinate transformation model so that the mechanism’s positioning error is minimized. In this Chapter, the fundamentals of mechanisms’ calibration are developed. The calibration algorithm and optimization methods are presented. Then, modeling problems linked to calibration processes are treated along with some tools to certify the calibration results.

3.1 3.1.1

Calibration Principle Error Sources

The repeatability of PKMs is generally what can be expected from the actuator positioning repeatability1 . However, the lack of absolute accuracy cannot be related to the actuator accuracy. The main error sources of machine static Cartesian inaccuracy that account for this are the following: ˆ geometrical errors, due to the unknown offsets in the actuator measuring systems, and to manufacturing and assembly errors, ˆ kinematic model simplifications in the control unit, ˆ thermal dilatation, due to temperature variations coming from the environment, from the actuator warming, and from the cutting process, ˆ elastic deformations, whose sources can be divided between the mechanism’s own weight and the forces and moments related to the cutting process, ˆ joint backlashes, which are of a geometrical origin but of a non-linear influence, ˆ stick-slip phenomena in passive joints.

My work focuses on the static accuracy of redundant parallel mechanisms, thus excluding the consideration of the cutting process influence on the temperature variations and elastic 1 Here, the repeatability under exactly the same conditions is considered. Indeed, the ability of PKMs to reach the same Cartesian point repeatedly is sensitive to both the movement direction and the platform orientation.

33

Chapter 3. Calibration Principle and Optimization Parameters

deformations. However, if an elastic model that takes into consideration the variations of TCP position due to the mechanism’s own weight can be found, this model should be also capable of taking the cutting forces and moments into account. Moreover, the temperature variations should be concentrated in the actuators, since the movements of the mechanism are slow or quasi-static and the warming in the bearings is avoided. However, the temperature elevation in the actuators should be specially considered for redundant parallel mechanisms, because of the internal constraints due to the geometrical errors. One of the advantages of redundant PKMs is the reduction of backlash effect by controlling the direction of forces in the mechanism, which reduces the variation of contact surface in the bearings. The effects of backlash phenomena are then considered to be smaller than in classical PKMs. They are thus neglected in this work.

3.1.2

Specificity of the Calibration of Parallel Mechanisms against Serial Ones

The development of serial mechanisms began much earlier than that of parallel ones and their acceptance in industry is much higher. A natural consequence of this is that the publications on the calibration of PKMs are less numerous. The reason is not only the age of the technology but also the fact that many research results on calibration of serial mechanisms can be applied to PKMs. The calibration principle remains the same. It consists in either measuring the end-effector situation or creating some geometrical constraints in order to get redundant, thus comparable, information about the state of the mechanism. The optimization process and the numerical difficulties associated with it are also similar. Since all the joints of serial kinematic machines are articulated, self-calibration methods are not suitable for them. For serial robots, and even more for machine-tools with serial kinematics, the parameters can be also identified one after another, as described in [Stone 86]. In [Daney 00], a method is proposed to identify the parameters of each kinematic chains of a Gough-Platform separately. However, in the case of PKMs, this separation is only possible with the condition that the degree-of-measurement is the same as the degree-of-freedom of the mechanism. In other terms, full-pose measurement is required.

3.1.3

Formulation of the Least-Squares Problem

The direct measurement of a parallel mechanism geometry can be a good help in improving its accuracy in the first steps of the assembly, cf. [Kauschinger 06], or even during its normal use, if it was appropriately designed. However, the assembly errors are usually not measurable, because the elements that are to be measured are unreachable. For example, the center point of spherical joints, which are the key-points of the kinematic transformations between machine-space and user-space co-ordinates, cannot be determined. The calibration principle is to gain the knowledge of the machine parameters of these kinematic transformations through a measurement of the internal or external state of the mechanism and identification of the required parameters by an optimization process. The general principle of calibration is based on the comparison between model values Wmod and their measured equivalents Wmeas . The model values are a function of some reference values Wref and the presumed machine parameters P through a computer calculation. The measured values are also a function of the same reference values but through the unknown behavior of the mechanism (model uncertainty) and its unknown parameters (model parameters 34

Chapter 3. Calibration Principle and Optimization Parameters

uncertainty). The reference values are also values that are obtained by measurements and are, therefore, perturbed. To give a clarifying example, for a classical calibration process where the TCP position is measured, the reference values are the actuator positions that are given by a measurement system with its associated precision. The advantage of this notion of reference values, as opposed to the commonly used notion of actuator values is to unify the formalism for the different calibration methods, such as the method with the inverse kinematic model, the one with the forward kinematic model and self-calibration methods. Vischer in [Vischer 96] even uses a more general formalism where the role of reference values and measurement values is not differentiated. This has the advantage of being more general but may be confusing for the explanations of the difference between the calibration methods. In a general way, we describe the model values as Wmod = g(P , Wref ).

(3.1)

The measurement values are obtained from the real mechanism considered to be a black box. We regard Wmod as being a column vector with m elements. m is then the number of measurement values, rather than the number of measurement poses. In order to be able to identify the parameters, m must be at least as great as the number of parameters. To improve the results quality, a significantly greater number of measurements is taken to attempt to cope with measurement noise. We have then an error function for each measurement value i i i i i fi (P ) = Wmod − Wmeas = gi (P , Wref ) − Wmeas = ∆Wi .

(3.2)

The function that needs to be minimized, the so-called cost-function, can be expressed as m

1 1X (fi (P ))2 = ∆W T ∆W . F (P ) = 2 i=1 2

(3.3)

Finding a local minimizer Pcal for this function, i.e. finding the parameter set that gives a local minimum value of F (P ) is then a least-squares problem. The functions fi are non-linear, which makes it, more precisely, a non-linear least-squares problem. The local minimizer on a small region δ is expressed as F (Pcal ) ≤ F (P ) for kPcal − P k ≤ δ.

(3.4)

Many methods exist for solving this specific problem, all of them being iterative. They are explained in Section 3.2.1.

3.2

Optimization Algorithms

The general process of the calibration is represented in Fig. 3.1. In Fig. 3.1, deterministic optimization processes alone are represented. For heuristic optimization process, the parameter set for the next iteration step is obtained by choosing one or more parameter sets among a bundle of them. For deterministic optimization processes, ∆P = h(∆W ) is the step that modifies the parameter set at each iteration step. Its value depends on the algorithm chosen and its parameters, as described in this Section.

35

Chapter 3. Calibration Principle and Optimization Parameters

measurement data Wmeas

reference values Wref

nominal parameters P0

Kinematic model g(P, W)

model values Wmod

Error vector for the least-square method f (Pn) = Wmod ¡ Wmeas

parameter correction Pn +1 = Pn + h(f ( Pn))

check stop criterion; criterion 0 obtained by line-search. The line-search consists in finding α that says how far from the current parameter set in the direction given by −∇F (P ) we should go so that the cost-function is minimized. This problem is an optimization itself. A high accuracy of α is not required because it is only a step inside an iterative method. The advantage of the steepest-descent method is its stable convergence. The bad condition number of the system can though lead to a convergence to bad parameters. The final convergence is linear and, often, very slow. Newton’s method or Newton-Raphson method is based on the second derivatives of F . It uses the Hessian H of F defined as ∂ 2F (P ) (3.7) Hi,j (P ) = ∂pi ∂pj The step direction hn is obtained as the solution of Hhn = −∇F (P )

(3.8)

∆Pn = αhn

(3.9)

With a line-search, the actual step is obtained as The method has a good quadratic final convergence. Its drawbacks are that 1) it is very sensitive to singularities in the Hessian matrix, 2) its convergence can be too local, and 3) the complicated analytic calculation of the Hessian matrix is required. The third disadvantage can be overcome by using a Quasi-Newton method where the Hessian matrix is numerically approximated and updated at each iteration step. The finite-difference method and/or appropriate updating methods can be used for this purpose. It can be also noted that the resolution of (3.8) can be advantageously solved by using either Cholesky’s orthogonal transformation or the Singular Value Decomposition. The problem is expressed as solving Ax = b. Instead of solving the problem by inverting A, the system can be solved by decomposition, which is more accurate. In [Schr¨oer 93], Schr¨oer gives a small review of these resolution methods for the calibration process and advises one to use the Singular Value Decomposition due to its better accuracy. 3.2.1.2

Gauß-Newton Method

This method is based on the first derivatives of the components of the vector function f . The linearized equation system to be solved is J∆P = −∆W

(3.10)

(JT J)hgn = −JT ∆W

(3.11)

The step direction hgn of the Gauß-Newton method is the solution of with J = J(P ) the Jacobian matrix obtained by ∂gi ∂fi = (3.12) Ji,j = ∂pj ∂pj The line search is also applied to calculate the correction step as ∆Pgn = αhgn . For the classical Gauß-Newton method α equals 1. Because it is specialized for least-squares problems, the Gauß-Newton optimization has a good convergence. In special cases, the convergence is quadratic. However, the speed of convergence depends on the values of ∆W . In particular, if one or more measurement errors are large, this may slow down the convergence. 37

Chapter 3. Calibration Principle and Optimization Parameters

3.2.1.3

Levenberg-Marquardt Method

The Levenberg-Marquardt method, [Levenberg 44], [Marquardt 63], is a damped Gauß-Newton optimization method. The step hlm is obtained by solving (JT J + µI)hlm = −JT ∆W

(3.13)

The parameter µ is the damping parameter. If µ has large values, the step corresponds to a short step in the steepest-direction. This is good if the current parameter set is far from the solution. If it has small values, the method is close to the Gauß-Newton method, with a quicker convergence. An algorithm is then applied to update the values of µ depending on the gain ratio to attempt to optimize the algorithm behavior. The updating strategy is not developed here and the reader may refer to [Lawson 74] or [Madsen 04]. Thus, the Levenberg-Marquardt method has advantages of both the steepest-descent method for the first iteration steps and Gauß-Newton method for the final convergence. The convergence is stable owing to the damping parameter µ. No specific line-search is needed. The correction step is then obtained as ∆Plm = hlm . The drawback of the Levenberg-Marquardt is a slow linear convergence in the case of a badly conditioned system. 3.2.1.4

Heuristic Methods

Heuristic methods, such as genetic algorithms, random strategy, and the simplex method, can be used as alternatives to deterministic methods for the identification of model parameters. They are very robust. However, they offer an extremely slow convergence and require a great number of function evaluations. One advantage of such methods is that they do not require the calculation of the derivatives of f . They offer a more global convergence and are very robust against singularities and erroneous measurements. Genetic algorithms were tested as optimization methods, [Dong 02], [Großmann 04]. However, they are frequently not recommended, for example in [Beyer 04], [Frayssinet 07]. 3.2.1.5

Conclusion

The two most efficient optimization algorithms for calibration processes are Gauß-Newton and Levenberg-Marquardt methods. Since Levenberg-Marquardt method combines the advantages of the a steepest-descent and Gauß-Newton method, it was chosen for all the calibration processes.

3.2.2

Constrained and Unconstrained Optimization

All the previously presented methods are unconstrained optimizations. Since the probable variations of model parameters are known, the information can be used in so-called trustregion optimization methods, which are constrained optimization algorithms. If the parameter correction step remains within a trust-region, the algorithms are equivalent to unconstrained optimization methods. If not, trust-region methods modify this step so that the parameter variation does not exceed a given radius. Powell proposed the so-called dog-leg method for the calculation of this step from a combination of the steepest-descent and Gauß-Newton directions controlled with the help of the radius of a trust-region, [Powell 70]. The problem with constrained optimization is that the model parameters that minimize the norm of the error vector ∆W are generally not the parameters that are the closest to the 38

Chapter 3. Calibration Principle and Optimization Parameters

geometrical reality, i.e. to the measurements made on the machine directly or to the estimation that can be reasonably done on the value of these parameters. This difference is explained by the simplifications that are carried out by the modeling, cf. [Deblaise 06a]. Another reason why constrained optimization methods are not widespread in the calibration algorithms is the difficulty of the choice of the trust-region radius. If it is chosen to be too small, a larger final norm of the error vector is obtained compared to the one that could be obtained without constraints. If it is too large, the constraints are only of very small influence.

3.3

Calibration-Compatible Modeling

3.3.1

Minimality, Completeness and Consistency of Models

A good modeling for describing the relationship between the reference values and the model values for a calibration process has the following three properties: it should be complete, minimal, and consistent. The completeness of a model is its ability to take into account a maximum number of error sources in order to be able to compensate for them. This notion is linked to the calibration process because the number of error sources that can be perceived depends on the calibration method and the measurement-system accuracy. The minimality account for the fact that model parameters should not be redundant since two redundant parameters induce proportional errors on the values measured during the calibration. Thus, they do not improve the completeness, they are rather responsible for a rank loss in the Jacobian matrix of the calibration, which prevents the parameters from being identified. The minimality of a model is checked with the observability analysis developed in Section 3.3.5. The consistency of a model, referred to as proportionality in [Everett 87] and [Vischer 96], reflects the fact that small changes in the geometry of the mechanism lead to small changes in the identified parameters. Consistency problems appear, for example, when the DenavitHartenberg modeling method with nearly parallel consecutive axes is used, [Hayati 83].

3.3.2

Sensitivity Analysis

The range within which the machine parameters can vary is assumed to be known. The aim of the sensitivity analysis is to determine the variation in the measurement values, Wmeas , when each parameter successively varies within its range. In order to compare the effects of parameters with different units and orders of magnitude, the variations in the measurement values must be normalized so that they are expressed without the unit of the varying parameter. The variation applied to parameter i is δpi = pmax − pnom , i i max nom where pi is the maximal value of the variation range and pi is the parameter nominal value. For the normalization, the Jacobian matrix of the calibration is multiplied by a n × n scaling matrix H s , which is built as Hs = diag(δp1 , δp2 , . . . , δpn )

(3.14)

with n the number of parameters. The scaled Jacobian matrix Js = JHs is obtained. For each pose and with some measurement systems, the measurements can be done along multiple directions. These directions will further be called measurement directions. Two 39

Chapter 3. Calibration Principle and Optimization Parameters

measurement values i and j correspond to the same measurement direction if the corresponding model values are obtained by the same functional gi , for instance, i.e. we have gi (P , Wref ) = gj (P , Wref ). The sensitivity index for each parameter k and for each measurement direction l = 1, . . . , r over the whole workspace is given with the 1-norm by P s |Jj,k | j∈Il , for l = 1 . . . r and k = 1 . . . n (3.15) σl,k = ml s with Il the row indexes of J s corresponding to the measurement direction l, Jj,k the element s at the j-th row and k-th column of J , and ml the number of elements of Il . The sensitivity index can be also given with the 2-norm as vP
u s 2 u Jj,k t j∈Il σl,k = , for l = 1 . . . r and k = 1 . . . n. (3.16) ml

If the measurement directions correspond to Cartesian translations along x and y, the parameter sensitivity can be also studied in terms of position deviation. The sensitivity index of parameter k on the TCP position deviation is 2 σTCP,k =

m/2 Pq i=1

s J2i−1,k


2

m

s + J2i,k


2 .

(3.17)

In this case, it is supposed that the Jacobian contains only lines corresponding to the xdirection for odd line numbers and to the y-direction for even ones. A parameter for which the sensitivity is negligible must be excluded from the calibration process.

3.3.3

Equation System Conditioning

As shown in Section 3.2.1, the linear equation system to solve for Gauß-Newton and LevenbergMarquardt methods has the form J∆P = −∆W . The conditioning of this system, i.e. the condition number of the matrix J, is an important value which is the expression of the amplification factor of errors in the right-hand side of the equation system. The condition number for the Euclidean norm is obtained from the Singular-Value Decomposition, SVD. Thanks to the SVD, the matrix J can be decomposed into the following matrices: J = UΣVT

(3.18)

with U and V being orthonormal matrices and Σ = diag(σ1 , σ2 , . . . , σn ),

(3.19)

the matrix of singular values of J sorted in descending order. σ1 is the largest singular value and σn is the smallest one. The geometrical interpretation of the singular values is that, if the errors on the vector ∆W are included in a hyper-sphere with radius 1, the influence on ∆P is included in a 40

Chapter 3. Calibration Principle and Optimization Parameters

hyper-ellipsoid whose axis lengths are the singular values. Thus, the larger the singular value, the more observable the parameter. There are several definitions of the observability index of the calibration, cf. [Nahvi 96], [Daney 00], [Sun 08]. The most commonly used one is the condition number of J defined as kJ =

σ1 . σn

(3.20)

A perfectly conditioned equation system has a condition number equal to 1. Some definitions for the conditioning are given with σp2 rather than σn to express the fact that, for the same condition number as defined in Eq. (3.20), the observability index is better for greater singular values, cf. [Nahvi 96]. This definition is however not widespread in robotics. If the condition number is too great the variations of measurement values cannot be associated with a parameter influence since two ore more parameters are of nearly linear depending influence and the measurements are perturbed. According to [Schr¨oer 93], a condition number of 100 is the maximum value for the reasonable calibration of real systems. The following two Sections show how the condition number can be improved by elimination of dependent parameters and normalization methods.

3.3.4

Jacobian Normalization

A normalization of the Jacobian matrix is necessary to take into account the different order of magnitude of the parameters and of the task-space variables, cf. [Hollerbach 96a]. Normalization also allows a smaller condition number, and, thus, a greater numerical result accuracy. 3.3.4.1

Task Variable Scaling

The measurement values can combine values of different kinds. They can be position, orientation-, and forces measurements, for example. Moreover, the measurement accuracy is not necessarily the same for all measured directions. A task variable scaling is then needed. It corresponds to a left multiplication of the Jacobian matrix J and the error vector ∆W in (3.10) by a scaling matrix B. The system to be solved is then BJ∆P = −B∆W .

(3.21)

After simplification, the modified Levenberg-Marquardt equation system is (JT GJ + µI)∆Plm = −JT G∆W ,

(3.22)

where G = BT B is the diagonal weighting matrix whose diagonal elements are 2 g12 , g22 , . . . , gm described below. The step for Gauß-Newton method is obtained with µ = 0. The a priori knowledge on the measurement accuracy can be used as weighting factors. Measurements with a higher accuracy will have a greater influence on the parameter correction than those with a lower one. If the measurement accuracy, including input noise and model errors, is σi for measurement i, a weight factor gi = 1/σi can be used. The weighting matrix has no effect on the final parameter set if all weighting factors are identical. The measurement errors are supposed to be non-correlated. If this is not the case, a weighting matrix based on the covariance matrix can be defined, cf. [Hollerbach 96a].

41

Chapter 3. Calibration Principle and Optimization Parameters

3.3.4.2

Parameter Scaling

The parameter scaling is used to normalize the different magnitudes of parameter variation on the variation of measurement values, cf. [Hollerbach 96a]. It is done by a right multiplication of the Jacobian matrix and the appropriate variable transformation H−1 ∆P . The modified equation system is (JH)(H−1 ∆P ) = −∆W .

(3.23)

For Levenberg-Marquardt method, the general equation system to solve for the calibration is then (HT JT JH + µI)H−1 ∆Plm = −HT JT ∆W .

(3.24)

A commonly used normalization is realized through the scaling matrix H given by H = diag(h1 , h2 , . . . , hp ) such that (cf. [Nahvi 94]) hi = kJic k−1

(3.25)

with Jic the i-th column of J. In the case when some parameters have a small sensitivity, very large scaling factors have to be used. This can result in amplification of uncertainties in J, which is especially important if this latter is retrieved by finite-differences, cf. [Schr¨oer 93]. Schr¨oer proposes then to use the parameter deviation that causes a given variation in the normalized measurement values as a scaling factor. According to him, these factors give better results than the column norms.

3.3.5

Determination of a Set of Identifiable Parameters

The study of parameter dependencies is called observability analysis. It allows the selection of linearly independent parameters, cf. [Besnard 01]. If one of the parameters has a linear dependency on one or more parameters, its corresponding column in the Jacobian matrix of the calibration is also dependent on the column of the parameters on which it is dependent. Thus, the rank of the Jacobian matrix is smaller than its number of column, the matrix is non invertible, in the sense of pseudo-inverse and the optimization fails. Due to measurement errors it must also be studied if parameter influences are close to be singular rather than strictly singular. The condition number allows to detect such behavior. Let us define a matrix B composed of n − 1 columns of the matrix A from the previous Section. An important result from [Lawson 74] is that the condition number of B is smaller or equal to this of A. As a consequence, the condition number related to the calibration of a specific model can by improved through the reduction of parameter number. The normalized Jacobian matrix is used for the observability study. It is given by JH = JH.

(3.26)

The number r of observable parameters is the number of singular values greater than a tolerance τ related to the measurement precision. r is the practical rank of JH . σ1 ≥ σ2 ≥ . . . ≥ σr ≥ τ ≥ σr+1 ≥ . . . ≥ σm

(3.27)

According to [Schr¨oer 93], in the case when the same measurement accuracy meas applies to all the measurements, the tolerance τ can be chosen as τ = meas kAk.

(3.28) 42

Chapter 3. Calibration Principle and Optimization Parameters

Let us define Vr as the matrix containing the first r columns of V. To determine which of the parameters are observable a QR-decomposition of VrT is carried out. The QR-decomposition of VrT gives VrT E = QR

(3.29)

where Q is a unitary r × r matrix, R is an r × m upper triangular matrix with decreasing diagonal elements, and E is the m × m column permutation matrix chosen so that R is obtained in that order. The vector Is of parameter indexes sorted in descending order of importance is given by Is = EI

(3.30)

with I = (1, 2, . . . , n)T . The set of identifiable parameters is then constituted by the r first parameters indexed in Is .

3.4

Calibration Lead-Through

3.4.1

Elimination of Measurement Outliers

One of the means to improve the quality of the parameter identification is to eliminate measurement outliers. The difficulty is due to the automated recognition of these outliers. The more measurement values are available, the easier and the more reliable this elimination is. This is linked with the confidence in the calculated standard deviation. The amount of data necessary for a reliable detection of outliers raises as the measurement system accuracy decreases. The detection of outliers occurs via the average and the standard deviation in the set of measurement errors ∆W . The study is carried out separately along each measurement direction i = 1 . . . r defined in Section 3.3.2. The average error for the direction i is µi =

1 X ∆Wi . mi i∈I

(3.31)

i

The corresponding standard deviation is defined as s 1 X σi = (∆Wi − µi )2 . mi − 1 ∈I

(3.32)

i

Statistics usually advise to exclude the measurement errors that deviate by 2.5 to 3 times the standard deviation from the average value. If the measurement error follows the Gaussian distribution, this would correspond to the elimination of 0.3 % of the values. According to Schr¨oer in [Schr¨oer 93], 1 to 2 % of the measurement poses have to be eliminated from experimental results using this rule. The utility of the elimination is essentially found when the measurement system accuracy is not sufficiently good compared to the error caused by wrong geometrical parameters.

43

Chapter 3. Calibration Principle and Optimization Parameters

3.4.2

Calibration Certification

3.4.2.1

Calibration Gain

One of the good means to assess the final gain of the calibration is to take a few extra measurement values that would be excluded from the optimization process itself. Then, in order to compare the machine parameters before (Pnc ) and after (Pcal ) the calibration, a gain can be defined for each measurement value i as ηi =

i i | | − |∆Wcal |∆Wnc i |∆Wnc |

(3.33)

i i i with ∆Wnc = gi (Pnc , Wref ) − Wmeas , the i-th measurement value error among m0 performed i i i is the measurement error obtained ) − Wmeas = gi (Pcal , Wref before the calibration. ∆Wcal after the calibration. The final gain is obtained as 0

m 1 X η= 0 ηi . m i=1

(3.34)

A final gain of 1 corresponds to perfectly identified parameters with a perfect model. A gain of 0 indicates that the parameters were not changed during the calibration process. Finally, a negative gain indicates that the measurement errors are larger after the calibration than before, indicating that the calibration failed. A weighted average value could be used instead of (3.34) for the calculation of the gain η to take into account the relative importance of the desired accuracy along some specific directions. For instance, one could wish a very good position accuracy whereas the rotational accuracy would be of second order of importance. The weighting can also be applied to transform the rotation errors into translational errors with a given tool length, assuming that the rotation center point is known.

3.4.3

Parameter Dependencies

In order to facilitate the model analyses and to avoid multiple descriptions of very similar models, the models are first developed without taking the possible redundancy into account. Thus, for a given mathematical complexity, for example, in term of polynomial degree, the model is designed to take into account the largest possible number parameters. For the study of simpler models, one may wish to reduce the number of parameters. In this case, either the value of the parameter is kept constant, or it is calculated at each step of the calibration as a linear dependency of other parameter values. In both cases, the parameter has to be excluded from the optimization process. The actual parameter correction is ∆P = D∆Pred ,

(3.35)

where ∆P is the correction of the whole set of model parameters as a column vector, D is the dependency matrix, ∆Pred contains the actual correction step for parameters that were identified, and zero for the excluded parameters as well as the parameters that are obtained by dependency to other parameters. The matrix D is defined in such a way that the line of excluded parameters is null, the line of an identified parameter i contains only zeros with the exception of Di,i = 1. Without loss 44

Chapter 3. Calibration Principle and Optimization Parameters

of generality, if a parameter variation is defined as ∆Pi = aj ∆Pj + ak ∆Pk , the i-th line of D is null except for indexes j and k, where the elements equal aj and ak . The following example can be considered. The complete modeling of the passive parallelogram structure that can be found on the Scissors-Kinematics or on Delta robots should include all its lengths, cf. [Savour´e 06]. If one wishes to consider that the parallelogram is perfect, a linear dependency can be introduced in the calibration algorithm. It ensures that the parameter that is excluded from the identification process is equal to the identified length so that the polygon remains a parallelogram. This allows one avoid implementing several models during the study step.

3.4.4

Final Improvement

After the calibration process has been carried out, the residual errors can be reduced by using compensation tables. For each point of the workspace, such tables give the value that should be added to the control values to attempt to obtain a better accuracy. The compensation values between the points where the position errors were measured are retrieved by interpolation, cf. [Bleicher 04], [Bringmann 06]. The reasons why compensation tables are not used instead of a calibration process are multiple. Firstly, they are known to lack good accuracy between the support points. Secondly, the memory cost for storing multi-dimensional tables in the controller forces the limitation of the correction along one or two main directions and with respect to a limited number of Cartesian directions, cf. [Schr¨oder 04]. Model-based optimization has a better global potential for accuracy enhancement. A table compensation should be used, however, as a complement because its efficiency is independent of what source of errors can be modeled.

45

Chapter 4. Geometrical Calibration of Redundant PKMs

Chapter 4 External and Autonomous Geometrical Calibrations of Redundantly Actuated Parallel Kinematic Machines In this Chapter, the Scissors-Kinematics is presented. The several transformation between machine- and user co-ordinates that will be needed throughout this dissertation are explained. The parameters of these transformations must be identified through calibration to improve the TCP positioning accuracy. Two calibration classes are used: external calibration and selfcalibration. New external calibration methods are developed to take the actuation redundancy into account. At last, the different calibration methods combined with several measurement systems are tested on the Scissors-Kinematics.

4.1

Presentation of the Scissors-Kinematics

The Scissors-Kinematics is a subset of a hybrid machine tool called Dynapod. The ScissorsKinematics itself is a redundantly actuated parallel kinematic mechanism. The whole system represents a hybrid milling machine tool with five degrees of freedom. The lowest part consists of a light-weight bridge made of metal foam moving on two guides along the x-axis and actuated by four sixteen-meter-long linear direct drives. The Scissors-Kinematics is actuated by four linear direct drives moving along the y-axis and mounted on this gantry bridge, but it is a two-degree-of-freedom mechanism. It possesses consequently a second-order actuator redundancy. A guide actuated by ball screws along the z-axis is mounted on the moving platform of the Scissors-Kinematics and carries a machining head for the rotation movements of the tool. The complete structure is represented in Fig. 4.1. The left part of the mechanism is a serial cutting head whose structure is not be considered in my dissertation. The Scissors-Kinematics is shown in Fig. 4.2, without the covers that now hide the mechanism. The Scissors-Kinematics itself is represented in Fig. 4.3. The z-carrier and the machining head are also represented in grey on this figure but they do not belong to the ScissorsKinematics. Fig. 4.4(a) and Fig. 4.4(b) represent the Scissors-Kinematics in the x-z and x-y planes, respectively. The kinematic structure of the mechanism is represented in Fig. 4.5. The first prismatic link corresponds, in reality, to four mechanically-bound linear actuators constituting a Gantry bridge. The Scissors-Kinematics itself is the marked part.

46

Chapter 4. Geometrical Calibration of Redundant PKMs

Figure 4.1: CAD representation of the Dynapod machine (Source: VRCP, Chemnitz).

Figure 4.2: The Scissors-Kinematics during the assembly.

47

Chapter 4. Geometrical Calibration of Redundant PKMs

Figure 4.3: CAD representation of the Scissors-Kinematics.

(a)

(b)

Figure 4.4: CAD views of the Scissors-Kinematics.

R

R

R

R

P

R

R

P

R

R

P

R

R

Base

P P

P

R

R

TCP

Scissors-Kinematics

Figure 4.5: Kinematic structure of the Scissors-Kinematics. The Scissors-Kinematics viewed as a planar mechanism is represented in Fig. 4.6. The absolute co-ordinate system is such that the y-axis is parallel to the guides, as already mentioned and that it is at the same distance to these two guides in nominal models. The z-axis is perpendicular to the mechanism plane. 48

Prismatic actuated joint Revolute passive joint Rod 2

y

Carrier 1

Carrier 2

Chapter 4. Geometrical Calibration of Redundant PKMs

Rod 1

Guide 2

Guide 1

Rod 5

Rod 3

Carrier 4

Carrier 3

Platform

Rod 4

0

x

Figure 4.6: Simplified top view of the mechanism. Each rod is connected to one carrier and to the moving platform with two revolute joints around the z-axis. Rods 4 and 5 constitute a parallelogram that maintains mechanically the orientation of the platform around the z-axis. These two rods are thus called the parallelholder.

4.1.1

Simple Model

4.1.1.1

Model Description

The first model of the Scissors-Kinematics is very simple. For this model, the parallel-holder is supposed to be perfect. This means that its geometry is described as a parallelogram, the distance between the revolute joints being equal in pairs. The consequence is that the platform has a constant angle to the guide 2, which support the parallel-holder, whatever the actuator positions are. For simplification, we consider that the platform remains parallel to the guide 2. The model parameters are then, cf. Fig.4.7 ˆ 4 strut lengths: l1 to l4 , ˆ 8 positions of the platform joint centers relative to the platform reference frame: xa1r , ya1r , xa2r , ya2r , xa3r , ya3r , xa4r , ya4r , ˆ 8 positions of the carrier joint centers relative to the drive reference frame: xb1r , yb1r , xb2r , yb2r , xb3r , yb3r , xb4r , yb4r , ˆ 2 guide origin x positions: d1 , d2 , ˆ 2 guide angles: γ1 , γ2 .

The total number of model geometrical parameters is then 24. Therefore, throughout my dissertation, this model is denominated as model 24. These parameters are described in Fig. 4.7. The nominal parameters are given in Table 4.1. 49

Chapter 4. Geometrical Calibration of Redundant PKMs

B2

Q2

B1

y b2r xb2r

xb1 r y b1 r

l2 l1

y A2

xa1 r A 1

xa2 r

y a2 r

P A3

Q1

y a1 r

y a3 r xa3 r

l3

xa4 r

y a4 r

A4

Q3 y b4r

l4

xb4r

B3

°

y b4 r Q4 xb4 r °

1

B4

0 G1

d1

d2

x

2

G2

Figure 4.7: Geometrical parameters for model 24. Table 4.1: Nominal parameters for model 24. i 2 3 4 1 li (mm) 950 950 950 950 xair (mm) -520 -520 520 520 yair (mm) 280 -280 -280 280 xbir (mm) 310 310 -310 -310 ybir (mm) 80 -80 -80 80 di (mm) -1460 1460 γi (rad) 0 0 4.1.1.2

Inverse Geometrical Model

For notation simplifications, we first express the joint center points in the absolute co-ordinate system, considering that the angle of the platform local y-axis to the axis of guide 2 is null. For the points Ai on the platform, we have ( xaia = x + xair cos γ2 − yair sin γ2 (4.1) yaia = y + xair sin γ2 + yair cos γ2 , i = 1..4. The loop-closure equations are fi = (xaia + dj − xbir cos γj + qai sin γj )2 + (yaia − xbir sin γj − qai cos γj )2 − li2 = 0, i = 1..4,

(4.2)

with qai = qi + yb1r and j = 1 for i = 2, 3; j = 2 for i = 1, 4. Developing these equations gives 2 qai +2((xaia + di − xbir cos γi ) sin γi − (yaia − xbir sin γi ) cos γi )qai +((xaia + di − xbir cos γi )2 + (yaia − xbir sin γi )2 − li2 ) = 0.

(4.3)

50

Chapter 4. Geometrical Calibration of Redundant PKMs

2 By identifying the second-order canonical form: qai + 2bi qai + ci = 0, we have

bi = (xaia + dj − xbir cos γj ) sin γj − (yaia − xbir sin γj ) cos γj

(4.4)

ci = (xaia + dj − xbir cos γj )2 + (yaia − xbir sin γj )2 − li2 .

(4.5)

and

The solution of (4.2) is then q qi = −bi + k b2i − ci − ybir

(4.6)

with k = 1 for i = 1, 2 and k = −1 for i = 3, 4. The rejected solutions correspond to different working modes [Chablat 98]. Their exclusion from the solution set is obvious. The assumption that the parallel-holder is perfect allows one to obtain an analytical formulation of the inverse geometrical model. The obtained transformation is noted as Qnr = igm(X, P24 )

(4.7)

24

with P24 standing for the set of parameters for this model and Qnr the actuator positions of the non-redundant part of the mechanism. 4.1.1.3

Forward Geometrical Model

Because of actuation redundancy, the equation system containing all the loop-closure equations is over-determined. For model 24, it has four equations for two unknowns, x and y. Thus, for inverse geometrical models, a subset of actuators must be chosen. For the ScissorsKinematics, it must contain the parallel-holder because this latter determines the orientation of the platform. The subsystem containing actuators q1 and q4 is chosen. It is called the non-redundant part of the mechanism. The geometrical simplifications of model 24 also allow an analytical solution of the forward geometrical problem. For notation simplification, let us define ( xaip = xair cos γ2 − yair sin γ2 (4.8) yaip = xair sin γ2 + yair cos γ2 , i = 1..4. We also express the co-ordinates of the joint center points B1 and B4 in the absolute coordinate system: ( xbia = d2 + xbir cos γ2 − (qi + ybir ) sin γ2 (4.9) ybia = xbir sin γ2 + (qi + ybir ) cos γ2 , i = 1, 4. The loop-closure equations of the non-redundant subsystem (4.2) are rewritten: ( l12 = (x + xa1p − xb1a )2 + (y + ya1p − yb1a )2 l42 = (x + xa4p − xb4a )2 + (y + ya4p − yb4a )2 .

(4.10)

The solution corresponds to the intersection of two circles of center (xi , yi ) and radius li , i = 1, 4, with xi = −xaip + xbia and yi = −yaip + ybia . The general solution for the intersection of two circles is given in Annex A. 51

Chapter 4. Geometrical Calibration of Redundant PKMs

We have N24 =

l42 − l12 + x21 − x24 + y12 − y42 , 2(y1 − y4 ) 

A24 =

 B24 =

x1 − x4 y1 − y4

2

x1 − x4 y1 − y4



+ 1,

(y1 − N24 ) − x1 ,

(4.11)

(4.12)

(4.13)

and 2 − l12 − 2y1 N24 . C24 = x21 + y12 + N24

Then, the correct solution is the one with the smallest x-value: p 2 −B24 − B24 − A24 C24 x= . A24 The y co-ordinate can then be calculated by   x1 − x4 . y = N24 − x y1 − y4

(4.14)

(4.15)

(4.16)

The forward geometrical model is then X = fgm(Qnr , P24 ).

(4.17)

24

4.1.2

Complete Planar Model

4.1.2.1

Model Description

For the complete geometrical model, the Scissors-Kinematics is still considered to be a planar mechanism. However, no further assumption is made concerning the geometrical model. Compared to model 24, five parameters must be added to obtain model 29 (cf. Fig. 4.8): ˆ l5 length of rod 5, ˆ xa5r and ya5r , the co-ordinates of joint center A5 in the platform co-ordinate system, and ˆ xb5r , yb5r the co-ordinates of point B5 relative to the center of actuator q4 .

For this model, the platform is no longer always parallel to guide 2. The angle is dependent on the actuator position, i.e. on the Cartesian position. The platform angle ψ is defined as the rotation angle between the platform local x-axis and the absolute x-axis. It is a parasite angle in the sense that it cannot be controlled independently.. The complete set of parameters with their respective index is then summed up in Tables 4.2 and 4.3. The set of parameters of the non-redundant subsystem is denoted as ξnrs , and the one of the redundant subsystem is ξrs . We have ξnrs = 1, . . . , 17 and ξrs = 18, . . . , 29. 52

Chapter 4. Geometrical Calibration of Redundant PKMs

B2

Q2

Ã

l2

y b1 r

l1

y A2

P A3

y a3 r

l3

y a1 r

A5

y a5 r xa5 r xa4 r

y a4 r

l5

xa3 r

A4

xb5 r B5

Q3 y b3r

y b5 r

l4

B3

y b4 r Q4 xb4 r °2

°1 B4

0 G1

Q1

xa1 r A 1

xa2 r

y a2 r

xb3r

xb1 r

B1

y b2r xb2r

d2

d1 redundant part

x

G2

non-redundant part

Figure 4.8: Geometrical parameters for model 29. Table 4.2: Complete parameter set Name l1 l4 l5 Index 1 2 3 Unit mm mm mm Value 950 950 950 Name xb1r xb4r xb5r Index 10 11 12 Unit mm mm mm Value −310 −310 −310 4.1.2.2

for geometrical models (non-redundant part). xa1r xa4r xa5r ya1r ya4r ya5r 4 5 6 7 8 9 mm mm mm mm mm mm 520 520 520 280 −280 280 yb1r yb4r yb5r d2 γ2 13 14 15 16 17 mm mm mm mm rad 80 −80 480 1460 0

Inverse Geometrical Model

For the resolution of the inverse geometrical model, the platform angle has to be determined from actuator q4 which drives the carrier supporting the parallelogram. The two loop-closure equations that need to be solved simultaneously are  f4 = (xf 4 + q4 sin γ2 + xa4r cos ψ − ya4r sin ψ)2     + (yf 4 − q4 cos γ2 + xa4r sin ψ + ya4r cos ψ)2 − l42 (4.18)  f5 = (xf 5 + q4 sin γ2 + xa5r cos ψ − ya5r sin ψ)2    + (yf 5 − q4 cos γ2 + xa5r sin ψ + ya5r cos ψ)2 − l52 with xf i = x − d2 − xbir cos γ2 + ybir sin γ2 and yf i = y − xbir sin γ2 − ybir cos γ2 . 53

Chapter 4. Geometrical Calibration of Redundant PKMs

Table 4.3: Complete parameter set for geometrical models (redundant part). Name l2 l3 xa2r xa3r ya2r ya3r xb2r xb3r yb2r Index 18 19 20 21 22 23 24 25 26 Unit mm mm mm mm mm mm mm mm mm Value 950 950 −520 −520 280 −280 310 310 80 Name yb3r d1 γ1 Index 27 28 29 Unit mm mm rad Value −80 −1460 0 With the simplification that the platform ψ is small, i.e. cos ψ ≈ 1 and sin ψ ≈ ψ, an analytical solution was found for the inverse geometrical transformation with model 29. However, the overall resolution involves solving of a cubic equation that degrades into a quadratic equation when the parameters are nominal. The problem is that the cubic equation lead to a numerical instability when the parameters were close to but different from the nominal parameters. The mathematical demonstration is not developed here for this reason. The inverse geometrical model must then be solved numerically. The fact that the platform angle ψ is unknown prevents the use of an equation similar to (4.3). The method is to determine ψ simultaneously with the position of the actuator q4 with a numerical optimization method. According to [Last 05], by using iterative numerical methods to solve the inverse geometrical model, real-time capability cannot be guaranteed anymore, as numerical methods are mathematically complex and there is a risk of divergence. The computational robustness should then be checked before implementation. Gauß-Newton iterative algorithm is used as optimization method. For this algorithm, the differentiation of the equations in (4.18) is needed: ∂fi = 2(xf i + q4 sin γ2 + xair cos ψ − yair sin ψ) sin γ2 ∂q4 − 2(yf i − q4 cos γ2 + xair sin ψ + yair cos ψ) cos γ2 = 2(xaia − xbia ) sin γ2 − 2(yaia − ybia ) sin γ2 , i = 4, 5

(4.19)

∂fi = 2(−xair sin ψ + yair cos ψ)(xaia − xbia ) ∂ψ − 2(xair cos ψ − yair sin ψ)(yaia − ybia ) = 2(y − yaia )(xaia − xbia ) − 2(x − xaia )(yaia − ybia ).

(4.20)

and

The Jacobian matrix for the optimization method is   ∂f4 ∂f4  ∂q4 ∂ψ   Jigm,29 =   ∂f5 ∂f5  . ∂q4 ∂ψ

(4.21)

The start values (q40 , ψ 0 = 0)T are retrieved from the simplified model. These start values are close to the values of the real model because the real machine parameters are close to the 54

Chapter 4. Geometrical Calibration of Redundant PKMs

nominal values. Thus, convergence problems should be avoided. These start values can also be retrieved from the preceding step in the machine control. The actuator values at step n + 1 is then     q4 q = 4 − Jigm,29 Fn (4.22) ψ n+1 ψ n with F = (f4 , f5 )T . The optimization process converges quickly and the solution is reached within five iterations. The actuator positions q1 to q3 are then obtained with Eq. (4.6) with the difference that ( xaia = x + xair cos ψ − yair sin ψ (4.23) yaia = y + xair sin ψ + yair cos ψ, i = 1..3. An analytical model was not found for this set of parameters. The iterative model is noted as, (4.24)

Qnr = igm(X, P29 ) 29

with P29 the complete set of parameters. The convergence of the algorithm was verified through a series of tests with random variations of parameters over the whole workspace. The solution was always found in either four or five iterations with the fact that the norm of F is smaller than 10−8 mm2 as stop criterion. 4.1.2.3

Forward Geometrical Model

The numerical solution is obtained with the Gauß-Newton optimization process with optimization criterion (f1 , f4 , f5 )T = (0, 0, 0)T . The start values (x0 y 0 ψ 0 )T are obtained from the forward geometrical transformation of model 24. For this optimization method, the Jacobian matrix expressing the variations of f1 , f4 and f5 in respect to x, y and ψ is needed: ∂fi = 2(xaia − xbia ), ∂x

i = 1, 4, 5,

(4.25)

∂fi = 2(yaia − ybia ), ∂y

(4.26)

∂fi = 2(y − yaia )(xaia − xbia ) − 2(x − xaia )(yaia − ybia ). ∂ψ

(4.27)

and

The Jacobian matrix is then   ∂f1 ∂f1 ∂f1  ∂x ∂y ∂ψ     ∂f ∂f ∂f   4 4 4 Jfgm,29 =  .  ∂x ∂y ∂ψ     ∂f5 ∂f5 ∂f5  ∂x ∂y ∂ψ

(4.28)

55

Chapter 4. Geometrical Calibration of Redundant PKMs

The TCP position is then obtained iteratively with     x x y   = y  − Jfgm,29 Fn ψ n+1 ψ n

(4.29)

with F = (f1 , f4 , f5 )T . The algorithm converges in four or five iterations to a solution for which the norm of F is smaller than the stop criterion 10−8 mm2 . The tests that were carried out confirmed the robustness of the optimization. The obtained transformation is fgm29 , such that X = fgm29 (Qnr , P29 ).

4.2

Model Identification through External Calibration

In this Section, the machine accuracy is improved through an external calibration where the TCP position is measured by an external measurement system.

4.2.1

Calibration Methods

The measurements for external calibration are done directly on the TCP with a measurement system that has to be mounted on the machine. Either the x and/or y TCP co-ordinates or the distance between the TCP and a fixed point are measured. These measurements associated with the actuator position values allow one to reach a calibration index greater or equal one. Classically, for the calibration of non-redundant machines, one or more components of the TCP position obtained from the co-ordinate transformation model are compared to their corresponding values obtained directly by the measurement system. This calibration is the so-called forward kinematic calibration. For redundantly actuated PKMs, the forward geometrical kinematic transformation used in the forward kinematic calibration can be calculated by using only a non-redundant part of the mechanism, [Ecorchard 06]. Therefore, the parameters of the remaining redundant part cannot be identified. One of the solutions is to perform the parameter identification for the various non-redundant subsystems by multiple use of the same measurement data. All subsystems used in the calibration process must be kinematically viable. The optimization for all subsystems must be performed simultaneously so that a single value is obtained for the parameters of the parallel-holder. However, even by using several times the same measurement data simultaneously, it is not possible to identify any parameter related to actuator q2 with a forward calibration. The needed co-ordinates transformation returns the TCP position from the parameters and the position of actuators q2 and q4 . This transformation is singular in the middle of the workspace. For other positions in the workspace, the proximity of a second assembly mode prevents reliable convergence The solution chosen for the identification of the parameters of all branches is twofold. In a first step, the forward calibration method allows the identification of the parameters related to actuators q1 , q3 , and q4 . In the second step, the results of this identification are used to create virtual measurements of the TCP position and platform orientation. The parameters related to actuator q2 can be identified. If the measurements can be performed along all movement directions of the TCP, i.e. if the measurements are full-pose, the inverse kinematic calibration can be used. For the Scissors-Kinematics, this corresponds to measuring both TCP x and y co-ordinates. With this 56

Chapter 4. Geometrical Calibration of Redundant PKMs

calibration principle, the inverse kinematic transformation is used, which includes all branches of the mechanism. Besides the fact that this transformation can often be expressed analytically for parallel kinematics, the big advantage of this method is that the parameters of each kinematic branch can possibly be identified separately, thus, improving the conditioning of the system to be solved for the optimization, cf. [Ecorchard 02]. The condition that allows the separate identification for each kinematic loop is, however, that each parameter appears in only one loop-closure equation. This is not the case with the Scissors-Kinematics, since the parameters related to the parallel-holder, i.e. the parameters related to branches 4 and 5, appear in all loop-closure equations. The identification phase must be subdivided into three subproblems that must be solved by using the same measurement data several times, in the same way as for the forward kinematic calibration. If the passive platform angle ψ can also be acquired through measurements, the parameter identification can be then carried out for each actuator independently1 . Several measurement systems were used for the acquisition of the TCP position: ˆ Laser-tracker from Faro, [Faro], Fig. 4.9(a) ˆ KGM 182, grid-encoder from Heidenhain, [Heidenhain], Fig. 4.9(b) ˆ Double Ball-Bar (DBB) from Heidenhain, Fig. 4.9(c) ˆ Laser-interferometer ML10 from Renishaw, [Renishaw], Fig. 4.9(d)

(a)Laser-Tracker.

(b)KGM 182 (source: [Heidenhain]).

The KGM 182, the DBB and the ML10 have a measurement error close to one micrometer. However, the measurement accuracy of the DBB is dependent on the setup conditions, because there is no electronic way of resetting the null-position. The measurement error of the lasertracker is approximately 5 µm in the laser direction, but for the perpendicular directions, it is given as 18 µm + 3 µm/m by the manufacturer. Although the three dimensional capability of the laser-tracker system is not exploited for the measurement of the Scissors-Kinematics, it still remains advantageous compared to other systems, owing to the fact that the whole workspace can be measured with one setup. 1

As far as I know, there exists no definition for this type of measurements. Full-pose measurements commonly refer to measurements of all controllable TCP movements. A degree of measurement greater than the TCP’s degree of freedom, i.e. when measurements include the parasite movements, is not defined yet.

57

Chapter 4. Geometrical Calibration of Redundant PKMs

(c)DBB (source: [Heidenhain])

(d)Laser-Interferometer

Figure 4.9: Used measurement systems. The laser-tracker and the grid-encoder are able to measure the TCP position in both directions. The inverse kinematic calibration method can be thus used with these measurement systems. The big advantage of the grid-encoder and laser-interferometer is their very good accuracy. However, both measurement systems must be set up several times in order to cover the whole workspace. For the grid-encoder, the reason for this is the limited measuring range, a disc of diameter 230 mm. For the laser-interferometer, it is the fact that the measurements are one-dimensional. For measurements along the direction x, the laser-interferometer must be set up for each y position. Therefore, the laser-interferometer can be involved in the forward kinematic calibration method exclusively. The double ball-bar measures the distance between the TCP and a fixed point. The measurements of the double ball-bar are then not full pose, because the x and y components cannot be separated. Thus, the calibration can only be carried out as a forward kinematic calibration. All measurement systems, except the double ball-bar, need to be aligned to the machine axes. This way, both the measured position and the measurement axes are relative. Usually a movement of the mechanism along the most accurate axis, commonly, the longest axis, is used to set-up the measurement system axis, computationally (e.g. with the laser-tracker) or physically (e.g. with the laser-interferometer). For the Scissors-Kinematics, the kinematic redundant movement of the Gantry bridge, i.e. the movement along the x-axis, is used to setup the measurement system and is considered to be an absolute direction reference. For the measurements along the y-axis with the laser-interferometer, the alignment with the Gantry movement is not possible. This means that the angle of guide 2 has no influence on these measurements. These measurements were completed by a measurement of the platform orientation with a laser-interferometer and the adequate optics. Naturally, the measurements can only be used with model 29, since the angle measurements cannot be performed with respect to an absolute reference frame. For model 24, the platform angle is constant in the workspace. 4.2.1.1

Expression of the Cost-Function for the Optimization

As seen in Section 3.1.3, we need to express some model values Wmod that are to be confronted with their corresponding measurement values Wmeas . In order to use the same formalism (Eq. (3.2)) for all calibration methods, the considered reference values Wref for external 58

Chapter 4. Geometrical Calibration of Redundant PKMs

calibration with the laser-tracker, the KGM, and the laser-interferometer for the redundant mechanism are the TCP positions whereas the measurement values Wmeas are the actuator positions. For the measurements with the DBB, the inverse calibration method cannot be used and, hence, the classical denomination for the model values, i.e. the TCP position, more exactly, the distance between the TCP and a fixed reference point, is used. An important point to consider for external calibrations is that measurements are never absolute. The measurement system always needs to be set up or reset at some unknown machine position. The only values that are known are the actuator positions corresponding to this position, called the reset position. For calibrations with the laser-tracker, with the KGM and with the laser-interferometer with full-pose measurement, each component of the model values vector Wmod is one of the components of (QT , ψ)T , with Q = igm(Xmeas + fgm(Qnr0 , P ), P ).

(4.30)

Xmeas are the relative (x, y) co-ordinates of the TCP directly obtained by the measurement system. Qnr0 are the non-redundant actuator position values at the corresponding reset position. For the first step of the calibration with the DBB, each model value is the distance between the TCP and an unknown position where the socket is fixed on the machine table. This distance is given as p (4.31) r = (x − x0 )2 + (y − y0 )2 with (x0 , y0 )T = fgm(Qnr0 , P ) and (x, y)T alternatively defined as (x, y)T = fgm(Qnr , P ) and (x, y)T = fgm0 (q3 , q4 , P ). fgm0 is the forward geometrical model that gives the TCP position with respect to actuator positions q3 , q4 , and the machine geometrical parameters P . 1 The second step of this calibration is an inverse calibration. Let us call Pcal the parameter identified after the first step of the calibration. The virtual measurements are calculated as 1 Xe = fgm(Qnr0 , Pcal ). The model values are then q2 = igm(Xe , P ).

4.2.2

(4.32)

Jacobian Matrix

For all deterministic optimization algorithms, the so-called Jacobian of the calibration is needed. It gives the variations of the model values Wmod with respect to the variations of parameters for all measurement positions. The development of the Jacobian matrix Jigm expressing the variation in actuator positions with respect to the parameter variations is given in Annex B.2 for one TCP position. The global Jacobian matrix of the calibration is then obtained as   J1igm  J2   igm  Jext =  ..  (4.33)  .  Jm igm with Jiigm the Jacobian matrix for pose i.

59

Chapter 4. Geometrical Calibration of Redundant PKMs

For the first step of the calibration with the double ball-bar, the Jacobian is   J1r  J2   r Jdbb =  ..   .  Jm r

(4.34)

with Jir the Jacobian matrix for pose i, developed in Annex B.3. For the second step of the calibrations with the double ball-bar and with the laserinterferometer, it is  1  Jfgm  J2   fgm  Jfgm =  ..  . (4.35)  .  Jm fgm

4.2.3

Sensitivity Analysis

The sensitivity analysis was done for all calibration methods to detect the parameters that have a small or no influence on the measurements. The ranges within which the machine parameters can vary are 0.1 mm and 1◦ for the parameters of length and angle, respectively. The bar plots of this Section represent the sensitivity index calculated with the 2-norm over the whole workspace, as defined in Section 3.3.2. For the Scissors-Kinematics, the whole workspace corresponds to a movement of the TCP along x from −150 to 150 mm and along y from 700 to 2700 mm. 231 points were taken over the whole workspace in all sensitivity analyses. 4.2.3.1

Model 24

sensitivity (µm)

The sensitivity analysis was carried out for the calibration with the laser-tracker and is presented in Fig. 4.10. dq1 dq2 dq3 dq4

400 200

d1 γ1

xa2r xa3r ya2r ya3r xb2r xb3r yb2r yb3r

l2 l3

d2 γ2

xa1r xa4r ya1r ya4r xb1r xb4r yb1r yb4r

l1 l4

0 Parameters Figure 4.10: Sensitivity for the calibration with the laser-tracker and model 24. It can be already noticed that all parameters are sensitive. Each parameter is sensitive at least in one measurement direction. The parameter sensitivity is clearly divided into groups corresponding to different loops. Thus, a parameter influences the actuator positions of the branch this parameter belongs to. The only parameter that is sensitive to the measurement on 60

Chapter 4. Geometrical Calibration of Redundant PKMs

all actuator positions is the angle γ2 of the right-hand side guide, to which the parallel-holder is linked. This prevents the division of the optimization problem into subproblems because the value of γ2 , which is included into all subproblems, would be unclear. The sensitivity study for the calibration with the grid-encoder is presented in Fig. 4.11. It can be mentioned that the profile of the graph is very similar to the preceding one. However, all parameters are less sensitive as compared with the laser-tracker. The reason is that the parameter variations cannot be measured over the whole workspace rather than over the circular area of the grid-encoder. sensitivity (µm)

300

dq1 dq2 dq3 dq4

200 100

d1 γ1

xa2r xa3r ya2r ya3r xb2r xb3r yb2r yb3r

l2 l3

d2 γ2

xa1r xa4r ya1r ya4r xb1r xb4r yb1r yb4r

l1 l4

0 Parameters Figure 4.11: Sensitivity for the calibration with the grid-encoder and model 24.

dr dq2

100

xa2r ya2r xb2r yb2r

l2

d1 γ1

xa3r ya3r xb3r yb3r

l3

d2 γ2

xa1r xa4r ya1r ya4r xb1r xb4r yb1r yb4r

50

l1 l4

sensitivity (µm)

The sensitivity study for the calibration with the double ball-bar is divided into two parts that were for convenience concatenated into one graph, cf. Fig. 4.12. The first part shows the parameter-error influence on the measurements with the double ball-bar. All parameters are sensitive. The influence of the parameters related to rod 2 is also well measurable. The sensitivity limit for these parameters is linked to the computational precision and not to the accuracy of the measuring system and this limit is very low (around 10−16 times smaller than the parameter values).

Parameters Figure 4.12: Sensitivity for the calibration with the double ball-bar and model 24. Fig. 4.13 is the result of the sensitivity study for the calibration with the laser-interferometer. In model 24, it is assumed that the parallel-holder is perfect. This means that no parameter variation has any influence on a value measured along the laser-interferometer axis, which corresponds to the y-axis rotated from the guide angle γ2 , as is mentioned above. Thus, the parameters are sensitive only on a measurement along the x-axis. All parameters are sensitive in this direction but the parameters xa4r and xb4r have a small sensitivity coefficient, 10 µm on the x-position for a parameter variation of 0.1 mm. 61

dx dq2

100

xa2r ya2r xb2r yb2r

l2

d1 γ1

xa3r ya3r xb3r yb3r

l3

d2 γ2

xa1r xa4r ya1r ya4r xb1r xb4r yb1r yb4r

50

l1 l4

sensitivity (µm)

Chapter 4. Geometrical Calibration of Redundant PKMs

Parameters Figure 4.13: Sensitivity for the forward calibration with the laser-interferometer with model 24.

62

Chapter 4. Geometrical Calibration of Redundant PKMs

4.2.3.2

Model 29

sensitivity (µm)

The results of the sensitivity analysis for model 29, presented in Fig. 4.14, show clearly that the parameters are more interdependent than for model 24. dq1 dq2 dq3 dq4

400 200

l1 l4 l5

xa1r xa4r xa5r ya1r ya4r ya5r xb1r xb4r xb5r yb1r yb4r yb5r d2 γ2 l2 l3 xa2r xa3r ya2r ya3r xb2r xb3r yb2r yb3r d1 γ1

0 Parameters Figure 4.14: Sensitivity for calibration with the laser-tracker and model 29. Indeed, the parameters that belong to the parallel-holder, i.e. all parameters with indexes 4 and 5, have an influence on all branches of the mechanism. Although it seems that some parameters could be identified separately it is not convenient for the same reason as with model 24. As with model 24, all parameters have a sensitive influence on the actuator positions but the influence of angle parameters is clearly greater than length parameters. The sensitivity analysis for the calibration with the grid-encoder is presented in Fig. 4.15. The parameters are less sensitive than for the calibration with the laser-tracker and model 29. The parameters of rod 4 are more sensitive than with model 24 but the sensitivity profile is similar. sensitivity (µm)

300

dq1 dq2 dq3 dq4

200 100

l1 l4 l5

xa1r xa4r xa5r ya1r ya4r ya5r xb1r xb4r xb5r yb1r yb4r yb5r d2 γ2 l2 l3 xa2r xa3r ya2r ya3r xb2r xb3r yb2r yb3r d1 γ1

0 Parameters Figure 4.15: Sensitivity for the calibration with the grid-encoder and model 29. The results of the sensitivity analysis for the calibration with the double ball-bar show that the parameters for the same calibration method are more sensitive than with model 24, cf. Fig. 4.16. For the forward calibration with the laser-interferometer, the parameters in model 29 are also more sensitive than the corresponding ones in model 24, as seen in Fig. 4.17. For the calibrations that are completed by platform angle measurements, the sensitivity of parameters’ errors on the four actuator positions is the same as without angle measurements. 63

sensitivity (µm)

Chapter 4. Geometrical Calibration of Redundant PKMs

dr dq2

100 50

l1 l4 l5

xa1r xa4r xa5r ya1r ya4r ya5r xb1r xb4r xb5r yb1r yb4r yb5r d2 γ2 l3 xa3r ya3r xb3r yb3r d1 γ1 l2 xa2r ya2r xb2r yb2r

0 Parameters

sensitivity (µm)

Figure 4.16: Sensitivity for the calibration with the double ball-bar and model 29. dx dq2

100 50

l1 l4 l5

xa1r xa4r xa5r ya1r ya4r ya5r xb1r xb4r xb5r yb1r yb4r yb5r d2 γ2 l3 xa3r ya3r xb3r yb3r d1 γ1 l2 xa2r ya2r xb2r yb2r

0 Parameters Figure 4.17: Sensitivity for the calibration with the laser-interferometer and model 29. Fig. 4.18 shows then only the influence on the angle measurements ψ. According to Renishaw documentation, the angular accuracy when measuring an angle around zero at a distance of five meters is approximately 1 µm/m, i.e. 10-3 mrad. Thus, the platform angle measurements should help to obtain better calibration results because all parameters related to the parallelholder are largely sensitive. The results in Fig. 4.19 showing parameter influence factors smaller than with the lasertracker confirm the importance of measuring the whole workspace without resetting the measuring system.

4.2.4

Observability Analysis

Through the observability analysis, the parameters are sorted with respect to their linear dependency. If two parameters are linearly dependent, i.e. two columns of the global Jacobian matrix of the calibration are linearly dependent, the one with the smallest sensitivity is sorted out of the calibration process and the parameter sorting process is done on the rest of the Jacobian matrix. These results are shown in two column tables in this Section. The first column, labelled ”Parameters”, or abbreviated ”Par.”, contains the parameter list sorted according to this principle. The second column contains the condition number of the normalized Jacobian matrix when all parameters up to the one of the corresponding line are included in the calibration process.

64

dψ

0.3 0.2 0.1 0 l1 l4 l5

xa1r xa4r xa5r ya1r ya4r ya5r xb1r xb4r xb5r yb1r yb4r yb5r d2 γ2 l2 l3 xa2r xa3r ya2r ya3r xb2r xb3r yb2r yb3r d1 γ1

sensitivity (mrad)

Chapter 4. Geometrical Calibration of Redundant PKMs

Parameters

dψ 0.20 0.10 0.00 l1 l4 l5

xa1r xa4r xa5r ya1r ya4r ya5r xb1r xb4r xb5r yb1r yb4r yb5r d2 γ2 l2 l3 xa2r xa3r ya2r ya3r xb2r xb3r yb2r yb3r d1 γ1

sensitivity (mrad)

Figure 4.18: Sensitivity for calibrations with the laser-tracker, platform angle measurements and model 29.

Parameters Figure 4.19: Sensitivity for the calibration with the grid-encoder, platform angle measurements and model 29.

65

Chapter 4. Geometrical Calibration of Redundant PKMs

4.2.4.1

Model 24

The observability analyses are summarized in Table 4.4 for several calibration methods in order to compare them in an easier way. ”LT 24” stands for the calibration with the laser-tracker and model 24, ”KGM” for the grid-encoder (Kreuzgittermessger¨at in German), ”DBB” for the double ball-bar and ”LI” for the laser-interferometer. For the calibration with the double ballbar and the laser-interferometer, only the parameters identifiable during the first calibration phase are listed because the condition number for the second phase of the calibration is not as relevant as for the first one. Table 4.4: Sorted geometrical parameters and the associated observability index for several calibration methods and model 24. LT 24 KGM 24 DBB 24 LI 24 Par. Condition Par. Condition Par. Condition Par. Condition number number number number γ2 1 γ2 1 γ2 1 γ2 1 γ1 1.3 γ1 1.4 γ1 2.3 γ1 3.7 l2 6.5 l2 10 l1 11 l1 7.1 ya1r 7 xa3r 16 l4 14 l4 58 xb1r 8.5 l1 26 d1 20 l3 130 ya3r 12 l4 50 yb1r 120 ya1r 950 xb3r 14 yb3r 74 xb1r 240 ya4r 1.4 · 105 xb2r 33 xa4r 91 l3 460 xb4r 2.9 · 105 l1 310 yb1r 410 ya3r 6300 yb1r 8.7 · 105 l4 360 l3 460 ya1r 2.3 · 105 yb4r 4.4 · 107 l3 530 d1 550 xa3r 3.5 · 105 xa1r 3.3 · 108 5 ya2r 550 yb4r 630 yb3r 9.9 · 10 xa4r 1.2 · 1010 xa2r 1.7 · 107 ya2r 1.4 · 106 xa1r 3.5 · 106 d2 1.5 · 1010 The lower observability indexes of the calibration with the laser-tracker compared to the calibration with the grid-encoder is explained by the fact the a single reset position is needed when measuring the laser-tracker, whereas the grid-encoder measuring system must be reset eight times to cover the whole workspace. The high condition number from the 11th parameter on for the calibration with the double ball-bar is due to the fact that the measurement system is one-dimensional. The great difference between the condition number of the calibration with the laserinterferometer and the one with the double ball-bar can be explained by the fact that the direction along which the measurement system is sensitive to changes with the double ballbar, whereas it is constant with the laser-interferometer. For example, when the double ball-bar is oriented along the y-axis, a variation on the measured distance corresponds to a variation in the direction y. On the opposite, the laser-interferometer measures along a direction which is constant relative to the machine bed. In Table 4.4, the calibration methods are ordered according to ascending reliability from left to right. According to [Schr¨oer 93], a condition number smaller than 100 should give reliable results during a real calibration. For the calibrations with the laser-tracker and the grid-encoder, eight parameters should be included. With the double ball-bar they should be seven. With the laser-interferometer, one includes the first four or five parameters listed in Table 4.4. However, the calibrations can be also carried out with a greater number of parameters in order to confirm or infirm this hypothesis. 66

Chapter 4. Geometrical Calibration of Redundant PKMs

4.2.4.2

Model 29

The observability analysis for the calibrations with model 29 are presented in Table 4.5. For all calibration methods the condition number is better than with model 24. The number of parameters identifiable with a reasonable reliability is nine, eight or nine, eight, and four or five for the calibrations with the laser-tracker, the grid-encoder, the double ball-bar and the laser-interferometer, respectively. Table 4.5: Sorted geometrical parameters and the associated observability index for several calibration methods with model 29. LT 29 KGM 29 DBB 29 LI 29 Par. Condition Par. Condition Par. Condition Par. Condition number number number number γ2 1 γ2 1 γ2 1 γ2 1 γ1 1.5 γ1 1.4 γ1 3.3 γ1 3.2 xa4r 6.2 yb4r 11 yb5r 7 l5 6.8 l3 7.3 xa4r 20 ya1r 12 l1 18 ya5r 11 ya1r 29 l4 42 l4 110 l2 15 l2 32 l1 100 xa4r 360 ya1r 16 xa1r 62 l3 260 l3 840 xb1r 35 l3 89 xa5r 700 yb4r 5500 xb5r 71 xa5r 110 l5 3000 ya4r 2.1 · 104 l4 670 l4 770 d2 6000 xb4r 2.2 · 105 l1 740 l1 820 xa4r 9200 yb1r 7.2 · 105 yb2r 760 ya3r 1000 ya5r 1.6 · 105 xa5r 2 · 106 5 l5 4300 l5 4600 xb4r 2 · 10 xb1r 2.5 · 107 xb2r 1.5 · 104 xb2r 1.8 · 104 yb4r 6.5 · 105 xb5r 1.1 · 108 6 6 6 ya4r 6.2 · 10 ya4r 1.8 · 10 xb5r 2.3 · 10 xa1r 4.7 · 108 4.2.4.3

Model 29 with Platform Angle Measurements

The observability study for the three calibrations completed with platform angle measurements show the same results as without the platform angle measurements, cf. Table 4.6.

4.3 4.3.1

Model Identification through Self-Calibration Self-Calibration Method

On classical PKMs, the parameter identification through self-calibration is usually achieved through additional sensors mounted on passive joints. This leads to the rise in cost without essentially improving the mechanism itself. On the opposite, on redundant PKMs, the sensors needed for a self-calibration are already available. On redundantly actuated mechanism, the redundant actuators can be switched off without changing the mechanism degree-of-freedom. Thus, a self-calibration can be carried out without the influence of the internal constraints due to redundancy. The self-calibration of the Scissors-Kinematics is carried out by switching off the controllers of actuators q2 and q3 , as described in Fig. 4.20. The associated measurements must nevertheless be available. Actuators q1 and q4 keep their normal function. 67

Chapter 4. Geometrical Calibration of Redundant PKMs

Table 4.6: Sorted geometrical parameters and the associated observability index for several calibration methods with model 29 and platform angle measurements. LT 29 KGM 29 LI 29 Parameter Condition Parameter Condition Parameter Condition number number number γ2 1 γ2 1 γ2 1 γ1 1.4 γ1 1.5 γ1 3.2 yb4r 4.4 yb4r 10 yb4r 7.4 l3 6.3 xa4r 17 xa4r 19 xa4r 11 l2 20 l5 110 l2 13 ya1r 28 l1 420 yb1r 16 xa1r 55 l4 860 xb1r 28 l3 83 ya5r 7700 xb5r 50 xb2r 110 l3 2.5 · 104 l4 370 l4 500 xb5r 8.3 · 105 l1 400 l1 520 xb1r 1.6 · 106 ya3r 530 ya3r 550 ya4r 3.2 · 106 l5 3000 l5 2000 xa5r 5.4 · 107 yb5r 9700 xb3r 8400 yb5r 2.1 · 108 xa2r 1 · 104 yb5r 1.1 · 104 ya1r 1.2 · 109 6 6 ya4r 4.8 · 10 ya4r 1.9 · 10 xa1r 2.1 · 109 For this calibration method, the actuator positions q2 and q3 as a function of actuator positions q1 and q4 are needed. This model will be called redundant geometrical model, because it gives the redundant actuator positions in respect to the non-redundant ones. As a first step, the platform position X and its angle ψ need to be calculated. As for the inverse and direct transformation models, two model complexities will be tested. We have X = fgm(Qnr , P ),

n = 24, 29.

(4.36)

n

Moreover, the parasite angle ψ is retrieved during the calculation of the platform position. For the model 24, we have ψ = γ2 . Then, position of actuators q2 and q3 are obtained with (4.6) and Eq. (4.23). The redundant geometrical models are then noted Qr = rgm(Qnr , P ),

n = 24, 29.

(4.37)

n

It is to be noted that the number of parameters is not minimal but kept constant to be consistent with the respective inverse geometrical model on which each model is based.

4.3.2

Jacobian Matrix

The development of the Jacobian matrix occurs in two steps. First the variations of Cartesian co-ordinates in respect to variations of parameters related to actuators q1 and q4 are expressed, i.e. to all parameters of the non-redundant subsystem. Then, the variations of q2 and q3 in respect to the parameter variations is expressed, thus integrating the first results. The development of the Jacobian matrix is given in Annex B.4. 68

Chapter 4. Geometrical Calibration of Redundant PKMs

measurements + actuation

measurements

q2

q1

y

q3

q4

0

x

Figure 4.20: Self-calibration principle. The matrix J23 (B.15) gives the influence of all the parameters on the redundant actuator positions for a measurement position i, its complete notation should be J23 (Qinr ), with Qinr = (q1 , q4 )T at position i. The Jacobian matrix of the calibration is then obtained by concatenating the Jacobian matrices at position i = 1, . . . , m, with m being the number of measurement values, i.e. twice the number of measurement poses:   J23 (Q1nr ) J23 (Q2 ) nr   Jself =  (4.38) . ..   . J23 (Qm nr )

4.3.3

Sensitivity Analysis

As shown in Section 3.3.2, the sensitivity analysis allows one to determine which parameter influence on carrier position measurements for q2 and q3 can be detected. For each measurement point i and for each parameter k, we have ( σ2,i,k = Js 2i−1,k (4.39) σ3,i,k = Js 2i,k , where σ2,m,k and σ3,m,k are the variations in the position of actuators q2 and q3 , respectively, for measurement point i, as a consequence of a variation dpk in parameter k. We have then the mean sensitivity for each parameter over the whole workspace as: sP  σ2,i,k    i   σ2,k = sPm (4.40)  σ  3,i,k   i σ = . 3,k m 69

Chapter 4. Geometrical Calibration of Redundant PKMs

4.3.3.1

Model 24

sensitivity (µm)

Fig. 4.21 shows the variation of actuator position as a consequence of a variation in each parameter successively, i.e. σ2,k and σ3,k . All parameters are sensitive. The parameters of the right-hand side of the mechanism, the guide position d1 and the guide angle γ1 are sensitive on the two redundant actuator positions, whereas the parameters with indexes 2 and 3 are sensitive only on the actuator they are related to. dq2 dq3

200 100

d1 γ1

xa2r xa3r ya2r ya3r xb2r xb3r yb2r yb3r

l2 l3

d2 γ2

xa1r xa4r ya1r ya4r xb1r xb4r yb1r yb4r

l1 l4

0 Parameters Figure 4.21: Sensitivity analysis for the self-calibration and model 24.

4.3.3.2

Model 29

sensitivity (µm)

In Fig. 4.22 the results of the sensitivity analysis are presented. All parameters can be seen to have a significant influence of the value of q2 and q3 . The strut lengths l1 to l5 have the greatest influence. The guide-angle errors also induce big variations in the measured values, especially γ2 associated with the parallel-holder. The parameters linked to rods 1, 2, and 3 produce the same influence as in the case of model 24, the other parameters, however, being are all more sensitive. dq2 dq3

300 200 100

l1 l4 l5

xa1r xa4r xa5r ya1r ya4r ya5r xb1r xb4r xb5r yb1r yb4r yb5r d2 γ2 l2 l3 xa2r xa3r ya2r ya3r xb2r xb3r yb2r yb3r d1 γ1

0 Parameters Figure 4.22: Sensitivity analysis for the self-calibration and model 29.

4.3.4

Observability Analysis

The observability analysis is carried out on both models so that the parameters that are linearly dependent can be excluded from the calibration process.

70

Chapter 4. Geometrical Calibration of Redundant PKMs

4.3.4.1

Model 24

As has already been mentioned, the number of parameters is the same for both the redundant geometrical model and the inverse geometrical model for the purpose of consistency and avoidance of multiple model definitions. The consequence of this is that the model is not minimal (cf. Section 3.3.1). The parameters which influence the measured values are linearly dependent on other parameters and must be excluded from the calibration process. For model 24, we have ψ = γ2 . Thus, we have ∂fi ∂fi =− ∂xbir ∂xair

(4.41)

∂fi ∂fi =− . ∂ybir ∂yair

(4.42)

and

The columns of the Jacobian matrix will then be linearly dependent. A classical drawback of self-calibration methods it that the position of the whole mechanism cannot be determined in its environment. Thus, the guide distance d1 cannot be determined at the same time as d2 . The QR-decomposition made it possible to determine further linear dependencies that are not obvious. Through QR-decomposition, the parameters are ordered in respect to their ascending relevance in the Jacobian matrix of the calibration. The associated condition number is then calculated. Table 4.7 shows the ordered parameters and the associated condition number for the self-calibration process. Table 4.7: Sorted geometrical parameters and the associated observability index for the selfcalibration method and model 24. Parameter Condition number γ2 1 γ1 1.5 l1 7.1 l3 10 l4 32 l2 94 d1 140 xa4r 1800 yb1r 1.9 · 104 ya3r 8.5 · 104 xa3r 1.3 · 105 The set of parameters for which reliable calibration results can be expected is (γ2 , γ1 , l1 , l3 , l4 , l2 , d1 ). With this set of seven parameters, the condition number of the normalized Jacobian matrix is around 140. However, calibration needs also to be carried out with larger sets of parameters in order to justify this selection. 4.3.4.2

Model 29

For the same reasons as for model 24, the parameter d1 is excluded from the calibration process. With the remaining parameters, the observability analysis was carried out with various 71

Chapter 4. Geometrical Calibration of Redundant PKMs

parameter sets obtained through a small random variation around the nominal parameters. The results are presented in Table 4.8. Table 4.8: Parameters for the self-calibration with model 29 sorted in decreasing order of influence and the associated condition number. Parameter Condition number γ2 1 γ1 1.5 l1 7.1 l5 20 l2 25 l3 51 d2 190 l4 640 ya4r 6500 ya1r 2.7 · 104 yb2r 7.3 · 104 The set of parameters for the calibration is then (γ2 , γ1 , l1 , l5 , l2 , l3 ).

4.4

Experimental Validation

The various calibration methods associated with the various models were tested on the ScissorsKinematics. An important remark for the test of the results of a calibration is that the parameter compatibility for the use in a redundant mechanism should be checked first. If the parameters are not compatible, the constraint in the mechanism can be excessive and can make some damages to it. To check the compatibility of the parameters, the redundant actuators q2 and q3 are switched to the passive mode. Then the value obtained from the redundant geometrical transformation with the calibrated parameters are compared to the current position read in the controller. The difference is ∆q2 = q2mod − q2meas

∆q3 = q3mod − q3meas

(4.43)

with qimod being the model values for actuator i obtained from Eq. 4.37 and qimeas its measured equivalent. If the difference between the model values and the measured values is reasonable, the four actuators are switched to the active mode and the current that is used for the drives to keep their position is recorded. This current is called redundancy current and is given as percentage of the nominal current. The nominal current, i.e. when the current equals 100 %, corresponds to a force of 8,600 N. For the Scissors-Kinematics, a parameter deviation of approximately 400 µm can already generate a current of 100 % of the nominal current in one of the actuators and is thus the maximal allowed variation. The maximal allowed deviation is, however, dependent on the combination of directions for the deviations in q2 and q3 and in the TCP position. If the deviations of q2 and q3 have an opposite direction, the redundancy current is much higher than if they have the same direction, for the same absolute values. The compatibility test was carried out in one position in the middle of the workspace. 72

Chapter 4. Geometrical Calibration of Redundant PKMs

For redundantly actuated mechanisms, the nominal parameter set can represent a danger for the mechanism with all position-controlled actuators. Parameter incompatibility between the non-redundant and the redundant parts of the mechanism is indeed responsible for excessive internal constraints, specially for the Scissors-Kinematics, which is a stiff machine. Therefore, the compatibility was checked for the nominal parameters. The absolute position differences for redundant actuators were ∆q2 = 0.414 mm and ∆q3 = 0.641 mm. The difference on q3 is too large. This means that the nominal parameters cannot be used on the machine if all four actuators are position-controlled. To solve this problem, the mechanism has been mechanically adjusted and was driven by a force control on the redundant actuators. A pre-calibrated parameter set was obtained, with which it was possible to drive the machine. The calibration processes, however, use the nominal parameters as starting parameters.

4.4.1

Calibration with the Laser-Tracker

The calibration with the laser-tracker can be done in one setup. The setup occurs by movements on the mechanism. A plane, an oriented line and a point are needed. The plane is the x-y plane in which all 3D co-ordinates will be projected because the z co-ordinates are ignored for the Scissors-Kinematics. The reference line must correspond to a movement of the Gantry bridge in order to align the Scissors-Kinematics to the rest of the machine. The reference point is the middle of the workspace. The measurements occur on a mesh grid with 5 x positions and 9 y positions and cover the whole workspace. By using the criterion seen in Section 3.4.1 with three times the standard deviation, there were no outliers. 4.4.1.1

Calibrations with Model 24

Table 4.9 shows the results of the compatibility study for the calibration with model 24. The actuator position difference as well as the redundancy current are sufficiently low for the application in the redundant mechanism. Both the current and the actuator position differences improve with the ascending number of parameters. Table 4.9: Compatibility test for the parameters model 24. Number of ∆q2 parameters (mm) 4 0.084 5 0.087 6 0.121 7 0.112 8 0.082 9 0.078 10 0.057 11 0.050

of the calibration with the laser-tracker and ∆q3 (mm) 0.047 0.050 0.051 0.063 0.058 0.048 0.081 0.067

current (%) 27 27 19 18 17 15 29 25

The results of the circular tests for the four parameters sets are presented in Fig. 4.23. The radius deviation corresponds to the absolute difference between the mean measured radius and the circle radius programmed in the controller (100 mm). The circularity is the difference between the maximal and minimal measured radii. The complete circular test for the best parameter set is shown in Annex, Fig. C.1. The circular deviation for this set is 38 µm and the circularity is 105 µm. 73

Chapter 4. Geometrical Calibration of Redundant PKMs

redundancy current (%)

0.200

50

0.164

44

0.128

38

0.092

32

0.056

26

0.020

4

5

6

7 8 number of parameters

9

10

current (%)

deviation (mm)

radius deviation (mm) circularity (mm)

20 11

Figure 4.23: Calibration results with the laser-tracker and model 24. 4.4.1.2

Calibrations with Model 29

The parameter compatibility test for the calibration with the laser-tracker and model 29 is shown in Table 4.10. There are very little differences with the same calibration method for model 24. The main difference is that there is no improvement with respect to the number of parameters. Table 4.10: Compatibility test for the parameters of the calibration with the laser-tracker and model 29. Number of ∆q2 ∆q3 current parameters (mm) (mm) (%) 4 0.079 0.070 15 5 0.086 0.068 14 6 0.105 0.043 15 7 0.122 0.051 17 8 0.077 0.039 13 9 0.112 0.046 17 10 0.103 0.043 16 11 0.090 0.054 15 12 0.088 0.045 14 13 0.089 0.042 14 14 0.076 0.078 17 The results of the circular tests presented in Fig. 4.24 confirm that the calibrations with model 24 and model 29 give similar results. The circular test for the parameter set with twelve identified parameters can be found in annex, Fig. C.2. The circular deviation and the circularity are respectively 38 µm and 116 µm.

4.4.2

Calibration with the Grid-Encoder

The diameter of the grid-encoder is 115 mm. For each y-position two positions of the gridencoder are then necessary to cover the x range of the Scissors-Kinematics. Eight positions were taken along y. For each position of the grid-encoder, a set-up is necessary. First, the 74

Chapter 4. Geometrical Calibration of Redundant PKMs

redundancy current (%)

0.200

65

0.168

56

0.136

47

0.104

38

0.072

29

0.040

4

5

6

7

8 9 10 11 number of parameters

12

13

current (%)

deviation (mm)

radius deviation (mm) circularity (mm)

20 14

Figure 4.24: Calibration results with the laser-tracker and model 29. measurement axes must be aligned with the machine axes. A movement of the Gantry bridge sets the x axis so that the x axis of the Scissors-Kinematics will be aligned with x axis of the bridge. Then, the null position is set in the middle of the measuring grid so that measurements are relative. A set of nine points regularly spaced on a square with 140 mm side is measured. No outliers needed to be excluded from the calibration. 4.4.2.1

Calibrations with Model 24

Table 4.11 shows clearly that the results of the compatibility tests for the calibration with the grid-encoder are not as good as with the laser-tracker for the same model 24. The actuator current is three times as high as the one obtained with the laser-tracker. Table 4.11: Compatibility test for the parameters of the calibration with the grid-encoder and model 24. Number of ∆q2 ∆q3 current parameters (mm) (mm) (%) 5 0.274 0.259 54 6 0.304 0.253 57 7 0.292 0.277 60 8 0.306 0.252 58 9 0.272 0.220 51 10 0.251 0.229 52 11 0.280 0.257 59 12 0.293 0.246 58 These results are contradictory to the very good accuracy results from the circular tests presented in Fig. 4.25. The mean deviation with the parameter set from the calibration with eight parameters is as low as 10 µm, cf. Fig. C.3 in Annex. This contradiction is an important result of the calibration of redundant mechanisms. It shows that good TCP accuracy is not synonymous with low redundancy current, i.e. small preload constraints. However, it is difficult to draw the conclusion whether a parameter set that gives small internal constraints necessarily gives a good TCP accuracy. During the measurements with the laser-tracker, a redundancy current compensation was active whereas it could be deactivated during the measurement with 75

Chapter 4. Geometrical Calibration of Redundant PKMs

the grid-encoder. The redundancy current compensation corrects the position of the redundant actuators in order to limit their energy consumption. This means that the measurements with the laser-tracker were done with less internal constraints than with the grid-encoder. This is a second important result of the comparison between the results of the laser-tracker calibration and grid-encoder calibration. Although the calibrations based on the TCP measurements lead to parameter sets for which the accuracy is better, the internal constraints obtained are in the same order as the internal constraints during the measurements. radius deviation (mm) circularity (mm) 0.200

deviation (mm)

0.150

0.100

0.050

0.000

5

6

7

8 9 number of parameters

10

11

12

Figure 4.25: Calibration results with the grid-encoder and model 24.

4.4.2.2

Calibrations with Model 29

The results of the compatibility test for the calibration with model 29 shown in Table 4.12 are similar to those with model 24. The redundancy current is around 60 % for all tested parameter sets (except for the set with nine identified parameters which can be considered to be an outlier). Table 4.12: Compatibility test for the parameters of the calibration with the grid-encoder and model 29. Number of ∆q2 ∆q3 current parameters (mm) (mm) (%) 4 0.299 0.254 56 5 0.294 0.252 57 6 0.283 0.246 57 7 0.248 0.231 47 8 0.231 0.215 46 9 0.289 0.262 55 10 0.311 0.296 66 11 0.331 0.261 59 12 0.336 0.261 59 It can be seen in Fig. 4.26 that the obtained TCP accuracy is good and that it improves with an ascending number of identified parameters. The results are similar to the results with model 24 so that the comments and conclusions also apply for model 29. 76

Chapter 4. Geometrical Calibration of Redundant PKMs

radius deviation (mm) circularity (mm) 0.250

deviation (mm)

0.200

0.150

0.100

0.050

0.000

4

5

6

7 8 9 10 number of parameters

11

12

13

Figure 4.26: Calibration results with the grid-encoder and model 29. The circular test for the best parameter set of this calibration method is in Annex, Fig. C.4. The obtained characteristics are a radius deviation of 8 µm and a circularity of 51 µm.

4.4.3

Calibration with the Double Ball-Bar

The diameter of the double ball-bar (300 mm) covers the range of TCP movements along the x-axis. Then, the measurements were carried out by taking 8 different y positions and a central x position. For each position of the double ball-bar, 37 measurement points were taken on the circle. When the threshold has a value of three times the standard deviation (Section 3.4.1), all measurement values are considered to be valid. 4.4.3.1

Calibrations with Model 24

Table 4.13 sums up the results of the calibration with the double ball-bar and model 24. The apparently better results of the compatibility test for the identification with 6 and 7 parameters did not lead to smaller redundancy current, because, for these two cases, the errors on q2 and q3 have opposite directions, whereas they have the same directions for the other parameter sets. The excessive mean redundancy current for all parameter sets is not compatible with the use on the machine. Table 4.13: Compatibility test for the parameters of the calibration with the double ball-bar and model 24. Number of ∆q2 ∆q3 current parameters (mm) (mm) (%) 6 0.370 0.112 >100 7 0.368 0.113 >100 8 0.355 0.369 77 9 0.353 0.311 69 10 0.368 0.348 73

77

Chapter 4. Geometrical Calibration of Redundant PKMs

4.4.3.2

Calibrations with Model 29

Table 4.14 shows the results of the calibration with the double ball-bar and model 29. Although the results are better than with model 24 with 6 and 7 parameters, the high redundancy current for all tested parameter sets prevents them from being used on the machine. Table 4.14: Compatibility test for the parameters of the calibration with the double ball-bar and model 29. Number of ∆q2 ∆q3 current parameters (mm) (mm) (%) 6 0.377 0.311 70 7 0.371 0.294 68 8 0.313 0.234 57 9 0.358 0.272 63 10 0.349 0.351 74 11 0.347 0.311 68 12 0.357 0.351 75

4.4.4

Calibration with the Laser-Interferometer

The measurements along x were taken with a 15-mm interval over 9 y-positions, the measurements along y with a 25-mm interval over 13 x positions. Two outlier measurements were excluded from the calibration. 4.4.4.1

Calibrations with Model 24

The results of the parameter compatibility for the calibration with the laser-interferometer and model 24 are shown in Table 4.15. In this Table, ”n/a” stands for ”not available”, i.e. the difference between model and reality is too great and, hence, the redundancy current cannot be measured, because the controllers of all actuators cannot be activated simultaneously. Table 4.15: Compatibility test for the parameters of the calibration with the laserinterferometer and model 24. Number of ∆q2 ∆q3 current parameters (mm) (mm) (%) 6 0.130 0.589 n/a 7 0.147 0.559 n/a 8 0.296 0.296 61 9 0.286 0.283 60 10 0.288 0.279 60 11 0.290 0.338 67 12 0.304 0.345 68

4.4.4.2

Calibrations with Model 29

Table 4.16 shows the results of the compatibility test for the calibration with the laserinterferometer and model 29. As with model 24 with the same measurement system, the 78

Chapter 4. Geometrical Calibration of Redundant PKMs

redundancy current is too high for the parameters to be used in the real machine. Table 4.16: Compatibility test for the parameters of the calibration with the laserinterferometer and model 29. Number of ∆q2 ∆q3 current parameters (mm) (mm) (%) 6 0.323 0.207 53 7 0.281 0.273 59 8 0.281 0.288 59 9 0.287 0.352 69 10 0.292 0.329 67 11 0.284 0.323 67 12 0.288 0.328 67

4.4.5

Influence of the Platform Angle Measurements

The calibrations with the laser-tracker, the grid-encoder and the laser-interferometer were completed with measurements of the platform angle performed with the laser-interferometer. For these three measurement systems, it is technically possible to measure the relative platform angle so that the angle measurements with the laser-interferometer reproduce the capabilities of the other measurement systems. For these three measurement systems, the principle of angle measurement is the same. The angle is obtained by the relative position variations of two points separated with a known distance. For the laser-tracker this can be done in two ways. Either the machine must follow the same path twice with two positions of the target which is fixed on the Scissors-Kinematics platform, or the measurements must be carried out with the time-of-flight mode and two targets are scanned alternatively when the machine stops at the measurement position. For the grid-encoder, a lever can be used to excenter the measurement head from the TCP. The lever distance must be measured on a co-ordinate-measuring machine. The easiest way to measure an angle deviation though is with the laser-interferometer. The optics are already available to measure an angle when the mechanism is moving parallel to the laser beam. The measurement range is small but is largely sufficient for the Scissors-Kinematics case. The platform angle was measured at 28 y positions with a middle x position and 21 x positions with three y positions (the minimum, middle and maximum of workspace). The measurements with movements along x axis were actually performed with the laser beam oriented along y by combining a x movement of the Scissors-Kinematics with the opposite movement of the Gantry bridge so that the TCP practically stays in front of the laser beam. 4.4.5.1

Laser-Tracker

Table 4.17 presents the results of the compatibility tests for the calibration with measurement of the extended platform position. The redundant actuator position differences between model and real machine are better for most of the compatibility tests compared to the results of the calibration without the angle measurement. However, the redundant current measurements are two and three times worse. As can be seen in Fig. 4.27, there is no improvement in the quality of the calibration with the laser-tracker when the platform angle is also measured. 79

Chapter 4. Geometrical Calibration of Redundant PKMs

Table 4.17: Compatibility test for the parameters of the calibration with the laser-tracker, platform angle measurements, and model 29. Number of ∆q2 ∆q3 current parameters (mm) (mm) (%) 4 0.039 0.038 25 5 0.044 0.035 26 6 0.115 0.039 48 7 0.129 0.032 48 8 0.133 0.028 51 9 0.091 0.031 37 10 0.101 0.041 42 11 0.092 0.042 43 12 0.092 0.007 29 13 0.106 0.009 34 14 0.090 0.033 38 15 0.110 0.017 39 radius deviation (mm) circularity (mm)

deviation (mm)

0.080

0.060

0.040

0.020

4

5

6

7 8 9 10 number of parameters

11

12

13

Figure 4.27: Calibration results with the laser-tracker, platform angle measurements and model 29. The shape of the circular test for the best parameter set with this method, cf. Fig. C.5, is also the same as without measurements of the platform angle. The obtained radius deviation is 37 µm and the circularity is 110 µm. That confirms the similitude of the results. 4.4.5.2

Grid-Encoder

Although the results of the compatibility test presented in Table 4.18 are very similar to those of the calibration with the grid-encoder without the platform angle measurements, the TCP accuracy results presented in Fig. 4.28 are far worse. Because of lack of knowledge on the behavior of redundant mechanisms regarding the calibration, no special care was taken of the internal constraints during the measurements. The fact that the redundancy current compensation was active during the measurements with the laser-interferometer and not during the measurements with the grid-encoder is the most plausible explanation for the deterioration of the calibration results. The importance of this last result must be emphasized. One of the greatest advantages of actuation redundancy is the ability to reduce the effects of backlashes 80

Chapter 4. Geometrical Calibration of Redundant PKMs

by applying an internal constraint which reduces the changes of contact surfaces in the joints. However, a large constraint would make the machine economically inapplicable because of the energy consumption. Thus, in order to obtain a correct accuracy and a reasonable internal constraints, a special care should be taken to drive the mechanism with the desired internal constraint during the whole of the measurement phase. The desired internal constraint should be sufficiently high to ensure that the constraint direction is constant in spite of the fact that the parameters, the model and the mechanics are not perfect. Nevertheless, it should be as low as possible. Thus, it is a priori a wrong idea to identify the parameters from a measurement without constraints, i.e. with inactive redundant actuators and apply the difference in the actuator values in order to create the internal constraint. It is more appropriate to apply a constant constraint by driving the redundant actuators with a force control. This way, the internal constraint is provoked by the parameter combination. Table 4.18: Compatibility test for the parameters of the calibration with the grid-encoder completed by platform angle measurements with model 29. Number of ∆q2 ∆q3 current parameters (mm) (mm) (%) 4 0.262 0.253 54 5 0.259 0.289 57 6 0.251 0.268 55 7 0.262 0.276 57 8 0.260 0.279 57 9 0.284 0.293 62 10 0.284 0.291 61

radius deviation (mm) circularity (mm)

deviation (mm)

0.160

0.140

0.120

0.100

0.080

4

5

6 7 8 number of parameters

9

10

Figure 4.28: Calibration results with the grid-encoder, platform angle measurements and model 29. A remarkable fact of the shape of the circular test for this calibration method, cf. Fig. C.6 in Annex, is that it is very similar to the shapes obtained from the calibration with the lasertracker. Although the accuracy results are not so good, they are 83 µm for the radius deviation and 218 µm for the circularity. This is probably due to the fact that the redundancy current conditions for the platform angle measurement were the same as for the measurements with the laser-tracker. They would influence the identification so strong that the shape of the identification would be transformed. Opposed to all other calibration methods, the augmentation 81

Chapter 4. Geometrical Calibration of Redundant PKMs

of the number of identified parameters is accompanied by a deterioration of the calibration results. This is a sign for the non-viability of the measurement data.

4.4.6

Self-Calibration

The measurements for the self-calibration method were taken by driving the machine along three lines along x (with a 5-mm raster for the line in the middle of the workspace and a 10-mm raster for the two lines on the workspace border) and one line along y (80-mm raster). No outliers needed to be excluded from the calibration process. 4.4.6.1

Calibrations with Model 24

Table 4.19 and Fig. 4.29 show the results of the measurements on the mechanism with the parameters identified with the self-calibration method and model 24. The results of the compatibility test are very similar to those of the calibration with the laser-tracker and model 24. Whereas the radius deviation has the same order of magnitude (around 80 µm) for calibrations with up to six parameters, the improvement for calibration with a greater number of identified parameters is much clearer for the self-calibration (10 µm) than for the laser-tracker (40 µm). It should be noted what appears to be a large step for the redundancy current between the calibration processes with six and seven parameters is in fact due to the scale of the right-hand axis of the plot. The fact that the circularity deteriorates when more than seven parameters are identified, whereas the radius deviation is practically constant, may be explained by the increasing condition number. The ratio between the parameters diverges from the ratio on the real machine, because of parameter dependencies. The circular shape is therefore deformed. Table 4.19: Compatibility test for the parameters of the self-calibration with model 24. Number of ∆q2 ∆q3 current parameters (mm) (mm) (%) 4 0.137 0.043 26 5 0.144 0.073 21 6 0.143 0.072 22 7 0.158 0.052 32 8 0.173 0.053 30 9 0.160 0.055 31 Fig. C.7 shows the circular test for the parameter set with seven parameters. The obtained radius deviation is 9 µm for a circularity of 92 µm. 4.4.6.2

Calibrations with Model 29

The results of the self-calibration with model 29 (Table 4.20, Fig. 4.30) are similar to those with model 24. For the calibration with ten and eleven parameters, a degradation of the circular test results could be observed, although this is not the case for the results of the compatibility test. The large condition number associated with calibrations with a greater number of parameters account for this accuracy deterioration. The parameter set with eight parameters yielded to a radius deviation of 9 µm and a circularity of 69 µm for the circular test shown in Fig. C.8. 82

Chapter 4. Geometrical Calibration of Redundant PKMs

redundancy current (%)

0.700

39

0.560

37.6

0.420

36.2

0.280

34.8

0.140

33.4

0.000

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5

6 7 number of parameters

8

9

current (%)

deviation (mm)

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32

Figure 4.29: Self-calibration results with model 24.

redundancy current (%)

0.700

60

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44

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36

0.140

28

0.000

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10

current (%)

deviation (mm)

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20 11

Figure 4.30: Self-calibration results with model 29.

83

Chapter 4. Geometrical Calibration of Redundant PKMs

Table 4.20: Compatibility test for the parameters of the self-calibration with model 29. Number of ∆q2 ∆q3 current parameters (mm) (mm) (%) 4 0.096 0.094 20 5 0.109 0.117 24 6 0.143 0.072 21 7 0.133 0.076 23 8 0.170 0.050 30 9 0.178 0.052 30 10 0.165 0.060 32 11 0.169 0.063 32

4.5

Synthesis and Conclusion

Table 4.21 sums up the results of all calibrations with geometrical models. LT 29+ and KGM 29+ correspond to the methods using platform angle measurements. Table 4.21: Summary of geometrical calibration results. calibration radius devi- circularity redundancy method ation (µm) (µm) current (%) LT 24 38 105 22 LT 29 38 116 15 KGM 24 10 39 56 KGM 29 8 51 55 DBB 24 n/a n/a 73 DBB 29 n/a n/a 68 LI 24 n/a n/a 63 LI 29 n/a n/a 63 LT 29+ 37 110 38 KGM 29+ 83 218 58 SELF 24 9 92 27 SELF 29 9 69 27 Although it can be found in the literature that a greater number of identified parameters improves the calibration, the difference between the results of the calibration with model 24 and model 29 are small for the Scissors-Kinematics. This may be explained by the fact that the parallel-holder was mechanically adjusted before the calibration. It could be seen, however, that for one description model, the accuracy after calibration improves when the number of identified parameters increases. This result shows that the quality of the measurements is of higher importance as the chosen modeling. Complexer modeling methods cannot compensate for measurement errors better than simpler ones. The particularities of the calibration of redundantly actuated were explained in this Chapter. Indeed, if the measurement is not full-pose, the calibration has to be carried out in two steps. In the first step, the parameters of the non-redundant subsystem can be identified whereas the parameters of the redundant subsystem(s) have to be identified either by creating virtual measurements with the help of the parameters identified by the first step. The re-use of the same measurement data is only possible if the redundant subsystems are kinematically viable. 84

Chapter 4. Geometrical Calibration of Redundant PKMs

The experimental results showed that the results of inverse kinematic calibrations are far better than those of forward kinematic ones. Very good accuracy results could be obtained with the grid-encoder measurements and the self-calibration, both methods have a very good accuracy and offer the possibility to measure two positions for each pose. The low accuracy of the results with the double ball-bar and the laser-interferometer are probably linked to the fact that the measurement systems are one-dimensional. The fact that the laser-interferometer can measure along both x and y directions, not simultaneously though, did not help improving the calibration results, because, firstly, the parameters are almost insensitive to relative measurements along y and, secondly, the measurements are not full-pose because the measurement values must be reset for each measured line along x and y. For these two measurement methods, the parameters related to rod 2 had to be identified by creating virtual measurements from the rest of parameters identified during a first calibration stage. The fact that the laser-tracker gave results with a lower accuracy and a lower constraint as compared to the grid-encoder is linked to the measurement conditions. For the laser-tracker, a redundant actuator position compensation was active in order to reduce the redundancy current. It was inactive during the measurements with the grid-encoder. This was confirmed by the platform angle measurements which negatively influenced the calibration with the grid-encoder, because the internal constraint conditions were different for the position measurements and for the angle measurements. This is an important fact that shows that the internal constraints also plays an important role in the calibration of redundantly actuated parallel mechanisms, so that it is difficult to calibrate such machines and guarantee a given internal constraint by using geometrical models.

85

Chapter 5. Static Elastic Deformation Models of Redundant PKMs

Chapter 5 Static Elastic Deformation Models of Redundantly Actuated Parallel Kinematic Mechanisms As shown in the preceding Chapter, the accuracy of calibration with geometrical models is limited by the fact that the constraints in redundant mechanisms have an influence on the calibration results but the constraints are not considered. Modeling methods have to be developed for the forward kinematic problem in order to accurately calculate the TCP position by taking into account the conjugate action of all actuators. One of the advantages of redundantly actuated PKMs is that a preload constraint can be used to reduce the effects of the backlash in the mechanism and, thus, improve its accuracy. The method for reducing the backlash is to guarantee an internal preload constraint that avoids great and sudden changes of the contact surfaces in the joints. The ideal solution is to have the constraint direction constant in the joint local reference frame. However, this constraint should remain minimal in order to avoid the acting of actuators against each other. One of the methods for the application of an internal preload constraint is to use force control on the redundant actuators or on the whole system. In [M¨uller 05], the author proposes a control method for the application of an internal preload which is based on such control. However, for a machine tool, a position control on all actuators is necessary for maintaining the accuracy and the dynamical properties. The difficulty is to deal with errors in the geometrical parameters used in the machine control, which are responsible for excessive constraint in the case of position control. The calibration plays an important role in this constraint as could be already seen from the results of Chapter 4. Three approaches are proposed: ˆ In the first approach, the calibration is carried out with measurements without constraints by using the redundant actuators as followers, i.e. without controller. The measurement can be external on the TCP or internal as self-calibration. The geometrical parameters are identified. During the normal use of the machine, an offset must be added to the redundant actuator positions calculated through the inverse geometrical model with the identified parameters in order to create the preload constraint. The induced TCP displacements must be compensated. A model to calculate these displacements is then needed. The use of the redundant actuators as followers may, however, not be possible on all mechanisms. ˆ The principle of the second approach is that, during the measurements for the calibration, a force control is used on the redundant actuators so that the internal preload can be

86

Chapter 5. Static Elastic Deformation Models of Redundant PKMs

directly controlled. The measurements can also occur externally or internally. During the use of the machine, the identified parameters give the actuator positions which are applied without modification on all actuators. If the identification succeeded, the preload constraint that was present during the measurements should be created, during the normal use, by the parameter combination itself. ˆ In the third approach, the calibration occurs with a position control on all actuators. The measurements are done on the TCP while an unknown constraint exists in the mechanism. The geometrical parameters are then identified using an elasto-geometrical model which takes into account the element elastic deformations due to the preload constraint. An offset must then be added to the redundant actuator positions to create the constraint and the induced TCP displacement must be corrected with the same elasto-geometrical model.

The second approach could not be tested on the Scissors-Kinematics for the reason that the force control on the redundant actuators could not be implemented. For the other two approaches, an elasto-geometrical modeling method is needed to deal with elastic deformations. This Chapter presents some solutions for the precise calculation of the TCP position of redundantly actuated PKMs. The first two Sections briefly present two existing solutions: the statistical method and the least-squares minimization method. In the third Section, the lumped model method is adapted to redundant PKMs. The next Section presents the finite-element modeling for non-redundant PKMs. In the last Sections, several methods are presented for the adaptation of the finite-element method to PKMs with actuation redundancy. These methods are as follows: the actuator-displacements method, the structure-division method, the joint-internal-forces method, and the thermo-mechanical method. The various methods are explained with the illustrative help of the Redundant Triglide shown in Fig. 5.1. The Redundant Triglide is a simplistic fictive planar mechanism. The presented methods are, however, valid for all PKMs with one or more actuated redundant branches. y

l3

l1 l2 h1

h2

h3

x

0

Figure 5.1: The Redundant Triglide. The Redundant Triglide is a 2-degree-of-freedom PR-PRR-PRR mechanism. The kinematic structure is shown in Fig. 5.2. The three actuated carriers q1 , q2 and q3 can move along a common rail. The TCP is linked to these carriers by three rods with length l1 , l2 and l3 and five revolute joints. The moving platform of the Redundant Triglide is considered to be a point.

87

P

R

P

R

R

P

R

R

platform

base

Chapter 5. Static Elastic Deformation Models of Redundant PKMs

P

actuated prismatic joint

R

passive revolute joint

Figure 5.2: Kinematic structure of the Redundant Triglide.

5.1

Statistical Method

In [Marquet 03] the authors use a statistical approach for the calculation of the TCP position of redundantly actuated mechanisms. The TCP position is the average of the values given by the forward kinematic model of each non-redundant subsystem, cf. Fig. 5.3. In order to model the mechanism behavior in a better way, the authors suggest using a weighted average, where the weight coefficients are obtained from the inverse of the condition number of the Jacobian of each subsystem. The Jacobian matrix considered here is the matrix linking variations on the TCP and variations on the actuator positions. y

l1

l2

q1

q3

x

0

y

l3

l2

+

q2

q1

TCP2

y

TCP1

0

l1

+ q2

q1

q3

x

0

TCP3

l3 q2

q3

x

Figure 5.3: Statistical method. Let us define hi the implicit function that gives the relationship between the platform position X, the machine parameters P and actuator i, i = 1, . . . , a with a the total number of actuators. We have hi (X, qi , P ) = 0 with ideal parameters. By differentiating these equations, we obtain Jx dX + Jq dQ + Jp dP = 0   ∂h1 ∂h1  ∂x ∂y   . ..  . with Jx =  . .   ,  ∂ha ∂ha  ∂x ∂h1  ∂q1  . Jq =   ..  ∂ha ∂q  1 ∂h1  ∂p1  . Jp =   ..  ∂ha ∂p1 

∂y ∂h1 ∂q2 .. . ... ∂h1 ∂p2 ... ...

(5.1)



...   , and  ∂ha  ∂qa  ...   .  ∂ha  ∂pn 88

Chapter 5. Static Elastic Deformation Models of Redundant PKMs

The actuator positions are fixed. We have then dQ = 0 and dX = (JTx Jx )−1 JTx Jp dP = Jred dP .

(5.2)

Jred is the Jacobian matrix of the parameters for the redundant forward kinematics model. If hi cannot be obtained analytically, the TCP error due to the parameter errors can be calculated by using an inverse and a forward kinematic transformations successively. For a given TCP position X, the error due to an error dP on the geometrical parameters is dX = fkm(igm(X, P + dP ), P ) − X,

(5.3)

where fkm is a forward kinematics model where all actuator positions are being taken into account and igm is one of the inverse geometrical models presented in Section 4.1. It can also be defined as dX = fkm(igm(X, P ), P + dP ) − X.

(5.4)

The advantages of this method is that, as is shown, the calculation of TCP displacements due to parameter errors can be calculated analytically. The problem of using of this method is that the condition number of the Jacobian of a subsystem on a singular position is infinite. Given that one of the objectives of actuation redundancy is to maintain the overall manipulability of the mechanism including the case when one of the non-redundant subsystems is at a singular position, such positions are not necessarily excluded from the workspace.

5.2

Least-Squares Minimization Method

The least-square minimization method is also evoked in [Marquet 03]. It consists in minimizing the so-called loop-closure expressions of all actuators simultaneously using a least-squares optimization method, for example, the Gauß-Newton method. The residuals ri of the b loopclosure equations are given by ri = ti (X, Q, P ) for i = 1, . . . , b,

(5.5)

where ti is the loop-closure equation for loop i. For non-redundant mechanisms we always have ri = 0, ∀i. With redundant mechanisms with arbitrary parameters, there can be one of more non-null loop-closure equations. The optimization occurs on the TCP position so that the least-squares optimization criterion c is minimal: b

1X 2 c= r . 2 i=1 i

(5.6)

The advantage of this method compared to the previous one is that the Jacobian matrices of each subsystem are not needed. The Jacobian used for the least-squares optimization method takes into account all subsystems simultaneously. The method is then numerically more stable. Let us note that the Jacobian matrix of the parameters, needed for the calibration, is obtained advantageously analytically, as in Eq. (5.2). The drawback of this method is that, as the statistical calculation method, it does not take into account the stiffness of the elements.

89

Chapter 5. Static Elastic Deformation Models of Redundant PKMs

5.3

Lumped Elastic Models

In this method the elements’ stiffness is taken into account. An elastic element is modeled by a rigid element associated with one or more virtual compliant joints whose translational or rotational displacements correspond to the deformation of the element in one direction. Fig. 5.4 illustrates how the bending and tension effects are described by compliant revolute or prismatic joints.

h

l0

l1

(a)Real elements.

revolute joint

prismatic joint

h

l0

l1

(b)Rigid elements and compliant joints.

Figure 5.4: Modeling of elastic elements subject to bending or tension. The method was mainly used for non-redundant mechanisms, e.g. in [Ecorchard 05], [Majou 04b]. The method was also used in [Jeong 04] for a PKM with an actuated redundant branch for the modeling of actuator stiffness. However, it was not found in the literature for the calculation of the TCP position of redundantly actuated PKMs. The following explains how the method can be modified to model the stiffness of the elements in the loops of redundant PKMs as well and, thus, to calculate the TCP position from arbitrary actuator positions and parameters. 1. A TCP position is calculated by using the non-redundant subsystem and the given set of parameters, P by Xnr = fgm nr (Qnr , P ),

(5.7)

where Qnr is the set of actuator positions of the non-redundant part of the mechanism. fgm nr is a forward geometrical model, which gives the TCP co-ordinates from the nonredundant actuator positions and the geometrical parameters. The redundant part of the mechanism as well as the corresponding parameters are ignored here. 2. The virtual compliant joint values of the redundant part are the expression of the parameter incompatibility and are calculated with Xnr , Q and P . Per redundant link, there must be at least one virtual joint able to represent the effects of a variation on each parameter. 3. The virtual compliant joints of the redundant subsystem and their associated stiffness give the efforts that are applied on the TCP, FTCP , cf. Fig. 5.5. ∆l3 , in Fig. 5.5, is the initial value of the compliant virtual joint. 4. The effort on the TCP and the stiffness of the compliant joints give the displacements δli in the virtual joints when all rods of the mechanism are considered. 90

Chapter 5. Static Elastic Deformation Models of Redundant PKMs y

virtual joint

l3

¢l3

F TCP

l1

®3

l2

q2

q1

q3 x

0

Figure 5.5: Method of the lumped elastic model - Steps 1-3 5. The parameters of the non-redundant part are corrected according to the values of the virtual joints of this subsystem, as shown in Fig. 5.6. 6. The TCP position is calculated by using the non-redundant subsystem and the corrected parameters, as in Eq. (5.7). y

l1

±l 1

l1 +±

l3 + ± l

l2

+

¢l3 ¡ ±

l3

±l

±l

2

2

q1

3

q2

0

q3 x

Figure 5.6: Method of the lumped elastic model - Steps 4-6 The advantage of this method compared to geometrical methods is that it takes into account the elasticity behavior of elements inside a loop, rather than the sole stiffness of actuators. However, the determination of the relevant virtual links that need to be included into the model must be preliminarily done using a more complex model such as a finite-element model. Moreover, the relevance of the elastic deformations may be position dependent. The model may need to be extended so as to consider the relevant deformations for all the positions in the workspace and thus may become more complex.

5.4

Finite-Element Modeling

A Finite-Element (FE) modeling with beam elements can be used to describe the elastic behavior of redundant parallel structures. The method takes into account all the element elastic deformations. The modeling has been developed in [Ecorchard 06] and [Deblaise 06a]. The method is first described for non-redundant parallel mechanisms and then adapted to redundant ones. However, the illustration will be done directly with the Redundant Triglide. The elastic model was designed to be as simple as possible in order to reach a very short calculation time. Since the Scissors-Kinematics is a two-dimensional mechanism, only planar elements are used. However, the proposed modeling is described for spatial mechanisms. The FE method is organized as follows: 1. Individual determination of the stiffness matrices of all rods and joints of the mechanism individually in the local beam or joint frame. 91

Chapter 5. Static Elastic Deformation Models of Redundant PKMs

2. Expression of the stiffness matrices into the global reference frame and assembly of the stiffness contribution of all elements to build the global stiffness matrix. 3. Resolution of the displacement of all nodes and in particular of the node associated with the TCP. The FE model of the Redundant Triglide with the node numbering is shown in Fig. 5.7. The number of nodes for finite-element models is noted s. We have s = 9 for the modeling of the Redundant Triglide. y 1

ideal joint

7

flexible joint

4

8 9

2 5 3

6

0

x

Figure 5.7: FE model of the Redundant Triglide.

5.4.1

Rod Modeling

Rods are modeled with 2-node beams. For each beam, a local reference frame