TIP TRAJECTORY TRACKING OF FLEXIBLE-JOINT MANIPULATORS

Transcription

1 TIP TRAJECTORY TRACKING OF FLEXIBLE-JOINT MANIPULATORS A Thesis Subitted to the College o Graduate Studies and Research In Partial Fulillent o the Requireents For the Degree o Doctor o Philosophy In the Departent o Mechanical Engineering University o Saskatchewan Saskatoon By HAMID SALMASI Copyright Haid Salasi, Deceber, 009. All rights reserved.

2 PERMISSION TO USE In presenting this thesis in partial ulilent o the requireents or a postgraduate degree ro the University o Saskatchewan, I agree that the Libraries o this University ay ake it reely available or inspection. I urther agree that perission or copying o this thesis in any anner, in whole or in part, or scholarly purposes ay be granted by the proessor or proessors who supervised y thesis work or, in their absence, by the Head o the Departent or the Dean o the College in which y thesis work was done. It is understood that any copying or publication or use o this thesis or parts thereo or inancial gain shall not be allowed without y written perission. It is also understood that due recognition shall be given to e and to the University o Saskatchewan in any scholarly use which ay be ade o any aterial in y thesis. Requests or perission to copy or to ake other use o aterial in this thesis in whole or part should be addressed to: Head o the Departent o Mechanical Engineering College o Engineering, University o Saskatchewan Saskatoon, Saskatchewan, Canada S7N L5 i

3 ABSTRACT In ost robot applications, the control o the anipulator s end-eector along a speciied desired trajectory is the ain concern. In these applications, the end-eector (tip) o the anipulator is required to ollow a given trajectory. Several ethods have been so ar proposed or the otion control o robot anipulators. However, ost o these control ethods ignore either joint riction or joint elasticity which can be caused by the transission systes (e.g. belts and gearboxes). This study ais at developent o a coprehensive control strategy or the tiptrajectory tracking o lexible-joint robot anipulators. While the proposed control strategy takes into account the eect o the riction and the elasticity in the joints, it also provides a highly accurate otion or the anipulator s end-eector. During this study several approaches have been developed, ipleented and veriied experientally/nuerically or the tip trajectory tracking o robot anipulators. To copensate or the elasticity o the joints two ethods have been proposed; they are a coposite controller whose design is based on the singular perturbation theory and integral aniold concept, and a swar controller which is a novel biologically-inspired controller and its concept is inspired by the oveent o real biological systes such as locks o birds and schools o ishes. To copensate or the riction in the joints two new approaches have been also introduced. They are a coposite copensation strategy which consists o the non-linear dynaic LuGre odel and a Proportional-Derivative (PD) copensator, and a novel riction copensation ethod whose design is based on the Work-Energy principle. Each o these proposed controllers has soe ii

4 advantages and drawbacks, and hence, depending on the application o the robot anipulator, they can be eployed. For instance, the Work-Energy ethod has a sipler or than the LuGre-PD copensator and can be easily ipleented in industrial applications, yet it provides less accuracy in riction copensation. In addition to design and develop new controllers or lexible-joint anipulators, another contribution o this work lays in the experiental veriication o the proposed control strategies. For this purpose, experiental setups o a tworigid-link lexible-joint and a single-rigid-link lexible-joint anipulators have been eployed. The proposed controllers have been experientally tested or dierent trajectories, velocities and several lexibilities o the joints. This ensures that the controllers are able to peror eectively at dierent trajectories and speeds. Besides developing control strategies or the lexible-joint anipulators, dynaic odeling and vibration suppression o lexible-link anipulators are other parts o this study. To derive dynaic equations or the lexible-link lexible-joint anipulators, the Lagrange ethod is used. The siulation results ro Lagrange ethod are then conired by the inite eleent analysis (FEA) or dierent trajectories. To suppress the vibration o lexible anipulators during the anoeuvre, a collocated sensor-actuator is utilized, and a proportional control ethod is eployed to adjust the voltage applied to the piezoelectric actuator. Based on the controllability o the states and using FEA, the optiu location o the piezoelectric along the anipulator is ound. The eect o the controller s gain and the delay between the input and output o the controller are also analyzed through a stability analysis. iii

5 ACKNOWLEDGMENTS This work would not have been possible without assistance and support o y supervisors, Proessors Reza Fotouhi and Peter N. Nikioruk, under whose supervision I accoplished y Ph.D. studies. I also acknowledge invaluable guidance o the advisory coittee ebers Proessors Richard T. Burton, Daniel X. B. Chen and Leon D. Wegner. Moreover, I would like to thank Mohaad Vakil who kindly helped e through y research. Also, a thankul acknowledgent is ade to ACTA press and IFAC or perission to reprint y previously published aterials. Financial support o this work provided by a University o Saskatchewan Graduate Scholarship and a NSERC Discovery Grant and is grateully acknowledged. DEDICATIONS This thesis is dedicated to y ather Reza Salasi, y other Fateeh Y. Panah and y dear wie Haleh Salasi. iv

6 COPYRIGHT PERMISSION OF THE PAPERS At the tie o the publication o this thesis, a paper A biologically-inspired controller or tip trajectory tracking o lexible-joint anipulators (Chapter 4), is under review. Other papers are accepted/published, and perissions to reprint these papers have been sought ro the publisher through eail or ro the website o the publishers. v

7 Perission to include paper Tip trajectory tracking o lexible-joint anipulators driven by haronic drives authored by H. Salasi, R. Fotouhi, and P. N. Nikioruk, and was originally published in IASTED International Journal o Robotics and Autoation, Vol. 4, No., 009. vi

8 Perission to include paper A anoeuvre control strategy or lexible joint anipulators with joint dry riction authored by H. Salasi, R. Fotouhi, and P. N. Nikioruk, and to be published in the Cabridge journal o Robotica, (doi:0.07/s ). vii

9 Perission to reprint the paper Vibration control o a lexible link anipulator using sart structure authored by H. Salasi, R. Fotouhi, and P. N. Nikioruk, and originally was published in the Proceedings o the International Federation o Autoatic Control (IFAC) World Congress, Seoul, South Korea, p The ollowing stateent has been copied ro the website o Elsevier Ltd., the oicial, sole publisher o IFAC publications, on Septeber 3, 009. As an author, you retain rights or a large nuber o author uses, including use by your eploying institute or copany. These rights are retained and peritted without the need to obtain speciic perission ro Elsevier. These include: the right to include the article in ull or in part in a thesis or dissertation (provided that this is not to be published coercially); Perission to reprint paper Dynaic odeling o a anipulator with lexible links and lexible joints authored by H. Salasi, R. Fotouhi, and P. N. Nikioruk, and published in the Proceedings o st Canadian Congress o Applied Mechanics, Ryerson University, Toronto, ON, Canada, June 3-7, 007, (CD Ro). The ollowing stateent has been copied ro the irst page o the proceedings. Reprints ro this publication ay be ade provided credit is given to the authors and the reerence ade to the Proceedings o st Congress o Applied Mechanics, Ryerson University, Toronto, Ontario, Canada, 007 (CANCAM 007). (accessed Septeber 9, 009). (accessed Septeber, 009). viii

10 TABLE OF CONTENTS ix

11 x

12 TABLE OF FIGURES Chapter : Fig.. Coponents o a two-link robot anipulator... 3 Fig.. (a) Experiental odule o the single rigid-link lexible-joint anipulator available in the Robotics Laboratory o the University o Saskatchewan, (b) scheatic o the top-view o the experiental setup showing the angles o the joint and the link (subscripts s stands or shoulder)... 6 Fig. 3. (a) Experiental odule o the two-rigid-link lexible-joint anipulator available in the Robotics Laboratory o the University o Saskatchewan, (b) scheatic o the top-view o the experiental setup showing the angles o the joints and the links (subscripts e and s stand or elbow and shoulder, respectively)... 7 Fig. 4. Experiental results obtained or steady state conditions and estiation unction obtained based on the LuGre odel or (a) shoulder joint otor, (b) elbow joint otor... 0 Fig. 5. Model based riction copensation strategy used or experiental setup o two-rigid-link lexiblejoint anipulator... Fig. 6. Experiental veriication o the links rotations or [ q, q ] = [.0 rad,.0 rad] and [ K, K ] = [7.93 N / rad,.94 N / rad] using the coposite controller, (a) shoulder link (b) s e elbow link trajectory.... Fig. 7. (a) Scheatic o a swar oveent; each particle is shown by a solid circle (b) an actual swar o birds; the centre o the swar is denoted by c Fig. 8. Block diagra o the swar torque controller or the tip-trajectory tracking o lexible-joint anipulators... 7 Fig. 9. Scheatic o lexible-link anipulator with piezoelectric actuator... Fig. 0. Delection o the anipulator when (a) the piezoelectric actuator was inactive, (b) the piezoelectric actuator was activated and located at.... Fig.. Scheatic o a two lexible-link lexible-joint anipulator (TFLFJ)... 3 s e xi

13 Fig.. Tip delection o the anipulator... 4 Chapter : Figure. Control strategy designed or tip trajectory tracking o two-rigid-link lexible-joint anipulator (TRLFJ) Figure. (a) Experiental odule o two-rigid-link lexible-joint anipulator available in the Robotics Laboratory o the University o Saskatchewan, (b) the scheatic o the top-view o the experiental setup showing the angles o the joints and the links Figure 3. Two suraces sliding on each other with relative velocity o q ; the asperities are odeled as elastic bristles attached to the rigid bodies Figure 4. Scheatic o the control procedure used to identiy the LuGre odel paraeters or experiental setup shown in ig Figure 5. Experiental results obtained or steady state conditions and estiation unction obtained based on the LuGre odel or (a) shoulder joint otor, (b) elbow joint otor Figure 6. (a) Fast state Z approaches the integral aniold ϕ due to the ast controller, and (b) slow state X is equal to the reduced-order odel, as long as the ast state is on the integral aniold Figure 7. Experiental veriication o the links rotations or [ q, shoulder, q, elbow ] = [.0 rad,.0 rad] using CSTT controller, (a) shoulder link trajectory (b) elbow link trajectory Figure 8. Experiental veriication o the links rotations or [ q, shoulder, q, elbow ] = [.5 rad,.0 rad ] using CSTT controller, (a) shoulder link trajectory (b) elbow link trajectory Figure 9. Experiental veriication o the links rotations or [ q, shoulder, q, elbow ] 3 = [.0 rad,.5 rad ] using CSTT controller, (a) shoulder link trajectory (b) elbow link trajectory Figure 0. Experiental veriication o the links rotations or [ q, shoulder, q, elbow ] 4 = [.5 rad,.5 rad ] using CSTT controller, (a) shoulder link trajectory (b) elbow link trajectory... 5 xii

14 Figure. Experiental results: noralized tracking error or shoulder and elbow links or inal rotations o [ q, shoulder, q, elbow ] = [.0 rad,.0 rad ], using (a) CSTT controller, (b) rigid controller... 5 Figure. Experiental results: noralized tracking error or shoulder and elbow links or inal rotations o [ q, shoulder, q, elbow ] = [.5 rad,.0 rad ], using (a) CSTT controller, (b) rigid controller... 5 Figure 3. Experiental results: noralized tracking error or shoulder and elbow links or inal rotations o [ q, shoulder, q, elbow ] 3 = [.0 rad,.5 rad ], using (a) CSTT controller, (b) rigid controller... 5 Figure 4. Experiental results: noralized tracking error or shoulder and elbow links or inal rotations o [ q, shoulder, q, elbow ] 4 = [.5 rad,.5 rad ], using (a) CSTT controller, (b) rigid controller... 5 Chapter 3: Figure. Control strategy designed or tip trajectory o FJM Figure. Two suraces are sliding on each other; the asperities are odeled as bristles attached to the suraces... 6 Figure 3. Scheatic o the control procedure used to identiy the LuGre odel paraeters Figure 4. Coparison o experiental results with riction torque obtained in Eq. (0) based on the LuGre odel Figure 5. Experiental odule single rigid-link lexible-joint anipulator used Figure 6. The otor velocity response corresponding to the applied triangle torque to obtain experientally a in Eq. (9) Figure 7. The rate o the energy o the controller torque τc applied to the lexible joint anipulator to obtain W ro Eq. () or anoeuvre = 0. rad during two seconds I 5 Figure 8. The rate o the dissipated energy by viscous daping to obtain W d ro Eq. () or anoeuvre = 0. rad during two seconds xiii

15 Figure 9. The rate o the dissipated energy by dry riction to obtain Wd ro Eq. (3) or anoeuvre = 0. 5 rad during two seconds Figure 0. Coparing siulation and experiental results o anipulator anoeuvre using coposite LG WE() controller and dierent riction copensation torques, (a) τ in Eq. (86), (b) τ in Eq. (83) (link angle q L and the irst desired trajectory () q d )... 8 LG Figure. Applied torque to the lexible-joint anipulator o otions in Fig. 0 (a) τ c s + τ, (b), τ + τ... 8 c, s WE() Figure. Coparison values o error nors RMSE and SSE or dierent riction copensating torques or the coposite controller or a SRLFJ experiental setup Figure 3. Coparing siulation and experiental results o the lexible joint anipulator anoeuvre using coposite controller with link angle q L or (a) desired trajectory q, (b) desired ( ) d trajectory ( 3 ) q d, (c) desired trajectory q ( 4 ) d Figure 4. (a) Rigid controller torque, τ c, 0, based on the assuption o rigid joint, (b) coposite controller torque τ c, s or a SRLFJ Figure 5. Coparing siulation and experiental results o anipulator anoeuvre with rigid controller torque (link angle q L and the desired trajectory () q d ) or a SRLFJ Figure 6. Tracking error, e = () q L q, between the actual link, L d q, and the desired trajectory o the link angle () q, or the rigid controller torque and coposite controller or a SRLFJ experiental d setup Figure AI.. Coparison results o inite-eleent analysis and theoretical ethod or a two-rigid-link lexible-joint anipulator, (a) shoulder link rotation, and (b) shoulder joint rotation... 9 Chapter 4: xiv

16 Fig.. Swar control strategy or tip trajectory tracking o a two-rigid-link lexible-joint anipulator.. 99 Fig.. (a) Experiental odule o a two-rigid-link lexible-joint anipulator available in the Robotics Laboratory o the University o Saskatchewan, (b) scheatic o the top-view o the experiental setup showing the angles o the joints and the links (subscripts e and s stand or elbow and shoulder, respectively) Fig. 3. (a) Scheatic o a swar oveent; each particle is shown by a solid circle (b) An actual swar o birds; the center o the swar is denoted by c Fig. 4. Block diagra o the swar torque controller or the tip-trajectory tracking o lexible-joint anipulators Fig. 5. Experiental results obtained or steady state conditions and estiation unction obtained based on the LuGre odel or a TRLFJ (a) shoulder joint otor, (b) elbow joint otor Fig. 6. Experiental results o swar control or two-rigid-link lexible-joint anipulator: Links rotations or ninth-order trajectory with inal rotation ; (a) shoulder link angle, (b) elbow link angle... Fig. 7. Experiental results o swar control or two-rigid-link lexible-joint anipulator: Links rotations or ninth-order trajectory with inal rotation ; (a) shoulder link angle, (b) elbow link angle... Fig. 8. Experiental results o swar control or two-rigid-link lexible-joint anipulator: Links rotations or ninth-order trajectory with inal rotation ; (a) shoulder link angle, (b) elbow link angle... Fig. 9. Experiental results o swar control or two-rigid-link lexible-joint anipulator: variation o the gains or the shoulder link; (a) proportional gains, (b) derivative gain, (c) integral gain, variation o the gains or the elbow link; (d) proportional gains, (e) derivative gain, () integral gain... 5 Fig. 0. Experiental results: noralized tracking error or swar and rigid controllers (a) shoulder link, (b) elbow link... 7 xv

17 Fig.. Experiental results: coparison o the end-eector error or swar and rigid controllers... 7 Fig.. Experiental results o swar control or the two-rigid-link lexible-joint anipulator: (a) shoulder link angle, (b) elbow link angle. ( )... 9 Fig. 3. Experiental results o swar control or the two-rigid-link lexible-joint anipulator: (a) shoulder link angle, (b) elbow link angle. ( )... 9 Fig. 4. Experiental results o swar control or the two-rigid-link lexible-joint anipulator: variation o the gains or the shoulder link; (a) proportional gains, (b) derivative gain, (c) integral gain; variation o the gains or the elbow link; (d) proportional gains, (e) derivative gain, () integral gain. (., initial gains or the shoulder and the elbow )... Fig. 5. Experiental results o swar control or the two-rigid-link lexible-joint anipulator: variation o the gains or the shoulder link; (a) proportional gains, (b) derivative gain, (c) integral gain; variation o the gains or the elbow link; (d) proportional gains, (e) derivative gain, () integral gain. (, initial gains or the shoulder, and the elbow )... 3 Chapter 5: Fig.. Model o a cantilever bea with piezoelectric actuator... 4 Fig.. A single lexible link robot anipulator including its tip ass and hub inertia Fig. 3. Block diagra o the controlling vibration o the anipulator Fig. 4. Controllability easure or dierent locations o the piezoelectric actuator Fig. 5. Nors o noralized tip delection o cantilever bea or optiu location o piezoelectric actuator Fig. 6.(a) Applied bang-bang controller torque, and (b) hub rotation Fig. 7. Case (Kc= 0): Single-link lexible anipulator, (a) tip delection w.r.t. shadow bea, and (b) FFT spectru o tip delection xvi

18 Fig. 8. Case 3 (Kc=4.00 ): Single-link lexible anipulator, (a) tip delection w.r.t. shadow bea, and (b) FFT spectru o tip delection Fig. 9. Single-link lexible anipulator, evaluation criteria or dierent gain values, (a) nors o vibration aplitudes, and (b) peak values o FFT spectrus at doinant requencies Chapter 6: Figure. Model o the i th lexible-link lexible-joint anipulator Figure. Two lexible-link lexible-joint anipulator (TFLFJ) Figure 3. Applied torques to shoulder and elbow otors... 5 Figure 4. Tip delection o the anipulator )... 5 Appendix II: Figure AII.. Rotation o the shoulder link o TRLFJ Figure AII.. Rotation o the shoulder otor o TRLFJ Figure AII.3. Rotation o the elbow link o TRLFJ Figure AII.4. Rotation o the elbow joint o TRLFJ Figure AII.7. Details o the subsyste shown in Fig. AII xvii

19 Chapter. Introduction

20 The irst chapter o thesis provides a brie background regarding lexible robot anipulators, the research objectives and a suary o the papers included in this thesis. Out o this study eleven research papers have been subitted/published in technical journals and the proceedings o international conerences []-[]. However, because o the need or brevity, only ive o these papers have been included in the ollowing Chapters -6.. Background The concept o robotics cae ro Robota, eaning ser labour. This concept was irst introduced by a science iction author, Karel Capek, in his play Rosu s Universal Robots where it was used to reer to autonoous capable o peroring a range o huan activities []. Ater about a century, robots are now utilized alost everywhere: oceans, actories, hospitals and even on Mars. An iportant class o robots are robot anipulators that are eployed in a variety o tasks, ro autootive industries to aerospace applications. As shown in Fig., a robot anipulator generally includes our ain coponents: i. Links: ideally links can be considered rigid or slow otion and sall interacting orces. However, or links with a sall width-length ratio, the lexibility ust be considered in both dynaic odeling and control o these anipulators. ii. Actuators: the otors, hydraulic or pneuatic pistons or other eleents that cause the links to ove are called actuators [3]. iii. Joints: transission systes such as belts and gearboxes which connect the links to the actuators. iv. End-eector: the end part o the anipulator that interacts with the environent. For instance it can be a gripper, welding tool, etc.

21 End-eector Links Joint Actuator Fig.. Coponents o a two-link robot anipulator The otion o the end-eector is controlled through the torques applied by the actuators to the links. Thereore, to ove the end-eector along a desired path, a controller is required to adjust the torques applied by the otors. In the case that the body o the robot (links and joints) is rigid, the design o the controller will not be very coplicated, and a large nuber o control strategies have been proposed to control the rigid robot anipulators. However, in reality lexibility exists in two coponents o the robot anipulators: links and joints. For anipulators that have links with a slender design and lightweight aterial, the lexibility o the links ust be considered in the dynaics and control procedure. The lexibility o the joints is produced by the otion transission/reduction systes such as the belts and haronic drives. Recent studies have shown that the joint lexibility has a signiicant eect on the perorance o industrial robot anipulators [4-8]. This eect becoes ore iportant in robot applications in which high-precision is required in the perorance o the anipulator [9-3]. For instance, any robots use a haronic drive as the transission syste to increase the applied torque to the links [4]. Haronic drives are ascinating to designers due to their high torque transissibility and sall size [5-6]. However, torsional elasticity in lexible parts 3

22 o haronic drives produces undesired oscillations during the perorance o the anipulator. Thereore, or robot anipulators driven by haronic drives, it is necessary to take into account the eect o the lexibility o the joints in the design procedure o the controller. On the other hand, considering lexibilities in the joints contributes to coplexity in the dynaics and control o the anipulator. The ain coplexity is due to the act that lexible-joint anipulators are under-actuated; that is, the nuber o actuators is lower than the degrees o reedo. Several controller strategies have been proposed so ar to overcoe the diiculty o under actuation in lexible-joint anipulators; however, iportant actors such as joint riction and inertia o the rotors are ignored in these control ethods. This deteriorates the perorance o the controller and aects the accuracy o the end-eector otion. Thereore, a ore coprehensive ethod which considers dierent aspects o dynaics and control o lexible-joint anipulator is needed especially in robots peroring high-speed high-precision tasks.. Research objectives This thesis ais to develop a coprehensive control strategy or the lexible-joint (elastic-joint) anipulators. The controller ust take into account the lexibility and riction in the joints, while it provides a precise otion or the anipulator s end-eector. Besides the control o lexible-joint anipulators, dynaic odeling o lexible-link lexible-joint and vibration suppression o lexible-link anipulators are other objectives o this research. Thereore, the objectives o this research can be suarized as ollows: - Precise tip trajectory tracking o lexible-joint anipulators during and at the end o anoeuvre (Chapters, 3 and 4) - Friction copensation in haronic drives (Chapters and 3) 4

23 3- Experiental validation o the proposed controller or lexible-joint anipulators (Chapters, 3 and 4) 4- Vibration suppression o lexible-link anipulators using piezoelectric actuators (Chapter 5) 5- Dynaic odeling o lexible-link lexible-joint anipulators (Chapter 6) A suary o the ethodologies used to accoplish these objectives is explained in Section.3. The details o these approaches can be ound in Chapters -6. Reark: Because o the anuscript nature o this thesis, equations, igures and reerences ight be repeated in the thesis..3 Suary o anuscripts The papers published or subitted by the candidate are cited in the reerence list []-[]. For brevity, ive o these papers have been selected and explained in this thesis. The irst three papers, given in Chapters -4, describe dierent control strategies used or the tip trajectory tracking o the lexible-joint anipulators. In these control strategies, the controller had two parts: the irst part was designed to copensate or the lexibility o the joints, and the second part was to copensate or the joints riction. Dierent cobinations o these controllers were exained and experientally veriied. Table shows the structure o the proposed controllers. The perorance o these controllers was tested using the experiental setups o a single-rigid-link lexible-joint (SRLFJ), shown in Fig., and a two-rigid-link lexible-joint anipulator (TRLFJ), shown in Fig. 3, available at the Robotics Laboratory o the University o Saskatchewan. 5

24 Table. Dierent control strategies eployed and experientally veriied anipulators Controller coponents Explained in or lexible-joint Coposite controller based on copensator the SPT * + LuGre-PD riction Chapter Coposite controller based on the SPT * + riction copensator Chapter 3 based on Work-Energy ethod Swar controller+ LuGre-PD riction copensator Chapter 4 * SPT: Singular Perturbation Theo ory Shoulder joint (a) (b) Fig.. (a) Experiental odule o the single rigid-link lexible-joint anipulator available in the Robotics Laboratory o the University o Saskatchewan, (b) scheatic o the top-view o the experiental setup showing the angles o the joint and the link (subscripts s stands or shoulder) 6

25 Elbow joint Shoulder joint (a) (b) Fig. 3. (a) Experiental odule o the two-rigid-link lexible-joint anipulator available in the Robotics Laboratory o the University o Saskatchewan, (b) scheatic o the top-view o the experiental setup showing the angles o the joints and the links (subscripts e and s stand or elbow and shoulder, respectively) To validate the developed control strategies, approxiately 0 experients were perored at dierent teperatures, huidities, lexibilities (elasticity) o the joints, trajectories and velocities. It is noteworthy that the experiental veriication was an indication that the proposed controllers can be ipleented in the applications. A list o the experients carried out using the experiental setup o the SRLFJ can be ound in Appendix I. In total, the results o 57 experients were recorded or the SRLFJ, soe o which are presented in Chapter 3. Three riction copensation strategies, the LuGre ethod, work-energy ethod, and LuGre-PD ethod, were exained at our dierent desired trajectories, and or low and high lexibilities o the joint ( K = 3.93N. and K = 0.33 N. ). Based on the experiental results, it was concluded that the best controller was a cobination o the coposite controller to copensate or the lexibility o the joints and the LuGre-PD ethod to copensate or the riction. This strategy was then eployed to control the TRLFJ shown in Fig. 3a. 7

26 For the TRLFJ, the experients were carried out using the coposite and swar controllers along with the LuGre-PD ethod. Soe o these experiental results are given in Chapters and 4, and also in [4]. The list o these experients is given in Appendix I. The paper included in Chapter 5 describes the application o a piezoelectric actuator to suppress the vibration o a lexible-link anipulator. This ethod has been used or vibration suppression o a cantilever bea and a single-link lexible link anipulator. The ith paper, given in Chapter 6, presents the dynaic odeling o lexible-link lexible-joint anipulators. Lagrange s ethod along with the assued ode shapes ethod is eployed to derive the dynaic equations. The siulation results are then veriied by inite eleent analysis. The ollowing sections give ore details o each paper presented in Chapters Tip trajectory tracking o lexible-joint anipulators driven by haronic drives In this paper, a new control strategy has been proposed or the tip trajectory tracking o lexible-joint anipulator []. The otion control o the end-eector o the anipulator is the ain concern in ost robot applications, such as spacecrat and autootive industries. In these applications, a desired trajectory is usually speciied in the task-space and the robot tip (endeector) is required to ollow this prescribed trajectory. A apping is irst deterined between the location o the robot end-eector and the rotations o the joints, and based on the desired trajectory o the end-eector, the desired rotations o the joints are coputed. A control approach is then designed such that the error between rotations o the joints and their desired trajectories are iniized. The proposed control strategy in [] had two ain advantages copared with other controllers developed or the tip trajectory tracking o robot anipulators. These advantages were: (a) taking into account the lexibility o the joints, (b) considering the eect o the joints 8

27 riction. The control strategy was a two-part controller. The irst part was a coposite controller to copensate or the lexibilities o the joints, and the second part was a riction copensation torque to copensate or the riction in the joints. The ollowing describes these two parts o the controller. The irst part o the controller was developed using singular perturbation theory along with the integral aniold concept. As discussed in Section., the ain proble in the trajectory tracking o lexible joint anipulators was that these systes were under-actuated. A serial lexible joint anipulator with n links had n degrees o reedo, n otor rotations and n link rotations, while it had only n actuators which were the otor torques. Eploying the singular perturbation theory allowed or the dividing o the dynaics o lexible-joint anipulator into two subsystes: one corresponding to the ast dynaics (high requency vibration due to lexibility) and the other to the slow dynaics (low-requency otion). For each o these subsystes, a controller was then designed. The second part o the proposed control strategy was to copensate or the riction in the joints. Two ethods were developed in this research or riction copensation. In the irst ethod, a linear eed-orward torque was designed using the Work-Energy principle []. This ethod was easy to ipleent and did not need any eedback ro the syste. However, it could not odel such riction characteristics as the Stribeck eect. To overcoe the disadvantages o the Work-Energy ethod, a ore general riction copensating strategy was proposed. This copensating torque was a cobination o a torque identiied based on the LuGre ethod 3 and a PD copensator. To identiy the riction in the joints based on the LuGre 3 The LuGre odel is a coprehensive riction odel developed based on the delection o the icroscopic asperities o the rubbing suraces. In this thesis, a sipliied or o this odel was obtained or the steady-state 9

28 ethod, 57 tests were carried out to deterine the riction at dierent velocities. By interpolating the data, the riction torques were obtained as unctions o the velocity. The experiental data and the estiated unctions or the shoulder and the elbow joint otors, are shown Fig. 4. (a) (b) Fig. 4. Experiental results obtained or steady state conditions and estiation unction obtained based on the LuGre odel or (a) shoulder joint otor, (b) elbow joint otor In ideal situations, the counterbalancing estiated joint riction torque shown in Fig. 4 was equal to the real riction o the joints, and, hence, the riction o the joints would be copensated copletely. However, practically there was always soe error in the estiation o joint riction torque, or exaple due to change in the oil teperature, load, huidity and actuator wear. To reove this error, a PD copensator along with the riction torque based on the LuGre ethod was used. Thereore, the riction copensating torque τ was τ = τ + K q q ) + K ( q q ) () L d ( d p d where τ was the joint riction torque identiied based on the LuGre ethod. Paraeters L and were the gains o the PD copensator, and and were, respectively, the desired conditions and eployed to estiate the riction o the joints. For brevity, the sipliied LuGre odel obtained or the steady-state conditions is reerred to as the LuGre odel in this thesis. 0

29 trajectory and its derivative. The irst ter o the riction copensating torque in () was identiied based on the LuGre odel. The scheatic o the entire riction copensation strategy which incorporated a PD copensator cobined with the LuGre riction odel is shown in Fig. 5. It was experientally ound that the riction was successully copensated or by eploying the proposed riction copensating, though it led to a steady-state error at the end o the anoeuvre. This steady-state error in the riction copensation was boundedd according to a stability analysis carried out based on the Lyapunov theory [4]. Also, it was shown that the range o the steady-state error was dependent on the riction paraeters and gain controllers. Fig. 5. Model based riction copensation strategy used or experiental setup o two-rigid-link lexible-joint anipulator The experiental setup used or veriication is shown in Fig. 3. As shown in this igure, the irst link (shoulder link) was coupled to the irst joint (shoulder joint) by eans o a lexible joint. At the end o the shoulder link a second haronic drive was connected to the elbow link via the lexible elbow joint. Also, in Fig. 3 angles and were rotations o the shoulder

30 and elbow otors, and angles and were rotations o the shoulder and elbow links, respectively. Both otors and both lexible joints were instruented with quadrature optical encoders whose resolutions were rad/count [7]. The experiental results indicated the superiority o the proposed control strategy copared to other typical controllers such as inverse dynaics torque control. For instance, as shown in Fig. 6, the tracking error between actual and desired rotations o the shoulder and elbow links was alost zero when the coposite controller was used or two dierent types o trajectory. The desired trajectories or the shoulder and elbow links were ninth order and third order polynoials, respectively. (a) (b) Fig. 6. Experiental veriication o the links rotations or [ q, q ] = [.0 rad,.0 rad] and [ K, K ] = [7.93 N / rad,.94 N / rad] using the coposite controller, (a) shoulder link (b) s e elbow link trajectory..3. A anoeuvre control strategy or lexible joint anipulators with joint dry riction Chapter 3 presents another control strategy or the tip trajectory tracking o lexible-joint anipulators []. The irst part o this control strategy was siilar to the coposite controller developed or the control o TRLFJ anipulator and explained in the previous section. The s e 4 Subscripts e and s stand or elbow and shoulder, respectively.

31 second part o the control strategy was a novel riction copensating torque whose design was based on the Work-Energy principle. Copared to the LuGre riction odel described in the previous section, the Work-Energy ethod had several advantages. The LuGre riction torque was obtained based on the assuption o steady-state conditions. Thereore, the pre-sliding regie in which stick-slip otion occurred at very low velocities was not odelled in the LuGre odel. Another disadvantage o the LuGre odel was that the stiction was assued to be the sae at the beginning and end o the otion o the anipulator, while in practice the stiction at the beginning o the anoeuvre was usually greater than that o at the end. In the LuGre odel, and ost riction odels developed to predict the joint riction, an experiental identiication procedure was necessary to deterine the riction odel paraeters. This identiication procedure involved several experients which needed to be carried out at dierent velocities and required a considerable aount o tie. However, the riction copensating torque based on the Work-Energy principle could be identiied in only two steps and, hence, the proposed odel was ore cost-eective in ters o the experiental tie required to identiy the paraeters o the controller. According to the Work-Energy principle, the work done during the otion o the lexible joint robot anipulator included the work done by the otor torque, W, the energies dissipated by the viscous daping, W, and the Coulob riction torque, W d d I. The total work done by the output controller was coposed o two parts, the work o the coposite controller torque τ c, which is W I c, and the work o the copensating torque o dry riction These works were written as WE τ d, which was W. I I t t t t WE τ dq = τ q dt = τ c + τ d ) q dt = τ c q dt W ( τ q dt = W + W = WE d I c I () 3

32 where and t were the inal rotation o the joint and the tie at which link reached its inal position, respectively. The energy dissipated by the viscous daping was W d = c q dq 0 = t 0 c q dt and the energy dissipated by the dry Coulob riction, τ c, was W d = τ c dq = 0 t 0 τ q c dt. The principle o work and energy or a general syste was K = + (3) W K where W expressed the total work done by the orces (conservative and non-conservative) on the syste ro the initial position to the inal position, and K and K were the total kinetic energy o the syste at the initial and inal positions, respectively. Since the velocity was zero at the initial and inal positions, K and K were zero and equation (3) reduced to W = 0, and thus W = W I + W I W d W d = c 0 (4) WE Fro (4), the work done by τ d was obtained as and thereore, W I W I + W c d + = W (5) d t 0 τ q dt = W + W + W (6) WE d The riction copensating torque, τ WE d I c d d, was then deterined ro (6). There were any candidates or the riction copensating torque unction which satisied (6). For convenience, the riction copensating torque was assued to be linear; that is Thus, equation (6) becae WE a + bt 0 < t < ( a / b) τ = (7) d 0 t > ( a / b) 4

33 t ( a + bt ) q dt = W I + W d + W (8) 0 c d Paraeter a was the torque required to overcoe the stiction and initiate the otion o the anipulator ro the rest position. This paraeter, reerred to as the breakaway orce, was obtained experientally by increasing the torque gradually, until the anipulator started to ove. Once the paraeter a was identiied, the paraeter b was coputed using equation (8) and the riction torque was deterined in (7). The proposed controller was veriied using the experiental setup o a single rigid-link lexible-joint anipulator shown in Fig.. According to the experiental results, the controller was successul in the tip trajectory tracking such that the tracking error during and at the end o the anoeuvre was ound to be sall. The experiental results are given in Chapter A biologically-inspired controller or tip trajectory tracking o lexible-joint anipulators In Chapter 4 a novel biologically-inspired controller is introduced or the tip trajectory tracking o the anipulators [3]. The ain concept o this controller was inspired by real biological systes. Populations such as swars o birds or schools o ish oten ove in coordinated but localized eorts toward a particular target [8]. In act, by siply adjusting the trajectory o each individual toward its own best location the swar inds its best position at each tie step [9]. For instance, when a swar o birds (or a school o ish) is oving towards a speciic destination, each bird (or ish) has to ollow a desired path so that the entire group oves towards the target. As shown in Fig. 7, the oveent o a lock o birds (or a school o ishes) can be odelled as a group o particles that oves ro an initial location (origin) towards a speciied destination. Each bird (or ish) is sybolically odelled by a particle, and the center o the swar is located at the center o the group. The swar s center can be easily 5

34 ound as the average o the positions o the particles in the space; that is and where denotes the Cartesian coordinate o the swar s center, denotes each particle s location and is the nuber o particles (e.g. or the odel o Fig. 7). The local error is deined as the error between the actual path o each particle and its desired path, and the global error shows the deviation o the group ro its desired path. Each particle has a eory and intelligence and, hence, is able to correct its position such that the local and global errors are iniized during the otion o the swar. Thereore, the swar oveent can be resebled by an optiization proble in which the objective unction constitutes the local and the global errors. The optiu oveent is achieved when the objective unction is iniized. c (a) Fig. 7. (a) Scheatic o a swar oveent; each particle is shown by a solid circle (b) an actual swar o birds; the centre o the swar is denoted by c. In this research study, the concept o swar oveent was applied to control the lexible-joint anipulators. For this purpose, each link was represented by a particle and the robot end-eector was represented by the swar s center. For the tip trajectory tracking, each link (particle) had to ove along its desired path so that the end-eector (swar s center) could ollow the swar desired trajectory. Thereore, the local error was deined as the dierence between the rotation o each link and its desired rotation and the global error was the dierence (b) 6

35 between the position o the end-eector and its desired position. The block-diagra o the swar control schee is shown in Fig. 8 which consisted o our principle parts: -Variable-gain PID controller: In this coponent, the applied torques to the otors were evaluated using the ollowing equation (9) where, and were, respectively, the derivative, proportional and integrativee gains o the PID controller, and the dierence between the rotation o each link and its desired rotation. The ain reason to choose the PID controller to copute torque was due to its siplicity and clear physical eaning. Especially, in industrial applications, siple controllers are preerred to coplex controllers, i the perorance enhanceent by eploying coplicated control ethods are not signiicant enough [3]. - Gain generator: This coponent produced proportional, derivative and integrative gains every tie-step. 3- Meory: Ater coputing gains in the gain generator, they were stored in a eory. These values were then eployed in the gain generator in the next tie-step to evaluate proportional, derivative and integrative gains. 4- Error evaluator: In this coponent, local errors (links errors) and global error (end-eector- error) were calculated based on the eedback signals o link rotations. Fig. 8. Block diagra o the swar torque controller or the tip-trajectory tracking o lexible- joint anipulators 7

36 The key advantage o this control strategy was that the positions o the joints and the robot end-eector were controlled siultaneously. In ost controllers developed or the lexiblejoint anipulators, the joints were controlled separately, and the error in the end-eector position was not taken into account in the control procedure. Thereore, it was possible that while the error in the joint rotations was kept sall, the error at the end-eector position becae very signiicant. This could happen or serial robot anipulators, because the end-eector position was a trigonoetric unction o the joints position, and not a linear unction o the rotations o the joints. To veriy the proposed approach, the experiental setup o the TRLFJ anipulator shown in Fig. 3 was eployed. The experiental results indicated that the controller was successul in the tip trajectory task or dierent inal rotations and dierent anipulation speeds. An experiental coparison between the swar controller and a typical controller used or rigid robots, inverse dynaic torque control, deonstrated the superiority o the proposed swar control strategy. Furtherore, to analyze the eects o the initial gains o the tie-variable-gain controller, a sensitivity analysis was experientally perored and it was shown that tracking and the steady-state errors did not change signiicantly or dierent initial gains o the controller. The details o the swar control strategy and the corresponding experiental results are given in Chapter Vibration suppression o a lexible link anipulator using piezoelectric actuator The active vibration suppression o a lexible link anipulator using a sart structure (piezoelectric actuator) is presented in Chapter 5 and [8]. The control strategy was a proportional control schee in which the applied voltage to the piezoelectric actuator was adjusted based on the delection o the anipulator so that the vibration o the bea was suppressed. One o the 8

37 advantages o this control ethod was that in contrast to any other controllers developed or vibration suppression o lexible link anipulators, the proposed control strategy was a velocityree schee and used only position as the eedback. It noteworthy that velocity easureent is generally not desirable as ost lexible anipulators have only strain gauges to easure the delection along the bea [30]. Also, taking the derivative ro the position signal to obtain the velocity signal was not a good approach since it resulted in generating noise in the velocity signal which deteriorated the perorance o the controller. A inite-eleent odel was irst developed or a cantilever bea with a piezoelectric actuator. To start the siulation, the bea was delected one centietre ro its horizontal position, and then it was released ro its delected position. Since the piezoelectric actuator was inactive, and the eect o the aterial daping was not considered in odeling procedure 5, the bea oscillated with a constant aplitude. The piezoelectric actuator was then activated, and the applied voltage to the piezoelectric actuator was adjusted proportionally to the delection o the bea. The delection o the bea was easured using a sensor placed at the location o the piezoelectric actuator. The actuator and sensor were located at the sae location to avoid instability due to the non-collocation o sensor and actuators [3]. It was shown in [3] that noncollocation o the sensor and actuator can lead to the instability o the controller. The siulation results or the cantilever bea indicated that the piezoelectric actuator was very successul in 5 The aterial daping was ignored so that the eect o the piezoelectric actuator in vibration suppression was only analyzed. Otherwise, the vibration was suppressed due to both piezoelectric actuator perorance and the aterial daping, and it was not possible to deterine which portion o the vibration was eliinated because o piezoelectric actuator. 9

38 vibration suppression o the cantilever bea such that the aplitude o vibration was reduced ro one centietre at the beginning to two illietres ater one second. Finding the optiu placeent o the piezoelectric actuator along the cantilever bea was another part o this study. The best location was ound based on the controllability o the syste states and using FE analysis. To deterine the optiu location in FE, two criteria were deined based on the delection o the bea as given in equations (4)-(5) in Chapter 5. The siulation results showed that the piezoelectric actuator had the best perorance when it was located at where was the distance o the piezoelectric ro the base o the anipulator and was the length o the link (See Fig. 9). The optiu placeent o the piezoelectric was then validated based on the controllability o the syste states. The control ethod was then extended to suppress the vibration the lexible-link anipulator shown in Fig. 9. The siulation results indicated that the piezoelectric actuator was successul in suppressing the oscillations o the anipulator during and ater the anoeuvre. For instance, as shown in Fig. 0, the delection o the anipulator was reduced considerably by eploying the piezoelectric actuator with controller s gain, and when the piezoelectric actuator was located at. The eect o the controller s gain on the perorance o the piezoelectric actuator was also analyzed. To deterine the best controller s gain, our dierent evaluation criteria were used. Three o these criteria,, and, were based on the delection o the anipulator, and deined respectively or the total anoeuvre tie ( ), beore the anipulator reached the inal rotation ( ), and ater the anipulator reached its inal rotation ( ). The ourth criterion was deined based on the aplitude o the doinant requencies contributing in the delection response. This criterion was deterined ro the 0

39 Power Spectru Density (PSD) o delection responses s or dierent gains. According to these our criteria, the best perorance was obtained when the controller s gain was. The details o this approach are given in Chapter 5 and in the paper published in the WIT Transactions on the Built Environent: Coputer Aided Optiu Design in Engineering X [9]. Fig. 9. Scheatic o lexible-link anipulator with piezoelectric actuator (a) (b) Fig. 0. Delection o the anipulator when (a) the piezoelectric actuator was inactive, (b) the piezoelectric actuator was activated and located at 0.3.

40 In another research study carried out by the candidate in [5], the eect o the tie delay between the input and output o the controller was investigated. The stability o the controller was analyzed or dierent gains and tie-delays. It was concluded that by increasing the controller s gain greater than a certain value the syste becae unstable. This value was called the critical gain and was proportional to the tie delay o the syste; that is, by increasing the tie delay, critical gains generally becae larger. This was consistent with the siulation results in which a lexible-link anipulator was odeled using a inite eleent odel. The siulation was perored or dierent tie delays and gains and it was shown that the vibration o the anipulator was successully suppressed provided that the proper tie delays and controller s gains were used..3.5 Dynaic odeling o a anipulator with lexible links and lexible joints Dynaic odeling o the anipulators with lexibility in the joints and links is presented in Chapter 6 and [0]. A scheatic o a anipulator consisting o two lexible links interconnected with lexible joints is shown in Fig.. In this igure, is the link delection, where and is the length o the i th link, the rae is attached to the rigidlike links and represents the rotating rae. To derive the dynaic equations or the anipulator shown in Fig., two ethods were eployed. The irst ethod was a cobination o the Lagrange ethod and assued ode shapes ethod (LAMM), and the second ethod was non-linear inite eleent analysis.

41 Fig.. Scheatic o a two lexible-link lexible-joint anipulator (TFLFJ) In the irst approach, the assued ode shapes ethod was used to approxiate the link delection as ollows (0) where is the nuber o ode shapes used or each link, and is the weight o the assued ode shape. To obtain the equations o the otion Lagrange s equations were used. For a syste with degrees o reedo Lagrange s equations were () where, and were respectively the kinetic and potential energies o the syste, and the torque applied to the i th otor, and is the i th generalized coordinate. By assuing two ode shapes or each link, two rotations or the otors and two rotations or the links, it was seen that eight degrees o reedo contributed to the dynaics o the TFLFJ. The kinetic energy o a lexible-link lexible-joint anipulator,, was coposed o the kinetic energies o the rotors,, links,, hubs,, and the payload ass, ; that is, () 3

42 The potential energy was the su o the elastic energies stored in the links and the potential energy due the lexibility o the joints (3) where and are respectively the rigidity o the i th link and stiness o the i th joint. Also, angles and are the otor and link angles easured with respect to the rigid rae shown in Fig.. By substituting the values o the kinetic and potential energies ro equations (4) and (5) into (3), the dynaic equation o syste can be obtained. The details o derivation o dynaic equations are given in Chapter 6 and Appendix II. The siulation results o the developed dynaic odel are shown in Fig.. As shown in this igure, the results obtained by LAMM approach were in an excellent agreeent with the FEA results. Fig.. Tip delection o the anipulator.4 Reerences Candidate s reerences:. H. Salasi, R. Fotouhi & P. N. Nikioruk, Tip trajectory tracking o lexible-joint anipulators driven by haronic drive, International Journal o Robotics and Autoation, 4(),

43 . H. Salasi, R. Fotouhi & P. N. Nikioruk, A anoeuvre control strategy or lexible joint anipulators with joint dry riction, Robotica, In press, doi: 0.07/S , H. Salasi, R. Fotouhi & P. N. Nikioruk, A biologically-inspired controller or tip trajectory tracking o lexible-joint anipulators, Subitted to International Journal o Robotics and Autoation, H. Salasi, R. Fotouhi & P. N. Nikioruk, On the stability o a riction copensation strategy or lexible-joint anipulators, Subitted to Advanced Robotics, H. Salasi, R. Fotouhi & P. N. Nikioruk, Stability analysis o tie delay control o lexible link anipulators using piezoelectric actuators, Subitted to Journal o Advances in Acoustics and Vibration, R. Fotouhi, H. Salasi, S. Dezulian & R. Burton, Design and control o a hydraulic siulator or a lexible joint robot, Advanced Robotics, (6), 008, M. Vakil, R. Fotouhi, P. N. Nikioruk & H. Salasi, A constrained Lagrange orulation o ulti-link planar lexible anipulator, Transactions o the ASME, Journal o Vibration and Acoustics,30(3), 008, H. Salasi, R. Fotouhi & P. N. Nikioruk, Vibration control o a lexible link anipulator using sart structures, Proceedings o 7 th World Congress International Federation o Autoatic Control (IFAC), Seoul, South Korea, 008, H. Salasi, R. Fotouhi & P. N. Nikioruk, Active vibration suppression o a lexible link anipulator using piezoelectric actuator, WIT Transactions on the Built Environent: Coputer Aided Optiu Design in Engineering X, 9, 007, H. Salasi, R. Fotouhi & P. N. Nikioruk, Dynaic odeling o a anipulator with lexible links and lexible joints, Proceedings o the st Canadian Congress o Applied Mechanics, Toronto, Canada, 007, (CD Ro).. H. Salasi, R. Fotouhi & P. N. Nikioruk, Gain analysis o the swar controller or the tip trajectory tracking o lexible-joint anipulators. To subit to IEEE/ASME International Conerence on Advanced Intelligent Mechatronics, Montreal, Canada, 00. 5

44 Other reerences:. N. Slack, The Blackwell Encyclopedic Dictionary o Operations Manageent (Blackwell Publishers, Oxord, UK, 999). 3. B. Siciliano & O. Khatib, Springer Handbook o Robotics (Springer, Berlin, Gerany, 008). 4. M. W. Spong, Control o lexible joint robots: a survey, Coordinate Science Laboratory, College o Engineering, University o Illinois, UILU-ENG-90-03, February H. Diken, Frequency-response characteristics o a single-link lexible joint anipulator and possible trajectory tracking, Journal o Sound and Vibration, 33(), 000, B. O. Al-Bedoor & A. A. Alusalla, Dynaics o lexible-link and lexible-joint anipulator carrying a payload with rotary inertia, Mechanis and Machine Theory, 35(6), 000, S. K. Spurgeon, L. Yao & X. Y. Lu, Robust tracking via sliding ode control or elastic joint anipulators, Journal o Systes and Control Engineering, 5(4), 00, A. Benallegue, Adaptive control or lexible joint robots using a passive systes approach, Control Engineering Practice, 3(0), 995, K. Khorasani, Adaptive control o lexible joint robots, Proceedings o IEEE International Conerence on Robotics and Autoation, Sacraento, CA, USA, 3, 99, M. C. Han & Y. H. Chen, Robust control design or uncertain lexible-joint anipulators: A singular perturbation approach, Proceedings o the IEEE Conerence on Decision and Control, San Antonio, TX, USA,, 993, D. Wang, A siple iterative learning controller or anipulators with lexible joints, Autoatica, 3(9), 995, C. J. B. Macnab & G. M. T. D'Eleuterio, Neuroadaptive control o elastic-joint robots using robust perorance enhanceent, Robotica, 9(6), 00,

45 3. N. Jalili, An ininite diensional distributed base controller or regulation o lexible robot ars, Journal o Dynaic Systes, Measureent and Control, 3(4), 00, H. D. Taghirad & P. R. Bélanger, H -Based robust torque control o haronic drive systes, Journal o Dynaic Systes, Measureent and Control, 3(3), 00, A. Albu-Schaer, C. Ott & G. Hirzinger, A uniied passivity-based control raework or position, torque and ipedance control o lexible joint robots, International Journal o Robotics Research, 6(), 007, M. Spong, K. Khorasani & P. Kokotovic, An integral aniold approach to the eedback control o lexible joint robots, IEEE Journal o Robotics and Autoation, 3(4), 987, Reerence anual o -DOF serial lexible joint, Quanser Copany, Toronto, revision, D. S. Morgan & I. B. Schwartz, Dynaic coordinated control laws in ultiple agent odels, Physics Letters A, 340, 005, A. Ratnaweera, S. K. Halgauge & H. C. Watson, Sel-organizing hierarchical particle swar optiizer with tie-varying acceleration coeicients, IEEE Transactions on Evolutionary Coputation, 8(3), 004, G. Besançon, Global output eedback tracking control or a class o Lagrangian systes, Autoatica, 36(), 000, W. B. Gevarter, Basic relations or control o lexible vehicles. AIAA Journal 8(4), 970, A. Preuont, Vibration control o active structures, (Kluwer Acadeic Publishers, Boston, 997) 7

46 Chapter. Tip Trajectory Tracking o Flexible-Joint Manipulators Driven by Haronic Drives 8

47 International Journal o Robotics and Autoation, Vol. 4, No., 009 TIP TRAJECTORY TRACKING OF FLEXIBLE-JOINT MANIPULATORS DRIVEN BY HARMONIC DRIVES H. Salasi, R. Fotouhi and P. N. Nikioruk Mechanical Engineering Departent, University o Saskatchewan, Saskatoon, Canada Eail: {haid.salasi, reza.otouhi, peter.nikioruk}@usask.ca Abstract: In this paper a new control strategy is introduced or tip trajectory tracking o lexible-joint anipulators driven by haronic drives. The proposed controller incorporates riction copensating torque (FCT) and coposite controller torque (CCT), which copensate or the riction in the haronic drives and the lexibility o the joints, respectively. The FCT is a new controller which includes a non-linear one-state LuGre odel and a proportional-derivative (PD) controller. The CCT s design is based on the singular perturbation ethod and integral aniold concept. An experiental setup o a two-rigid-link lexible-joint (TRLFJ) anipulator was used to veriy the proposed controller. The experiental results deonstrated the eectiveness o the controller at slow and ast otions o the anipulator or dierent trajectories. In particular, the tracking error was reduced signiicantly by using the proposed controller copared to the results obtained using traditional ethods such as the inverse dynaics ethod or the control o lexible-joint anipulators. Key Words: Flexible joint anipulator, haronic drive, joint riction 9

48 International Journal o Robotics and Autoation, Vol. 4, No., 009. Introduction Because o their sall size and high controllability, electric otors are ideal or use in robot anipulators, but have the disadvantage o having high velocity and low torque. To overcoe this drawback, electrically actuated robots use a gear transission to decrease the operating speed and increase the torque []. Aong gear transission systes haronic drives have such unique eatures that they have captured designers attention. They have zero backlash, high torque transissibility and copact size []. On the other hand, it has been deterined experientally [3] that torsional elasticity in haronic drives results in lightly daped oscillatory vibrations, which liit the application o the haronic derives especially in robot anipulators peroring high precision tasks such as those used in the anuacture o circuit boards. The perorance o haronic drives also deteriorates due to the high riction between the teeth o lexsplines and circular splines. This riction, which is called joint riction, is one o the ain liitations in achieving high accuracy anipulation tasks. Two ain concerns in designing controllers or robot anipulators driven by haronic drives are thereore joint riction and joint lexibility. However, ost control strategies developed so ar (e.g. [4]) or robot anipulators do not address these concerns and consequently using these control ethods ay lead to a signiicant trajectory tracking error which is especially crucial in high-precision tasks. In other works, the dynaic equations o the otion o the links and joints are written as two separate dynaically decoupled equations without considering the reaction o the joint torques on the links [5]. In these odels, the equations o rotations o the links and otors are decoupled while the orces related to the rotations o links are assued as zero. One exaple o this is ignoring the reaction o the elbow torque on the shoulder link or a two link serial anipulator. 30

49 International Journal o Robotics and Autoation, Vol. 4, No., 009 In this paper, a new control strategy or the Tip Trajectory tracking (CSTT) o lexible joint anipulators is presented. While this controller overcoes the shortcoings o the previously developed controllers, it also provides high accuracy in robotic anipulation tasks. In particular, it is experientally shown that the controller is very precise in the trajectory tracking o anipulators driven by haronic drives. Two ain coponents o this controller are: A riction copensating torque (FCT) which is designed to copensate or the riction eect, and a coposite controller torque (CCT) which is designed to copensate or the lexibility eect in the joints. The FCT is based on the LuGre odel which is a dynaic odel o nonlinear riction, and to copensate or the lexibility in the joints, the CCT is designed based on the singular perturbation theory and the integral aniold concept. The outline o this paper is as ollows. An overall picture o the experiental setup and the developed control strategy are given in Section. The procedures used to obtain controller coponents, the FCT and the CCT, are described in Sections 3 and 4. Section 5 includes the experiental results obtained using dierent control strategies or dierent trajectories, and Section 6 contains the conclusions.. Description o the Experiental Setup Fig. shows the scheatic o the control strategy that was used in the control o the tworigid-link lexible-joint anipulator (TRLFJ). The desired trajectory and q d and its derivatives, q d q d were the inputs to the controller and the rotation and velocity o the otor and link were used as eedback to the controller. The positions o the otor and link, q and L q, were easured using the encoders ipleented in the experiental setup and the velocities were then obtained by taking the derivatives o the position signals using a irst-order low-pass ilter which 3

50 International Journal o Robotics and Autoation, Vol. 4, No., 009 attenuated high-requencies noise in the velocity signal. This ilter was used which cancelled out high-requency noises due to dierentiation o the position signal. The applied torque τ had two parts; the riction torque τ and the coposite torque τ c, i.e. τ = τ + τ. The procedures o obtaining these torques are described in the ollowing sections. To calculate the current required by the otor, the resultant torque τ was ultiplied by the inverse o the torque constant o the c otor, K t, and the current was then ed into the D/A card. To obtain the velocity signal, The otor torque constants or the experiental setup were K t shoulder /, = 8.9 N. A and K elbow t, =.3 N. / A. The experiental setup which was used or experientation is shown in ig.. As shown in this igure, the irst link (shoulder link) was coupled to the irst joint (shoulder joint) by eans o a lexible joint. At the end o the shoulder link was a second haronic drive which was connected to the elbow link via the lexible elbow joint as shown in ig. (b). In this igure, angles and were rotations o the shoulder and elbow otors, and angles and were rotations o the shoulder and elbow links, respectively. Both otors and both lexible joints were instruented with quadrature optical encoders whose resolutions were rad/count [6]. Other physical paraeters o the anipulator are given in Table. 3. Design o FCT The design procedure o the riction copensating torque (FCT) is explained in this section. The riction properties o the haronic drives were odeled and experientally It is noteworthy that the DC peranent agnet otors usually have a linear relationship to otor torque. 3

51 International Journal o Robotics and Autoation, Vol. 4, No., 009 identiied in Sections 3. and 3., and the ethod used to obtain the FCT is explained in Section Friction Modeling A well-known riction odel which describes dierent aspects o the riction torque is the LuGre odel [7]. In this odel, on the icroscopic scale, suraces sliding on each other are odeled as two rigid bodies that ake contact through their elastic bristles which odel the rando asperities o the suraces (see ig. 3). The proposed odel has the atheatical or: dy dt where y is the average delection o the bristles (see ig. 3), q = q, 0 g( q ) y σ () dy τ = σ 0 y + σ + cq, () dt suraces, σ 0 the stiness o the bristles, σ the daping o the bristles, and q the relative velocity between c the viscous daping coeicient. Also, τ is the riction torque between the sliding suraces. The unction g( q ) represents the Stribeck eect. Since the average delection o the bristles can not be easured directly, inding the exact solution to predict the riction behavior is not possible. However, () and () can be sipliied or the steady-state condition (sliding regie) in which the deoration rate o the bristles is zero ( dy / dt = 0 ). Thus, ro () or steady-state conditions: q σ 0 yss = g( q ) = g( q )sgn( q ) (3) q where y is the steady-state value o the bristles deoration and sgn( q ) = or q > 0 and ss sgn( q ) = or q < 0. Substituting (3) in () results in: 33

52 International Journal o Robotics and Autoation, Vol. 4, No., 009 τ = sgn( q ) g ( q ) + c q (4) A typical proposed or or the unction g(q ) which describes the Stribeck eect is g τ where τ, c and τ, s are the Coulob riction and stiction ( q / s ) ( ), (,, ) v q = c + τ s τ c e torques respectively, and vs is the Stribeck velocity. By replacing this unction in (4), the riction torque based on the LuGre odel is obtained as: ( q / vs ) τ = [ τ, c + ( τ, s τ, c ) e ]sgn( q ) + c q (5) The irst two ters o the riction torque in (5) represent the Coulob riction and Stribeck eect, respectively, and the last ter accounts or the viscous daping. To deterine the riction torque as a unction o the velocity in (5) our paraeters ψ = τ, τ, v, c ] ust be [, s, c s identiied. The experiental procedure which was designed and used to identiy these paraeters is described in the ollowing section. 3.. Paraeter Identiication o Friction Model The scheatic o the experient procedure which was used to identiy the riction torque in joints is shown in ig. 4. The dynaic equations o the otors are expressed as: J = τ τ (6) q where J is the atrix o ass oent o inertias o the otors. Torques τ and τ are the vectors o the applied torque and the joint riction torque, respectively, and 34 q is the vector o accelerations o the otors. The otor position was controlled by a proportional-integratorderivative (PID) controller to ollow a triangle wave position reerence. Since the reerence trajectory is a constant-velocity trajectory, i the otor ollows the reerence trajectory, the otor acceleration, q, will becoe zero and as a result the applied torque will be equal to the riction torque in the above equation, i.e. τ = τ. Thereore, the riction torque was deterined

53 International Journal o Robotics and Autoation, Vol. 4, No., 009 by easuring the applied torque. In total, 57 experients were carried out at dierent velocities to estiate the riction unction or the shoulder and elbow joints. The experiental data o the shoulder and elbow joint otors are shown in ig. 5. These data were then interpolated (curveitted) using the unction τ q ) given in (5), and the riction torques were obtained as unctions ( o the velocity or the shoulder joint otor: τ τ = e = e ( q / ) ( q / 0.99 ) q q,, q q > 0 < 0 (7) and or the elbow joint otor: τ τ = e = e ( q / ) ( q / ) q q,, q q > 0 < 0 (8) According to the unctions o the riction torques given in (7) and (8), the Stribeck eect in the elbow joint otor is greater than in the shoulder joint otor. However, the shoulder joint has larger viscous daping and stiction than the elbow joint Friction Copensation in Haronic Drives There are dierent ways to copensate or the riction in robot anipulators. A siple well known ethod is that o adding a high requency dither signal to the control signal, but this ethod cannot be used in applications with high precision [8]. Another non-odel-based riction copensating ethod is using a sti PD controller which is suitable or regular tasks, however, it results in steady-state errors and ight lead to the stick-slip oscillations at low velocities [9]. The ethod which was used in this investigation was a odel-based riction copensation strategy. In this ethod, riction was copensated by applying an equivalent torque in the opposite direction o the riction torque. The equivalent torque was ound based on (7) and (8). Ideally, the counterbalancing estiated riction torque, τ, was equal to the real riction o the 35

54 International Journal o Robotics and Autoation, Vol. 4, No., 009 syste, τ, and, hence, the riction o the syste would be copensated copletely. However, practically there was always an error between the estiated riction torque and the actual riction torque, because the riction torque deined in (7) and (8) was obtained based on the assuption o steady-state conditions. Also, the eects o the oil teperature, load, huidity and actuator wear were not considered in odeling joints riction [0]. To overcoe these deiciencies o the riction torques obtained in (7) and (8), a linear PD controller was added to the LuGre odel. Using a PD controller along with the riction copensating torque given in (7) and (8) guarantees that the error in riction copensation is bounded and the syste reains stable provided that the gains are chosen properly. In this research, the gains o PD controller or shoulder joint were chosen as K = 500 and K = 40. These values or elbow joint were K = 5 and K = 8. p v p v 4. Design o Coposite Controller Torque The second part o the controller was a coposite controller which was designed to copensate or the eect o the lexibility o the joints. The singular perturbation theory and integral aniold concept were used in the design procedure o the coposite controller torque (CCT). Using singular perturbation theory allows dividing the dynaics o lexible-joint anipulator into two subsystes: one corresponding to the ast dynaics (high requency vibration due to the lexibility) and the other to the slow dynaics (low-requency otion). The standard governing equations o a singularly perturbed syste are []: 0 n = ( X, Z, ε, t), X ( t 0 ) = X, X R (9) X 0 ε Z = g ( X, Z, ε, t), Z ( t 0 ) = Z, Z R (0) 36

55 International Journal o Robotics and Autoation, Vol. 4, No., 009 where ε represents a sall paraeter, X and Z are state vectors o diensions n and, respectively, n R, g R, and X and Z represent the derivatives o X and Z with respect to tie, respectively. The above equations represent a two-tie-scale syste in which the state Z is the ast paraeter o the syste because ε is a sall paraeter and Z is very large in (0). The state X is the slow state o the syste because it has a slower tie derivative than state Z. Another useul concept which can be used in the context o the singular perturbation theory is the integral aniold. By eans o the integral aniold, it is possible to reduce the order o the singular syste presented in (9) and (0) ro ( n + ) to n. The ipleentation o integral aniold concept or singularly perturbed systes includes the ollowing steps: Step (Integral aniold ipleentation): The ast states, Z, are approxiated as a set o unctions o the slow states and the singular perturbation paraeter, i.e. Z Z = ϕ( X, ε ) where ϕ is an approxiation unction or the ast states. Step (Coposite controller design): The coposite controller, τ c, includes the slow and ast controllers. The slow controller, τ c, s, is designed to control the reduced-order odel and the ast controller, τ c,, guarantees that the approxiation error ade in step between the ast states and the integral aniold, Z ϕ exponentially goes to zero and the ast states are on the integral aniold (see ig. 6(a)). As long as Z is on the integral aniold ϕ, the slow state X approaches the reduced-order odel (see ig. 6(b)). Thus, provided that the ast state Z is restricted to ove along the integral 37

56 International Journal o Robotics and Autoation, Vol. 4, No., 009 aniold, the slow controller can only be designed or the reduced-order odel to control the slow state X. In the ollowing the application o the singular perturbation theory and the integral aniold concept is used to design a coposite controller or lexible joint anipulators. 4.. Singularly Perturbed Equations o Flexible-Joint Manipulators The dynaic equations o a serial anipulator consisting o n -rigid links connected by ( n ) elastic joints are described by the ollowing set o dierential equations: D ) q + C ( q, q ) q + K ( q q ) = H ( τ τ ) () ( ql L l L L L L J q K ( q q ) = τ τ () L where D q L R n n ( ) is the inertia atrix o the links and l L L L C ( q, q ) q represents the Coriolis and centriugal vectors. The diagonal atrix K = diag k, k,, k ) represents the stiness o ( n the joints. n n J R is the inertia atrix o the otors, τ is the vector o the applied torques to the otors, and τ is the riction joint torques. Also, n n ql R and R q represent, respectively, the rotations o the links and the otors. The coponents o the atrix H are deined as H = when i = j, and H = 0 when i j i, j =,,, n. As deined in ij ij Section, the applied torque τ is the suation o the riction copensating torque, τ, and the coposite controller torque τ c ; i.e. τ = τ + τ. Assuing that the riction copensating torque τ counterbalances the real riction o the joints τ copletely; that is, τ = τ, the joints riction can be eliinated ro the dynaic equations o the anipulator. Thus: L L l L L L c D( q ) q + C ( q, q ) q + K( q q ) = Hτ (3) L L c J q K( q q ) = τ (4) c 38

57 International Journal o Robotics and Autoation, Vol. 4, No., 009 To deine the singular perturbation paraeter, all the joint stiness k i, i =,,, n are assued to have the sae order o agnitude and they can be written as a ultiple o a single large paraeter k ; that is k ~ = k k, i =, n. The elastic joint torques are then written as: i i, ~ ξ = K q q ) = kk( q q ) (5) ( L L ~ ~ ~ where K = diag( k,, k ).Without loss o generality (by rescaling the ξ variables i necessary), n the atrix K ~ is chosen as ~ K = I. The singular perturbation paraeter can now be deined as µ = / k, and (5) becoes µξ = ql q. Ater soe atheatical anipulation, the statespace representation o the singularly perturbed syste (3) and (4), is obtained as: X = A + A Z A τ (6) + 3 c Z = B X + B Z B τ (7) ε + 3 c where X [ q L q L ] T =, Z [ ξ ε ξ ] T =, µ = q On n On n A, A = D Cl q L D On n L ε =,, On n On n On n On n I n n On n A3 =, B = D H, B = On n D C, B 3 = l ( J + D ) O and O is n n D H J a null atrix and I is the identity atrix. In the ollowing section, the integral aniold concept is eployed to obtain the coposite controller τ c. The purpose o coposite controller τ c is to track the link rotation q L on the desired trajectory q d. 4.. Ipleenting Integral Maniold or FJMs An integral aniold or a lexible joint anipulator in the space n R is: n Mε : Z = ϕ( X, τc, s, ε, t); X R (8) 39

58 International Journal o Robotics and Autoation, Vol. 4, No., 009 I the ast states Z ove on the integral aniold (see ig. 6(a)), then Z can be substituted by the integral aniold ϕ in (7), and thereore: ε ϕ = + (9) B X + Bϕ B3τ c, s Also, the reduced-order odel o the syste is obtained by substituting the integral aniold in (6): (0) X = A + Aϕ + A3 τc, s The procedures used to construct the slow and ast controllers are explained in the ollowing section Construction o the Coposite Controller Torque or Flexible-Joint Manipulators The design o the coposite controller torque (CCT) was based on the singularly-perturbed dynaic equations o the lexible-joint anipulators and had two parts: the slow controller, τ, c, s and the ast controller τ c,. The slow controller was designed to control the slow subsyste representing the low-requency otion o the anipulator. Thus, when the joint lexibility approached ininity, the slow subsyste reduced to a rigid anipulator. The ast controller, τ, was selected to suppress the high-requency vibration o the anipulator. Thus the coposite controller τ c can be written as the suation o slow and ast controller torques as: c, τ = + () c τ c, s τ c, The paraeter η is the dierence between Z and the integral aniold ϕ, thereore: η = Z ϕ () Taking the derivative o () and cobining with (7), (9) and () results in: ε η = ε + (3) Z εϕ = Bη B3τ c, Fro (6), () and (3) the equations o the slow and ast subsystes in state space are: 40

59 International Journal o Robotics and Autoation, Vol. 4, No., 009 X = A + A + A η A τ (4) ϕ + 3 B η B3τ c, c ε η = + (5) Equation (4) describes the low-requency otion and (5) describes the high-requency otion o the robot anipulator. It is worth noting that once the ast state, Z, eets and stays on the integral aniold ϕ (that is Z = ϕ ), paraeter η and its derivative η will be zero according to () and (5). This results in τ becoing zero in (5) since B 0, and the coposite c, 3 controller torque having only the slow part, i.e. τ = τ. The slow controller or lexible-joint anipulator can be obtained as: τ c c, s c, s = τ c,0 + ε ( D H J ) ϕ0, (6) where τ c, 0 is the rigid controller and it can be designed based on the conventional ethods available or the rigid anipulators such as the inverse-dynaics approach []. Also ϕ 0, is the second tie derivative o ϕ 0,. The vector ϕ 0, includes the irst n coponents o the vector ϕ 0 R n where: 0 3 c, ϕ = B ( B X + B τ ) 0 (7) The ast controller ust be designed so that the deviation between the integral aniold and paraeter Z asyptotically approaches zero. I the ast controller is chosen to have the stateeedback orat as c, = B η where B 4 is a constant atrix and the paraeter η is the τ 4 dierence between Z and the integral aniold ϕ, (5) becoes: η = ( B + B B η (8) ε 3) 4 I the eleents o atrix B 4 are chosen such that the eigenvalues o the square atrix ( B + B ) 4 always have negative real parts, the deviation o Z ro its aniold will 3B asyptotically goes to zero and, inally, Z will be on its invariant aniold. 4

60 International Journal o Robotics and Autoation, Vol. 4, No., Experiental Veriication The two-rigid-link lexible-joint (TRLFJ) anipulator shown in ig. was used to veriy experientally the perorance o the new controller described in the previous sections. The physical paraeters are given in Table.The desired link trajectory was chosen as: q d = [90( t / t ) 35( t / t ) + 540( t / t ) 40( t / t ) + 6( t / t ) ] q (9) where q d, q and t were, respectively, the desired trajectory o the rotations o the links, the inal rotation o the link and the tie o total anoeuvre. The proposed trajectory was designed such that it satisied the initial conditions o q d ( 0) = 0 and q d ( t ) = q, and its irst our derivatives were zero at t = 0 and t = t. 5.. Trajectory Tracking Results Since the riction is very sensitive to the velocity, our dierent sets o trajectories were experientally exained to deterine the perorance o the controller at dierent velocities and accelerations. The inal positions o the shoulder and elbow links were: [ q [ q, shoulder, shoulder, q, q, elbow, elbow ] ] 3 = [.0 rad,.0 rad], [ q = [.0 rad,.5 rad], [ q, shoulder, shoulder, q, q, elbow, elbow ] ] 4 = [.5 rad,.0 rad] = [.5 rad,.5 rad] (30) The anoeuvre tie was the sae or all sets o trajectories, i.e. t = sec., hence, increasing the inal positions resulted in ore rapid otion o the anipulator. Thereore, choosing dierent inal positions enabled the perorance o the controller to be deterined at both slow and ast anoeuvres o the anipulator. The rotations o the shoulder and elbow links are shown and copared with the desired trajectories in Figs. 7 to 0. As shown in these igures, the links ollowed the desired trajectories accurately. The tracking error values are shown in Section 5.. Also, it was experientally ound i the riction was not considered in the design 4

61 International Journal o Robotics and Autoation, Vol. 4, No., 009 procedure o the controller, the anipulator was not able to start its otion because o high stiction in the haronic drives. 5.. Coparing New Control Strategy with the Rigid Controller To analyze the eectiveness o the proposed new control strategy or Tip Trajectory tracking (CSTT), the trajectory tracking results were copared with the results obtained when the rigid controller, τ c, 0, was cobined with the riction copensating torque τ. The rigid controller was obtained using the inverse-dynaic approach and based on the assuption o rigid joints (i.e. K ) (see appendix II or details). To have a reasonable easure or coparing the results, the tracking error percentages were noralized as: error [( q q ) / q ] 00 (3) = L d This error indicates the dierence between experiental and the desired values o the link rotation. Also, the axiu o the noralized error during the anoeuvre is deined as: error = ax ( q q ) / q (3) ax The results or the our sets o trajectories given in the previous section, equation (9), are shown in Figs. to 4. It can be observed that the tracking error was signiicantly reduced by using the CSTT copared with the results obtained using the rigid controller. For instance, according to the values o the axiu errors in Table, the perorance o the controller iproved 6% or the shoulder and 3% or the elbow by using the CSTT when the inal rotations o the links were [ q, shoulder, q, elbow] = [ rad, rad]. Also, it is shown in Figs. (b) and 4(b) that by increasing the inal rotation o the shoulder link ro.0 rad to.5 rad, the rotations o the links, especially the elbow link, becae unstable using the rigid controller. Also, the responses o the links were unstable when the inal rotation o the elbow link was increased L d 43

62 International Journal o Robotics and Autoation, Vol. 4, No., 009 to.5 rad in ig. 3(b). That eans that the rigid controller was unable to control the rotations o the links or high-speed anoeuvres. In contrast to the results o the rigid controller, the results obtained using CSTT showed that the tracking error reained in an acceptable range o ± % or high-speed and slow anoeuvres o the lexible-joint anipulator. Also, it can be observed in Figs. 3(a) to 6(a), that the elbow and shoulder links oscillated with requencies close to their natural requencies when they reached the inal rotation, q. The shoulder link oscillated with a requency o.08 Hz the elbow link oscillated with a requency o 3.Hz. Furtherore, it can be seen in Figs. 3(a) and 4(a) that the response o the elbow link in the irst part o the anoeuvre ( 0 < t ) was very dierent than that o the second part ( < t ) when the CSTT was used. This property can be explained based by the act that the riction was dierent during acceleration (increasing velocities) than deceleration (decreasing velocities), due to rictional lag or hysteresis [7]. In the irst part o the response, whereas the velocity was increased (acceleration), the riction was copensated eectively. However, in the second part, where the velocity was decreased (deceleration), one ay conclude that the estiation o riction was not very accurate. A suary o the axiu values o errors are tabulated in Table () or the rigid and the CSTT controllers or dierent trajectories. 6. Conclusions A new control strategy was developed or the tip trajectory tracking o lexible joint anipulators which are driven by haronic drives. The proposed controller included two parts, the riction copensating torque (FCT) to copensate or the riction eect and the coposite controller (CCT) to copensate or the lexibility o the joints. The FCT was a cobination o the LuGre dynaic odel and a PD controller. The paraeters o the dynaic odel were 44

63 International Journal o Robotics and Autoation, Vol. 4, No., 009 deterined through constant-velocity closed-loop experients, and the gain values o the PD were chosen such that the controller was stable. The integral aniold in the context o the singular perturbation theory was used to design a new coposite controller, the CCT, which included two parts, slow and ast controllers. The perorance o the proposed controller was experientally conired at dierent anipulation speeds or several trajectories. Also, coparing experiental results using the developed controller with the results o a rigid controller deonstrated the superiority o the coposite controller over the rigid controller which is usually used. For instance, the axiu error was reduced signiicantly using the CSTT copared to the results o the rigid controller. Furtherore, when the inal rotations o the links were increased ro.0 rad to.5 rad, the results o the rigid controller becae unstable while the tracking error o the CSTT reained in an acceptable range. 7. Reerences [] H.D. Taghirad, & P.R. Bélanger, H -Based robust torque control o haronic drive systes, Journal o Dynaic Systes, Measureent and Control, 3(3), 00, [] A. Albu-Schaer, C. Ott, & G. Hirzinger, A uniied passivity-based control raework or position, torque and ipedance control o lexible joint robots, International Journal o Robotics Research, 6(), 007, [3] L. Sweet, & M. Good, Redeinition o the robot otion-control proble, IEEE Control Systes Magazine, 5(3), 985,

64 International Journal o Robotics and Autoation, Vol. 4, No., 009 [4] J.G. Yi, J.S. Yeon, J.H. Park, S.H. Lee, & J.S. Hur, Robust control using recursive design ethod or lexible joint robot anipulator, IEEE International Conerence on Robotics and Autoation, Roa, Italy, 007, [5] F. Ghorbel, & M.W. Spong, Integral aniolds o singularly perturbed systes with application to rigid-link lexible-joint ultibody systes, International Journal o Nonlinear Mechanics, 35(), 000, [6] Reerence anual o -DOF serial lexible joint, Quanser Copany, Toronto, revision, 006. [7] C. Canudas de Wit, H. Olsson, K.J. Astro, & P. Lischinsky, A new odel or control o systes with riction, IEEE Transactions on Autoatic Control, 40(3), 995, [8] M.A. Michaux, A.A. Ferri, & K.A. Cuneare, Eect o tangential dither signal on riction induced oscillations in an SDOF Model, Journal o Coputational and Nonlinear Dynaics (3), 007, 0-0. [9] R.H.A. Hensen, Controlled echanical systes with riction, Ph.D. Thesis, Eindhoven University o Technology, Eindhoven, The Netherlands, 00. [0] P. Lischinsky, C. Canudas-de-Wit, & G. Morel, Friction copensation o a Schilling hydraulic robot, Proceedings o IEEE International Conerence on Control Applications, Hartord, CT, USA, 997, [] P.V. Kokotovic, H.K. Khalil, & J. O Reiley, Singular perturbation ethods in control: analysis and design, (New York: Acadeic Press, 986). [] Y. Feng, S. Bao, & X. Yu, Inverse dynaics nonsingular terinal sliding ode control o two-link lexible anipulators, International Journal o Robotics and Autoation, 9(), 004,

65 International Journal o Robotics and Autoation, Vol. 4, No., 009 Figure. Control strategy designedd or tip trajectory tracking o two-rigid-link lexible-joint anipulator (TRLFJ) Elbow joint Shoulder joint (a) (b) Figure. (a) Experiental odule o two-rigid-link lexible-joint anipulator available in the Robotics Laboratory o the University o Saskatchewan, (b) the scheatic o the top-view o the experiental setup showing the angles o the joints and the links 47

66 International Journal o Robotics and Autoation, Vol. 4, No., 009 Table Physical paraeters o the experiental setup o TRLFJ Paraeter Sybol Shoulder Elbow Equivalent inertia o the downstrea echanis o the link ( D Mass oent o inertia o the otor ( J Joint stiness K Length o the link l Figure 3. Two suraces sliding on each other with relative velocity o bristles attached to the rigid bodies q ; the asperities are odeled as elastic The equivalent inertia or the shoulder includes the ass oent o inertias o the shoulder and elbow links, and the elbow otor with respect to the shoulder joint. Also, the equivalent inertia or the elbow includes the ass oent o inertia o the elbow link with respect to the elbow joint. 48

67 International Journal o Robotics and Autoation, Vol. 4, No., 009 Figure 4. Scheatic o the control procedure used to identiy the LuGre odel paraeters or experiental setup shown in ig. (a) (b) Figure 5. Experiental results obtained or steady state conditions and estiation unction obtained based on the LuGre odel or (a) shoulder joint otor, (b) elbow joint otor (a) (b) Figure 6. (a) Fast state Z approaches the integral aniold ϕ due to the ast controller, and (b) slow state X is equal to the reduced-order odel, as long as the ast state is on the integral aniold 49

68 International Journal o Robotics and Autoation, Vol. 4, No., 009 (a) (b) Figure 7. Experiental veriication o the links rotations or [ q, shoulder, q, elbow ] = [.0 rad,.0 rad] using CSTT controller, (a) shoulder link trajectory (b) elbow link trajectory (a) (b) Figure 8. Experiental veriication o the links rotations or [ q, shoulder, q, elbow ] = [.5 rad,.0 rad ] using CSTT controller, (a) shoulder link trajectory (b) elbow link trajectory (a) (b) Figure 9. Experiental veriication o the links rotations or [ q, shoulder, q, elbow ] 3 = [.0 rad,.5 rad ] using CSTT controller, (a) shoulder link trajectory (b) elbow link trajectory 50

69 International Journal o Robotics and Autoation, Vol. 4, No., 009 (a) (b) Figure 0. Experiental veriication o the links rotations or [ q, shoulder, q, elbow ] 4 = [.5 rad,.5 rad ] using CSTT controller, (a) shoulder link trajectory (b) elbow link trajectory (a) (b) Figure. Experiental results: noralized tracking error or shoulder and elbow links or inal rotations o [ q, shoulder, q, elbow ] = [.0 rad,.0 rad ], using (a) CSTT controller, (b) rigid controller (a) (b) Figure. Experiental results: noralized tracking error or shoulder and elbow links or inal rotations o [ q, shoulder, q, elbow ] = [.5 rad,.0 rad ], using (a) CSTT controller, (b) rigid controller 5

70 International Journal o Robotics and Autoation, Vol. 4, No., 009 (a) (b) Figure 3. Experiental results: noralized tracking error or shoulder and elbow links or inal rotations o [ 3 q, shoulder, q, elbow ] = [.0 rad,.5 rad ], using (a) CSTT controller, (b) rigid controller (a) (b) Figure 4. Experiental results: noralized tracking error or shoulder and elbow links or inal rotations o [ 4 q, shoulder, q, elbow ] = [.5 rad,.5 rad ], using (a) CSTT controller, (b) rigid controller 5

71 International Journal o Robotics and Autoation, Vol. 4, No., 009 Table 3 Experiental results: Maxiu tracking error ( error (3)) during the aneuver or the shoulder and ax elbow links using the CSTT and rigid controller Shoulder ( 0 ) Elbow ( 0 ) [, shoulder, elbow q, q ] CSTT Rigid CSTT Rigid [.0,.0 rad] rad [.5,.0 rad] rad * * [.0,.5 rad] rad. 3. * * [.5,.5 rad] * The response was unstable rad * * 53

72 International Journal o Robotics and Autoation, Vol. 4, No., 009 Biographies Haid Salasi received the B.Sc. and M.Sc. in Mechanical Engineering ro the Shari University o Technology, Tehran, Iran, in 003 and 005, respectively. He is currently working toward the Ph.D. degree in the Mechanical Engineering at the University o Saskatchewan, Saskatoon, Canada. His research interests include Robotics and Vibration Control. Reza Fotouhi obtained his PhD in Mechanical Engineering ro the University o Saskatchewan in Canada. He is currently a proessor o Mechanical Engineering at University o Saskatchewan in Canada. His research interests include Robotics (dynaics and control), Structural Dynaics and Vibrations, Coputational Mechanics, and Bioechanics. Peter N. Nikioruk is Dean Eeritus, College o Engineering, at the University o Saskatchewan, Canada. Previously, he was Dean o Engineering or 3 years, Head o Mechanical Engineering or 7 years and Chair o the Division o Control Engineering or 5 years. He holds the B.Sc. degree in engineering physics ro Queen s University in Canada and Ph.D. degree in electrical engineering ro Manchester University in England. He received the D.Sc. degree, also ro Manchester University, in 970 or research in control systes. He served as chair or eber o ive Councils in Canada, was recipient o seven Fellowships in Canada and England and seven other honors, and eber o several Boards. His ields o research are Adaptive and Control Systes. 54

73 Chapter 3. A anoeuvre control strategy or lexible joint anipulators with joint dry riction 55

74 Robotica, In press, (doi: 0.07/S ), 009. A anoeuvre control strategy or lexible joint anipulators with joint dry riction H. Salasi, R. Fotouhi and P. N. Nikioruk Mechanical Engineering Departent, University o Saskatchewan, 57 Capus Drive, Saskatoon, Canada, S7N 5A9 Keywords: Flexible joint anipulator, dry riction, integral aniold controller Abstract: A new control strategy based on the singular perturbation ethod and integral aniold concept is introduced or lexible joint anipulators with joint riction. In controllers developed so ar based on the singular perturbation theory, the dynaics o actuators o lexiblejoint anipulators are partially odeled, and the coupling between actuators and links are ignored. This assuption leads to inaccuracy in control perorance and error in trajectory tracking which is crucial in high-precision anipulation tasks. In this paper, a coprehensive dynaic odel which takes into account the coupling between actuators and links is developed and a coposite controller is then designed based on the singular perturbation theore and integral aniold concept. To overcoe the joint riction, a novel ethod is introduced in which a linear eed-orward torque is designed using the principle o work and energy. Finally, the experiental setup o a single rigid-link lexible-joint anipulator in the Robotics Laboratory at the University o Saskatchewan was used to veriy the proposed controller. Experiental results eploying the new controller showed that the trajectory tracking error during and at the end o the otion o the robot anipulator was signiicantly reduced. Corresponding author 56

75 Robotica, In press, (doi: 0.07/S ), Introduction A wide variety o robots is now eployed in dierent areas such as spacecrat and vehicle anuacturing. In ost o these robot anipulators, a transission syste is eployed to increase the torque applied to the links and consequently to apliy the robot power. Most o controllers developed so ar to control robot anipulators were based on the assuption o rigid joints; that is the transission was assued to be copletely sti and that there was no dierence between the angular positions o the driving actuator and that o the driven link [-6]. However, it was experientally shown that the assuption o rigid joints generally liits a robot s ability to peror high-speed high-precision anipulations [7-], and lexibility o the joints ay produce lightly daped oscillatory otions. Especially, the eect o joint lexibility becoes ore signiicant in robot anipulators driven by haronic drives. While haronic drives have zero backlash, high torque transissibility and copact size [3-4], there is high torsional elasticity in haronic drives which liit their capability [5]. Thereore, a ore accurate control strategy in which the joint lexibility is taken into account is essential, especially in applications such as robots used in the anuacture o precision coponents such as circuit boards. However, because o the existence o inherent nonlinearities in the dynaic equations o lexible anipulators, developing an accurate odel is diicult. In addition to joint lexibility, another iportant actor which aects the perorance o robot anipulators is joint riction. It has been proven that joint riction can lead to tracking errors and undesired stick-slip otion, and aects both the dynaic and static behaviours o a robot anipulator at the start, during and at the end o the otion [6-]. Despite this, joint riction is usually neglected in odeling and control strategies developed or lexible joint anipulators, and ost o the odels developed or lexible joint anipulators ignore joint riction. For instance in [3] and [4] a control strategy is designed based on singular perturbation theory or tip trajectory tracking o lexible joint anipulators. However, in these works the joint riction was not considered in the dynaics o the anipulator. In this paper a novel control strategy is introduced or the tip trajectory tracking o a lexible joint anipulator which includes two ain parts: 57

76 Robotica, In press, (doi: 0.07/S ), Friction copensating torque which is designed so as to copensate or the riction eect. - Coposite controller which is designed to copensate or the lexibility eect in the joints. To copensate or the riction eect in the joints, a new riction copensation strategy is introduced in this paper. The key advantage o this strategy is that the riction copensating torque can be identiied easily in two steps. In ost riction odels developed to predict the joint riction, an experiental identiication procedure is necessary to deterine the riction odel paraeters. This identiication procedure involves a large nuber o experients which need to be carried out at dierent velocities and requires considerable aount o tie. In the proposed odel, the paraeters o the copensation strategy are identiied in only two steps and, hence, the proposed odel is ore cost-eective in ters o the experiental tie required to identiy the paraeters o the controller. To copensate or the lexibility in the joints, a coposite controller is designed based on the singular perturbation theory along with the integral aniold concept. In singular perturbation odels developed or lexible-joint anipulators, e.g. in [9] and [4], the inertias o the rotors are ignored, however, ignoring any paraeters in the dynaics o a syste leads to inaccuracy in the perorance o the controller since controllers whose design is based on the singular perturbation theory are odeled-based. This error speciically becoes ore iportant when high precision in control is expected. In this paper, a ore accurate dynaic odel, which takes into account the inertias o rotors including gear reduction, is presented. While considering the eect o rotors inertias leads to a ore detailed and ore coplex dynaic odel, it iproves the precision o the controller in trajectory tracking. Finally, the developed control strategy was experientally veriied by the experient setup o a single rigid-link lexible-joint (SRLFJ) anipulator in the Robotics Laboratory at the University o Saskatchewan. Also, in order to assess the eectiveness o the work-energy principle based 58

77 Robotica, In press, (doi: 0.07/S ), 009. odel, experients were perored and copared using two riction copensating strategies: a new ethod established based on the work-energy principle [5] and the LuGre ethod [8].. Methodology Figure shows the scheatic o the control strategy that was used in the control o lexible joint anipulators. The desired trajectory q d and its derivatives, q d and q d were the inputs to the digital controller, and the rotation and velocity o the otor and link were used as eedback to the controller. The applied current was ed into the digital controller and the output digital signal to a D/A card. The analog current signal was then apliied and applied to the plant which is a lexible joint anipulator (FJM). The positions o the otor and link, q and L q, were easured using the encoders ipleented in the experiental setup and the values o the velocity were then obtained by taking the derivative o the position signals using a ilter. The applied torqueτ = τ + τ c had two parts; the riction copensating torque τ and the coposite torqueτ c. The procedures o designing these torques are described in ollowing sections. To calculate the current required by the otor, the resultant torque τ was ultiplied by the inverse o the torque constant o the otor, K t, and the current was then ed into a D/A card. The value o the otor torque constant or the experiental setup was K t = 8.9 N. / A 3. The angle easures the rotor angle ater gear reduction, however, or the purpose o abbreviation, it is reerred to as the otor angle in this paper. 3 It is noteworthy that DC peranent agnet otors usually have a linear relationship to otor torque. 59

78 Robotica, In press, (doi: 0.07/S ), 009. Figure. Control strategy designed or tip trajectory o FJM 3. Friction copensation in haronic drives In this section two dierent riction odels are developed or the experiental setup o single rigid-link lexible-joint anipulator. In the irst odel, joint riction, τ, is approxiated as a LG unction o velocity based on the LuGre odel, τ [8]. For this purpose, the joint riction values are obtained at dierent velocities and the joint riction torque is then estiated as a unction o velocity by the curve-itting ethod. In the second odel, the riction copensating torque is designed based on the work-energy principle as a eed orward linear copensating WE torque, τ. 3.. LuGre odel A typical riction odel which describes dierent aspects o riction torque is the LuGre odel. In this odel, on the icroscopic scale, suraces sliding on each other are odeled as two rigid bodies that ake contact through elastic bristles which odel the rando asperities 4 o the suraces (See Figure ). The bristles start to delect when one surace (upper surace) starts to ove and the orce is then applied tangentially to the bristles o the lower surace. I the 4 Regardless o how sooth a surace is, it has any rando icro-scale irregularities which are called asperities. 60

79 Robotica, In press, (doi: 0.07/S ), 009. delection is suiciently large, the suraces will slide on each other. The proposed odel has the atheatical or [8] τ LG dy dt = q σ q y 0 g( q dy = σ y + σ + c dt 0 where y is the average delection o the bristles and suraces. σ 0, σ and ) q,, q is the relative velocity between c are the stiness o the bristles, daping o the bristles and viscous daping coeicient, respectively. Function g q ) represents the Stribeck eect which LG characterizes the transition between the static riction (stiction) and viscous riction, and τ is the riction torque odeled based on LuGre ethod. Eqs. () and () are nonlinear equations describing the riction behavior based on the delection o bristles. Since the average delection o the bristles can not be easured directly, inding the exact solution to predict the riction behavior is not possible. However, Eqs. () and () can be sipliied or the steady-state condition (sliding regie) in which the deoration rate o bristles is zero ( dy / dt = 0). ( Figure. Two suraces are sliding on each other; the asperities are odeled as bristles attached to the suraces Thus, ro Eq. (), σ y q = g( q ) = sgn( q q ) g( q 0 ss ) where y ss is the steady-state value o the bristles deoration and the sign unction is deined as 6

80 Robotica, In press, (doi: 0.07/S ), 009. sgn( q ) = For steady-state condition, dy / dt= 0, the riction torque is obtained by substituting Eq. (3) in q q > 0 < 0 Eq. () τ LG = sgn( q ) g( q ) + c q A typical proposed or or the unction g q ) is [8] ( where LG τ c g LG LG LG ( q / s ) ( ) ( ) v q = τ + τ τ e c LG and τ s are the Coulob riction and stiction torque respectively, and vs s c is the Stribeck velocity. Replacing Eq. (6) in Eq. (5) leads to τ LG ( q, ψ ) = [ τ LG c + ( τ LG s τ LG c ) e s ( q / v ) ]sgn( q ) + c q where ψ is a vector which includes our constant paraeters o the LuGre riction odel; that is, LG LG ψ = [ τ τ v c ]. The irst and second ters o the riction torque in the above equation s c s describe the Coulob riction and Stribeck eect respectively and the last ter accounts or the viscous daping. To characterize riction torque as a unction o velocity in Eq. (7), our LG LG unknown paraeters, ψ = τ τ v c ], ust be identiied. A scheatic o the experiental [ s c s procedure which was used to identiy riction odel paraeters is shown in Figure 3. The otor position was controlled by a PID controller to ollow a triangular wave position reerence. The dynaic equation o otor is expressed as J q = τ τ where J,τ, τ and q are the ass oent o inertia o the otor, applied torque, joint riction and acceleration o the otor respectively. I the otor ollows the reerence trajectory, which is a triangular trajectory with constant velocity, the otor acceleration, q, will becoe zero and as a result the applied torque will be equal to the actual riction torque in the above equation, i.e. τ = τ torque.. Thereore, the riction torque can be deterined by easuring the applied 6

81 Robotica, In press, (doi: 0.07/S ), 009. Figure 3. Scheatic o the control procedure used to identiy the LuGre odel paraeters In total, experients were carried out in dierent steady-state velocities to estiate the riction torque. The experiental data were then interpolated (curve-itted) with the LuGre riction torque LG τ given in Eq. (7). For this purpose, the nonlinear optiization technique (MATLAB unction lsqcurveit.), was used to iniize the quadratic cost unction I which was deined as I = N ( i) LG ( i) [ τ ( q ) τ ( q, ψ)] i= (i) where τ ( ) represents the actual joint riction torque which is easured experientally during q (i) LG ( i) constant otor velocity q. The torque τ ( q, ψ) is the joint riction torque which is odeled based on the LuGre odel and is given in Eq. (7), and N is the nuber o the easured data- (i) points. The cost unction I represents the error between experiental data points, τ ( ), and LG ( i) the estiation unction o LuGre odel, τ ( q, ψ). Thereore, by iniizing cost unction, LG LG optiu values o ψ = τ τ v c ] can be ound. These values were obtained as [ s c s ψ =[ ] and ψ =[ ] or positive and negative velocities, respectively. By replacing the optiu values o the paraeters in the LuGre odel in the sliding regie, the riction torques can be expressed as unctions o velocity q τ τ LG LG = [ e = [ e ( q / 0.90 ) ( q / 0.99 ) ]sgn( q ]sgn( q ) + 3.6q ) +.93q ( N. ) ( N. ) q q > 0 < 0 LG Figure 4 shows the experiental, τ, and estiated results o joint riction, τ. 63

82 Robotica, In press, (doi: 0.07/S ), 009. Figure 4. Coparison o experiental results with riction torque obtained in Eq. (0) based on the LuGre odel 3.. Friction odel based on work-energy principle In this section, a riction copensation strategy based on the work-energy principle is proposed. As shown in the previous section, the LuGre ethod, which is an approxiate ethod or odelling the riction, requires a tie-consuing identiication procedure ncluding a large nuber o experients. Also, the riction torque deined in (0) is obtained based on the assuption o steady-state conditions. Thereore, it cannot odel the pre-sliding regie in which the stick-slip otion occurs at very low velocities. Furtherore, the riction can change with oil teperature, load, huidity and actuator wear []. However, these eects are not taken into account in the riction odel presented or the shoulder joint using the LuGre ethod. Another disadvantage o the LuGre torque given in Eq. (7) is that it assues that the stiction at the beginning and end o the otion o the anipulator are the sae, while in practice the stiction at the beginning o the anoeuvre is usually greater than that at the end. Here, a new approach is introduced or riction copensation which is identiied in only two steps. According to the work-energy principle, the work done during the otion o the lexible joint robot anipulator includes the work done by the otor torque, W I, the energies dissipated by viscous daping, W, and the Coulob riction torque, d W d. The total work done by the output 64

83 Robotica, In press, (doi: 0.07/S ), 009. controller is coposed o two parts, the work o the coposite controller torque τ c, which is W Ic WE and the work o the copensating torque o dry riction τ, which is W d I. These works can be written as where I t t t t WE τ dq = τ q dt = τ c + τ d ) q dt = τ c q dt W ( τ q dt = W + W = and t are the inal rotation o the joint and the tie at which link reaches its inal WE d I c I position, respectively, and W viscous daping is I c = t τ cq dt and t W I = 0 0 τ WE d q dt. The energy dissipated by the W d = c q dq 0 = The energy dissipated by the dry Coulob riction, W d = τ c τ c dq 0 = The principle o work and energy or a general syste is K W + = t 0 c q dt, can be written as t 0 τ K c q dt where W expresses the total work done by the orces (conservative and non-conservative) on the syste ro the initial position to the inal position, and K and K are total kinetic energy o the syste at the initial and inal positions, respectively. Since the velocity is zero at the initial and inal positions, K and K are zero and Eq. (4) reduces to Thus W = W = WI + WI Wd Wd = c 0 0 WE and the work done by τ d can be obtained as W I = W I + W c d + W d Hence, t 0 τ WE d q dt = W I c + W d + W d 65

84 Robotica, In press, (doi: 0.07/S ), 009. WE The riction copensating torque, τ, can now be obtained ro Eq. (8). There are any d candidates or the riction copensating torque unction which satisies Eq. (8). For convenience, the riction copensating torque was assued to be linear or, that is Thus, Eq. (8) becoes WE τ d a + bt = 0 0 < t < ( a / b) t > ( a / b) t ( a + bt) q 0 dt = W I c + W d + W d Paraeter a is the torque required to overcoe stiction and initiate otion o the anipulator ro the rest position. The paraeter b can then be obtained using above equation. The riction torque obtained in Eq. (9) includes the dry riction eect. To obtain the total riction torque, the viscous riction eect, c, ust be added to the torque in Eq. (9), and hence q τ WE a + bt + c = c q q 0 < t < ( a / b) t > ( a / b) WE where τ is the joint riction torque which includes dry and viscous eects. While the riction torque obtained based on the work-energy principle in Eq. () has ewer paraeters and a sipler or copared to the LuGre odel in Eq. (0), it is ore accurate in copensating or the riction. For instance, the experiental results in Section 5. or a SRLFJ show that by using WE τ the trajectory tracking error is signiicantly reduced in contrast to the results obtained by using the LuGre ethod. 4. Coposite controller In this section, a coposite controller is designed or tip trajectory tracking o lexible-joint anipulators. In the coposite controllers developed so ar to control o lexible joint anipulators, the dynaic coupling between the actuators and the links is ignored (e.g. [4] and [0]). This leads to the act that the o-diagonal ters in the ass atrix in the dynaic odel becoe zero and the dynaic odel o lexible joint anipulator is sipliied. However, ignoring the coupling between the actuators and the links leads to the error in the perorance o the controller since in singular perturbation theory the controller is odel-based and any error in 66

85 Robotica, In press, (doi: 0.07/S ), 009. the dynaic odel causes the deiciency in the control perorance. To reove this error, all coupling eects between the actuators and the links are considered in the dynaic odel developed in this paper and consequently it provides ore accuracy in the control o lexiblejoint anipulators. 4.. Singularly perturbed systes The standard governing equations o a singularly perturbed syste are [6-7] X 0 = ( X, Z, ε, t), X ( t ) = X, 0 ε Z = g( X, Z, ε, t), Z ( t ) = Z, 0 0 X R Z R where ε represents a sall paraeter, X and Z are state vectors o diensions n and, respectively, n R, g R, and X, Z, 0 X and n 0 Z are the tie derivatives and initial conditions o states X and Z, respectively. State Z is called the ast state o the syste because ε is a sall paraeter and the tie variation o Z is very large (Eq. (3)). In contrast, state X is called the slow state o the syste because it has a slower tie variation copared with state Z. For Eqs. () and (3), which represent the ( n+ ) diensional state-space equations o X and Z, depending on the paraeter ε, there is an integral aniold diensional set o equations and is deined as M ε : Z ϕ ε n = ( X, ); X R, Z R M ε which represents the - where ϕ is a unction o state X and tie. Since Z in Eq. (4) is the solution o Eq. (3), it has to satisy the ollowing integral aniold condition ϕ ε Z = ε ( X, ϕ( X, ε )) = g( X, ϕ( X, ε )) X I Eq. (5) is satisied, ast states Z can be replaced by the integral aniold ϕ( X, ε) in Eq. (). This leads to X 0 = ( X, ϕ ( X, ε ), ε, t), X ( t ) = X, 0 X n R 67

86 Robotica, In press, (doi: 0.07/S ), 009. The resultant odel in Eq. (6) is called the reduced-order odel since it includes only the slow states X and has an order o n copared to the original syste Eqs. () and (3) which has an order o ( n+ ). Thus, as long as the ast state Z oves on the integral aniold, that is Z = ϕ( X, ε), instead o the original syste Eqs. () and (3), the reduced-order odel Eq. (6) can be used to describe the dynaics o the slow states. On the other hand, i Eq. (5) is not satisied there will be the dierence η between Z and ϕ, i.e. η = Z ϕ( X, ε). In the control approach based on the integral aniold, two controllers are designed. The irst one, called the ast controller, is designed to reove the dierence between the ast state and the integral aniold, and as a result η 0 and Z = ϕ( X, ε). The second controller, the slow controller, is designed to control the slow states in the reduced-order odel (Eq.(6)). Thereore, using the singular perturbation theory along with the integral aniold concept allows or the decoposing syste into ast and slow subsystes, and controllers are then designed or each subsyste separately. Fast and slow subsystes can be obtained ro the singularly perturbed equations o the syste, Eqs. () and (3), in ters o X and η X = ( X, ϕ ( X, ε ) + η ), X ( t = X ϕ( X, ε ) ε η = ε Z ε ϕ( X, ε ) = εz ε X = X ϕ( X, ε ) g( X, ϕ( X, ε ) + η) ε ( x, ϕ ( X, ε ) + η X ), 0 ) 0 η ( t 0 ) 0 = η 0 where η is the vector o the initial condition o the ast states η. In act, Eq. (7) describes the slow variations o the state X, and Eq. (8) represents the dynaic o the ast state η. It is clear that i η = 0, Eq. (8) will be the sae as Eq. (5), which eans that the aniold condition is satisied and Z oves on its aniold. 68

87 Robotica, In press, (doi: 0.07/S ), Singularly perturbed odel o lexible joint anipulators The dynaic equations o a serial anipulator consisting o n -rigid links connected by ( n) elastic joints are described by the ollowing set o dierential equations D( q L ) q L + C l ( q L, q L ) q L + K ( q L q ) = H ( τ τ ) J q K ( q q ) = τ τ L where D q L R n n ( ) is the inertia atrix o the links, l L L L C ( q, q ) q represents the Coriolis and centriugal eects, n n K R, deined as K = diag k,, k ), represents the stiness o the ( n joints, n n J R is the inertia atrix o the otors, and τ is the vector o applied torques through the otors, respectively. Also, n n ql R and R q represent, respectively, the rotations o the links and the otors. The coponents o atrix H are deined as H = when i = j, and H = 0 when i j. The correctness o dynaic equations (9)-(30) has been ij veriied using inite-eleent analysis (FEA) or the two-rigid-link lexible-joint anipulator. The details are given in Appendix I. ij As stated earlier, the applied torque τ includes two parts; coposite torque τ c and riction copensating torque τ. That is τ = τ c + τ As a result, the joint riction can be eliinated ro the dynaic equations o the anipulator and hence, D( q ) q + C ( q, q ) q + K( q q ) = Hτ L L l J q L In the ollowing, the application o the singular perturbation theory and the integral aniold concept is applied to control lexible joint anipulators. Based on the singular perturbation approach, the syste is irst decoposed into two subsystes, slow and ast, and controllers are then designed or each subsyste. The slow subsyste represents the low-requency otion o the lexible anipulator where the working requency range depends on the lexibility o the joints. For instance, i the joint lexibility goes to ininity, the slow subsyste reduces to a rigid 69 L L L K( q q ) = τ L c c

88 Robotica, In press, (doi: 0.07/S ), 009. anipulator in which the anipulator anoeuvre will be a rigid body otion (zero requency). The ast subsyste deals with the otion o the anipulator at higher requency vibration. I all the spring constants k i are assued to be o the sae order o agnitude, they can be written as a ultiple o a single large paraeter k, ~ k = k k, i =, n i i, and the elastic joint torque is, ~ ξ = kk( q L q ) where ~ K ~ ~ = diag( k,, k n ) Without loss o generality (by rescaling the ξ variables i necessary), the atrix K ~ is chosen as ~ K = I becoes. The singular perturbation paraeter can now be deined as µ = / k, and Eq. (35) Then by using Eq. (36), Eqs. (3) and (33) becoe q L = D µξ = ql q ( C q l L + ξ Hτ ) c q = J ( ξ + τ ) c Setting x = q and subtracting Eq. (38) ro (37), one obtains x = D ( C x + ξ Hτ ) l µ ξ = J [ ξ + ( JD H I ) τ ] D ( C x + ξ ) Eqs. (39) and (40) represent the singularly perturbed odel o the lexible joint anipulators. These equations can be written as where x = a x, x ) + a ( x) ξ a ( x) τ ( + 3 µ ξ = b x, x ) x + b ( x) ξ b ( x) τ ( + 3 c c c c l ) a ( x, x = D C x, l a ( x) = D, a3 ( x) = D H 70

89 Robotica, In press, (doi: 0.07/S ), C l D x x b ), ( =, ) ( ) ( + = D J x b, 3 ) ( = J H D x b Moreover, the state-space representation o the singularly perturbed syste, Eqs. (4) and (4), can be obtained by choosing x x =, x x =, ξ = z and ξ = ε z where µ ε =. Thus c X A Z X A X A X τ ) ( ) ( ) ( = c X B Z X B X X B Z τ ε ) ( ) ( ) ( = where T n x x X ] [ =, T n z z Z ] [ = ), ( = n x x a x A, n n n n n n n n O x a O O A ) ( =, n n n n x a O A = 3 3 ) ( n n n n n n n n x x b O O O B ), ( =, n n n n n n n n O x x b I O B ), ( =, n n n n x x b O B = 3 3 ), ( 4... Recovery o the rigid robot anipulator In the ollowing it will be shown that by setting 0 = µ the equation o otion o the rigid anipulator will be recovered. In an ideal rigid anipulator, joints stiness ust tend to ininity, i.e. K, and as a result 0 µ. Thereore, the lexible joint anipulator in Eqs. (4) and (4), ust behave as a rigid anipulator when 0 = µ. Setting 0 = µ in Eqs. (4) and (4), leads to c x a x a x x a x τ ξ ) ( ) ( ), ( = c x b x b x x x b τ ξ ) ( ) ( ), ( = Solving (46) or ξ and substituting the result in (45), leads to c c x a x b x x x b x x b a x x a x τ τ ) ( ] ) ( ), ( )[ ( ) ( ), ( = Eploying the deinitions o a, a, b, b and 3 b, Eq. (47) becoes c c l l H D H D J x C D D J D x C D x τ τ ] ) ( [ ) ( = and ater soe algebraic anipulations, Eq. (48) yields (see Appendix II or details)

90 Robotica, In press, (doi: 0.07/S ), 009. x = ( J + D) [ C l x + ( H + I ) τ ] which are the dynaic equations o the counterpart o the lexible joint anipulator when the joints are rigid [8]. c 4... Integral aniold ipleentation According to the deinition o the integral aniold in Eq. (4), an integral aniold or a lexible joint anipulator in the space M n R is ε : Z = ϕ ( X, τ, ε, t); X R c n I Z oves on the integral aniold, Z can be substituted by the integral aniold ϕ in Eq. (44), and thereore The reduced order odel o the syste is then ε ϕ = B X ) X + B ( X ) ϕ B ( X ) τ X = A ( + 3 X ) + A ( X ) ϕ A ( X ) τ ( + 3 c c As discussed in Section Two, as long as Z is on the integral anioldϕ, the original governing equations o the syste, Eqs. (43) and (44), reduce to its reduced-order odel, Eq. (5). Thus, provided that the ast state Z is restricted to ove on the integral aniold, the controller can be only designed to control the slow state X in Eq. (5). It is a very signiicant advantage o using the integral aniold concept, because the controller will be designed or the reduced-order odel o diension n, while the original syste has a diension o n +. The design procedure o the controller is discussed in the ollowing section Coposite controller design procedure Here, a coposite controller torque which consists o slow and ast controllers is proposed. The slow controller, τ c, s, is designed to control the slow subsyste which presents the low-requency otion o the anipulator. I the joint lexibility goes to ininity, the slow subsyste reduces to a rigid anipulator and the slow controller will be equal to the controller torque or the rigid anipulator, τ c, 0, which can be obtained by using the inverse-dynaic approach [9]. The ast 7

91 Robotica, In press, (doi: 0.07/S ), 009. controller, τ c,, is selected to suppress the high-requency vibration o the anipulator. Thus the coposite controller τc can be written as the suation o slow and ast controller torques as τ c = τ, ( X, Z, X, Z c s ) + τc, ( X, Z, X, Z ) The dierence between Z and the integral aniold ϕ, is η = Z ϕ ( X, τ c,, ε, t) s and replacing Eqs. (44) and (5) in Eq. (54) results in εη = ε Z εϕ ( X, τ c, s, ε, t) = B ( X ) X + B ( X ) Z + B ( X ) τ { B ( X ) X + B ( X ) ϕ + B ( X ) τ } 3 3 c c, s = B ( X ) η + B ( X ) τ It is worth noting that once Z is on the integral aniold, η will be zero and as a result will be zero and the control torque has only the slow controller, i.e. τ ( X, Z, X, ) 3 c, τ =. c c, s Z τ c, Fro Eqs. (43) and (55) the equations o the slow and ast subsystes in state space are X = A X ) + A ( X ) ϕ + A ( X ) η A ( X ) τ ( + 3 ε η = ( η + τ B X ) B3 ( X ) c, c Eq. (56) describes the low-requency otion o the robot anipulator and Eq. (57) represents the high-requency otion o the robot anipulator. Designing slow controller In this section, the slow controller τ c, s is designed using perturbation ethod. For this purpose, the integral aniold ϕ and the slow controller τ c, s are expanded in ters o ε = ϕ0 + εϕ + ε + ϕ c, s ϕ τ = τ + ε τ + ε τ, + c, s c,0 c, c 73

92 Robotica, In press, (doi: 0.07/S ), 009. For a anipulator with rigid joints (i.e. ε 0), the slow controller reduces to τ c, 0 which can be designed based on the conventional ethods available or the rigid anipulators such as the inverse-dynaic approach [9]. In order to deterine the other parts o the slow controller, (i.e. τ τ, ), Eqs. (58) and (59) are substituted into integral aniold condition Eq. (5). c,, c, Thereore, ε( ϕ + εϕ + ε ϕ + ) = B ( X ) X + B 0 B ( X )( τ 3 c,0 + ετ Equating the coeicients with the sae power o ε leads to ϕ + ( X )( ϕ + εϕ + ε ϕ + ) + c, 0 + ε τ c, 0 = B ( X ) X + B ( X ) 0 B3 ( X ) c,0 ϕ ϕ + τ 0 = B ( X ) B3 ( X ) c, ϕ ϕ + τ = B ( X ) B3 ( X ) c, τ + ) ϕ B X ) ϕ + B ( X ) τ,3, i = ( i+ 3 c, i+ i = Fro Eqs. (6) to (64), ϕ 0, ϕ and ϕ i i =,3, will be ound in ters o τ c, i, iteratively. Fro Eq. (6) ϕ 0 = B ( X)[ B ( X) X + B3 ( X) τ c, 0] According to the deinition o atrices B, B and B 3, the last n coponents o vector ϕ 0 R n are zero; that is, ϕ 0 can be written as ϕ ϕ 0 = [( 0) n O n ] n n where ϕ 0 represents irst n coponents o vector ϕ 0 R. I the second part o integral aniold, ϕ, is written as ϕ ϕ ϕ = [( ) n ( ) n ] n using the deinition o the atrices B, B and B 3, Eq. (65) leads to ϕ 0 = ϕ = b ( X ) ϕ + b τ 0 3( X ) c, By setting τ 0, ϕ becoes zero in Eq. (68). Also, substituting ϕ in Eq. (63), results in c, = 74

93 Robotica, In press, (doi: 0.07/S ), = ϕ ϕ = b ( X ) ϕ + b 3 ( X ) τ c, where ϕ and ϕ represent, respectively, the irst and second n coponents o vector ϕ R n. I τ c, is now chosen as τ c, = b3 ϕ then ro Eq. (69) ϕ 0 and consequently ϕ 0. Also, the ters containing ϕ i = 3,4,, = will becoe zero by setting τ c, i = 0 i = 3,4, =, in Eq. (64). Thereore i ϕ i = 0 i = 3,4, Consequently, ro Eqs. (58) and (59), the ters o ϕ and τ c, s are expressed in the ollowing or ϕ = ϕ 0 + εϕ τ c, s = τ c,0 + ε b3 ϕ Designing ast controller As discussed in Section 4.., the ast controller ust be selected so that the dierence between the integral aniold and paraeter Z asyptotically goes to zero; that is η 0. I the ast controller τ c, is chosen to have the state-eedback orat τ 4η c, = B where B 5 is a constant atrix, then Eq. (57) becoes η = [ B ( X ) + B3 ( X )] B η ε 4 I the eleents o atrix B 4 are chosen so that the eigenvalues o the square atrix [ 3 B4 B ( X ) + B ( X ) ] always have negative real parts, the deviation o Z ro its aniold will asyptotically go to zero and, inally, Z will be on its invariant aniold. 75

94 Robotica, In press, (doi: 0.07/S ), Experiental veriication In order to veriy the design o the controller described in the previous section, the experiental single rigid-link lexible-joint anipulator (SRLFJ) available at Robotics Laboratory o the University o Saskatchewan was used. Figure 5 shows the experiental odule o the SRLFJ that was used. Figure 5. Experiental odule single rigid-link lexible-joint anipulator used 5.. Experiental setup The experiental odule o the SRLFJ included a servo haronic drive and a rigid link, as well as two parallel linear springs to represent the joint stiness. Since the parallel springs applied a torque proportional to q q ), they were odeled as a torsional spring. The nuerical values ( L o the experiental setup paraeters were j L = 0.09kg., where j L, j = 0.00 kg., c 3.6 N.. s / rad =, k = 0.3 N. / rad, l = 0. 3 j and l are, respectively, the ass oent o inertias o the link and otor, and the length o the link. The oents o inertia o the link and otor were calculated using the CAD odel, and the inoration provided by the anuacturer o the otor, respectively. The viscous daping o the otor was deterined ro the torque ipulse test. Four dierent trajectories were chosen to veriy the perorance o the controller at dierent velocities and accelerations. These trajectories were 76

95 Robotica, In press, (doi: 0.07/S ), 009. q () d = [70( t / t ) 9 35( t / t ) ( t / t ) 7 40( t / t ) 6 + 6( t / t ) 5 ] q () d = [ 0( t/ t ) q (3) d 7 = [6( t / t ) q (4) d + 70( t/ t ) 5 6 5( t / t ) = [ ( t / t 84( t/ t ) 4 3 ) + 3( t / t 5 + 0( t / t ) ] + 35( t/ t ) 3 ) ] 4 ] where the inal position was = 0. rad and the tie required to reach the inal position was 5 t = sec. In all o these trajectories the otion o the anipulator started ro ( i) ( i) q (0) = q d d (0) = 0, i =,,3, 4 and reached the inal position where q ( i) d ( t ( i) ) =, q ( t ) = 0 i =,,3,4. Furtherore, the irst and the second proposed trajectories d () were designed such that d i i q / dt = 0, i =,3, 4, and i () i d q / dt = 0, i =, 3 at t = 0 and at t = t d, respectively. Also, or the third trajectory the acceleration o the anipulator was set to d zero at t = 0 and t = t. The experiental results or each o these desired trajectories are shown in Section Friction copensating torque Based on the approach discussed in Section 3, two copensating riction torques were designed based on the work-energy principle and the LuGre ethod. To design a riction copensating torque based on the work-energy principle, as discussed in Section 3., two paraeters, a and b, ust be identiied to deterine the riction copensating torque WE τ = a bt in Eq. (9). d + Paraeter a, the breakaway orce, was obtained experientally by increasing the torque gradually, until the anipulator started to ove, as shown in Figure 6. This torque was about as. N.. To obtain the paraeter b ro Eq. (0), the work o the coposite controller torque, and the dissipated energies by coulob dry riction and viscous daping during otion o the anipulator were calculated using Eqs. (), () and (3) respectively. The values o work Wd and 7-9 as d W I c, W were obtained experientally by calculating the areas under the curves in Figures 77

96 Robotica, In press, (doi: 0.07/S ), 009. W I c =.0377 J, Wd = J, Wd = J Fro Eq. (7), the work o the copensating riction torque ust be I =0. 574J Also, two other riction copensating torques were selected as W Finally, by solving nuerically Eq. (0), coeicient b can be obtained as b = N / s Thereore, based on the work-energequation () the riction copensating torque including the viscous riction becae principle explained in Section 3. and according to τ τ WE () WE() = t q =.t q N N τ WE(3) = 0.8 t q N to ensure that the paraeter was was selected properly in Eq. (8). Another copensating torque τ LG q e ( / 0.90) = q. which was obtained based on the LuGre odel explained in Section 3.. The siulation and experiental results are shown in Figure 0 or riction torques based on the work-energy () principle and the LuGre riction torque or the desired trajectory q. The results or other N d ( ) ( 3 ) desired trajectories, q, q d d ( 4 ) and q, are shown later in this section. d Figure 6. The otor velocity response corresponding to the applied triangle torque to obtain experientally a in Eq. (9) 78

97 Robotica, In press, (doi: 0.07/S ), 009. Figure 7. The rate o the energy o the controller torque τc applied to the lexible joint anipulator to obtain W I ro Eq. () or anoeuvre = 0. 5 rad during two seconds Figure 8. The rate o the dissipated energy by viscous daping to obtain W ro Eq. () or anoeuvre d = 0. 5 rad during two seconds Figure 9. The rate o the dissipatedd energy by dry riction to obtain W ro Eq. (3) or anoeuvre d = 0. 5 rad during two seconds 79

98 Robotica, In press, (doi: 0.07/S ), 009. As shown in Figure 0a by using the riction copensating torque LG τ whose design was based on the LuGre odel, the error was signiicant, especially there was a considerable steady-state error at the end o trajectory. This error occurred since in reality the rictions at the beginning and at the end o the otion were dierent due to hysteresis [30] (see Figure 6), while riction copensating torque LG τ did not include the hysteresis eect and was the sae or the beginning and end o the otion o the anipulator (see the corresponding torque in Figure a). There was also a signiicant error between desired trajectory and link rotation when riction WE () WE (3) torques τ and τ, Eqs. (84) and (85), were used. In act, by eeding less energy into the WE () syste, using τ, the anipulator oved behind the desired trajectory and the error reained WE (3) as a negative value at the end o the otion o the robot anipulator. In contrast, by using τ, the input energy into syste was ore than the required energy to overcoe the riction and viscous daping o the syste cobined. Consequently, the link oved beyond the desired trajectory and the error becae as a positive value at the end o the otion. The experiental WE () WE (3) results or torques τ and τ are not shown here or brevity. The tracking error or WE() τ is shown in Figure 0b, where the tracking error at the end o the anipulator anoeuvre was approxiately zero; and during the anipulator otion this error reained in an acceptable range. The corresponding applied torque is shown in Figure b. Furtherore, according to the siulation results in Figure 0, the tracking error was alost zero or both cases, the LuGre odel and the WE odel. That indicates that the riction copensating torque was able to copensate or the joint riction alost copletely, and the theoretical odel developed or lexible-joint anipulator with joint-riction given in Eqs. (3) and (33) was accurate. In the experientation, however, there were other actors which were not possible to odel the exactly (such as the eect o position and load on the riction), and consequently, there were error in the riction copensation in the experients. 80

99 Robotica, In press, (doi: 0.07/S ), 009. To have a better assessent o the results, the root ean square error (RMSE) and steady-state error (SSE) are deined, respectively, as RMSE t = ( e t 0 dt) / SSE = e( t ) 00 () where e = q L q d. The RMSE shows the error between the link angle and the desired trajectory during the otion o the robot anipulator and the SSE represents the steady-state error at the end o the anipulator anoeuvre. A suary o the results in ters o the RMSE and values o SSE are illustrated in Table or dierent riction copensating torques, and the results are then WE() copared in Figure. It can be observed that the RMSE and SSE or τ have the iniu value o the RMSE and SSE and it can be chosen as the best candidate. 8

100 Robotica, In press, (doi: 0.07/S ), 009. (a) (b) Figure 0. Coparing siulation and experiental results o anipulator anoeuvre using coposite LG WE() controller and dierent riction copensation torques, (a) τ in Eq. (86), (b) τ in Eq. (83) (link angle q L and the irst desired trajectory () q d ) (a) (b) LG Figure. Applied torque to the lexible-joint anipulator o otions in Fig. 0 (a) τ, + τ, (b) τ + τ c s c, s WE() Table.Values o error nor RMSE and SSE or dierent riction copensating torques or the coposite controller or a SRLFJ experiental setup Friction copensating torque (N.) RMSE ( 0 ) 8 SSE (%) (Eq. 83) (Eq. 84) (Eq. 85) (Eq. 86)

101 Robotica, In press, (doi: 0.07/S ), 009. Figure. Coparison values o error nors RMSE and SSE or dierent riction copensating torques or the coposite controller or a SRLFJ experiental setup To veriy that the designed riction copensating torque, τ WE(), acted accurately at dierent, trajectories, the perorance o WE() τ ( ) ( 3 ) was exained or other desired trajectories q, d q d ( 4 ) and q given in Eqs. (77) to (79). The results are copared in Figure 3. As shown in this d igure, the anipulator ollows the desired trajectories accurately. Thereore, it can be concluded that the controller works with dierent trajectories, ro the sooth ninth order to less sooth third order Coposite controller In this section the eectiveness o the coposite controller designed or SRLFJ is studied. The results o coposite controller were copared with the results o the rigid controller τ c, 0. The rigid controller τ c, 0 the coposite controller was based on the assuption o a rigid joint and is shown in Figure 4a, and τ c, s is shown in Figure 4b. The corresponding response is copared with the desired trajectory in Figure 5 or the rigid controller. It can be seen that there is a signiicant tracking error between the desired trajectory and the link angle during the otion o 83

102 Robotica, In press, (doi: 0.07/S ), 009. the link ( t < t ) and a noticeable SSE at the end o the anipulator anoeuvre ( t > t ). However, by using the coposite controller torque these errors were signiicantly reduced (see Figure 6). For instance, the value o SSE in Eq. (88) was coputed as 0.06/0.50 = 5.% when the rigid controller was used, but decreased to 0.4% when the coposite controller was used. Furtherore, the value o RMSE, which indicates the tracking error during the otion o the link, decreased ro rad or the rigid controller, to.55 0 rad or the coposite controller. In act, the rigid controller was not able to reduce the tracking error during the anoeuvre, to reach a sall value beore the end o the anoeuvre ; the SSE reained large ater because o the stiction. In contrast to the rigid controller, the coposite controller had a better perorance in tracking; the tracking error relatively was sall during the anoeuvre, and the SSE reained sall ater. In both approaches, the torque WE() based on work-energy principle, τ, was eployed to copensate or the riction, as this torque provided a better result copared to the LuGre torque. 84

103 Robotica, In press, (doi: 0.07/S ), 009. (a) (b) Figure 3. Coparing siulation and experiental results o the lexible joint anipulator anoeuvre using coposite controller with link angle desired trajectory ( 4 ) q d (c) q L or (a) desired trajectory ( ) q d, (b) desired trajectory ( 3 ) q d, (c) 85

104 Robotica, In press, (doi: 0.07/S ), 009. Figure 4. (a) Rigid controller torque, τ c, 0, based on the assuption o rigid joint, (b) coposite controller torque (a) τ c, s or a SRLFJ (b) Figure 5. Coparing siulation and experiental results o anipulator anoeuvre with rigid controller torque (link angle q L and the desired trajectory () q d ) or a SRLFJ Figure 6. Tracking error, e = q () L q, between the actual link, q L d, and the desired trajectory o the link angle () q, or the rigid controller torque and coposite controller or a SRLFJ experiental setup d 86

105 Robotica, In press, (doi: 0.07/S ), Conclusion In this study, a general control strategy or tip trajectory tracking o lexible-joint anipulators with riction was presented and experientally validated. The irst part o the controller was a linear eed-orward torque whose design was based on the work-energy principle and it was developed to copensate or the riction in the joints. The design o the second part o the controller was based on the singular perturbation theory and the integral aniold, and it was developed to copensate or the joints lexibility. In contrast to other controllers whose designs were based on the singular perturbation theory, the dynaic coupling between the links and actuators was considered in the dynaic odeling o the anipulators in this study. As a result, the odel presented in this paper provided ore accurate dynaic odeling and control o lexible-joint anipulators. The siulation and experiental results veriied the eectiveness o the designed controllers. The irst part o experients was perored to deterine the eiciency o the riction copensating strategy. Coparing this strategy with other ethods such as LuGre ethod conired the eectiveness o the proposed strategy. For instance, the steady-state error, SSE, was signiicantly reduced ro 9.7% by using the LuGre ethod to 0.48% by using new proposed riction copensating torque. Also, the accuracy o the controller was veriied at dierent trajectories to ake sure that the controller acts precisely at dierent velocities and accelerations. The second part o the experients was carried out to deterine the eectiveness o the coposite controller. In these experients, it was ound that the tracking error was considerably reduced during the otion o the robot ar such that the value o RMSE was decreased ro rad or the rigid controller to.55 0 rad or the coposite controller. The axiu value o the steady-state error at the end o the anipulator was only 0.48% using the coposite controller, while using rigid controller this igure was 5.%. Thus, using the new proposed controller would be recoended or robot anipulation tasks in which high-accuracy in trajectory tracking is desired. 87

106 Robotica, In press, (doi: 0.07/S ), Reerences. R. Fotouhi, H. Salasi, S. Dezulian and R. Burton Design and control o a hydraulic siulator or a lexible joint robot, Advanced Robotics 3(6) (009).. M. W. Spong, Control o lexible joint robots: a survey, University o Illinois, Coordinate Science Laboratory, College o Engineering, UILU-ENG-90-03, February H. Diken, Frequency-response characteristics o a single-link lexible joint anipulator and possible trajectory tracking, Journal o Sound and Vibration 33(), (000). 4. B. O. Al-Bedoor and A. A. Alusalla, Dynaics o lexible-link and lexible-joint anipulator carrying a payload with rotary inertia, Mechanis and Machine Theory 35(6), (000). 5. S. K. Spurgeon, L. Yao and X. Y. Lu, Robust tracking via sliding ode control or elastic joint anipulators, Proceedings o IMechE, Part I (Journal o Systes and Control Engineering), 5(4), (00). 6. A. Benallegue, Adaptive control or lexible joint robots using a passive systes approach, Control Engineering Practice 3(0), (995). 7. K. Khorasani, Adaptive control o lexible joint robots, Proceedings o IEEE International Conerence on Robotics and Autoation, Sacraento, Caliornia (April 99), vol. (3), pp M. Readan, Flexible Joint Robots, CRC Press, Boca Raton, Florida, M.C. Han and Y. H. Chen, Robust control design or uncertain lexible-joint anipulators: A singular perturbation approach, Proceedings o the IEEE Conerence on Decision and Control,, 6-66 (993). 0. D. Wang, A siple iterative learning controller or anipulators with lexible joints, Autoatica, 3(9), (995).. C. J. B. Macnab and G. M. T. D'Eleuterio, Neuroadaptive control o elastic-joint robots using robust perorance enhanceent, Robotica 9(6), (00).. N. Jalili, An ininite diensional distributed base controller or regulation o lexible robot ars, Journal o Dynaic Systes, Measureent and Control 3(4), 7-9 (00). 3. A. Albu-Schaer, C. Ott and G. Hirzinger, A uniied passivity-based control raework or position, torque and ipedance control o lexible joint robots, International Journal o Robotics Research 6(), 3-39 (007). 4. M. Spong, K. Khorasani and P. Kokotovic, An integral aniold approach to the eedback control o lexible joint robots, IEEE Journal o Robotics and Autoation 3(4), (987). 88

107 Robotica, In press, (doi: 0.07/S ), L. Sweet and M. Good, Redeinition o the robot otion-control proble, IEEE Control Systes Magazine 5(3), 8-5 (985). 6. H. Abdellati, B. Heiann, On copensation o passive joint riction in robotic anipulators: odeling, detection and identiication, IEEE International Conerence on Control Applications, Munich, (October 006) pp G. Liu, A.A. Goldenberg and Y. Zhang, Precise slow otion control o a direct-drive robot ar with velocity estiation and riction copensation, Mechatronics 4(7), (004). 8. C. Canudas de Wit, H. Olsson, K. J. Astro and P. Lischinsky, A new odel or control o systes with riction, IEEE Transactions on Autoatic Control 40(3), (995). 9. M. K. Ciliz, Adaptive control o robot anipulators with neural network based copensation o rictional uncertainties, Robotica 3(), (005). 0. P. Toei, Robust adaptive riction copensation or tracking control o robot anipulators, IEEE Transactions on Autoatic Control 45(), 64-9 (000).. J. C. Piedboeu, J. de Caruel and R. Harteau, Friction and stick-slip in robots: siulation and experientation, Multibody Syste Dynaics 4(4), (000).. M. Grotjahn, M. Daei and B. Heiann, Friction and rigid body identiication o robot dynaics, International Journal o Solids and Structures 38(0), (00). 3. B. Subudhi and A. S. Morris, Singular perturbation based neuro-h control schee or a anipulator with lexible links and joints, Robotica 4(), 5-6 (006). 4. F. Ghorbel and M. W. Spong, Integral aniolds o singularly perturbed systes with application to rigid-link lexible-joint ultibody systes, International Journal o Nonlinear Mechanics 35(), (000). 5. J. H. Ginsberg, Advanced engineering dynaics, Cabridge University Press, Second Edition (995). 6. P. V. Kokotovic, H. K. Khalil and J. O Reiley, Singular perturbation ethods in control: analysis and design, Acadeic Press, New York, (986). 7. B. Siciliano and W. J. Book, A singular perturbation approach to control o lightweight lexible anipulators, The International Journal o Robotics Research 7(4), (988). 8. A. Ferrara and L. Magnani, Motion control o rigid robot anipulators via irst and second order sliding odes, Journal o Intelligent and Robotic Systes 48(), 3-36 (007). 9. S. Torres, J. A. Mendez, L. Acosta and V. M. Becerra, On iproving the perorance in robust controllers or robot anipulators with paraetric disturbances, Control Engineering Practice 5(5), (007). 89

108 Robotica, In press, (doi: 0.07/S ), B. Arstrong-Hélouvry, Control o Machines with Friction, Kluwer Acadaic Publishers, Norwell, USA (99). 90

109 Robotica, In press, (doi: 0.07/S ), 009. Appendix I. Dyaic equations o two-rigid-link lexible joint anipulator Equations (9) and (30) were derived using the Euler-Lagrange ethod. For instance, or a tworigid-link lexible-joint anipulator, equations (9) and (30) were written as where,,,,,,,,,,, (A) For, paraeter was the applied torque to joint i; was the ass o link i; was the ass o the anipulator s tip; was the length o link i; was the distance o the center o ass o link i ro joint i axis. Paraeters,, and were respectively ass oents o inertia o link i, anipulator s tip, hub o joint i, and rotor o joint i. Also, in equations (A) and (A) rotations o links and rotations o joints were easured in the absolute coordinate raes. The rae o the irst link (shoulder link) was a Cartesian coordinate ixed to the irst joint (shoulder joint). The rae o the second link (elbow link) was also a Cartesian coordinate syste ixed to the second joint (elbow joint), and parallel to the shoulder rae during the otion. (A) 9

110 Robotica, In press, (doi: 0.07/S ), The nuerical values used or veriication were,,,,,,,,,,,. The correctness o dynaic equations o (A) and (A) were validated using inite-eleent analysis (FEA). The results or rotations o shoulder link and shoulder joint are copared in Figure A.. According to this igure, the results o FEA and theoretical approach conir each other. Figure AI.. Coparison results o inite-eleent analysis and theoretical ethod or a two-rigid-link lexible-joint anipulator, (a) shoulder link rotation, and (b) shoulder joint rotation Appendix II. Deriving the dynaic equations o the lexible joint anipulator or 0 µ = Eq. (48) can be written as ] ) ( [ ) ( ) ( ) ( ) ( ) )]( ( [ ) ( ) ( ) ( ) ( ] ) [( ) ( ] ) ( [ ) ( ] ) ( [ ] ) ( ][ ) ( [ c l c c l c c l c c c l c c c l c c l l I H x C D J H D J D J x C D J H D J D J D J D D J x C D J H D H D J J D D J x C JD D D J D J H D H D D D J J D D J x C JD D J D H D H D J x C D J D J D D x C D x τ τ τ τ τ τ τ τ τ τ τ τ τ = = = = = = (b) (a)

111 Chapter 4. A biologically-inspired controller or tip trajectory tracking o lexible-joint anipulators 93

112 Subitted to International Journal o Robotics and Autoation, July 009. A BIOLOGICALLY-INSPIRED CONTROLLER FOR TIP TRAJECTORY TRACKING OF FLEXIBLE-JOINT MANIPULATORS H. Salasi, R. Fotouhi and P. N. Nikioruk Mechanical Engineering Departent, University o Saskatchewan, Saskatoon, Canada Eail: {haid.salasi, reza.otouhi, peter.nikioruk}@usask.ca Abstract: This paper presents a new biologically-inspired control strategy or tip trajectory tracking o lexible-joint anipulators. The social behavioural pattern o swars such as bird locks and ish schools is the inspiration to develop a new controller or lexible joint anipulators in this study. This controller, reerred to as swar control, is a sel-organized robust technique which is able to control the rotations o the joints and the position o the robot end-eector siultaneously. Thereore, in contrast to other studies in which the joints are individually controlled, and the perorance o the controller is independent o the position o the end-eector, the swar control takes into account the errors in the positions o the endeector and the joints at the sae tie. In this control schee, the paraeters o the controller are updated every tie-step based on their values at the previous tie step and also the eedbacks o the positions o the links and the end-eector. Along with the swar control, a riction copensating torque which is based on the LuGre ethod is eployed to counterbalance the eect o the riction in the joints. Veriication o the proposed controller is perored using the experiental setup o a two rigid-link lexible-joint. The experiental results show that the controller is successul in tip trajectory tracking at several dierent desired trajectories and at several dierent speeds, such that the tracking error and the steady-state errors are alost zero 94

113 Subitted to International Journal o Robotics and Autoation, July 009. during the anoeuvre o the anipulator and ater reaching to the inal desired position. Furtherore, experiental coparison with the coputed torque schee deonstrates the superiority o the swar control strategy as the axiu end-eector error is reduced by a actor o three by using the swar control schee. Keywords: Flexible joint anipulators, biologically-inspired swar control, tip trajectory tracking.. Introduction In ost robot applications, such as spacecrat and autootive industries, the control o the endeector o the anipulator (e.g. welding tool) is the ain concern. In these applications, a desired trajectory is usually speciied in the task-space and the robot end-eector (the tip) is required to ollow this prescribed trajectory. To achieve this task, a apping is deterined between the robot end-eector and the rotations o the joints, and based on the desired trajectory o the end-eector, the desired rotations o the joints are coputed. A control strategy is then designed such that the error between rotations o the joints and their desired trajectories are iniized. Thereore, the position o the end-eector is indirectly controlled. This ethod has been widely eployed in dierent control strategies. They include eedback linearization [], singular perturbation techniques [,3], integral aniold approach [4], adaptive control [5-8], iterative approaches [9], robust [0] and neuroadaptive ethods []. However, the ain drawback o these control ethods is that the joints are controlled separately, and the error in the end-eector position is not taken into account in the control procedure. Thereore, it is possible that while the error in the joints rotations are kept sall, the error at the end-eector position becoes very signiicant. This can happen because, or serial robot anipulators, the end- 95

114 Subitted to International Journal o Robotics and Autoation, July 009. eector position is a trigonoetric unction o the joints position, and not a linear unction o the rotations o the joints. The ain concept o the swar control strategy which is introduced in this paper is inspired by the real biological systes. Populations such as swars o birds or schools o ishes oten ove in coordinated but localized eorts toward a particular target []. In act, by siply adjusting the trajectory o each individual toward its own best location the swar inds its best position at each tie step [3]. For instance, when a swar o birds (or a school o ish) is oving towards a speciic destination, each bird (or ish) has to ollow a desired path so that the entire group oves towards the target. In this paper, we eploy the swar oveent concept to propose a new biologically-inspired control strategy or the tip trajectory tracking o lexiblejoint anipulators. In this ethod, the positions o the joints and the robot end-eector are controlled siultaneously. Several approaches have been developed so ar based on the swar intelligence. Kennedy and Eberhart [4] introduced the particle swar optiization (PSO) technique which is a coputationally eicient ethod o inding the optial solution or continuous nonlinear unctions. This ethod is used in [5] or reactive and power control considering voltage stability. In other work, PSO is eployed to optiize eiciently ultiple achining paraeters siultaneously or illing operation [6]. Other applications o the swar control can be ound in [7]. In this study, we develop a novel swar control strategy applicable to the lexible-joint anipulators. The lexibility o the joints ay arise ro several phenoena such as elasticity in the transission syste, belts and bearings [5], and it ay produce lightly daped oscillatory otions during the anoeuvre o the anipulator. Especially, the eect o joint lexibility becoes ore signiicant in robot anipulators driven by haronic drives. While haronic drives have zero backlash, high torque transissibility and copact size [8-96

115 Subitted to International Journal o Robotics and Autoation, July ], there is high torsional elasticity in haronic drives which liit their capability [0]. Thereore, a ore accurate control strategy in which the joint lexibility is taken into account is essential, especially in applications such as robots used in the anuacture o precision coponents (e.g. circuit boards). Another iportant eect which is taken into account in the design o the controller is joint riction. It has been proven that joint riction can lead to tracking errors and undesired stick-slip otion [-9]. Despite this, joint riction is usually neglected in odeling and control strategies developed or lexible joint anipulators, e.g. [30] and [3]. In this study, to counterbalance the eect o the riction in the joints, a copensating torque is used along swar control. The design o riction copensating torque is based on the non-linear dynaic LuGre odel and its paraeters are identiied through closed-loop steady-state experients. To veriy the perorance o the proposed controller an experiental setup o a two rigid-link lexible-joint anipulator (TRLFJ) available at the Robotics Laboratory at the University o Saskatchewan is eployed. Experient results are presented or several trajectories and the perorance o the swar control is copared to the coputed torque schee which is usually utilized or control o the anipulator with sti joints. The structure o the paper is as ollows. Section o the paper details the characteristics o the experiental setup o the TRLFJ, and gives an overall picture o the proposed control schee. The dynaics o the lexible-joint anipulators are given in Section 3, and Section 4 describes the application o the swar control in the tip trajectory tracking o lexible-joint anipulators. Section 5 explains the design procedure o riction copensating torque, and Section 6 presents the experiental results obtained using the experiental setup o TRLFJ. 97

116 Subitted to International Journal o Robotics and Autoation, July Experiental setup The block diagra o the control strategy which was used in this study to control a two-rigid-link lexible-joint anipulator (TRLFJ) is shown in Fig.. The desired trajectory and its derivatives, and, were the inputs to the controller. The positions o the otor,, and link,, were easured using encoders ipleented in the experiental setup and the velocities were then obtained by taking the derivatives o the position signals using a low-pass ilter. Eploying the low-pass ilter reoved the high requency noises which ight deteriorate the controller perorance. Also, since applying high-requency signals to DC otors would eventually daage the otor brushes a band liit o was considered or the requency o the signals applied to the otors. In our experientation, the cutting requency o the ilter was set to () which was uch saller than the band liit. The applied torque had three parts; the inverse-dynaic torque, the riction copensating torque and the swar torque ; that is,. The procedures to obtain these torques are described in Sections 4 and 5. To calculate the current required by the otor, the resultant torque was ultiplied by the inverse o the torque constant o the otor,, and the current was then ed into a D/A card. The otor torque constants or the experiental setup were and where the subscripts and stand or the shoulder and elbow, respectively. The experiental setup which was used is shown in Fig.. The anipulator included two rigid links, which were driven using two DC otors. In order to increase the torques applied to the links, two haronic drives were used, in the shoulder and elbow joints, with gear speed It is noteworthy that DC peranent agnet otors usually have a linear relationship to otor torque. 98

117 Subitted to International Journal o Robotics and Autoation, July 009. reduction ratios o :80 or the shoulder joint and :00 or the elbow joint, respectively. As shown in Fig., the shoulder link was coupled to the shoulder joint by eans o a lexible joint. At the end o the shoulder link a second haronic drive was connected to the elbow link via the lexible elbow joint. Both otors and both lexible joints were instruented with quadrature optical encoders whose resolutions were rad/count [3]. Other physical paraeters o the anipulator are given in Table Fig.. Swar control strategy or tip trajectory tracking o a two-rigid-link lexible-joint anipulator Elbow joint Shoulder joint Fig.. (a) Experiental odule o a two-rigid-link lexible-joint anipulator available in the Robotics Laboratory o the University o Saskatchewan, (b) scheatic o the top-view o the experiental setup showing the angles o the joints and the links (subscripts e and s stand or elbow and shoulder, respectively) 99

118 Subitted to International Journal o Robotics and Autoation, July 009. Table. Physical paraeters o the experiental setup o TRLFJ [3] Paraeter Sybol Shoulder Elbow Inertia o the downstrea echanis o the link 3 Mass oent o inertia o the otor Joint stiness Length o the link 3. Dynaics o lexible-joint anipulators The dynaic equations o a serial anipulator consisting o -rigid links connected by -elastic joints (e.g. = or the two-link anipulator shown in Fig. ) are () () where is the inertia atrix o the links and represents the vectors o Coriolis and centriugal torques. The diagonal atrix represents the stiness o the joints. The atrix is the inertia atrix o the otors, is the vector o the applied torques to the otors, and is the vector o the actual riction joint torques. Also, and represent, respectively, the rotations vectors o the links and the otors. The coponents o the atrix are deined as when, and when. 3 The inertia or the shoulder includes the ass oent o inertias o the shoulder and elbow links, and the elbow otor with respect to the shoulder joint. Also, the equivalent inertia or the elbow includes the ass oent o inertia o the elbow link with respect to the elbow joint. 00

119 Subitted to International Journal o Robotics and Autoation, July 009. In case all joints becoe rigid, the stiness o the joints will approach ininity; that is, and the rotations o the links and the joints will be the sae; that is. Thereore, dynaic equations () and () are sipliied as (3) where is the identity atrix. A proposed ethod or the tip trajectory tracking o the rigid anipulator is applying the ollowing torque to the joints (4) in which the applied torque incorporates three parts. The irst part is the inverse-dynaic torque which copensates or the eects o the inertial and centriugal orces; that is (5) The second part is a PID controller which is (6) where, and are, respectively, the derivative, proportional and integrative gains o the PID controller, and the third part is a riction copensating torque which is designed to counterbalance the eect o the riction in the joints. Assuing that the riction copensating torque counterbalances the real riction o the joints ; that is, the joints riction is eliinated ro the dynaic equations o the anipulator, and cobining (4)-(6) into (3) leads to the error equation (7) where. Dierentiation o (7) results in a 3 rd order ordinary dierential equation and since the inertia atrix cannot be zero (8) 0

120 Subitted to International Journal o Robotics and Autoation, July 009. It can be shown provided that the gains o the PID controller are selected properly, the syste in (8) will be asyptotically stable and the links error will exponentially approach zero. The aoreentioned ethod has been successully used in [33] or tip trajectory tracking o anipulators in which the lexibility o the joints can be ignored. However, or tip trajectory tracking o anipulators in which the lexibility o the joints is crucial, the lexibility o the joints ust be taken into account, and a ore coplicated controller is required. Siilar to the rigid controller torque given in (4), our proposed controller torque or lexible-joint anipulators has three parts (see Fig. ) (9) where the irst part o the applied torque is the inverse dynaic torque and it can be obtained ro (5). The second part is a new biologically-inspired controller which is designed based on the swar concept and is explained in the ollowing section, and the third part is the riction copensating torque which is based on non-linear dynaic LuGre, and is explained in Section Swar control The ain concept o the swar control is based on the social behavioural pattern o organiss such as bird locks and ish schools. When a lock o birds (or a school o ish) is oving towards a destination, each bird (or ish) has to ollow a desired path so that the entire group oves towards the target. This is shown in Fig. 3. Each bird (or ish) is sybolically odelled by a particle, and the center o the swar is shown at the center o the group. The swar s center can be easily ound as the average o the positions o the particles in the space; that is and where denotes the Cartesian coordinate o the swar s 0

121 Subitted to International Journal o Robotics and Autoation, July 009. center, denotes each particle s location and is the nuber o particles (e.g. or the odel o Fig. 3). The local error is deined as the error between the actual path o each particle and its desired path, and the global error shows the deviation o the group ro its desired path. Each particle has a eory and intelligence and, hence, is able to correct its position such that the local and global errors are iniized during the otion o the swar. Thereore, the swar oveent can be resebled by an optiization proble in which the objective unction constitutes the local and the global errors. The optiu oveent is achieved when the objective unction is iniized. c (a) (b) Fig. 3. (a) Scheatic o a swar oveent; each particle is shown by a solid circle (b) An actual swar o birds; the center o the swar is denoted by c To apply swar control to the lexible-joint anipulators, each link is represented by a particle and the robot end-eector is represented by the swar s center. For tip trajectory tracking, each link (particle) should ove along its desired path so that the end-eector (swar s center) can ollow the desired trajectory o the swar. Thereore, or lexible-joint anipulators, the local error is deined as the dierence between the rotation o each link and its desired rotation 0 03

122 Subitted to International Journal o Robotics and Autoation, July 009. and the global error is the dierence between the position o the end-eector and its desired position; that is () where and (shown in Fig. b) are the coordinates o the tip o the anipulator in the coordinate syste shown in Fig. (b), and, and are the coordinates o the desired position o the anipulator s tip. The block-diagra o the swar control schee is shown in Fig. 3 which constitutes our principal parts:. Variable-gain PID controller: in this coponent, the applied torques to otors are evaluated using the ollowing equation, () where, and are, respectively, the derivative, proportional and integrative gains o the PID controller. These gains are updated every tie-step in the gain generator. The ain reason to choose the PID controller to copute torque lies in its siplicity and clear physical eaning. Specially, in industrial applications, siple controllers are preerred to coplex controllers, i the perorance enhanceent by eploying coplicated control ethods are not signiicant enough [34].. Gain generator: this coponent produces proportional, derivative and integrative gains every tie-step. These gains are then used in the PID controller, equation (). Updating o the gains are based on the local and global errors and also their values at the previous tie-step; that is, (3) (4) (5) 04

123 Subitted to International Journal o Robotics and Autoation, July 009. where and are updating coeicients or the proportional part o the controller in (3). Coeicients and are or the derivative part, equation (4), and coeicients and are used to update the integrative part in (5). Variable is the tie step that is used to update the gains. Updating gains ake the swar controller siilar to an adaptive gain scheduling control schee in which the paraeters o the controller are odiied depending on local and global errors, and the controller adjusts its paraeters based on behaviour o the syste. To start coputing the gains in (3)-(5), they ust be initialized; that is, the initial values o,, and ust be speciied at. The initial starting values can be deterined ro the 3 rd order dierential equation (8) such that all roots o the characteristic equation o (8) are negative, and, hence, the stability o the controller is guaranteed. Although equation (8) has been obtained with the assuption that the joints are rigid, however, it can be used as a good starting point to estiate the gains. The values o the gains are then adjusted based on the local and global errors using (3)-(5) during the anoeuvre o the anipulator. 3. Meory: Ater coputing gains in the gain generator, they are stored in a eory. These values are then eployed in the gain generator in the next tie-step to evaluate proportional, derivative and integrative gains. 4. Error evaluator: in this coponent, local errors (links errors) and global error (endeector-error) are calculated based on the eedback signals o links rotations using equations (0) and (). Thereore, generally speaking the swar control strategy is a sel-organized robust technique in which the applied torques are regulated every tie-step based on the local and 05

124 Subitted to International Journal o Robotics and Autoation, July 009. global errors observed in the links rotations and the end-eector s position. The value o the swar torque obtained in () is then added to the inverse dynaic torque and the riction copensating torque, and the resultant torque is applied to the otors (see Fig. ). Fig. 4. Block diagra o the swar torque controller or the tip-trajectory tracking o lexible-joint anipulators 5. Friction copensation The design procedure o the riction copensating torque (FCT) is explained in this section. The ain principles o the LuGre odel and identiication procedure o the riction properties o haronic drives are explained in Sections 5.. Details o the design o the riction copensating torque are given in Section Experiental estiation o the riction paraeters A riction odel which describes dierent aspects o the riction torques in haronic drives is the LuGre odel [35]. In this odel, the proposed riction torque or steady-state conditions has the atheatical or (6) where is the joint riction torque. Ters and are the Coulob and stiction riction torques, respectively, and and are the Stribeck velocity and viscous daping coeicient, 06

125 Subitted to International Journal o Robotics and Autoation, July 009. respectively. The irst two ters o the riction torque in (6) represent the Coulob riction and Stribeck eect, respectively, and the last ter accounts or the viscous daping. To deterine the riction torque as a unction o the velocity our paraeters ust be identiied. These paraeters are,, and c which or each otor ust be deterined experientally. An experiental procedure which was designed and used to identiy these paraeters is discussed in detail in [4]. In total, 54 experients were perored or the shoulder and elbow otors to estiate the riction torque as a unction o velocity. The experiental data o the shoulder and elbow joint otors were then interpolated (curve-itted) using the unction o given in (6). Using the experientally identiied paraeters, the riction torques were expressed as unctions o the velocity or the shoulder joint otor as, (7), (8) and or the elbow joint otor as, (9) (0) The experiental data and the estiated unctions given in (7)-(0) or the shoulder and the elbow joint otors are shown in Fig. 5. As can be seen in this igure, or the shoulder joint, the overall riction torque varied ro -4 to 4 N.., when the range o the velocity was to (rad/s). For the elbow joint, the change was uch saller ro -0.7 N.. to 0.7 N.. or the sae range o velocity. Also, according to the unctions o the riction torques, the Stribeck velocity in the elbow joint otor was greater than in the shoulder joint otor. However, the shoulder joint had larger viscous daping and stiction than the elbow joint. 07

126 Subitted to International Journal o Robotics and Autoation, July 009. (a) (b) Fig. 5. Experiental results obtained or steady state conditions and estiation unction obtained based on the LuGre odel or a TRLFJ (a) shoulder joint otor, (b) elbow joint otor 5.. Friction copensation strategy Ideally, the counterbalancing estiated joint riction torque ound in (7)-(0), is equal to the real riction o the joints, and, hence, the riction o the joints will be copensated copletely. However, practically there is always an error between the estiated joint riction torque and the actual riction torque. Soe o these error sources were as ollows:. The riction torques were obtained based on the assuption o steady-state conditions, and the pre-sliding regie in which stick-slip otion occurs at very low velocities was not considered.. Although the riction can change with oil teperature, load, huidity and actuator wear [36], the eects o these paraeters were not taken into account in the riction odel presented or the shoulder and elbow joints. To reove the aoreentioned errors a PD controller along with riction torque based on (7)- (0) was used. Thereore, the riction copensating torque was, () 08

127 Subitted to International Journal o Robotics and Autoation, July 009. in which the irst ter was identiied based on (7)-(0), and constants and were proportional and derivative gains o the riction copensating torque. The stability analysis using the Lyapunov theory o this riction copensating strategy has been discussed in [37]. According to this analysis, the controller was stable provided that the appropriate paraeters and were selected. Also, it has been shown in [37] while eploying the proposed ethod in () could avoid hunting 4, it led to a steady-state error in riction copensation. The range o this error was a unction o riction paraeters, values o paraeters and, and inertias o otors. 6. Experiental results To veriy the perorance o the new controller described in the previous sections, the two-rigidlink lexible-joint (TRLFJ) anipulator, shown in Fig., was used. The physical paraeters o the experiental setup are given in Table in Section. The desired trajectory or rotations o links was considered to be the ninth-order polynoial () where, and were, respectively, the desired trajectory o the rotation o both shoulder and elbow links, the inal rotation o the link and the tie o total anoeuvre. This trajectory was designed such that it satisied the initial conditions o, and. Thus, since the trajectory had to satisy ten initial conditions, it was 4 Stick-slip oscillations around target position caused by the dierence between Coulob riction and Stiction is reerred to as hunting [38]. 09

128 Subitted to International Journal o Robotics and Autoation, July 009. chosen as a polynoial o ninth order. Experients were also carried out or other lower-order desired trajectories, however, or brieness they are not included in this paper. 6.. Trajectory tracking Several desired trajectories were tested to ensure that the controller was able to peror eectively at dierent trajectories and velocities. According to the technical anual o the experiental setup, the axiu rotations both links could reach was. Thereore, the experients were perored or inal rotations less than with steps o ; that is, inal rotations were chosen as and in dierent experients. The shape o the trajectory was the ninth-order polynoial given in (), and the anoeuvre tie was The initial values o the gains chosen were or the shoulder and or the elbow. As discussed in Section 4, the initial values o the gains were selected such that the stability o syste given in (8) or rigid anipulator was guaranteed. While these initial values were ound based on the assuption o the rigid-joints, yet, they were good starting points or our controller. This was proven through a sensitivity analysis that was perored to investigate the eect o initial values o the gains on the perorance o the controller. Details o this analysis are given in Section 6.4. The updating coeicients and used in the gain generator, equations (3)-(5), to update the gains o the swar control were and or the shoulder. For the elbow link these paraeters were selected as and. Experients were perored or several trajectories with dierent inal rotations, however, or brieness, only the trajectory tracking results or the inal rotations, and are shown in Figs. (6)-(8). As shown in these igures, the links ollowed the desired trajectory precisely such that the 0

129 Subitted to International Journal o Robotics and Autoation, July 009. tracking error between the actual and desired trajectories was alost zero or dierent inal rotations o the anipulator. This indicated that the swar controller was able to peror successully at dierent trajectories.

130 Subitted to International Journal o Robotics and Autoation, July 009. (a) (b) Fig. 6. Experiental results o swar control or two-rigid-link lexible-joint anipulator: Links rotations or ninth-order trajectory with inal rotation ; (a) shoulder link angle, (b) elbow link angle (a) (b) Fig. 7. Experiental results o swar control or two-rigid-link lexible-joint anipulator: Links rotations or ninth-order trajectory with inal rotation ; (a) shoulder link angle, (b) elbow link angle (a) (b) Fig. 8. Experiental results o swar control or two-rigid-link lexible-joint anipulator: Links rotations or ninth-order trajectory with inal rotation ; (a) shoulder link angle, (b) elbow link angle

131 Subitted to International Journal o Robotics and Autoation, July 009. To have a better understanding o the experiental data, several error easures were considered: noralized tracking error percentage, steady-state error, axiu noralized error and the tip error. The percentage o noralized tracking error was deined as, (3) where, and were, respectively, the desired trajectory o the rotation, the inal rotation o the link and the position o the link. The noralized tracking error showed the dierence between the experiental and desired values o the links rotation during the anoeuvre o the anipulator. The axiu noralized error was, (4) and the tip-trajectory-tracking error was obtained based on the distance dierence between the actual and the desired positions o the anipulator s tip. The tip position was easured during the aneuver o the anipulator using the rotations o the links as ollows (5) where and were respectively the coordinates o the tip o the anipulator in the coordinate syste shown in Fig. (b). Paraeter was the length o each link, and was the rotation o the links. Also, as entioned beore subscripts s and e denoted the shoulder and elbow respectively. Once the actual position o the tip was ound ro (5), the tip-trajectorytracking error was coputed using (). Steady-state error was easured as the dierence between the actual and desired trajectories when the response o the syste becae stable, and the noralized steady state error was equal to the steady state error divided by the inal rotation. A suary o the error values are tabulated in Table or dierent desired trajectories. 3

132 Subitted to International Journal o Robotics and Autoation, July 009. According to these values, or all cases the noralized axiu tracking error was less than while the steady-state error was less than. The end-eector error was at ost when the inal rotations o the links were. This error was or, and when. Thereore, based on these values, one ay conclude that the controller acted very successully in trajectory tracking task. The tie variation o the controller gains or the shoulder link is shown in Fig. 9, when the inal rotation o the anipulator was. It can be observed that the gains were kept constant ater a anoeuvre tie o. The reason or doing this was that because o the steady-state (constant) error at the end o the anoeuvre, the gains o the controller were linearly increased in equations (3)-(5). Thereore, according to () the applied swar torque was increasing constantly and ater a while the syste becae unstable. To avoid this, a switch was utilized in the controller so that the gains were kept constant as the anipulator reached its inal position, and hence, the controller tuned into a typical PID controller with constant gains at the end o the anoeuvre ( ). Table. Experiental results o the swar control or two-rigid-link lexible-joint anipulator: coparison o error values or dierent inal rotations Shoulder Elbow Shoulder Elbow Shoulder Elbow Max. error (rad) Steady-state error (rad) Noralized ax. error (%) Noralized Steady-state error (%) Max. end-eector error ()

133 Subitted to International Journal o Robotics and Autoation, July 009. (a) (d) (b) (e) (c) () Fig. 9. Experiental results o swar control or two-rigid-link lexible-joint anipulator: variation o the gains or the shoulder link; (a) proportional gains, (b) derivative gain, (c) integral gain, variation o the gains or the elbow link; (d) proportional gains, (e) derivative gain, () integral gain. 5

134 Subitted to International Journal o Robotics and Autoation, July Rigid controller versus swar controller In this section, the perorance o the swar controller is experientally copared to the rigid controller. The rigid controller torque, given in (4), was obtained using the inverse-dynaic approach which was a typical control ethod or robots with the rigid joints (i.e. ). In these experients the desired trajectory o the links was the ninth-order trajectory given in (). Final rotations o both links were, and the anoeuvre tie was The noralized tracking error percentage, given in equation (3), is copared or the rigid and swar controls in Fig. 0. It can be observed that the tracking error was signiicantly reduced, or both shoulder and elbow links, by using the swar controller copared to the results obtained using the rigid controller. For instance, according to the values o the axiu tracking errors in Table 3, the perorance o the swar controller was ive ties ( ) better than the rigid controller or the shoulder, and about three ties ( ) better or the elbow link. The steady-state errors were also decreased by applying the swar control; especially or the elbow, this error was reduced ro or the rigid controller to or the swar controller. Furthereore, as shown in Fig. the end-eector error was considerably decreased by eploying the swar control as the axiu error was reduced ro or the rigid controller to 3.6 or the swar controller. 6

135 Subitted to International Journal o Robotics and Autoation, July 009. (a) (b) Fig. 0. Experiental results: noralized tracking error or swar and rigid controllers (a) shoulder link, (b) elbow link Fig.. Experiental results: coparison o the end-eector error or swar and rigid controllers Table 3. Suary o experiental results or swar and rigid controllers Rigid controller Swar control Shoulder Elbow Shoulder Elbow Max. error (rad) Steady-state error (rad) Noralized ax. error (%) Noralized Steady-state error (%) Max. end-eector error ()

136 Subitted to International Journal o Robotics and Autoation, July Eect o the speed o the anipulator In this section the perorance o the controller is experientally analyzed or dierent anipulation speeds. The desired trajectory o the links was the ninth-order polynoial given in () with the inal rotation o. In the experients presented in Section 6. or (see Fig. 7), the aneuver tie was, the axiu speed o the links reached, and the axiu speed o the end-eector was. To test the controller at a higher speed, the aneuver tie o the anipulator was decreased to while the inal rotation o the anipulator was kept constant as beore; that is. In this experient, the axiu speed o the links was recorded, and the axiu speed o the end-eector was. The corresponding experiental results are shown in Fig. or the shoulder and elbow links. As shown in this igure and as indicated in Table 4, the tracking error was very sall or both links and the axiu error at the end-eector was approxiately. The experients were then repeated or ore rapid otion o the anipulator as the aneuver tie was decreased to.5 or the inal rotation o. For this case, the axiu speed o each link reached, and the axiu speed o the end-eector was about which was relatively high 5. The experiental results are shown in Fig. 3 where the tracking error between the rotations o the links and the desired trajectories was reained very sall. As indicated in Table 4, or this case the axiu error at the end-eector position reached. Thereore, according to the results presented in this section, the controller was able to work successully in tracking task at dierent speeds while 5 Note that or typical siilar size robot anipulators the axiu end-eector speed is [7] 8

137 Subitted to International Journal o Robotics and Autoation, July 009. both tracking error or the links and the end eector reained very sall during the aneuver o the anipulator. (a) (b) Fig.. Experiental results o swar control or the two-rigid-link lexible-joint anipulator: (a) shoulder link angle, (b) elbow link angle. ( ) (a) (b) Fig. 3. Experiental results o swar control or the two-rigid-link lexible-joint anipulator: (a) shoulder link angle, (b) elbow link angle. ( ) Table 4. Experiental results o swar control or the two-rigid-link lexible-joint anipulator: coparison o error values or dierent anipulation speeds or Max speed 0.9 (, Fig. 7) Max speed.3 rad/s (, Fig. ) Max speed.46 rad/s (, Fig. 3) Shoulder Elbow Shoulder Elbow Shoulder Elbow Max. error (rad) Steady-state error (rad) Noralized ax. error (%) Noralized Steady-state error (%) Max. end-eector error ()

138 Subitted to International Journal o Robotics and Autoation, July Sensitivity analysis In this section, the inluence o the initial values o the gains on the behaviour o the tievariable-gain controller is investigated. Ideally, the controller should be copletely robust and insensitive to the initial values. This eans that or an ideal controller the response o the syste did not change by assigning dierent initial gains. However, in practice, it is acceptable i the controller can work successully or a liited range o the initial gains. In the experiental results presented in the previous sections the initial values o the gains were selected as or the shoulder and or the elbow. These initial values were selected such that the syste given in (8) or a rigid anipulator was stable. For sensitivity analysis, the initial values o the gains were increased by 0%. Thereore, new initial values were or the shoulder and or the elbow. The variable gains are shown in Fig. 5. Siilar to Fig. 7, the tracking errors were still sall and the links ollowed the desired trajectories accurately in this experient. The initial values o gains were decreased by 0% with respect to the original values; that is, the new initial values were or the shoulder and or the elbow. The corresponding experiental results or the variable gains are shown in Fig. 6. According to Table 5, the tracking errors still reained very sall, and it can be observed even i the initial values o the gains were changed by, the errors did not change considerably. In particular, the end-eector error was very sall or all cases. This error was between to as shown in Table 5. Thereore, according to the experiental results one can conclude that the controller was robust to the initial conditions and the perorance o the controller was alost insensitive to the changes in initial conditions o the controller paraeters. It is noteworthy to ention that while the experiental results presented 0

139 Subitted to International Journal o Robotics and Autoation, July 009. here indicated the robustness o the controller or o the initial gains, however, the controller was stable or a wider range o initial gains. For exaple, it was experientally proven that the controller was stable or initial gains ro to or the shoulder, and to and or the elbow.

140 Subitted to International Journal o Robotics and Autoation, July 009. (a) (d) (b) (e) (c) () Fig. 4. Experiental results o swar control or the two-rigid-link lexible-joint anipulator: variation o the gains or the shoulder link; (a) proportional gains, (b) derivative gain, (c) integral gain; variation o the gains or the elbow link; (d) proportional gains, (e) derivative gain, () integral gain. (, initial gains or the shoulder and the elbow )

141 Subitted to International Journal o Robotics and Autoation, July 009. (a) (d) (b) (e) (c) () Fig. 5. Experiental results o swar control or the two-rigid-link lexible-joint anipulator: variation o the gains or the shoulder link; (a) proportional gains, (b) derivative gain, (c) integral gain; variation o the gains or the elbow link; (d) proportional gains, (e) derivative gain, () integral gain. (, initial gains or the shoulder, and the elbow ) 3

142 Subitted to International Journal o Robotics and Autoation, July 009. Table 5. Experiental results o swar controller or the two-rigid-link lexible-joint anipulator: coparison o error values or dierent initial values o the gains (, original initial gains, ) Original initial gains 0% increase o 0% decrease o initial gains initial gains Shoulder Elbow Shoulder Elbow Shoulder Elbow Max. error (rad) Steady-state error (rad) Noralized ax. error (%) Noralized Steady-state error (%) Max. end-etor error () Conclusions In this paper, a novel biologically-inspired swar controller was proposed or the tip trajectory tracking o lexible-joint anipulators. The ain concept o the controller was inspired by the social behaviour pattern o swars in which every particle corrected its position toward its best location so that the entire swar ound its best position. This idea was adopted or the control o a set o lexible-joint anipulators. The local error was deined as the dierence between the rotations o links and their desired positions, and the global error was deined as the dierence between the end-eector position and its desired one. The ain objective o the swar control was to iniize the local and global errors siultaneously. For this purpose, the paraeters o the controller were updated every tie step based on their previous values, restored ro the eory ebedded in the controller, and the local and global errors. To veriy the proposed approach, the experiental setup o two-rigid-link lexible-joint anipulator was eployed. The experiental results indicated that the controller was successul in the tip trajectory task or dierent inal rotations and dierent anipulation speeds. An experiental coparison between swar control and a typical controller used or rigid robots, inverse dynaic torque control, 4

143 Subitted to International Journal o Robotics and Autoation, July 009. deonstrated the superiority o the proposed swar control strategy. Furtherore, to analyze the eects o initial gains o the tie-variable-gain controller, a sensitivity analysis was experientally perored and it was shown that the tracking and the steady-state errors did not change signiicantly or dierent initial gains o the controller. 8. Reerences. M. W. Spong, Modeling and control o elastic joint robots, Journal o Dynaic Systes, Measureent and Control, 09(4), 987, A. Konno, & L. Dean, A singularly perturbed ethod or pole assignent control o a lexible anipulator, Robotica, 0(6), 00, Y. R. Hu, & G. Vukovich, Position and orce control o lexible joint robots during constrained otion tasks, Mechanis and Machine Theory, 36(7), 00, H. Salasi, R. Fotouhi, & P. N. Nikioruk, Tip trajectory tracking o lexible-joint anipulators driven by haronic drives, International Journal o Robotics and Autoation, 4(), A. Benallegue, Adaptive control or lexible joint robots using a passive systes approach, Control Engineering Practice, 3(0), 995, F. Ghorbel, J. Y. Hung, & M. W. Spong, Adaptive control o lexible-joint anipulators, IEEE Control Systes Magazine, 9(7), 989, S. S. Ge, Adaptive controller design or lexible joint anipulators, Autoatica, 3(), 996, J. H. Oh, & J. S. Lee, Control o lexible joint robot syste by backstepping design approach, Proceedings o IEEE International Conerence on Robotics and Autoation, Albuquerque, NM, 997, D. Wang, A siple iterative learning controller or anipulators with lexible joints, Autoatica, 3(9), 995, S. Jain, & F. Khorrai, Robust adaptive control o lexible joint anipulators, Autoatica, 34(5), 998,

144 Subitted to International Journal o Robotics and Autoation, July C. J. B. Macnab, & G. M. T. D'Eleuterio, Neuroadaptive control o elastic-joint robots using robust perorance enhanceent, Robotica, 9(6), 00, D. S. Morgan, & I. B. Schwartz, Dynaic coordinated control laws in ultiple agent odels, Physics Letters A, 340, 005, A. Ratnaweera, S. K. Halgauge, & H. C. Watson, Sel-organizing hierarchical particle swar optiizer with tie-varying acceleration coeicients, IEEE Transactions on Evolutionary Coputation, 8(3), 004, J. Kennedy, & R. Eberhart, Particle swar optiization, Conerence Proceedings o IEEE International Conerence on Neural Networks, Perth, Western Australia, 4, 995, H. Yoshida, Y. Fukuyaa, S. Takayaa, & Y. Nakanishi, A particle swar optiization or reactive power and voltage control in electric power systes considering voltage security assessent, Conerence Proceedings o IEEE International Conerence on Systes, Man, and Cybernetics, Rio de Janeiro, Brazil, 6, 995, V. Tandon, H. El-Mounayri, & H. Kishawy, NC end illing optiization using evolutionary coputation, International Journal o Machine Tools and Manuacture, 4(5), 00, R. C. Eberhart, & Y. Shi, Particle swar optiization: developents, applications and resources, Proceedings o the IEEE Congress on Evolutionary Coputation (CEC 00), Seoul, Korea,, 00, A. Albu-Schaer, C. Ott, & G. Hirzinger, A uniied passivity-based control raework or position, torque and ipedance control o lexible joint robots, International Journal o Robotics Research, 6(), 007, M. Spong, K. Khorasani, & P. Kokotovic, An integral aniold approach to the eedback control o lexible joint robots, IEEE Journal o Robotics and Autoation, 3(4), 987, L. Sweet, & M. Good, Redeinition o the robot otion-control proble, IEEE Control Systes Magazine, 5(3), 985, H. Abdellati, & B. Heiann, On copensation o passive joint riction in robotic anipulators: odeling, detection and identiication, IEEE International Conerence on Control Applications, Munich, 006,

145 Subitted to International Journal o Robotics and Autoation, July A. Lotazar, & M. Eghtesad, Application and coparison o passivity-based and integrator backstepping control ethods or trajectory tracking o rigid-link robot anipulators incorporating otor dynaics, International Journal o Robotics and Autoation, (3), 007, G. Liu, A. A. Goldenberg, & Y. Zhang, Precise slow otion control o a direct-drive robot ar with velocity estiation and riction copensation, Mechatronics, 4(7), 004, C. Canudas de Wit, H. Olsson, K. J. Astro, & P. Lischinsky, A new odel or control o systes with riction, IEEE Transactions on Autoatic Control, 40(3), 995, R. A. Al-Ashoor, Robust adaptive Cartesian control o robot anipulators using bounded uncertainties approach, International Journal o Robotics and Autoation, 0(), 995, M. K. Ciliz, Adaptive control o robot anipulators with neural network based copensation o rictional uncertainties, Robotica, 3(), 005, P. Toei, Robust adaptive riction copensation or tracking control o robot anipulators, IEEE Transactions on Autoatic Control, 45(), 000, J. C. Piedboeu, J. de Caruel, & R. Harteau, Friction and stick-slip in robots: siulation and experientation, Multibody Syste Dynaics, 4(4), 000, M. Grotjahn, M. Daei, & B. Heiann, Friction and rigid body identiication o robot dynaics, International Journal o Solids and Structures, 38(0), 00, B. Subudhi, & A. S. Morris, Singular perturbation based neuro-h control schee or a anipulator with lexible links and joints, Robotica, 4(), 006, F. Ghorbel, & M. W. Spong, Integral aniolds o singularity perturbed systes with application to rigid-link lexible-joint ultibody systes, International Journal o Non- Linear Mechanics, 35(), 000, Reerence anual o serial lexible joint anipulator, Quanser copany, Docuent No:69, Toronto (005). 33. M. W. Spong, & R. Ortega, On adaptive inverse dynaics control o rigid robots autoatic control, IEEE Transactions on Autoatic Control, 35(), 990, B. Siciliano, & O. Khatib, Springer Handbook o Robotics, Springer (008) 7

146 Subitted to International Journal o Robotics and Autoation, July C. Canudas de Wit, H. Olsson H., K. J. Astro, & P. Lischinsky, A new odel or control o systes with riction, IEEE Transactions on Autoatic Control, 40(3), 995, P. Lischinsky, C. Canudas-de-Wit, & G. Morel, Friction copensation o a Schilling hydraulic robot, Proceedings o IEEE International Conerence on Control Applications, Hartord, CT, USA, 997, H. Salasi, R. Fotouhi, & P. N. Nikioruk, On the stability o a riction copensation strategy or lexible-joint anipulators, Subitted to Advanced Robotics, March R. H. A. Hensen, M. J. G. van de Molengrat, & M. Steinbuch, Friction induced hunting liit cycles: A coparison between the LuGre and switch riction odel, Autoatica, 39(), 003,

147 Subitted to International Journal o Robotics and Autoation, July 009. Biographies Haid Salasi received the B.Sc. and M.Sc. in Mechanical Engineering ro the Shari University o Technology, Tehran, Iran, in 003 and 005, respectively. He is currently working toward the Ph.D. degree in the Mechanical Engineering at the University o Saskatchewan, Saskatoon, Canada. His research interests include Robotics and Vibration Control. Reza Fotouhi obtained his PhD in Mechanical Engineering ro the University o Saskatchewan in Canada. He is currently a proessor o Mechanical Engineering at University o Saskatchewan in Canada. His research interests include Robotics (dynaics and control), Structural Dynaics and Vibrations, Coputational Mechanics, and Bioechanics. Peter N. Nikioruk is Dean Eeritus, College o Engineering, at the University o Saskatchewan, Canada. Previously, he was Dean o Engineering or 3 years, Head o Mechanical Engineering or 7 years and Chair o the Division o Control Engineering or 5 years. He holds the B.Sc. degree in engineering physics ro Queen s University in Canada and Ph.D. degree in electrical engineering ro Manchester University in England. He received the D.Sc. degree, also ro Manchester University, in 970 or research in control systes. He served as chair or eber o ive Councils in Canada, was recipient o seven Fellowships in Canada and England and seven other honors, and eber o several Boards. His ields o research are Adaptive and Control Systes. 9

148 Chapter 5. Vibration Control o a Flexible Link Manipulator Using Sart Structure 30

149 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p Vibration Control o a Flexible Link Manipulator Using Sart Structures H. Salasi*. R. Fotouhi**, P. N. Nikioruk *** *, **, *** Mechanical Engineering Departent, University o Saskatchewan, Saskatoon, Canada; eail:haid.salasi@usask.ca, reza.otouhi@usask.ca, peter.nikioruk@usask.ca Abstract: The active vibration suppression o a lexible link anipulator using a sart structure (piezoelectric actuator) is investigated. For this purpose, a Finite Eleent (FE) odel is developed or the odal and transient analysis o a cantilever bea and a lexible link anipulator. The novelty o this work is in the developent o an accurate inite eleent odel o a piezoelectric and bea/anipulator. Also, the eect o the placeent o the piezoelectric actuator along the bea, based on the controllability o the syste states and using FE analysis, is investigated. To avoid syste instability, a collocated sensor-actuator pair is used and a proportional control strategy is eployed to adjust the voltage applied to the piezoelectric actuator so as to control vibrations. For the lexible link anipulator, it is shown that the vibration is well suppressed during and at the end o a aneuver by locating the piezoelectric actuator at the optiu location. The eect o the controller gain on the vibration behavior o the syste is investigated and the optiu controller gain is ound using two ain evaluation criteria; these are the contribution o the doinant requencies in the response and the error nors o the vibration aplitudes.. INTRODUCTION Designing and utilizing robot anipulators having higher load capacities is always desired. However, vibration is an iportant actor that restricts the perorance o such devices especially in applications where accurate positioning is very iportant. In the past decade 3

150 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p dierent approaches have been used or vibration suppression. Active vibration control is one o the best approaches to suppress vibration. One o the ethods o active control is using piezeoelectrics as actuators (Lewis and Inan, 00). Piezoelectric actuators have been successully used or vibration suppression in soe works. Khajehpour and Golnaraghi (997) developed a nonlinear controller or vibration control o a cantilever bea using piezoelectric actuators. The optiu placeent o the actuators or a cantilevered plate was proposed in (Peng et al., 005). Eect o the placeent o the piezoelectric actuator on the odal and spatial controllability o a structure was analysed in (Moheiani and Ryall, 999) based on a perorance index H. This index represents the nor o the input-output characteristics o a dynaical syste and can be used to ind the optial placeent o the actuator/sensor or plates. However, in ost o the odels developed or the control o lexible structures, the controller is designed or a particular range o requency, and it is coon practice to reove the higher odes o vibration which are lying out o the desired range o requency (Clark, 997). This approach leads to truncation errors and the closed-loop perorance will be considerably dierent ro the predicted theoretical odel. In act, by ignoring the higher odes in the assued ode shapes ethod, the zeros o the syste are located ar ro where they should be and as a result the developed odel will be dierent ro the original one. One o the ethods used to reduce the truncation error is inite eleent analysis (FEA) (Theodore and Ghosal, 995). Since a large nuber o the ode shapes o the syste are considered in FEA, the truncation o the error, due to ignoring the higher odes, is iniized in inite eleent odels provided that a reasonably enough nuber o eleents are used. The ain contribution o this paper is in the developent o an accurate odel o a piezoelectric and bea/anipulator using the inite eleent (FE) ethod and inding the optial placeent o the piezoelectric actuator along the lexible structure. Veriying the FE odel by analytical calculations and error analysis can be considered as other less iportant contributions o this paper. It is believed that in the FE odel, i the tie integration and iterative solver provide accurate solutions, the coputer siulations o the anoeuvre o the anipulators, even very lexible ones, will be quite reliable and closer to the experiental easureents. To check the accuracy o the FE odel, the natural requencies o the FE odel are calculated and veriied 3

151 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p using the theoretical approach. The optial placeent o the piezoelectric is then ound or the cantilevered bea, based on the controllability o the syste, and is then copared with the results o the FEA. In the ollowing, the piezoelectric is utilized or the vibration suppression o the lexible anipulator during the anoeuvre and ater reaching the desired position. The eect o the gain on the controller perorance is also investigated.. MATHEMATICAL FORMULATION. Piezoelectric Actuator A cantilever bea with a piezoelectric actuator, shown in Fig., was used in the study described in this paper. For perectly bonded piezoelectric actuators and assuing an Euler- Bernoulli bea, the oent induced by the applied voltage on the piezoelectric actuator is given as () where is the odule o elasticity o the piezoelectric eleent, the piezoelectric actuator constant, the thickness o the bea and the thickness o the piezoelectric actuator. and are respectively the applied voltage to the top and botto suraces o the piezoelectric actuator, and is the eective bending oent applied to the bea with an equivalent area oent o inertia. By letting, equation () becoes () I the applied voltage to the botto surace o the piezoelectric actuator is zero ( ), then ro equation () will be proportional to the applied voltage on the top surace,. I the bea is odeled as a Euler-Bernoulli bea with delection, where is easured ro the ixed end o the bea and is tie, the partial dierential equation o the syste becoes (3) where and are the odule o elasticity and density o the bea respectively., and, as shown in Fig., are the length o the bea, length o the piezoelectric actuator, and distance o the piezoelectric actuator ro the ixed end respectively, and is the Dirac unction. The 33

152 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p delection o the bea can be expressed using assued ode shapes (4) where is the th noralized ode shape, is the aplitude o the i th noralized ode shape, and is the nuber o the assued ode shapes. Substituting equation (4) into equation (3), ultiplying by and integrating, equation (3) becoes (5) where the dot indicates the tie derivative and represents the derivation with respect to. Using the orthogonality property o ode shapes and inclusion the odal daping ratio or the i th noral ode, equation (5) is written as where where (6) (7) (8) Equation (6) can be written in the state-space or (9) (0) and,, is a zero atrix o size, and is an identity atrix o size. Note that as shown in equation (8) the atrix depends on the location o the piezoelectric,. The optiu placeent o the piezoelectric can be obtained by iniizing the energy o 34

153 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p the control orce. In act, it is desired in iniizing the energy required to steer the initial state to the inal state. It eans that at the inal state all odes are well suppressed; that is and or. Thereore the inal state,, ust be zero. The value o the iniu energy unctional o the control voltage, which steers the initial state to zero, can be written as (Klaka, 99) () where is a controllability Graian atrix which is the solution o the ollowing Lyapunov equation () The controllability easure µ can be introduced as the reciprocal o the axiu value o the control energy or all initial states taken ro the unit sphere, that is, ; thus (3) where denotes the axiu eigenvalue o which is the inverse o the iniu eigenvalue o,. To ind the optial placeent o the piezoelectric actuator, the control energy in equation (3), ust be iniized or dierent locations o the piezoelectric actuator. In other words, ost controllability can be obtained when has its axiu value. Based on this approach, the optial placeent o a piezoelectric actuator will be ound in Section 4. or a cantilevered bea.. Dynaics o Manipulator The anipulator shown in Fig. has a hub at the base with a ass oent o inertia, a bea o length, a payload with ass and a ass oent o inertia. The coordinate syste is the global coordinate syste and rotates with angular velocity where the angle is the rotation o the base. A torque is applied by the hub (otor) and the ar rotates around its base during the interval tie o. Ater reaching the desired angle, the torque is reduced to zero and the ar behaves as a cantilever bea. Thus, the siulation procedure ust be perored in two steps or a rotating lexible anipulator and a 35

154 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p cantilever bea. A piezoelectric actuator is used or vibration suppression o the anipulator during the rotation o the ar and ater reaching its desired position..3 Controller Design A proportional controller was used in which the applied voltage to the piezoelectric actuator was proportional to the axial strain. A block diagra o the controller is shown in the Fig. 3 where,, and are the gain, strain at the idpoint o the piezoelectric actuator, potentioeter output voltage and applied voltage to the piezoelectric actuator, respectively. To avoid instability due to the non-collocation o sensor and actuators, the actuator and sensor were located at the sae location; that is the strain ε was easured at the location o the piezoelectric actuator. The set point (or error) was selected as zero. An iportant issue in vibration control o lexible structures is the collocation o sensor and actuator. In this study, a collocated sensoractuator pair is used. The collocation o sensor and actuator guarantees that the syste is positive real at least or lower requency odes. However, this guarantee does not apply to higher requency odes, because the collocation principle does not apply to odes o wavelength coparable to the size o piezoelectric actuator. In addition, coputational delays at high requencies can drive soe higher requency odes unstable (Falangas, 994). Thereore, the collocation o the sensor and actuator does not necessarily lead to the stability o the controller. Another actor aecting the stability criteria is the proper location o the piezoelectric which is discussed later in this paper. 3. FINITE ELEMENT MODEL Three types o eleents ro the ANSYS Sotware eleents library were used to odel the bea/anipulator. The bea was constructed using ten PLANE 8 eleents spaced equally along the bea. This eleent had eight nodes with two degrees-o-reedo (DOF), and translations in the x and y directions at each node. Since PLANE 8 did not have a rotational degree o reedo, two BEAM 3 eleents having three degrees o reedo, translations in the nodal x and y directions as well as rotation in the z-direction, were used or the base rotation. The eleent PLANE 3, which odels the piezoelectric actuator, was used as an actuator to suppress the vibration. The physical properties o the bea and piezoelectric actuator shown in Figs. and, are given as ollows 36

155 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p where, and are, respectively, the length o the bea, length o the piezoelectric actuator, and distance o the piezoelectric actuator ro the ixed end, and t b and are respectively the thicknesses o the bea and piezoelectric actuator.,, and are the odulus o elasticity, area oent o inertia, Poisson ratio and density o the bea respectively, and, and are the payload ass, ass oent o inertia o the hub and ass oent o inertia o the payload. 4. SIMULATION RESULTS In the siulation study, both odal and transient analyses were carried out or the cantilever bea and or the robot lexible link anipulator. The siulation was perored in three steps. In the irst step, the ree vibration and odal analysis o the cantilever bea was studied. The natural requencies o the anipulator which were deterined theoretically and using inite eleent analysis (FEA) are illustrated in table. The results obtained ro the theoretical approach and FEA are in good agreeent. In the next step, the eect o placeent o the piezoelectric actuator on the vibration was studied and the optial location o the piezoelectric actuator along the bea was deterined. Finally, the active vibration suppression o the robot lexible link anipulator during and at the end the anoeuvre was successully accoplished, and the eect o the gain on the vibration behaviour o the syste was deterined. In this study, it was assued that the base was ixed and the bea behaved as a cantilever. 4. Optiu location o the Piezoelectric Actuator Based on the approach described in Section., the optiu placeent o the piezoelectric was ound or the cantilevered bea. For this purpose, the irst two ode shapes were considered and the eigenvalues o the controllability Graian atrix,, were ound or dierent locations o the piezoelectric actuator using the Lyapunov equation (). Values o versus, are plotted in Fig. 4. It can be seen that the location had the 37

156 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p axiu value and provided the ost controllability. To veriy the optial placeent o the piezoelectric actuator, was perored using the inite eleent odel or dierent values o location o the actuator, the ollowing evaluation criteria were used the siulation. To ind the best (4) (5) where the nors and represent the values o the noralized tip delections or the tie intervals o and, respectively. The is the tip delection o the bea at the tie step i, is the nuber o tie steps at and is the nuber o tie steps at. Table illustrates the values o these nors or the dierent cases investigated. The values o and are plotted versus in Fig. 5. According to this igure, corresponds to the best location o the piezoelectric actuator. This inding is also consistent with the indings o (Peng, 005) or a siilar bea. 4. Flexible link robot anipulator To veriy the eectiveness o the piezoelectric actuator in suppressing the vibration o a robot anipulator, a single lexible anipulator was analyzed. In this case, the anipulator could rotate about its base. The physical properties and the diensions were the sae as those o the cantilever bea, except the values o the ass oent o inertia o the hub and the tip-ass which were selected as and respectively. The torque applied to the hub was o a bang-bang nature causing, as shown in Fig. 6.a, the anipulator initially to accelerate, then decelerate and inally to lock at its desired inal position, where it continued to vibrate as a cantilever bea. As shown in Fig. 6.b, the hub rotated approxiately 0.8 rad in one second and it was locked then at 0.8 rad. The natural requencies o the anipulator and cantilever bea were obtained theoretically and copared in table against the FEA results. Fig. 7.a illustrates the tip delection with respect to the shadow bea (see Fig. ) without the controller being active ( ). To ind the doinant requencies o the syste, the 38

157 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p FFT o the tip delections in tie was evaluated. This is illustrated in Fig. 7.b which indicates three doinant requencies. The irst was the ain excitation requency which was.0 Hz. The second requency was approxiately 7 Hz which corresponded to the irst natural requency o the cantilever bea and the third requency, which was approxiately 60 Hz, corresponded to the irst non-zero natural requency o the lexible link anipulator. To suppress the vibration a piezoelectric actuator was placed at the optiu location,, as reported in Section 4.. The siulation was carried out or three dierent values o the controller gain, and, and these are reerred to as Cases, 3 and 4 respectively. The vibration was well suppressed during and at the end o the anoeuvre as shown in Fig. 8.a or Case 3 ( ). The FFT o the tip delection in tie is shown in Fig. 8.b or this case. According to this igure, the peak values o the doinant requencies, especially the ain excitation requency and the irst natural requency o the cantilever bea, were signiicantly reduced or Case 3 in coparison with Case (Fig. 7). Three evaluation criteria, or (during the anoeuvre), or (at the end o the anoeuvre) and or (the total response), were deined so as to copare the results. The values o these nors were the noralized tip delections o the anipulator ar and were calculated using equations siilar to those reported or the cantilever bea (equations (4) and (5)). These nors are copared in Fig. 9.a or dierent gain values. As shown in Fig. 9.a, the gain value o Case 3 ( ) had the sallest values o nors and in coparison with other gain values. Thus the aplitude o vibration was saller or Case 3 than other cases, ater the anipulator reached its desired rotation during, as well as during the total response (0 ). Another index which was used to copare the results was the values o the PSD peaks at the doinant requencies. This index is shown in Fig. 9.b or our dierent cases. Also, or the three doinant requencies, overall Case 3 shows the best result in suppressing the vibration. 5. CONCLUSIONS Finite eleent analysis (FEA) was used in this paper or odeling a cantilever bea and a lexible robot anipulator. The optiu values or the controller gain were ound and the optiu location o the piezoelectric actuator was deterined or the cantilever bea based on 39

158 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p iniizing the energy o the control orce and was veriied by FEA. The controller was stable because o using a collocated sensor-actuator pair in the optiu position and because o using ull non-linear transient dynaic analysis using FEA. Also, it was concluded that Case 3 with a controller gain o Kc= 4.00e5 and the location o the piezoelectric actuator 30% o the bea length ro the base, produced the best results as ar as suppressing the vibration was concerned. These indings were veriied by analytical calculations and will be veriied experientally in near uture. 6. REFERENCES ANSYS Sotware ANSYS Inc., Canonsburg, PA, USA ( Chen C. Q. and Shen Y. (997). Optial control o active structures with piezoelectric odal sensors and actuators, Sart Materials and Structures 6(4) Clark R. L. (997). Accounting or out-o-bandwidth odes in the assued odes approach: iplications on collocated output eedback control, Journal o Dynaic Systes, Measureent, and Control 9(3) De Luca A. and Siciliano B. (99). Closed-or dynaic odel o planar ultilink lightweight robots, IEEE Transactions on Systes, Man and Cybernetics (4) Devasia S., Meressi T., Paden B. and Bayo E. (993). Piezoelectric actuator design or vibration suppression: placeent and sizing, Journal o Guidance, Control, and Dynaics 6(5) Falangas E. T., Dworak J. A. and Koshigoe S. (994). Controlling plate vibrations using piezoelectric actuators, IEEE Control Systes Magazine 4(4) Khajepour A. and Golnaraghi M.F., (997). Experiental control o lexible structures using nonlinear odal coupling: orced and ree vibration, Journal o Intelligent Material Systes and Structures 8(8)

159 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p Khorrai F., Zeinoun I. and Toe E. (993). Experiental results on active control o lexiblelink anipulators with ebedded piezoceraics, Proc. o IEEE International Conerence on Robotics and Autoation (Atlanta, GA, USA) 3-7. Klaka J. (99). Controllability o dynaical systes, Boston, Kluwer Acadeic Publishers. Lewis J. A. and Inan D. J. (00). Finite eleent odeling and active control o an inlated torus using piezoelectric devices, Journal o Intelligent Material Systes and Structures () Manning W. J., Pluer A. R. and Levesley M. C. (000). Vibration control o a lexible bea with integrated actuators and sensors, Sart Materials and Structures 9(6) Moheiani S. O. R. and Ryall T. (999). Considerations on placeent o piezoceraic actuators that are used in structural vibration control, Proc. o the 38th IEEE Conerence on Decision and Control (Phoenix, AZ, USA) 8-3. Peng F., Ng A. and Hu Y. (005). Actuator placeent optiization and adaptive vibration control o plate sart structures, Journal o Intelligent Material Systes and Structures 6(3) Theodore R. J. and Ghosal A. (995). Coparison o the assued odes and inite eleent odels or lexible ultilink anipulators, International Journal o Robotics Research 4() 9-. 4

160 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p Table. The natural requencies o the lexible anipulator and cantilever bea Flexible link anipulator Cantilever bea Set 3 Theoretical (Hz) FEA (Hz) Theoretical (Hz) FEA (Hz) Table. Evaluation criteria or dierent locations o actuator Fig.. Model o a cantilever bea with piezoelectric actuator 4

161 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p Fig.. A single lexible link robot anipulator including its tip ass and hub inertia Fig. 3. Block diagra o the controlling vibration o the anipulator Fig. 4. Controllability easure or dierent locations o the piezoelectric actuator 43

162 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p Evaluation nor Fig. 5. Nors o noralized tip delection o cantilever bea or optiu location o piezoelectric actuator (a) (b) Fig. 6.(a) Applied bang-bang controller torque, and (b) hub rotation. (a) (b) Fig. 7. Case (Kc= 0): Single-link lexible anipulator, (a) tip delection w.r.t. shadow bea, and (b) FFT spectru o tip delection 44

163 Proceedings o 7 th World Congress o Int. Federation o Autoatic Control (IFAC), Seoul, Korea, July 008, p (a) (b) Fig. 8. Case 3 (Kc= ): Single-link lexible anipulator, (a) tip delection w.r.t. shadow bea, and (b) FFT spectru o tip delection. 6.E-05 Evaluation nor PSD Index 5.E-05 4.E-05 3.E-05.E-05.E-05 Case Case Case 3 Case 4 0.E+00 (a) (b) st doinant req. (Hz) nd doinant req. (7Hz) 3rd doinant req. (60Hz) Fig. 9. Single-link lexible anipulator, evaluation criteria or dierent gain values, (a) nors o vibration aplitudes, and (b) peak values o FFT spectrus at doinant requencies. 45

164 Chapter 6. Dynaic odeling o a anipulator with lexible links and lexible joints 46

165 Proceedings o the st Canadian Congress o Applied Mechanics, Toronto, Canada, June 007 Dynaic odeling o a anipulator with lexible links and lexible joints Haid Salasi, Reza Fotouhi, Peter Nikioruk Mechanical Engineering Departent, University o Saskatchewan, Saskatoon, Canada, S7N 5A9 haid.salasi@usask.ca, reza.otouhi@usask.ca, peter.nikioruk@usask.ca. Introduction A wide variety o robots is now eployed in dierent areas such as spacecrat and vehicle anuacturing. However, designing and utilizing lighter robots with lexible ars has always been desired. Flexible anipulators have a nuber o advantages: they require less aterial, weigh less, consue less energy, are ore transportable and aneuverable, cost less and have a higher payload to robot weight ratio []. In this paper, the cobination o the Lagrange ethod and assued ode shapes ethod (LAMM), has been used or the dynaic odeling o lexible anipulators considering lexibility in the joints and links. The odel is then veriied by ull non-linear inite eleent analysis (FEA).. Matheatical Model. Kineatics o lexible anipulator A anipulator consisting o lexible links interconnected with lexible joints was considered. The odel o the i th lexible-link lexible-joint anipulator is shown in Figure. The rae 47

166 Proceedings o the st Canadian Congress o Applied Mechanics, Toronto, Canada, June 007 is attached to the rigid-like links and represents the rotating rae. Angles and are the otor and link angles respectively easured with respect to the rigid rae. The velocity o an arbitrary point along the bea,, can be written as () where is the absolute velocity o the coordinate syste, is the relative velocity o point with respect to the coordinate syste, is the absolute angular velocity o the coordinate syste, is the position vector o the point, and is the link delection, where and is the length o the i th link. The assued ode shapes ethod can be used to approxiate the link delection as ollows () where is the nuber o ode shapes used or each link, and is the weight o the assued ode shape. Figure. Model o the i th lexible-link lexible-joint anipulator 48