[n recent years, interest in mobiIe robotics has ken growing. The study of ...... New NN on-line weight updating niles were derived that may guuantee the overrill.
by
SHENG LIN
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mechanical and Industrial Engineering
University of Toronto
@3qy-ighr by Sheng LIN 2001
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My parents, my unch, my 6rottiers andskters In the Teoplé's @pu6liic of China
Robust and Intelligent Control of Mobile Manipulators Abstract In recent years, there has been considerable research performed in the area of mobile manipulators subject to kinematic constraints. This thesis addresses the position control problem of mobile manipulators subject to kinematic constraints. The main objective is to develop practical controllers that c m be easily implemented on 3 mobile münipulator in the presence of parameter mceaainty and unmodeied disturbances. such as terrain
irregularity and unknown payload. In this thesis, two mobile manipulator controllers are developed: (i) a neural networkbased hierarchical intelligent controlier; and (ii) a hierarchical robust controller. c ~ h aiming at different control purposes. The proposed hierarchical intelligent controllcr is based on rndicil bnsis fimctions. As such it has very good learning ability and adaptability, thus it is especiaily suitable for applications where mobile robots move in unknown and changing environments. When the adaptability is less demanding (e.:.. the environment is partially or completely known), or when the availüble systsm re5ourct.x such as size of embedded memory and on-board computational capability are restrictr J. the hierarchical robust controller is suggested as an alternative. It has a simple control structure and a low computational load, Both proposed controllers require the measurement of velocity signals. In prricticr. velocity sensors may have to be ornitted due to considerations of size and weight. In t h i h case the velocity signals are deterrnined by a first-order numerical difterentiiition of available position signals. This may introduce significant noise in the control signal ;inil degrade its performance. To resolve this problem, in this thesis we propose
3
nt.1~
reduced-order adaptive observer-based velocity measurement technique requiring unly position measurements. Rigorous stability and performance andysis of the developed theory is also provided in the thesis. In addition, extensive experiments were conducted on a four-dezree-ofCreedom robotic manipulator. The experimeneal resuIts validate the efiectiveness of the proposed controllers.
III
Acknowledgements i wish to chank my supervisor, Professor Andrew A. Goldenberg, for his continuou?;
support. insightful supervision and warmhearted encouragement throughout this reserirch program. i lemed from him not only how to make a successful research career. but A u how to live a meaningfd life. My Departmental Defense Committee members (Profs. Andrew A. Goldenber:. William L. Cleghorn and R. Ben Mrad) have provided valuable suggestions on the contents of this thesis. Fmal comments by Prof. William A. Gruver of Simon Fraser University (External Examiner) and Prof. Chu1 B. Park (Mechanical and Industriai Engineering Department) were invaluable. 1own many thanks to each one of them.
I am also indebted to Dr. Tian Lin and Dr. iraj Mantegh for their valuable advicr in choosing my research topic when 1 fust started this program. Their advice has srivsd nie from many blunders. 1 am very much thankful to Mr. William Wael Melek. Mr. John Yeow, Mr. Edrnir
Hasanaj, Mr. Neii Tischler, Mr. David Pitts, Ms. Wen Chen and Mr. Yuan Zhrins for their help and for the wondehl discussions we have had on various issues diiring the p s t years. i am honored that Figures 2.3, 3.2 and 6.2 are in William's hand. Moreover. he has provided many valuable suggestions for revising my thesis. Thanks rire also diiti to
Ms. Elizabeth Catdano for her kùid help, and to di my coileagues in the Robotics and Automation Laboratory at the University of Toronto for their support and for providing a friendly environment. I would k e here to express particuIar appreciation to Dr. Baoli Ma
rit
Beijing
University of Aeronautics and Astronautics, P. R. China, whose attractive introduction on robotics and enthusiastic encouragement successfully persuaded me to join the robotic research comunity. Findiy, but most imponantly, 1 wodd extend my profound and loving gratitude to
my parents, my undes, my brothers and sisters, and especialIy to Ms. Yoko Aizriwri. for their patience and support. It is to them that this thesis is dedicated.
Contents Abstract......................................................................................................III Acknow ledgements .................................................................................... IV Contents ...................................................................................................... V List of Figures ...........................................................................................VI1 List of Tables ............................................................................................. IX Notations .................................................................................................... S 1 Introduction ......................................................................................1 1.1 Background ....................................................................................... 1 1.2 Problem Statement ............................................................................4 1.3 Literature Survey...............................................................................5 1.3.1 Modeling and Control of Mobile Manipulators ...............................................i 1.3.3 Neural Nehvork-based Robotic Control............, , , . ......................., , .9 1.3.3 Robrrsr Robotic Control............................................................................... I I 1.3.4 Observer-bmed Velocity Measurement for Roboric Control ........................12
1.4 Contributions................................................................................... 13 " . 1.5 Organization.................................................................................... 14
2 Mathematics Background .............................................................. 15 2.1 introduction .................................................................................. 15 . 2.2 Mathematical Pretirninaries ......................................................... 13 2.3 Background on Neural Networks....................................................18 2.3.1 Mathematical Mode1 of (1 Newon ................................................................19 2.3.2 Multilayer Perceprrons anci Radial Basîs Fttncrion Nenvorks.................. 3) 11 2.3.3 Cornparisons Benveen MLPs and RBF Nmvorks ......................................
?-' 2.4 Dynarnic Modeling of Mobile Manipulator ..................................... -Li 2 - 4 1 Dynamic Mode1 of Consrrained Mobile Manipirlaror.................................24
2.4.2 Dynarrric Mode1 of Vehicle Base .................................................................-3 2.4.3 Dyncrmîc Mode1 of M(miprr1ator A m .........................................................30
2.5 Summary .........................................................................................
3 1
3 Hierarchical Intelligent Control of Mobile Manipulator...,.. 34 3.1 Introduction .....................................................................................34 3.2 A Hierarchical Intelligent Controuer for Mobile Manipulator ..........3 ï 3.3 Decision Level Design ....................................................................33 3.4 Leaming Level Design .................................................................... 39 3.4.1 Lyapirnov Function Design of Vehicle Base ................................................. 40 3.4.2 Lyapcrnov Frrnction Design af Manipulator Arm .......................................... 42 3-43 Lyaprrnov Function Design of Mobile Manipulutor ..................................... 43 3.44 RBF Nehvork-based On-line Estimator Design .......................................... 46
Contents
3.5 Adaptation Level Design ................................................................. 49 3.6 Execution Level Design................................................................... 51 -3 3.7 Sumrnary ......................................................................................... 34 Hierarchical Robust Control of Mobile Manipulator....................... 53 4.1 Introduction..................................................................................... 3~ 4.2 A Hierarchical Robust Controller for Mobile Manipulator .............. 54 4.3 Robust Damping Control of Manipulator Arm ................................ 56 4.4 Robust Damping Control of Vehicle Base ....................................... _IL) 4.5 Robust Damping Control of Mobile Manipulator ............................ 65 -?
4.5.1 Dynamic Parclmeters P~irtinilyKnown ........................................................ fi5 4.5.2 Dynatnic Paranterers Complereiy Unknom.................................................hi)
4.6 Summary ..................................................................................
71
5 Reduced-order Adaptive Velocity Observer..................................74
5.1 Introduction..................................................................................... 7-1 5.2 A Velocity Observer-based Robotic Controlier ............................... 7h 5.3 Reduced-order Adaptive Velocity Observer Design ........................ 7h 5.4 Combined Observer/Controller Design ............................................ 79 5.5 Summary ......................................................................................... 8,3
6 Experimental Validation ................................................................. 83 6.1 Introduction................................................................................. 83 6.2 Experimental Setup ................................ , . . ................................84 6.2.1 IRIS RoboTwin ........................................................................................... 84 6.2.2 Dynamic Mode! of RoboTwin ................................................................... SS 6.2.2 Eiperimen t Design ...................................................................................... 8. 6.3 Neural Network ControIler .............................................................. XS
-
6.4 Robust Damping Controuer.............................................................97 6.5 Observer-based Controller............................................................. 107 6.6 S u m m q ...................................................................................... 112 7 Contributions and Future Research ........................................... 114 7.1 Contriutions ................................................................................. 114 7.2 Future Research............................................................................. LI6
Bibliography ................................................................................... 118 Appendix .......................................................................................... 130
List of Figures FIGUREi .I: MR-1 IN A ~ O ........................................................................................... N 2 FIGURE 2.1.
MATHEMA~ICAL MODEL OF A NEURON
FIGURE 1.1.TWO-LAYER NEW
..............
11 .--
. . , , . . . . . . . . n . n . . . n n . . . . . . . . . . . . . . . . . . . . . .
NETWORK ...................................................................
F~GURE 2.3. A WHEELED MOBILE PLATFORM ....................
13 -2
.....................................31
................:7 FIGURE 3.2. A bibi-BASED INTELLIGENT CONTROLLER FOR MOBILE bIANLPUL4TOR ..........2't )
FIGURE3.1.
~ C H I C ; V . I N T E L L I G E N T C O N T R O L O F M O B ü I MANIPUWTOR
FIGURE4.1 : HJERARCMC AL. INTELUGENTCONTROL OF MOBaE MANIPUUTOR ................ 35 FIGURE 4.2.A m C CONTROLLER FOR MOBILE M A N I P U T O R ........................................ 36 FIGURE 4.3. RDC CONTROL OF W P L F L A T O R ARM ........................................................
F~GURE 4-4: mC CONTROL OF MOBiLE PiATFORM ................... .,.......
--
I
.....................62
FIGURE 4.5: WC CONTROL OF M O B U MANPULATOR WlTH PARTIAL KNOWN DYNAMICS PARAMETERS .............................. .,. ......................................... (17 FIGURE 4.6: W C CONTROL OF MOBILE MANIPüIATOR WiTH DYNrLMiC P.4RAMETERS COMPLETELY LNKNOWN.............................................................................. 70
FIGURE 5.1:ARCHlTECTURE OF THE PROFOSED OBSERVER-BASED ROBOT CONTROLLER ...76 F~GURE 5.2. AN OBSERVER-BASED W
R ESTbIATED STATE FEEDBACK CONTROLLER ....Y O
FIGURE 6.1 :ROBOTWIN(Ris FACIL~TY) ..,.......
s4
FIGURE 6.2. SOFIWARE ARCHITECTURE OF ROBOTWIN ...............................
Sb
FIGURE6.3. DESIREDTWECTORIES OFTHEFOUR JOINTS ( W C ) .................................... 'II FIGURE6.4. P O S ~ERRORS N O F J O W 1 (WC)............................................................ LII: FIGUE 6.5. POSKIONERRORS OF JOUVT2 (WC) ............................................................ 9 3 FIGURE 6.6. P O S ~ OERRORS N OF JOINT 3 (NNC)............................... FIGURE 6.7. P O S ~ OERRORS N OF JOINT^ (WC)...............................
, . . .
, . . .
FIGURE6.8. DRIVINGT O R Q OFOF ~ JOW 1 (NNC)...................................................... 95 FIGURE 6.9. DRNINGTORQL~ES OFOF JOINT^ ( W C ) ...................................................... 05
FIGURE6-10: D
~ TORQUES G OF OF JOINT 3 (NNC) ....................................................
96
FIGURE6:11: D ~ G T O R Q UOFOFIOINT~ ES (WC)........................ G...........................9~ FIGURE6.12. DESIREDTRAJECTORIES OF THE TWO JOINTS (REGULARTRAJECTORIES).... 100
List of Figures FIGURE 6.13:POSITIONERRORS OF JOKNT 1 (RDC-r)..................................................... 100
FIGURE 6.14. V E L O C ERRORS ~ OF I O N I (RDC.0. ................................................... 101 FIGURE 6.15. DRIVINGTORQUES OF JOINT 1 (DC-E).................................................... 101 FIGURE 6.16. POSITIONERRORS OF JOINT 2 (RDC-U ..................................................... 102
FIGURE 6.17. V E L O CERRORS ~ O F ~ O I N T(m-I) ~ ................................ ........
102
FIGURE 6.18. DRIVINGTORQUES OF JOINT 2 (RDC-r) .................................................... 103 FIGURE 6.19: DES-
.
TRNECTORES OFTHE TWO l O N S ( ~ R E G U W RW E C T O R I E S I 103
(RDC-I) ................................................... I0-I F~GURE 6.1 1: VELOC~TYERRORS OF JOINT 1 (RDC-1) ................................................. 104
FIGURE 6.20. POSITIONERRORS OF JOINT L
F ~ G U R6.22. E DRIVINGTORQES OFJOINT 1 (RDC-II) ................................................... IO5
F~GURE 6.73. POSITION ERRORS OF J O W 2 ( RDC-fl)....................................................IO5 FIGURE 6.74. V E L O C ERRORS ~ OF
JOINT^ (DC-LI)................................................... 100
....i06
FIGURE 6.25: DRIVING TORQUES OF JOINT 2 (RDC-JI)...............................................
FIGURE6.26. DESREDTRAJECTORES OF THE TWO JOINTS (OBC).................................. 1(ICI FIGURE6.27. P O S ~ OERRORS N OF JOINT 1 (OBC)........................................................10') FIGURE6.78. V E L O CERRORS ~ OF JOINT 1 (OBC)......................................................i l ( ] FIGURE 6.29. D W ~ GTORQUES OF JOINT 1 ( O 8 C ) ................................................... 110 FIGURE 6.30. P O S ~ O ERRORS N O F J O U V T ~ (OBC) ..................................................... 1 1 1
FLGURE 6.3 1: VELocrrY ERRORS OF JON 2 (OBC) ...................................................... I I I FIGURE 6.32. DRNINGTORQtlES OF JOINT 2 (OBC) ....................................................... II7
List of Tables TABLE 6.1: E X P E R ~ N T AROBOT L PARAMETERS ( W C ) ............................................. 8 9
TABLE 6.2: PERFORMANCE COMPAFUSON BFCWEEN STC AND W C ...............................
‘10
TABLE 6.3: PERCENTAGE iMPROVEEIlENT IN PERFORMANCE OF M C OVER STC............... 90 TABLE 6.4: EXPERLMENTAL ROBOT PARitMEïERS ( W C ) .................................................L~7 TABLE 6.5: PERFO~IANCE COE~IPARISON BETWEEN STC AND RDC ( I) ............................1)S TABLE6.6: PERCENTAGE IMPROVEMENT INPERFORMANCEOF RDC OVER STC ( 1) ..........ils
TABLE 6.7: PERFORMANCE COMPARKON BETWEEN STC ANI3 RDC (II) ............... TABLE6.8: PERCENTAGE IMPROVEMENT [N PERFOWCE
OF RDC OVER STC ( lI).........C)C)
TABLE 6.9: E X P E ~ E N T AROBOT L PARAMETERS ( O B 0 ............................................ 107
TABLE 6.10: PERFORMANCE COMPARISON BETWEEN Pm AND OBC ..............................1OS TABLE 6.1 1: PERCENTAGE MPROVEMENT IN PERFORMANCE OF Pm OVER OBC ............ 1 OS
Notations Kinematic constraint mat& Transpose of mtrix A Centripetai and Corioüs matrix
Continuous function space Center and width of a gaussian function Input transformation matrix
Position error, velocity error and acceleration error Observer tracking error Observer estimation error
Smooth function
Friction and gravitational vector Radial basis function n x n identicy rnacrix Controi gains Lyapunov funct ions
Inenial mass macrix Generalized courdinates Constrained generaiized coordinates
Free genenlized coordinates Desired positions Desired velocities Desired accelerations Filtered cracking error
Real number space Real n-vector space Nuii spiice of consmint muix A(qv)
Notations
Tirne Trace of a matrix Centripetai and Corioiis matrix Auxiliq velocity vector Weight matrices Desired task-space trajectory Cartesian coordiiates and orientation of the mobile plat form Position tracking error of mobile platform Reference smooth feedback velocity vector Leaming rate Positive constant Bounded and unknown vector approximation error Bound of approximation error Lagrange multiplier vector Positive numbers Nonlinear activation hnction Torque vector Disturbances Bounds of disturbances Actuated torque vector of the free coordinates Actuated torque vector of the constrained coordinates Robust damping vector
RBF basis set Unknown nonlinear dynarnics Nom of a vector or mtnx Frobenius n o m of a matrîx
CHAPTER 1
IlIIII.
1 ntroduction
1.1 Background Mobile robots have found wide-ranging appIiciitions in the past two decades, Typic:il applications are: Explosives Disposai Robot (ES1 Company Protile. 7000). Clir~hirir Robots (Ami 1996), Leg- Wieeled Robots (Dai 1996), Mobile Posr Syste1r1.s iVsstli 1Wh 1. Cuopercrrive Mobile Robots (S hiroma 1996), Autonornorrs Giiided Vrhiclrs ( Xrrii 1Wi 1 that cm perform transponation tasks in factories, warehouses. office buildings i King 1991), Healtlz Cure Robots (Fiorini 1997), and autonomous robots in h i i ~ d o i i ~ çnvironments. such as underwater robots (Canudas et al. 1998) and space rtinger telerobots (Seraji 1998). [n recent years, interest in mobiIe robotics has k e n growing. The study of mobile robots spans many different research domains, such as planning, navigation. control.
;incl
coordination of locomotion and manipubtion. The navigation (Koren and Borenstein 1991. Singh and Keller 1991) and planning (Laumond 1987, Barraquand and Latombe 1991) of mobile robots have k e n investigated extensively over the past decade. However, the work on dynamic control of mobile robots with nonholonomic constrriinth. such as wheeled mobüe manipulators, is more recent (Canudas and Roskam 199 1. Samson and Ait-Abderrahim 1991, Hootsmanns and Dubowsky 1991, Yamamoto 19941.
In this thesis, dynamic control of mobile robots subject to kinematic constrriints. specificaiiy mobile manipulators, will be the main concern. A mobiie manipuIator is a robotic arm mounted on a moving base (see Figure 1-11.
The base mobility substantidly increases the size of the robot workspace. rurd enables better positionhg of the miinipulator for efficient task execution. Typicd examples of
Chanter 1: Introduction
-7-
mobile manipulators are tracked robots, compound robots and wheeled robots. Such mobile manipulator systems have been suggested for various applications. for instrincti. tasks involving explosives, hazardous envkonments such as waste management. outdoor exploration and space operations.
Figure 1.1: MR-1 in action (Courresy of Engineering Services. [nc.. ri.wr..rsir.cwn~~
The moving base of a mobile nianipulator is subject to holonontic ( r . , ~ .trrickecl . robots) or nonholonomic (e.g., wheeled robots) conscraints (Murray er cil. 1993. Criniidn~ er cd. 1996, Zhang 1993): its end-effector may also be subject to kinematic constr;iint.+
when it is required to follow certain task-space profiles or moving surfaces (Yrimtimoto 1994), rendering the control of the mobile manipulator very challenging. Furthermore. the cornplex system structure, the highiy coupled dynamics between the mobile base ancl the mounted manipulator am, and the mobility of the wheeled mobile base are some of the features that substantially increase the dficulty of system design.
There are generaily two types of approaches for the control of mobile manipiilacon: decentralized control and centralized control. in the decentralized control approxh. the vehiçle base and manipulator arm are conuolled separately, neglecting the dynzlmic interaction between the two, Such strategy is appropriate when the coupled dynamics ih not significant (e-g., when the mobile robot moves at a relatively low speed), or when the
Chapter 1:Introduction
7
- 3 -
vehicle base and manipufatorarm exhibit significantly differenc dynamic charrictrristics. such as tirne-response property. Thus, control design of mobile manipulators is rebat ively
easy since both the conmi problem of vehicie bases and that of manipulator ürms havtl k e n weii understood (Kolmanovsky and McClamroch 1995, Canudus rr cil. 19961. However, when mobile robots move at a high speed or on uneven terrain. neglrctins the coup1ed dynamics may significantly degrade the contrai performance and cause undesirable arm-shaking phenomenon or even tip-over instability ( Dubowsky üncl Vtincc.
1989).
In the centraiized control approach, the mobile manipulator is regarded as a redundrinr robot, where the redundancy is htroduced by the motion of vehicle base. X centrrrlizrcl controiier is developed for the combined vehicle/rnanipuiator systern. and the redtindanr control technique is adidopted in rnanipuhbility and dexteriry analysis. coordiniition ot' locomotion and manipulation. and optimal configuration design. However. ignoring rhc kinematic constraints may result that the developed controiier is unable to implemcnt prescribed tracking tasks, In addition. neglecting the significant difference of dynaniic ctiaracteristics between the vehicle base and manipulator arm may make the controt
design not optimal, and the derived high-order controUer unnecessarily cornplex and verdifficult to hplement in real systerns because of its heavy computation overioad.
The mobile manipulator control strategy developed in this thesis is difkrent from both the centralized and decenrralized conwol approaches. In our
SCheme.
rnodels rhr
describe the vehicle and mruiipulator sub-systems are f ~ s derived t fiom the rinilied dynamic mode1 of constrained mobile manipulators usine the Lyapuno v dr sig ri technique. Based on them, vehicle and manipulator sub-controllers are devebped whiçh control the vehicle brise and mruiipulator arm separately, taking into account the coupIect dynamics between the two and the kinemsitic constraints an robot motions. such a.ç nonhobnomic kinematic constraints on the vehicle base and constraints on the end-
effector when it is required to foiiow certain task-space profdes. The proposed controltrr inhents the advantages of both centralized and decentraiized control approaches, rit
the
sarne time avoids the drawbacks of either of them.
In the foliowing sections, the problern attacked in this thesis, ix.,motion control of constrained mobile manipulators with dynamic iincertahties and disturbances. ts t'irst
-4-
Chapter 1: Introduction
deftned. Then, a comprehensive literature survey on the relevant reselirch m a is provided. The theory developed in the subsequent chapters is also briefly introduced. The outiine of the thesis is given in the final section.
1.2 Problem Statement In the literature, control problems of mobile mnipulator are largely simplitled b\: eithedor: (i) ignoring the kiematic constraints on its motion ( c g . . nonholonomic constraints on wheeled vehicle bases): (ü) assuming that a precise knowled_oe ot' the dynamics of mobile rnanipulator is available: and (iii) neglecting the vehicle dynrimics and dynamics of interaction between the vehicle and manipulator. The integration of thc nonholonomic constraints and the dynamics of a mobile manipulator in the contri11 problems has not been studied extensively. Moreover, even less attention ha k e n ziven to ~ h econtrol and robustness of mobile manipulators in the presence of dynamic uncertainties and disturbances. Furthemore, most existing schemes demand
ri
preçiht.
knowledge of the systern states, and in real applications. sorne states. such u: velocit!. signal. may not be available for measurernent. Ali these issues could make the esisting control schemes inappropriate for real applications. This thesis addresses the position control problem of mobile manipulators sribjrcr to
kinematic constraints (holonomic and nonholonomic) in the presence of tinkiiowrt parameters and disturbances, such as terrain irregulariîy and unknowrr payloarl. The
main objectives of the research are the following: Development of a neural network-based hierarchicai controller for mobile manipulators that takes into account the kinematic constraints and providcs adaptabüity for rnovement in unknown and changing environmenrs when unknown dynamics due to parameter uncenainties and disturbances is present. ys t guaraniees closed-loop stability of the overdi systern; Development of a hierarchical robust controiier for mobile manipularors with dynamics uncertainties which is especiaiiy suitable for appiications when the system resources, such as sire of embedded memory and computation capabilicy of on-board compter, are restricted, or when the mobiIe robot moves in partiallv
Chaoter 1 :Introduction
- 7- -
or completely known environments where the learning and adaptation capabilit ies of the controller are less cruciai: 3j Design of a reduced-order adaptive velocity observer for robotic control when
o d y the position signals are available for merisurement and feedback. without requicing the knowledge of robot dynamics parameters.
1.3 Literatuie Survey In this section, a survey of the previous work reIated to the research topics ment ionticl above is provided.
1.3.1 Modeling and Control of Mobile Manipulators A mobile manipulacor is a constrained mechanical system comprising a rnoving plntform subject to kinematic constraints, and a multiple degrees of tieedorn i DOFi manipulator arm mounted on it, Modeling of a general robot arm and various types o i mobile robots can be found in Lewis et al. ( 1993), Gotdenberg er cd. ( 19991 and Canudah e t ai. ( 1996).
Lew and Book ( 1992) derived a closed form of the motion equat ions for
serially connected mnipulators that c m reduce the number of computations significrintly
and show the structure of the coupting dynamics between the two mrinipulritors. Yamamoto (1994) developed a complete dynamic model of a mobile manipulator subjcct CO
nonholonomic constraints using the Langrange approach. Yun ( L998) providrd a
unified modeiing scheme for both holonomic and nonholonomic systerns. Thc instaneaneous bernatics and dexterity of mobiie manipulators was considered in Tchon and Muszynski (2000). Pitts, Emami and GoIdenberg (1999) descnbed a step by step validation procedure of a mobile mmïpulator simulation which inciuded performing basic model simulations, a sensitivity analysis, and a stochastic output anaiysis. Motion control of mobile robots can be divided into open-loop control and closerlIoop control. Open-loop conml requires the generation of proper motions that crin makc the mobiie robot achieve the desired objectives without using feedback. However, oprnIoop concml may not be capable of dealing with the applications when the robot needs ro interact with the surroundings, such as moving in unknown terrain or c h q i n ~
Chaptsr 1 : tntroduction
-
6-
environment full of dynamic obstacles. In this thesis, we are interested in the closed-loop stable control of mobile robot systems. A wheeled mobile manipulator is a nonholonomic system the base of which is
subject to nonholonomic kinemritics constraints. Nonholonomic systerns have brlen studied extensively during the last two decades (Goldstein 1981. Kolmanovsky rinrl McClamroch 1995). Brockett (1983) proved that for a nonholonomic system without drift, there exists no smooth static state feedback control Iaw that uymptotically stabilizes the system Therefore, alternative approaches have k e n proposed to rivoid violating Brockett's theorem for the control of mobile robots subject to nonholonvmic constraints, such as tirne-varying state feedback control (Samson 1992, Pomet 1993) and discontinuous feedback control (Bloch and McClamroch 1992. Bloch et cri. 1990). Bloch and McClamroch (1992) fmt demonstrated that a nonholonomic system cannot ht. stabilized to a single equilibrium point by a smooth feedback. but is small-tirne locdly controllable. Campion et ni. (1991) showed chat the system is controllable regrirclless ot' the structure of nonholonomic constraints. Murray and Sastry ( 1993) defined a class
tif
nonholonomic systerns that can be steered using sinusoids (chained systerns) and guvc conditions under which a class of two-input systems can be converted into this fortti. Mantegh, Jenkin and Goldenberg (1998) proposed a behavior-based controller to solvt. the path planning and obstacle avoidance problem of mobile robots. Jiang and Nijmrijcir ( 1999)
introduced a recursive technique for the tracking of nonholonomic systerns in
chained form. houe. Murakarni and Ohnishi (3000)described a stable motion control of a mobile manipulator against external forces. More details can be found in Zhang ( 19931. Murray et ai. ( 1993) and Kolmanovsky and McClarnroch ( 1995). Most mobile robot conmliers introduced in the literature employ velocity signal> a\ the conuol inputs. The control problem is simplified by neglecting the vehicle dpnarnics and considering o d y the s t e e ~ system. g The problem of integrating the nonholonornic kinematic controller and the dynamic controller of a mobile robot has been little studied so f'r (Samson 1991). There h a k e n even less investigation into the control and robustness of mobile robots in the presence of dynamic uncertainties and disturbances. However, in red applications! a mobile robot is driven by torques applied on the wheels
and joints. Thus, in order to acbieve the desired control objective, proper torque input
Chapter 1:Introduction
-
7-
must be guaranteed by taking into account mobile robot dynamics so that the preciw velocity tracking is ensured. This leads to the integation of the kinematic controllsr and dynamic controuer (Fierro and Lewis 1998). Carriker et al. (1989) discussed the coordinathg mobility and manipulation of a mobile manipulator, where the task planning is fomuIared as a nonlinear optimizrition problem. Hootsmans and Dubowsky (199 1) considered the large motion control ot' mobile manipulators includùig vehicle suspension characteristics. and dcveloped a control method based on an extended Jacobian to compensate for the dynamic interactions between the manipulator and vehicle base. A performance-functional based controiier was proposed in Miksch and Schroeder ( 1992) using
a direct task-space control. Liu and Lewis ( 199 1) developed ri dcccntralizecl
robust controller for trajectory tracking of the mobiIe manipulator end-et'fector, Ser-ji (1989, 1998) presented a computationaily efficient approach for on-line coordinatcd control that has k e n implemented in the planetary rover, where the base mobility ancl
arm manipulation degrees-of-freedom are treated equally as the joints of a kinrmatic redundant composite robot. Yamamoto and Yun (1993. 1994) considered the coordination of locomotion and mmipuIation of a mobile manipulator when its rndeffector foilows a moving surface, taking into account the dynamic interaction bet~vren the base and the arm. Xu et al. (1994) introduced a hierarchical structure for the contrill of a self-mobile space manipulator, dowing the controI to be executed on various Irvel.\ riutonomously or by tele-operation. Simmons (1994) advocated developing cornples mobile manipulator systems by layering reactive behaviors onto deliberative componrnts. Lizarralde and Weil (1995) developed a quaternion-based coordinated control of a subsert mobile manipulator with only position meuurements. Shibata et (11. ( 1995) proposd
a conuoller for mobile manipulators based on equivdent m s miitrix. Thompson rr d. ( 1995) compared
the performance of different control structures based on various sensors
for an agricultural mobile manipulator. Hatano et al. (1996) considered the control performance of a mobk manipulator adaptive control for traveling operatiom. Minami et al. (1996) discussed the dynamic modeling and compensation of disturbance torque by ,mvity for mobile manipulators travelling on an inclined surface. Yamamoto and Yun (1995) provided a stability analysis
-s-
Chanter 1:Introduction
of a mobile manipulator under force control. Khatib et al. (1996) considered vehicldarni coordination and multiple mobile manipulator decentralized cooperation. [noue et
tri.
(1996) proposed an impedance controller of mobile manipulators with the stabilit? to externd force acting on the work-piece. MacKenzie and Arkin (1996) described an implementation of a behavior-bad mobile manipulator capabIe of autonomousIy uansfemng a sample from one dnim to second in unstructured environments, integating the arm and base as
ri
;i
cohesive unit.
Tahboub (1997) developed a robust adaptive mobile manipulator controller iiw movement on uneven terrain. Chung et al, (1997) hvestigated the tracking problem by using input-output feedback linearization: the authors funher proposed an interricrion mobile manipulator controuer, comprishg a robust adaptive manipulator controller ;inci an input-output iinearizing controuer for mobile plritform (Chung et cd. 1998). Lini rr (11. (1997) considered the coIlision-tolerant end-effector position control problenr. Furtwaengler et (11. (1998) developed a dynnrnic controller for mobile manipulators operating in busy environments using coIlision-free dynamic bands. Colbaugh ( I1NS) investigated the problem of stabilizing mobile manipulators with unstructureci dynamic~. Umeda et al. (1999) proposed a hybrid positiodforce controller based on cooper;itivt. task sharing. Dixon et al. (2000) developed a robust uacking and replat ion controller for mobile robots, The stability of a mobile robot, especidy when it rnoves in outdoor uneven terrain.
ih
a critical issue that demnds serious consideration in the development of control systetm (McGhee and Iswandhi 1979. Messuri and Klein 1985, Dubowsky and Vance 19S9. Sugano et al. 1993, Ghasempoor and Sepehri 1995, Adams 1999). Most approricheh either require the precise knowledge of dynamics of the mobile manipulator. or simplify the dynamic model by ignoring cornplex dynamics issues. such as vehicle dynamics. payload dynamics, dynamic interactions between the base and the arm. or unknown disturbances such as dynamic effect caused by terrain irregularîty. These published schemes are inappropriate for real applications for the roUowing reasons: Eirst. due to ita complexity, the precise dynamic model of a mobile manipulator is nomaIl! unobtahable; Secondly, when the mobile manipulator moves at a relatively hiph speecl. i g n o ~ the g vehicIe dynamics and the dynamic interactions between the arm and the base
Chapter 1: Introduction
-9-
may cause unbearable vibrations of the system (Dubowsky and Vance 1989): Findly. when the robot moves on uneven terrain (e-g., for outdoor exploration). ignoring disturbances generated by terrain irregularities may lead to tip-over (Sugano et trl. 1993. Rey and Papadopoulos 1997).
1.3.2 Neural Network-based Robotic Control Conventionally, there are three different approaches to addressing the position control of a robot manipulator with dynrtmic uncertainties: adaptive control. intelligent conrrnl [neural network control, hzzy control. learning control, etc.), and robust control. tn tlic adaptive control approach (Slotine and Li 1987), the nonlinear dynamics of a ri$
rrihor
with unknown system parameters is expressed linearly as the product of a regresaar rnatrix and an unknown parameter vector. A parameter update law is then user1 to estimate the unknown parameters that are assurned to be constant or slowly vtirying. Sucti a scheme rnay no longer work when the unknown parameters change rapidly or when thc
Iinerir parameterkation property does not hold. Moreover, computation of the regrcwiw matrix is a time-consuming task that is difficult to implement in real time. In the recent years, neural networks (MVs), with their strong learning capability. have proven to be a suitable tool for controllmg complex nonlinex dynamic systems r Hornik et cd. 1990, Barron 1993, Sanner and Slotine 199 1, Narendra and Parthasarathy L990,
Lewis et al. 1999, Ge et al. 1998, Kamiel and Inbar 2000). Neural networks rire closdy modeled on biological processes for information processing, including speciticalIy the nervous system and its basic unit, the neuron. Signais are propagated in the form
tif
potential differences between the inside and outside of ceiis. Surveys of NN are given. tix instance, by Lippmann (1987) and Simpson (1992). Many other studies rire rilw available, such as Haykin ( 1994) and Lewis er al. (1999). In this thesis. we are interestcd only in the applications of NN in closed-loop controI of dynamic syscerns. The FoUowing properties of neurd networks d
e them suitable for control purpobrs
(Narendra and Parthasarathy 1990, Lorentz and Yuh 1996): Universal approximation capabüity Neural networks have k e n shown to have strong ability of approximating any continuous fuaction over a compact domüin.
Chapter 1: Introduction
-
IO
-
This property d e s them particularly suitable for the control of comples nonlinear dynamic systems; O
Multivariable nature The potential ability of neural networks to correct& mrip functions with multi-inputs and multi-outputs make them interesting for the control of multivariable systems;
ParaUel structure The parrile1 structure of neural networks îàcilities the construction of parailel implementation of conuol systems:
Hardware implementation NeuraI networks can be easily implemented
in
hardware because of their parailel smcture. A number of integrzited circuit.\ specifically designed for neural networks are avaiiable for purchase A foundation for neural networks in control has k e n provided by the seminal resiilt.\
of Narendra et al. (1987, 1989. 1990, 1993), Werbos (1974. 1989) and others. The ~ist.5of NN in closed-loop control are dramaticdy distinct fiom their use in open-lotir applications, which are rnainly in digital signai processing, including classif'ication. pattern recog-dion. and approximation of non-dynamic functions. In contrrist. in c l o d loop control of dynamic systems most applications have k e n ad hoc. with open-loop techniques employed in a na'ive yet hopeful rnanner to solve problems associated witli dynamic neural net evolution within a feedback loop, where the NN must provide stabilizing controls for the systern as weii as maintain aii its weights bounded. Several researchers have provided rigorous mathematicai analyses of NN in closed-loop contmi applications. The basic idea behind NN-based closed-Ioop control is to use the NN sstimator to identiQ the unknown nonlinear dynamics and cornpensate for it. Also. the XN-baccl approach can deai with the control of noniinear systerns that may not be Iinearly parameterizable, as required in the adaptive approach. With regard to neural networks, they have k e n widely adopted in the modehg and control of robotic manipulators (Narendra er al. L990, Lewis et ai. 1999. Ge et al. 19981. Fukuda and Shibata (1996) proposed rt two-level hierarchicril inteliigent control for robotic motion by using fuzzy logic, artificid intelligence and neural network methods. Polycarpou (1998) provided a systematic methodology for the identification of nonlineu systems using neural networks. Jin et al, (1994) developed a theoretical basis for the
- II
Chapter 1: Introduction
-
stable design of NN-based rnanipulator control. Gorinevsky er al. (19971 applied ncuril networks to path planning and tracking problems of mobile robots. Lewis rt
trl.
r 1995 i
proposed an NN control scheme chat guarantees closed-loop performance in tem.; (if small cracking errors and bounded controls. Fierro and iewis ( 19%) further app tisd rhis
methodology to the control of a nonholonornic mobile robot where unmode leci bounded disturbances and/or unstructured dynamics of the vehicle are considered.
1.3.3 Robust Robotic Control There are significant disadvantages of using NN-brised on-line control. The systriri resources, such as the size of embedded memory and computation capability of on-board computer. are restricted. As a result. such NN control schemes may be inappropriate for
m n y real applications. Also, it is difficuit to guarantee the convergence perfornirinct. r i f an NN contmller, which depends subsrantially on the learning rates for weight runing: when the learning rates are too low. the convergence of the NN controller
ïniiu
bc
unbearabiy slow; when they are coo hi& it rnay cause instability. In addition. most SS contro1 schernes require a preliminary leminp stage for the NN weight rnritriçcs. .-\Il these problems rnay prevent NN controllers from king utilized in reaI appiicririons. In fact. only a few of the reponed results have been able to guarantee closed-loop stabiIity of on-iine Ieming; even fewer results have been verified experimentally usin2 NN otiline leaniing controuers. Robust contml is control of fued structure thar parantees the stability and performance of uncertain systems (Lewis cr al. 1993, Qu 1998). [ts design genrrdly requües onIy bounds on the k g e s t possible size of the uncertainties. This implies that ir is capabie of cornpensating for both stnrctured and unsmctured uncertaincies.
ti
major
advantage over adaptive control. Compared with adaptive control and Ieunin,v çontroi. other advantages of robust contro1 are i~scomputational simplicity in implemcntarion because there is no adaptation process, its strong capabiiity of cornpensating for timt-
varying parameters and for unstniçtured noniinex uncertainties, and its guwanted stabiiity. Various robust conuol schemes have k e n introduced for the cracking and contrai of mobiIe robots (Canudas et aL 1996). One key step in developing robust controllers is to
Chapter 1:Introduction
- 12 -
determine sxpiicit expressions for the bounding iünctions of the uncertainties i Lw% e r al. 1993, Qu 1998). However, this is not trivial, Fiding the coefficients of the bounding functions requires knowledge of the uncertainties, such as ranges of parameter variations. maximum variation of load, and size bound on disturbances: these may not be availribtr. Furthemore, when the mode1 uncertainties or the disturbances are significanc. high-gain concrol input is required in order to garantee closed-loop stability. These drmbricks ni;iy prevent robust control from king adopted in real applications.
1.3.4 Observer-based Velocity Measurement for Robotic Control Most controiier design requires the feedback of certain states of the systrrn. which demnds
ii
precise knowledge of the system states. Conventionaliy. stritç kedbtick
controllers for robotic manipulators require both position and veIocity measurernetitl;. Robot systems are generally equipped with high precision sensors to obtriin posirion information. In practice, however, velocity sensors are mostly omitted dur:
to
considerations of size and weight. Even when the velocity sensors rire available. noibc generated by sensors irnparts severe limitations in the reachable closed-Ioop band~vitlth and constrains the use of high-gain controllers (Bona and Indri, 1998).
Velocity can be determined by a frst-order numerical differentiation of the tivailtiblc position signal. However, the approximation of the obtained velocity signal contains quantimion noise. which may not be adequate for control purposes and may caiihc undesirable oscillations in the manipulator response. Thus, in recent yeürs. dtèrnativc solutions have been introduced to estirnate the velocity signals by inserting obseners inru the control loop. During the past decade, significant progress has k e n made in linear state fiecibrick
(Qu and Dorsey 1991, Zhang et al. 1997) and velocity observer-based robatic control. Nicosia and Tomei (1990) first introduced an observer into the feedback loop co estim~te manipulatorjoint velocities where local asymptotic stability of the closed-loop systsm is ensured, which was tested experimentaliy in Nicosia, Tornambe and Valigi i 1990). Tornambe (1992) considered the high-gain obsemers for nonlinear systems. which was recently reconsidered to be applied in conjunction with a PD control law by Heredia and Yu (2000). Canudas and Fiiot (1991) proposed an adaptive observer-controiier chrit uses
-
Chagter 1:Introduction
13-
sliding observers; however, sliding mode switching may cause an undesirable chatterin2 phenomenon. Zhu et al. (1992) presented an algorithm combining variable struct~irc control and observation for treating the parameter uncertainties. In Berghuis and Nijmeijer (1994), a linear observer was developed on the büsis of a previous passivity-based combined controuer-observer strategy (Berghuis and Nijmeijrr 1993), where di the bounds of mode1 uncertainties are assumed known. and müny rohorspecific quantities must be determined in advance. Erlic and Lu (1995) cornbinecl a reduced-order adaptive velocity observer with an adaptive controUer for robot trtljectory controi. Kim and Lewis (1996) developed a dynamic recurrent neural network observer. where the inertia matrix is assumed known and a complex multi-layer neural network is empIoyed to compensate for the unknown dynamics. Bona and indri ( 1998) providsd a performance analysis and experimental irnplernentation of various types of observsn kir robot manipulators. The problem of force/motion control of constrained mtlnipulritor~ without veIocity rneasurements is addressed in Huang and Tseng ( 199 1). Prinrslep and Stotsky f 1993) and Lotis and Panteley (1999). Tan er al. (1998) developed
ri
hign
scheme for constructing a nonlinear observer and a nonlinear controller chat gumntee\ global stability of the closed-loop system for a class of nonlinex systems. Tt is well known that for a combined observer and controller nonlinear system.
rin
observer that asymptotically reconstructs the state of the system does not plirrintet. ttic stability of a given stabiluing state-feedback controler when the estimated state provtdsd by the observer is used, Le-, in general a nonlinear separation principle is not valid (Berghuis and Nijmeijer 1993). Due to the nonlinear structure of the robot systrm. the design of combined observer/conuoUer schernes with the guarantee of closed-hop stability is generaily formidable. Nevertheless, one advantage of using such appïoiich
ih
that one may be able to tune the parameters of both controller and observer mort:
efficiently (Berghuis and Nijmeijer 1993).
1.4 Contributions This research work provides the foiiowing main contributions:
Chaoter 1: Introduction
-
11 -
1) A neural network-based hierarchicai intelligent controller for mobile manipulators
with dynamics uncenainties is proposed (Chapter 3). The intelligent controller provides good Iearning and adaptation capabilities and is especially suitable for movements in unknown and changing environments where the robot is necessaru to be autonomous without human operators;
2) A robust damping controuer is introduced (Chapter 4). The robust controller
is
based on rt three-level hierarchy. The proposed conmller is capable of dcalin~ with dynamic uncenainties and is especially suitable for applications tvherc thc robots move in partidly or completely known environments. and the learnins ancl adaptation capabilities of the conuoller are less crucial:
3) A reduced-order adaptive velocity observer-based velocity measurement schrme is proposed for robotic control when only the position signals are rivailable for measurement, without requiring the knowledge of robot dynamics paramctcrs (Chapter 5). 4) Extensive çxperiinents have been conducted on a reai mrinipulator m. i-r.. IRIS
RoboTrvin. The experimental results illustrate the reIevance of the propo~ed theory (Chapter 6).
1.5 Organization This thesis is organized as foiiows. Chapter 2 provides the mthematical preliminario and background on the neurai networks that are necessary for the theoretical drvelopi~ieiir throughout. Aiso. a general dynamic mode1 of mobile manipulators subject to kinernatic constraints is presented. In Chapter 3, the NN-based hierarckcal intelligent controller h r mobile rnariipulators is developed, The Robust Damphg Conml ( R D 0 tçchniquc
ib
introduced in Chapter 4, and based on RDC a hierarchical robust controuer is proposed. Chapter 5 iilustrates the velocity observer design and velocity observer-based lincrir feedback controlier. Experimentai results are presented in Chapter 6 to illustrate the developed theory. Chapter 7 summarizes the main contributions of this thesis. rind provides some suggestions for hure work References and Appendix are included at the end.
Mathematical Background
A"
2.1 Introduction In this chapter, sorne mathematical preliminaries necessary for the theorerical development in subsequent chapters are h t presented. Then a brief discussion of neural network is pcovided, covering minly the topics that will be necessary in a discussion t ) i NN applications in closed-loop control of dynamic systerns. Finally. the dynmic modd
of mobile manipulators subject to kinematic constraints is formulated in
ti
wriy that
iy
convenient for the tùrther theoretical derivations using the Lyapunov design approaçh.
2.2 Mathematical Preliminaries This section presents the mathematical preliminaries that are necessary t'or the derivations found later in this thesis. UnIess noted, al1 the results may be found in Khrilil i t 996) ruid Lewis et ni. ( 1999).
Detinition 2.1 (Compact Set): Let 32" denote the space of real n-dimensional vector. A subset S d i " is said to be open i
for every vector XES, one c m f i d an E-neighborhood of s:
-41
N ( x , E= ) [ Z E W Illi - c E } . A set S is closed if and oniy if its complemenr 3'' ir
open. A set S is bortnded if there is r>O such that ~ ~ . r ~r l
.
(
2.3 1
where the trace tr(A) satisfies tr(A) = r r ( ~ ' )for any A E WXn . For any B E 'r\""" ancl
C E %'lx"'
. tr(BC)=tr(CB).Suppose that A is positive definite. then for any B E %""" . t r ( ~ A l f2)0 ,
i 2-41
with equality $f B is the mxn zero mtrix. and
Theorem 2.1: Consider non-antonornomdyamic qstems of thef o m
i = f(.r,t), ta,,,
(2.6)
where x E Sn- Assume that the origin is nn eqrrilibririm point. Let L(.r. t ) : 3'' x % + 'i\ be a scalar rime-vqingftinction slrch t h t L(0,t )=O,
and S be a compcict sirbser rd -3:'
.
Then the system (2.6) is said to be
a. Lyapunov Stable IfI for systern 12.6). there e-rists n fiinction L(,x.r) wirh c-onriniiotrs partial derivatives, srcch tharfor x in a compact set S c 3". L(.r,t) is positive defrnite, L(.r,t) > O ; ~ ( xt ), is negativr semidefinite,
L( x. t ) lO ,
then the eqirilibrium poinr is stable in the sense of Lyupunov (SISL). b. Asymptotically Stable I f l ftirrhermore, condition (2.8) is strengthened ro
- 17 -
Chaoter 2: Mathematical Backaround
L(X,
t ) is negative dfinite, L(.r,r) < 0 ,
(7-0)
tiren the eqriilibrirrm point is aq~nptoticstable (AS).
c. Globally Stable I f the eqrrilibriiim point is SISL (stable in the sense of Lwipiirroi.) orAS, $ S c %", and in addition, the radial unboiindednessholds:
-.
L(x,r) -,
V t as
II.tj +
(2.10~
m.
riten the stubili~is global. d. Uniformly Stable
If the eqirilibriiim poinr is SISL or AS, and
in addition Li .L r i.v
tlecrescenr, Le., there exists n rime-invarinnt positive definirefirnctiorr Ll(.r) mcit tii~rt L(x,r)5 L,(x), v t 2 0 ,
O./~J
riten the stability is riniforrn (e-g., independenr afro). The equilibrium may be both uniformly and globdiy stable. For instance: Definition 2.3 (Uniforrn ultimzlte boundedness, WB):
Consider the dynarnic system .i- = J(.r,r) with
XE
31" . Let the initial time bc ri,. ancl
the initial condition be x,, e x ( t o ) .The equilibrium point .r, is said to be unifornil-. ultimately bounded if there exists a compact set S c91n so that for al1
.Y,,
E
S
thcw
-
exists a bound B aml a time T(B,.q,) ,such that Ilx(t) rcllS B for al1 r 2 r,, + T .
In the above defiition, the term rrnifonn indicates chat Tdoes not depend on t,,. The term idtirnate indicates that the boundedness property holds after a time Itipse T. If
S = S n the , system is said ro be globally UUB. It is noted that both AS and SISL are too strong requirements for closed-loop control
in the presenct of unknown disturbances (Narendra and Annaswamy 1987). In prxticd closed-loop systerns, the bound 8 depends on the disturbance magnitudes and orher factors. However, B c m be made sufficiently srnail if the controlier is properly designeci. Thus, UUB stability is sufficient for most control purposes (Lewis et cd. 1999). A Lyapunov extension version of the Barbalat's Iemma is presented in the following.
Chaoter 2: Mathematical Background
- [S -
.
Recall that one may check for the boundedness of e(x. r) which implies
t
h
Li s.1 1
is uniformiy continuous. Barbaiat's extension can be used to show that certain sratrs ot' ii
syscem actually go to zero, though the standard Lyapunov analysis has revealed only that
the system is SISL (i.e.. States are bounded). In practical applications, there are often unknown disturbances or modelin,u srrors presented in the dynamic systems (2.6). ie., systems of the fom
X= f ( . r , t ) + d ( t ) .
i1.1-:1
w ith d(t ) an unknown bounded disturbance.
The next theorein shows thac UUB is gumnteed if the Lyapunov derivritivrl
.1.
negative outside some bounded region of 3 " .
Theorem 2.3: For system (2.13). if rliere e-rists ajincrion L(.r,t) with continirorw pcrrrid der-irirriws
srrch rharfor x in a compact S c 3", L(.r,r) is positive definite, L(.r, t ) > 0 ,
!or some R > O ntch rhat the bdl of radius R is conrnined in S. rhen the xysrrnr i s L'L'B. cind the n o m of the stare is bounded ro rvirhin n neighborhood of R.
2.3 Background on Neural Networks This section provides some background knowledge on neural networks thac is necessary for the theoreticd development in Chapter 3, We first present che mathemritictil
Chapter 2: Mathematical Background
- 19-
rnodel of a single neuron. Then, two most commonly used neural nets: ntrrltilqer perceptrons (MLPs) and radial basisfrtnction (RBF)network, are introduced. Finally.
ti
cornparison between iMLPs and RBF will be given to show the reasons why in Chapter 3 we decided to choose RBF network, rather than MLPs, for the intelligent concroI desien of mobile manipulators,
2.3.1 Mathematical Model of a Neuron A mathematical model of the neuron is depicted in Figure 2.1. which shows rht.
dendrite weights w , , the firing threshold
tu,,
signals, and the nonlinear activation frtncrion
the sumrnation of weighted incorning O(.)
. The inputs to rhe neuron are
t tic
signals xi(t),.r,(r), ..., .r,(t), and the output is the scalar y(r). The relations krwsen [tic inputs and the output c m be expressed as
-
The activation function is selected differently in different applications. A p w r a l cIass of monotonically non-decreasing functions taking on bounded values at -
and
+ = is known as the signtoid hnctions.
inputs
Figure 2.1: Mathematical model of a neirron. The expression for the neuron output y(t) can be streamlined by defming the colurnn vector of input signals F[t)E 3" and the colum vector of NN weights ii;(t)E 3" as
Chapter 2: Mathematical Background
LI
.~(r> =
.r2
.-*
- r J T , W(S> =[v,
v2
- 10 -
*a.
v,,
IT -
(7-151
The augrnented input column vector .r(r)~%"" and NN weight column vector
~ ( rE) Sn+'is defmed as
Hence, (2.14) c m be written in matrix notation y(,) =&r).
2.3.2 Multilayer Perceptrons and Radial Basis Function Networks 2.3.2.1 Multilayer Perceptrons
Figure 2.2 shows a typical hlly connected MLP, Tt ha three layers: ( i) the input iayrr that is c o ~ e c t e dto the outside worId: (ii) the hidden layer where processinp
(P.ily ultiniately bounded; and tii) the velocity tmckitig rrror c m he kcpr crrbitrariiy mal1 by incrensing riir controller gain p in ( 4 . 3 ) .
P m f : Let us choose the Lyapunov function as -
Notice that
M
is symmetnc positive defniee as shown in (2.44). therefore. V. is ;in
effective Lyapunov function. Differentiating (4.27) we cm obtain
Substituting (4.22) into (4.28) and using the dynarnic propenies of mobile platform listed in Section 2.4 yields V:
=zTg-(~+~a+~+~d)}
where q represents the generaiized coordinates of the vehicIe,
go, j,
and TB,are positive
constants (see Section 34), and the unknown vector A Ïs defined as
,c,
A~ = ~ M A ~ I I , ~ ~ ~ w A,&,r,,). ~~~~c~
(4.301
Chapter 4: Hierarchical Robust Control of Mobile Manipulator
- 64 -
We can show that A is bounded as each element of the vector is a bounded quüntity
(sec'
the properties Iisted in Section 2.4.2).
Substituthg the control law (4.25) into (4.39),we obtain
Y, = - P . - + z T -P
CpZlk~~'}+llz~l~~p
II-Il' lkI l 2 IlW l l l Il~ +
Let us choose the Lyapunov hnction for the overall mobile platform sysreni (combining the steering system (342) and vehicle dynmücs (2.44))as
v = v, tv,,
i3.32 i
where V Iand V2 are defined in (4.19) and (4.27), respectively. Differentiating (4.32) dong the system trrtjectories and using the results of (4.20) aiid
(4.31 ) wc obtain
In (4.31).
is a bounded qunntity; thus. V decreases monotonicdly until the so lurions
reach n compact set determined by the right-hand-side of (4.33).
Frorn (4.33)we obtitin
Therefore, when
we c m guarantee that
v CO. From the defmition of q~ in (4.26), it is straightfonvard to
show chat (4.35) implies that
kll~fi. From (4.35)and (4.36) it is obvious that when
(4.36 i
Chapter 4: Hierarchical Robust Control of Mobite Manipulator
ive obtain
v C O . Thus, the position tracking error e ( t ) and velocity error :(t)
-65-
are both
olobdly uniformiy ultimately bounded- The size of the residual set may be d e m a s r d bu
C
01
increasing the controlIer gain p in the control law (4.35).
4.5 Robust Damping Control of Mobile Manipulator In this section we develop the RDC for a mobile manipulacor. The deveIoprnent
ih
based on the dynamic equations of mobile rnanipulator described by (2.18). (?.?Ji and the Lyapunov design result (3.25) achiiieved in Chapter 3, using the RDC technique introduced in the previous sections. The design of the RDC controlIers stricts frorn i 3.25). For convenience, we repeat (3.25) in the Coiiowing:
where V4 is the Lyapunov hnction for the overali mobile manipulator syscerns detintxi in
(3.L7).The following two cases will be considered in the upcoming subsections: 1) The dynamic parameters of the mobile manipulacor are ptinially knoivn.
specifically, we assume that only the inertia matrices in (4.38) are known
ti
prion'; and
1) The dynamic parametes in (4.38) are completely unknown.
4.5.1 Dynamic Parameters Partially Known We first consider the control probiem of constraïned mobile manipulators when the parameters of the dynmic mode1 (2.29) are oniy partialiy known. Wirhout Loss ut' gneraiicy, we assume that only the Ïnertia tewns in (4.38). Le.,
m,,. Mi,. hi1, caitl Ji .. --
ore h o w n a priori- The cenhipetul and Coriolis rems, friction. pmtri~~~'rcruitrrici
distrcrbance temsare all asstimed to be tdnown. This assumption is reasonable. since in
Chapter 4: Hierarchical Robust Control of Mobile Manipulator
-60-
redity the inertia matrix of a robotic rnanipulator is relatively easy to obtain to an
acceptable precision. From the second term in the right-hand-side of (3.38) and Property 3.6 we c m obtriin
:'k. - ~ l , ~ - ~- p~ -~ f{ l+~ ~ ~~ ~~ kkk~)]} e( r i zrk,. - M,,d - sT { ~ + M12k(r~ ke)}} ~ q +
11dE.ll.l kl +L
cc12b
~
~
Ik, + kellCI +h 11~11 +Cl
YVI
=cT+,. - m l l d - ~ T { ~ 1 2 +q~r d, ~ k (k re )-} } + ( l r l ~ ~ ~ q ~ .
g k the b v n d of the mai..
=
h
S(qv1, i . r Siq,. i 5 gT.
are positive numbers defined in Section 2.4. A, and
i4
, . ti inci
r,,
y, are defined in the hllowing:
(cl16, c r C l l b * < l c r ? "Ni 1t.f) * d = (IIail lql . 1i.r +~~1lll~1l11411 1) A: =
3 )
(50
?
4.40 )
CL I
We c m show that A, is baunded as each element of the vector is a bounded quanrity ( Property 2.6), and
q, is the known RDC vector of the vehicle subsystem.
Sirniiarly, €rom the third term in the right-hand-side of (4.38) and Property 1.6 ive c m obcriin
?Crr
+ M,,kr+(C,
-~,,k)ke+& --
+i~~,~ci+~,,~or+C,,~u}
< r T L r r+ M,,(id + k ( r - h ) ) + ~ ~ , ( ~ à + ~ a ) } +
IldIbm14d + kell lql f C d ,IWI IqI +Cr +544 + rvz
= r T k r r+ M Ï ( i j d + k ( r - & H f M ~ , ( S ~ + S C X ) } + ~ ~ { A $ ~ ,
i 4.42 I
tvhere the unknown bounded vector A, and the J3DC vector cp2 are defineci in the Foiiowing: A
= T
9.
i,,ii+q,&
C
= ( I I +~kei~oqil. O~IIII~I . 11411. 1) -
where the positive constants O and k, > O are the controller gains, cp, (ind cp, cire rlrr
RDC vectors respectively deflned in (4.51) and (4.54), then the trcrcki~tyerrors closeri-[oop l t e m s i r e globally irnformly irltimately boirnded.
P m f : Substituting (4.55), (4.56) into (4.38) yields -
of rjtv
Chapter 4: Hierarchical Robust Control of Mobite Manipulator
where k,, = m W l ,k, 1 In (4.57).
.1
- 71 -
1 1 = a~ b~,ll.l[~, ~ I }.
I ~ 1 Ab a~ bounded quantity: therefore. V, decreases monotonicnlly until
the solutions reach a compact set determined by the right-hand-side of(J.57). The size of the residual set c m be decreased by increasing km,. According to the standard Lyapiinov theory and nzeoretri 2.2, this demonstrates t h t the control input (4.55) and (4.56) ma? guarantee global uniform ultimate boundedness of al1 tracking mors,
[71
4.6 Summary In this chapter, a hierarchical robust control architecture was developed for rht. control of mobile manipulators in the presence of unknown dynamics and unstructuretl disturbances. The architecture consists of three layers: (i) decision level: ( ii) robust ness level: and (üi) execution level. The design of the robustness level was illustrated in demil. A new robust control technique called Robust Damping Control (RDC) was introdi~ctcl. RDC controllers were developed at the robustness Ievel to control the vehicle ancl manipulator, The proposed RDC controllers have the capability of compensating h r unknown dynamics and unstnictured disturbances, and crtn be easily implemented in mobile mnipulator systems. This is a very useh1 fearure for applications. The proposed RDC controllers require kwer design parameters. with no requüemt.nt on the knowledge of mobile manipulator dynamics and the bounds of uncrrtaintitx which rnakes their design and implementation much simpler. Moreover. knowledge o t' the dynamic parameters of the mobiie manipulator is not required. Therefore. it
ih
possible to design a universai RDC controller for the mobile manipulator such thrit when the dynamic properties of the robot change, the RDC controller cari stiii guarantee becter performance without modifying the control structure. The proposed architecture is especiaily suitable for applications when the available system resources of mobile manipulators. such as size of embedded mernory and on-line computation ability of the on-board computer. are restricted, or when the robot moves in a partially or completel!
known environment where learning and adaptation capabilities are less demanding.
Cha~ter4: Hierarchical Robust Control of Mobile Mani~ulator
Experimental results presented in Chapter 6 proposed RDC controiier.
-73
WU illustrate the performance of
-
the
Reduced-order Adaptive velocity Observer
5.1 Introduction In the previous two chapters, a neurai network-based hierarchicai intelligent controiier and an RDC-based hierarchical robust controiier. were developed for
tlit
control of mobile manipulators in the pptesence of unknown dynamics and unmodelt.d disturbances. In both schernes. the positions and velocities of al1 joints are assumeci to Iw available for measurernents. However, in sorne practical robotic application?, dit-ect measurernent of the velocity signais may not be available. This is due to the t k t
tli;ir
velocity sensors are ofien omitted because of weight and size considerations. Even ~ v h e n the velocity sensors are avdabIe (such as tachorneters), the noise generated during the measurement process impacts severe Limitations in the reachable closed-loop bandwirlth and hence constrains the use of high-gain controllers. The velocity of a joint cm be atternpted by
ri
first-order differentiation of the
available position rneasurement. However. the approximation of the obtained vslocity measurernent contains quantization noise, which is also undesirable. In our experirnent3 presented in Chapter 6, we implemented the proposed NNC and RDC controllers for controlling a rnanipulator a m In the process, we obsewed that the quantization noise of velocity signais had a significant effect by undermining the overaii tracking performance of the robotic system, Therefore, it is of great interest to h d an alternative technique such that a better velocity signai c m be provided-
An alternative technique to using numerical differentiation is to impIement an observer for the reconstruction of the veIocity signais (Nicosia S. and P. Tomei 1990.
Chapter 5: Reduced-order Adaptive Velociîy Observer
- 75 -
Berghuis and Nijmeijer 1993, Eriic and Lu 1995, Bona and Indri 1998. Heredia and
Yti
1000).The basic idea behind the observer design is to use an observer dynamics proces.+ to ernulate the system dynamics such that the system state (Le.. the velocity signa1 q crin
be reconstructed. However. vebcity observer-based controi schemes require
t htl
knowledge of robotic dynamics rtnd/or the bounds of uncertainties, which are dit'tlcult [ci obtain in practice. This issue is undesirable and needs to be overcome.
In this chpter, a reduced-order adaptive observer is introduced for vrlocity measurement in robotic control. A velocity observer-based linear feedback controtler
i3
also developed. The main advantages of our velocity observer-based control scheme ovcr most conventional observer-based controllers are: {i) it can adaptively reconstrrict t he velocity signal without using variable structure and switching to mintain stability: i ii 1 it is reduced-order and h e m renders a much smailer tirne-deIay: and (iii) it doss not require the knowledge of the dynamic parameters of robotic manipulators f i r . . the inrrtiti matrix M(q). cenrriperal and Coriolis mtrix C(q,q), friction and gravity veçtor Fc 'pi 1 . and disturbance torque T, in Equation (2.49)). Moreover, the bounds of the unciimint i c h
are not required except for some knowledge of the iower bound of the inertia matris Mq).
In the toiiowing sections, we f i t briefly introduce the architecture of the propoxd velocity observer-bied controiler. Then, the design of the reduced-order ridaptite velocity observer wiil be presented. The development of the velocity observer-bascd controiier, which embodies both the observer and controiier with a proof of the closeclloop stability of the overaii scheme, is given. The experimentai results of the propo.d scheme are presented in Chapter 6. For convenience of theoreticai development, in the sequei a bold chrirricter s represents a vector or matrix, and x represents its n o m The n o m of a vector or a matris is defiied in Section 2.2.
5.2 A Velocity Observer-based Robotic Controller The architecture of the proposed velocity observer-based control scheme is shoivn in Figure 5,l.The observer-based conuoiier comprises two loops: (i) inner observation loop:
- 76 -
Chapter 5: Reduced-order Adaptive Velocity Observer
and (ü) outer feedback control bop. In the inner observation loop. the observer reconstructs the joint velocity signal q using the available position merisurement q ancl ihe previous conuol torque
r .Then, from the measured position q and estimated velociry
we obtaïn the position error e and observer velocity tacking error
/ . In
the outer
loop, the position and velocity errors are used to derive the next required controt torqiit.
r of the robotic manipulator. 7 ' f'
-
A
PD-like Connoller
1Contmi ~ o o (p
?
Manipulator
Position feedback
Figure 5.1: Arcliirecriire of the proposed observer-based robotic çolttroller
5.3 Reduced-order Adaptive Velocity Observer Design In this section, we propose a linear observer for velocity rneasurernents in robotiç
control applications. Let us consider the dynamics of an n-link hiiy actuated robutic manipuIator (Lewis et al. 1993)
M(q)q+ C(q,q)q + F(q& +rd=r ,
(5.11
The parameters and propeaies of (5.1) have already been discussed in Section 2.4.
In the development of the velocity observer-based robotic controlIer. onIy
the
rneasutement of the position vector q is required. The knowledge of the inertili matris M(q), centripetal and Coriolis matrix C(q, q) , gravity and friction tenn F(q.q and
- 77 -
Cha~ter5: Reduced-order Ada~tiveVelocitv Obsenrer
disturbance rd are not required. The gravity and friction term is assumed ro be boundsd by IP(q,q)ll S go +clllqll (Lewis et al. 1993) and the disturbance rd is ÿssumed to he
ci and
bounded by lkdllCre, where the bounds Ga,
rB are some positive constants
which are ali assumed to be unknown. Given a reference manipulator trajectory q, ( t )E %" which is assumed to be seconr! order differentiable and bounded by
with p, some unknown positive constant, the objective of the robotic control is to richieve the tracking of a prescribed reference trajectory, To make the theoretical development more concise, iet us use rhe notations x, = q .
X:
= q- .
,I
X,
=q,, tind
a
x' = q,, . Therefore, the dynamic equation (5.1) can be reformulated as
xi =xt
Let us defuie the actual position and vetocity tracking errors as follows:
d
e2 = x 2 - x 2
in (U), the position xi is assumed to be available for measurement. and the position error is assumed to be bounded by Ileill'pl, with pl u n h o \ s positive constant. This assumption is reasonable, as in practice the range of the joint position xi is always limited. The measurement of the velocity ~ ( i - e .q, in (5.1)) is not required: it is instead estimated in real time using a iinear veiocity observer.
Chapter 5: Reduced-orderAdaptive Velocity Observer
-
7s -
In the inner observation loop, the observer dynamics also has its own state variables.
- ) . Let us define the estimates (fi,,X,) of the systsm state which are denoted as (2, ,i, (x, ,x, ) as (Nicosia and Tomei 1990)
.
In (5.6) the observer gain T, = 4 I where
4 >O
Defme the position and velocity observation errors as
-
*
Xz =xl - X 2
We propose the foiiowing updating law to update the states of the observer (i,.i. i in real time:
where Mo =mol with 1 the unit matrix and m, a positive constant to be determined. m l the observer gains
= A,I with i = 1-1 and Ai > 0.
The velocity observer, represented by Equations (5.6) and (5.81, has the tollott.in_i features: (i) it is an adaptive observer. thus the system states of the robotic dynamics can be adaptively reconstructed; (ii) it does not use variable structure and switching
to
maintain stability, and as a result, the signal produced by the observer is smooth and the excitation of higher frequency "unrnodeled" dynamics is less likely. and hence. chattering is largely eliminated; and (iii) it has a first order dynamics response as opposed to a
second order system response provided by the fulI-order observer-based controller. therefore, it renders a much smaiIer timedday.
5.4 Combined Observet/Controller Design In this section, we combine a linear controller with the veIocity observer (i-e., (5.6)
and (5.8)) proposed in the previous section. The observer error dynarnics and tracking error dynamics are k s t derived. Then, the hear controlIer is proposed. At the final stase
Chapter 5: Reduced-arder Adaptive Velocity Observer
- 79 -
of design, we show that both observer error dynamics and tracking error dynamics are ultimate1y u n i f o d y bounded using the proposed observer-based control schemc
From (5.3), (5.7) and (5.8) ive obtain
ki = -x ~ - ~ , P , ,
(
5.9
and
i 2= K i r+ p - ~ ; ' -r r2x,- r3%, = ( M -' - Mi1)7+ B -(-
- r,r,)îi,-r3'3X2.
Equations (5.9) and (5.10) represent the observer estimation error dynamics.
,
Differentiating the position tracking error e yieIds
e , = e l =ê2+z2.
(5.1 I l
Since the actud velocity signai x, , and hence the acturil velocity tracking crror e Z. is not availabk for mensurement, let us inuoduce the observer veloci~ytracking èrror (denoted as
el
in Figure 5.1) which is defined as the difference between the desird
velocity x: and the estimated ve!ocity provided by the observer 9 , (denotsd ris il in
Figure 5.1) ê2 = i l - x l .d
(5-13
Differentiating (5.12) and substituting (5.6), (5.8) into it we obtain
6, = X 2 - x-2d
-
=z, + r3zl-x, d =~
; ' +r (T,- T3îl)Xi+ T3X,- -if -
(5.L31
In order to eliminate the thkd term in the last row of (5.10), and the second term in
the Last row of (5.13), we choose the observer gains in (5.6) and (5.8) to satisfy
=I',T,, orconespondingly&
i2
é,
=AS,then(5.10) and (5.13) -~;l)?+fl
= ~ o '+rT,P,
-r&,
- x,- d .
From (5.9 and (5. L 1) we obtain x2 =Z2 +êr + xd,.
are simplified to
( 5 -1-1) t5.15,
Chapter 5: Reduced-order Adaptive Velocity Obsewer
-80-
Hence, tkom (5.16) and the properties listed in the previous section we c m show char for the dynamics tenu fi (defmed in (5.4), the foiiowing property holds:
llall O, i = 0,1,2,3 are some positive constants that are assumed to be unknown. Figure 5.2 illusuates the detaiIed design of the velocity O bserver-bsed cont r d scherne chat has k e n shown in Figure 5.1.
Figure 5.2: An observer-basrd linrar esritnarrd srarefredbuck corrrroli~r
The foiiowing theorem shows that the velocity observer-based linear controller s h o w
in Figure 5.2 guarantees the closed-loop stability of the combined observer and contraller system
Theorem 5.1: Given m, in the observer Eqitation (5.8) chosen srrffciently large cind the oitsrn-rr gains
4- designed properly, $the
control torqire in (5.1) is
.r = Mo(-K,e, -Kzêt),
Chapter 5: Reduced-order Adaptive Velocity Obsemer
-SI -
the combi~zedobserver und conrroller qsrern described by Eqitarions (5.61, (5.8) m l (5.18) is riltimately rtniformly borrnded. in the conrrol laiv (5.18). borh
KLmu1 K2tuu
constant and diagonal gain matrices. The proof of the theorem is given in ilppetidk D. The proposed observer-basecl control scheme has a rather simple tinear structure. To the best of our knowlecige. the proposed observer-based controiier given by (5.6), (5.8) and (5.18) requires the most relaxed condition on the knowledge of the robotic dynamics (i.e., it is s h o w within the proof of Theorem 5.1 that only the lower bound g of the inertia matrix is assumed to be known as required in condition (a.25) (see Appendir E)).Moreover, in practice the precise knowledge of
is generally not required since the inequality (a.25) can be easily
satisfied by choosing the positive number nt, in (5.8) to be suffciently large. The controiier (5.18) is actualiy a PD-ke independent joint controller. However, it has a more significant performance in cornparison with the conventional ones. it is tvell known that the classical independent joint control can be considered effective only when the nominal dynamics is known and when the unknown dynamics caused by gravit!.. friction and external disturbances are not significant (Lewis et al. 1993). When the nominal dynamics is not available, andfor the unknown dynarnics is considerable. and hence c m not be ignored, the classical joint conmi may no longer provide good tracking performance. On the other hand. the proposed PD-like controlIer (5.18) is capable of dealing with the dynarnic uncenauities without r e q u i ~ gthe knowledge of the nominal dynamics, and hence it is useful for realistic applicarions.
5.5 Summary The adaptive velocity observer-based controiier proposed in this chspter designed to have a reduced-order dynamics; chat is, the observer has fîst order dynamics. Thus, compared with the conventionai illi order observer-based controllers. it exhibits a much faster dynamic response. Also, as we wiii show in Chapter 6, the velocity signal provided by the observer is smooth and reduces undesirable (and in some applications.
Chapter 5: Reduced-urder Adaptive Velocity Obsenrer
- S2 -
unavoidable) chattering. Moreover, the simple PD-like controiier structure ciin be c~isily
implemented in practical systems. The proposed observer-based controiier has the capability of cornprnstiting for unknown dynarnics, such as unknown gravity terms and unmodeled disturbances due w fiction, backlash erc. Such capabiiity is generaily not provided by the cli~~siciil independent joint control schemes, such as computed-toque PD control. Experimenrs performed on a two-DOF robotic a m aiso ilIustrate that the proposed controiler cleariy outperforrns a PD control loop in terms of smooth and desirable responsr' i tlit experimental results are presented in Section 6.5). Finally, it is possible to combine the veIocity observer with the NNC and RDC controllers developed in the previous chapters. In such an arrangement, the requirements
for the measurement of velocity signais cm be reduced.
CHAPTER 6
111111. Experimentai Validation 6.1 Introduction AFter developing the theory in chis research, we conducted extensive simulütionb
tu
verify the effectiveness of the proposed control schemes in Chapters 3.4 and 5. Howcwr. it is well known that good simulation performance of a control law does not guartinret. irb applicability in real systems due to friction and backlash between joints. iinrnoclt.l~cl dynrirnics, difficulty in parameter tuning, and other limitations. Among the
itiany
learning, adaptive and robust control schemes that appeared in the literature. only
~i
limited number have k e n verified by real experiments. In this chapter, we present an experimentd analysis conducted on a four d e p e r d freedom experimental faciiity (i-e.,IRlS RoboTivin manipulator arm) using the p r o p o d control schemes. The main objective is to illustrate the strength of the p r o p o d methodologies as opposed to conventionai Iinear and robust controllers whttn the parameters of dynarnic mode[ of the RoboTwin, which can be generally describecl hl. Equation (5. l), are completely unknown. in other words, in our experimenrs the dynamir: parameters of RoboTwin (Le., M(q). C(q,q), F ( y , q ) , r, in (5.1)) and their boundh were di assumed to be unknown. O d y the angular displacement of each joint were required and measured using a hi&-precision incremental optical encoder (HPNEDS5000). The velocity signais were obtained us&
s h p i e Fm-order numerical
differentiation of position rneasurements, except for OBC (observer-based controller) where the velocities were estimated using the velocity observer approach proposed in Chapter 5.
- 53 -
Chaoter 6: Eitoerimental Validation
Figure 6.1: RoboTwin (IRISfacilitv)
6.2 Experimental Setup 6.2.1 IRIS RoboTwin The experiments were carried out on the IRIS facility RoboTwi11 (Fisure 6.1 !. which
is a modular and reconfigurable robot (Kircanski and Goldenberg 1997). RO~IOTII~II is L; dual x m that comprises at Ieast two robotic manipulators, each having several rnodufurotary joints. Each joint is driven by a DC brushless motos connected to the output link through a hmonic drive gear, and is insmmented with a high precision incrementrii opticai encoder and a tension compression load-cell torque sensor. The harmonic drive consisting of wave generator, fiexspline and circular spline
rire
placed conccrntriçall~
around the robot mis. The selection of h m o n i c drive as the means of transmission ii, based on its superior features including a high reduction ratio in a compacc single step
unit and virtuaiiy zero backlash. For data coiiection, a high sampling-Erequency is desirable so as not to Loose any of the signai ccharacteristics. A Iower band of the sarnpling fiequency
can be obrtiined
using Shannon's sampling theorem which States that in order to recover a continuou'i tirne signai, the sampling fiequency should not have any Gequencies above the Syquist üequency mJ2 (Slotine and Li, 1991). in Kircanski and Goidenberg (1997). a Nyquist kequency
O€
about 20-30k
wris
idenufied as a rnemhD@ui bandwidth for position
Cha~ter6: Ex~erirnentalValidation
-85-
control of RoboTwin. Therefore, in our experiments we chose the sampling frequency a. as 50Hz that is accordingly sufficient for our appiication. The extraction of the curves of joint displacements is performed by introducing low-pass filtering of measured position signals provided by the position sensors in order that high-frequency noises containcd in the measured signals may be reduced. The cut-off frequency of the second-order Butterworth Iow-pass filter is suitably set to 5OHz (Kircanski and Goldenberg 1997).
in this research, we only consider the time-domain characteristics of the çontml systen The experimental results can be hrther analyzed in frequency domriin using: i ii Bode plots to determine the gain margin and phase margin of the control systems: i iii Nyquist criterion and the effects of additionai poles and zeros on the Nyquist locus: or (iii) specifications to describe the quality of the systems in the frequency domriin. i.e.. peak resonance, resonant frequency, bandwidth and cut-off frequency (Slotine and Li. 199 1).
Figure 6.2 illustrates the software architecture of the RoboTwin experimental sysreni. The real-tirne controiier of the RoboTwin has also k e n designed
CO
be modular and
exprtndable. It is based on a nodal architecture with a PC486 host and an AiilD29050 RiSC coprocessor as the CPU of each secondary node. The RiSC coprocessor board is instrumented with a RISC processor with a built-in floating-point unit and seprirate embedded rnernory. Each node is capable of controiiing 8 joints at 1KHZ while r.wuting over 1000 FLPS operations per joint in each sampling intervai for the C-coded pro,Oram>. High digital and analog i/0 speed is achieved by the use of parallel i/0 boards connrcted to the EISA bus with 16Kbyte cornmon rnemory windows, Communication between the
host cornputer and NSC coprocessors is performed through the comrnon memory windows. The V 0 system consists of: (9 DAC board with 6 analog output channels: ( iii ADC board with 8 il0 channels and 1Zbit sampling rate; and (iii) 96 bit paraIIel dizital
VO board for transmitting data fiom the opticai encoders. Therefore, RoboTwin is an ideal environment for evaluating various control methods in the study of physicrcl phenomena such as nontinear stiffness, hysteresis, backlash, fkiction. parameter variationpayload and dynamic effects of system (Kircanski and Goldenberg 1997).
-56-
Chapter 6: Expenmental Validation
grna INTERFACE
.+-.--~--P-b-
RlSC PROCESSOR
HOST COMPUTER
A
EISA BUS
IO DIGITAL
ENCOOER INTERFACES
El IRIS RoboTwin
Figure 6.2: Sofivare archirecrrrre of RoboTwin
6.2.2 Dynamic Model of RoboTwin In the experiments, we used one arm with a total of four actuated joints. The dynrirnic mode1 of the m can be described as:
M ( q ) q + C ( q , q ) q + F(q&+r,
=r.
(6.1)
where q ~ % ' denotes the joint variable vector, M(~)E%'"' represents the inertiri matrix, C(q,q)E 3'"' is the centripetal and Coriolis matrk representing dynamiç couplhg between the joints, F(q,q)E 3' represents the gavity and friction vector,
rd E 3.' denotes unknown disturbance inciuding unsuuctured dynamics and unknown
Chapter 6: Experimental Validation
payIoad dynamics, etc.,
T E 3'
- 87 -
is the torque vector applied to the joints by the niotors
through the harmonic drives.
In each joint, the torque s is proponional to the motor torque r,, by the factor of star ratio N,which c m be written as foiiows:
When taking into account the flexibility of the gearing system the dynamics generatecl by the appiied torque T c m be obtained in the foiiowing (Kircanski and Goldenberg 1997 1:
where q,, and
4,. are the motor shaft angular displxement and velocity. i, represcnts
the harmonic drive stiffness. and
B,
is the viscous coeficients.
The motor torque r,, is proponional to the motor current as (Kircanski and Goldenberg 1997) f ,II
= kn, LI,, *
t 6.41
where km is the motor torque constant. imis generated fcom the motor voltage ri,,, which
ih
provided by the DAC unit through the amplifier ittt
=
k.&
-
(6.51
6.2.3 Experiment Design In our experiments, the dynamic parameters of the arm, i.e., M ( q ) , Ci y . 4 ) . Fi q.41 and
rd in the dynamic mode1 (6.1), were al1 assumed to be complecely unknown. The
joint positions were measured using high precision incremental opticai encoders. and rhc joint velocities were estimated using simpIe Fust-order differentiation of position measurement. which was found to contain wide-band noise and affected the concrol performance in the experiments. In the experimental system, friction and bricklash existing in the joints and M a g e s were &O found to have a significant influence on the tracking performance, which greatly substantated the ciifficuity of control design.
ChaDter 6: Ex~erimentalValidation
- XS -
Three sets of experiments were carried out to ve@ the proposed NNC. RDC and observer-based controiier (OBC). For the purposes of comparison, additional experiments were performed by using the satrtrarion-rype conrrol (STC,Lewis er trl. 1993) t in comparison with NNC and W C )and PD controUer (in comparison with OBC). In oiir experiments, integral control was added ont0 the independent joint PD control since t h m was no compensation for gravity and nominai dynarnics as required for classical joint control (Lewis et al. 1993). AU controuers (i.e., W C , RDC,STC and OBC)had exaçtly the same PiD gains; therefore, the experirnentaf results directly show their improvements over PID or STC.The initial PD gains for each joint were designed independently iising a classical joint control technique (Lewis et al. 1993) to ensure good pertormünce. i.p.. minimum overshoot, minimum offset error and fast rise time (Kircanski and Goldcnbsrg.
1997). The structure of STC is defined by the foUowing (Lewis er al. 1993):
where e is the position error, k, is the derivative gain. k, is the proportional gain. k ,
1s
the integral gain. and
where the parameters
E,
&, al and 6,
are positive scdars that must be determincd
uniquely for each specific robot m. In our experiments. these parameters were designcd based on simulations and experimentai tests. In Lewis er al. (1993), it was proven that the conuoi law given by (6.6) and (6.7) guarancees the uniform ultimate boundedness of the closed-loop trackins errors. However, in the experiments, the stabiiity of the closed-Ioop system was found to hc: highly sensitive to aU these parameters: a smaii variation in any of them criused instability, e.g., arm-shakîng. No such pathologicd phenornena were obsewed in the experiments when the proposed NNC and W C conrrotlers were used. However. due to t h inaccuracy of estimated vehcity signais caused by using position measurement
Chapter 6: fxpenmental Validation
-
Y9 -
differentiation, in the experiments of NNC and RDC high-frequency noise \va3 introduced into the control loop.
6.3 Neural Network Controller Fust. it is emphasized that to the best of our knowledge no satisfactory experimental result has k e n reponed so far in the titerature on NN-based stable closed-loop robotic conrrol without a preliminary leaming stage of the NN weights. Four joints were used in this set of experirnents. Three groups of experiments were implemented: (i) a weli-tuned PID controiier; (ü) the STC controiier (Le.. PiD plu3 an STC compensator); and (üi) the NNC controller (i.e..
PID plus an NNC compensatori.
respectively. Mi three controllers were designed to have the same PID controller: hence. the experirnental results directly show the effects of STC and NNC. It was shown in the experiments that the STC achieved a siightly better performance in comparison with the PID. Thus, to Save space, only the results of STC and NNC will be presented tis follow\.
Table 6.1: Erperîmental robor purumeters
Table 6.1 shows the experimental robot parameters for the PID and STC. Thc parameters of the STC controller were tuned for improved performance from simulritioni, and extensive experiments. The parameters were found to be highiy sensitive to the stability of the closed-loop system, making their tuning very difficult. The derivative
gains of the P D controiier were chosen smaii in order to degrade the signiticant influence of estimation errors in the velocity signais, which was due to the inaccurticy of the numerical differentiation procedure. A RBF network was constructed, with 20 input neurons, 80 hidden-layer neurons and 4 output neurons, to e s t h t e on-line the unknown manipulator dynamics (3.28). The
- 90 -
Chapter 6: Experirnental Validation
RoboTrvin is a Fied-base robotic rnanipulator, so only the rnanipulator dynamics I3-28i is present. Gaussian hnctions d e h e d in (3.34) were chosen as M F bais functions. in order to reduce the on-line computation load, di RBF functions had f~vedwidths set
rit
[-5-51 that were set in accordance with the ranges of the driving torques of al1 joints. and
fixed centers that equivalently divide the NN approximation zones determineci by the widths. The input pattern of the NN conrrolier was chosen as (qrd,4,,.q,,,,e.ri whicti consists of the positions and velocities of the four joints of the manipulacor rirrn. Thr coupled dynamics transmitted from the vehicle base in (2.48) was absent becriuse of the fixed-base of RoboTwin. The output of the RBF net represented the compensation torclues for nonlinear dynamics of al1 joints. The weights of the RB€ net were simply initirilizrd at zero and were updated in a tirnely manner at each control cycle using the proposrd learning law (3.37) with the system sarnpling tiequency set at SûûHz and the Iramin: race of the RBF net (i.e., pz in (3.37)) chosen as 0.35 which had k e n selected b ~ s e don simulations and experimental tests. Error STC Controller 1 NNC Controller Measurernents Max 1 Max 1 Sumof 1 Max 1 Max 1 Sumof Sinusoida1 squared Positive Negative Negative squared Positive - Trajectories error (rad) error (rad) error (rad2) error (rad) error (rad) error (radz) , 0.061 9 -0.0079 Joint 1 0.0044 0.1376 0.0031 -0.0119 0.2026 Joint 2 0.0101 -0.0113 0.3517 0.0092 -0.0168 0.1068 Joint 3 1 0.0098 -0.0058 -0.0070 0.1686 0.0082 0.0204 -0.0046 0.0345 -0.0057 0.0045 Joint 4 1 0.0064
Table 6.2: Pe@ormance cornparison brn~,et.nSïC und N K
1 1
Joint No.
1
Percentage improvement in performance of NNC over STC: I
Joint 1 Joint 2 Joint 3 Joint 4
5s % 43 % 37 % 41 %
Table 6.3: Percentage improvemenr in perjomtance of NNC over STC
1
- 91 -
Chapter 6: Experimental Validation
The experimentai results show that by using NNC the magnitude of borh ma.uirnum positive and negative tracking errors of di joints are smailer than those of STC. and the sum of squared errors decrease by about 30
- 608 (see Tables 6.2 and 6.3). Thi.
variation is very significant. The results also show that the tracking performilnce
ih
improved over t h e (i.e., the tracking error decreases as learning process procerdhi. which verifies the learning capability of the proposed NN controlier. while STC does nor provide such capability. Figure 6.3 shows the desired sinusoidal trajectories tracked by the four joints. Fis 6.4 to Fig 6.7 present the joint position errors by using STC and NNC. respectively. F i p m 6.9 to 6.12 show the required torques of the two (also represent the energy consumtd bu the system) which are almost the same. Due to the inaccuracy of estimateci vrlocity signais caused by using numericai differentiation, significant noise was introduced into the control loop, which affects the tracking performance substantiaily.
In sumrnary, the experimentai results demonstrate that by using h ! C .
ri
better
performance was achieved both in terms of joint position and vslocity tracking mors in comparison with the conventional robust controlier STC. This can be mainly attributed to the foilowing facts: (i) NNC can learn the nonlinear dynamics of the systsrn by adaptiveiy tuning the NN weights and therefore generating more unçrrtainty compensation signals, while STC has a faed controller structure and does not have thc adapration ability: and (ii) NNC has a universal controlier structure. that is, independent of the dynamics property of the robotic manipulator system. While for STC.the boundins
function (Le., v, in (6.6)) should be designed specifically for each robotic arm* The desig of the bounding function requires certain knowIedge of the robot (cg.,
E
,
S,,. S, and S.
in (6.6)). However, in practice such quantities may not be calculated properly due
to the
incornplete knowIedge of the rnanipuiator dynarnics. In our experiments. these parameters were fust estimated off-line using simulations, and then were adjusted in the experiments for achieving a better performance. Yet the resulting parameters are stiH nor optimal since in the experiments they were found to be very sensitive to the behavior of the robotic arm (Le., a s
d variation of one parameter may cause chattering).
Chapter 6: Experimental Validation
Joint 3 40
Figure 6.3: Desired rrajecrories of theforrr joints
- 93 -
Chapter 6: €xperimental Validation
Figure 6.5: Position crrors ofjoint 1 {rop: 5°K;bottom: NNCI
Figure 6 5 : Position errors ofjoint 2 (top: STC; borrom: NNC)
- 93 -
Chapter 6: Experimentai Validation
0.01
Joint INNC)
,
Figure 6.6: Position errors ofjoinr 3 Ifop: STC; borrorn: NAT) Joint q
8~ 10.'
SQ
I
x 10 I
'
Joint #NNQ
Figure 6.7: Position errors ofioinr 4 (top: STC; botrom: NNC)
-93-
Chapter 6: Experirnental Validation
Joint l(SIC)
Figure 6.8: Driving rorqnes ofjoinr 1 (rop: STC; borront: IVNC) Joint Z(STC)
Joid üNNC)
Figure 6.9: Driving torqrres ofjoinr 2 (rop: STC;bottoni: NNC)
- 95 -
Chapter 6: Experimental Validation
Figure 6.10: Driving torques ofjoinr 3 (top: STC; borrom: NNC)
Figure 6.11: Driving torqties ofjoinr 3 (rop: STC; bottoni: NNC)
-96
-
Chapter 6: ExperimentaI Validation
-97-
6.4 Robust Damping Controller The RoboTwin is a fwed-base manipuIator arm; therefore, the RDC controller desi~n foliows the design procedure provided in Section 4.3. Two joints were used in this set of experiments. Similarly, three proups of experiments were performed: (i) a well-tuned PtD controuer; (ü) the STC controller (Le., PID plus an STC compensator); and (iii) the RDC controller (Le., P D plus a RDC compensator), respectively. The RDC control law
ih
given by Equation (4.3, with k, in (4.2) [5.0, 2.01 (for two joints). Ai1 three controllers were designed to have the same P D controiier: therefore, the experirnental resultb directly show the effects of RDC and STC, Again. it was shown in the experimrnts thar the STC achieved a slightiy better performance in cornparison with PtD.Thus. only the results of STC and RDC wiil be presented in the following. The experimental robot parameters of this set of experiments are presented in Table 6.4. Similarly, the parameters of the STC controller were tuned for irnproveri
petiormance From simulations and extensive experiments. UnIike STC. RDC has onlone extra RDC gain p (see the RDC controUer law (4.5)), which is dynamic independent and may be determined easily. This is aiso one sipificant advantage of RDC over STC and many other conventionai robust conmiiers.
Table 6.4: Erperimenraf robor parameters
Figures 6.12 to 6.18 give the experimental results fiom tracking a regulur sinusoidril curve using STC and RDC controiier, respectively. It is shown chat by using RDC, the magnitudes of both maximum positive and negative tracking errors of the two joints tire comparably smaiier than those of STC, and the sums of squared errors of the joints decrease about 66% and 46%, respectively, whiie the required control torques of the two (representingthe energy consumed by the system) are aimost the same. Figure 6.19 to 6.25 present the expecîmental results fiom tracking an irregul~rr sinusoidai cuve using the same controilers as above. The new trajectory comprises a
-9S-
Chapter 6: fiperimental Validation
fast-speed portion and a slow-speed portion. Frorn the figures Ive cari see that for the faster trajectones nodinear dynamic effects are dominant. It is shown that STC now causes undesirable chattering (e-g., significant noise signais present in both position errors and velocity errors), while for the RDC controiier. the overai1 systern behrivior remains srnooth, Tables 6.5 and 6.7 present the comparison of error measurements beween STC ;incl
RDC for each experiment. Tables 6.6 and 6.8 show the percencage iimprovemrnis of the sum of the square errors for each joint that RûC provides compared with thrit of STC. which proves that RDC is much more robust than STC. Again. by using RDC. the magnitudes of both maximum positive and negative tracking errors of the two joints rire comparably smaiier than those of STC, and the surns of squared errors of the joints decrease about 49% and 408, respectively.
In summary, the experirnentai results show that RûC achieved better performance both in t e m of joint position and velocity cracking errors for both Fast and smooth trajectones in cornparison with STC.
Error 1 ROC Controller STC Controller Measunments ~ a x 1 Max I Sumof I Max 1 Max I Sumof Regular Positive Negative Positive Negaove squand quand, f rabctories error (rad) error (rad) error (rad ) error (rad) error (rad) error (rad2) Joint 1 0.0038 0.0059 -0.0036 0.0076 -0.0025 0.0026 Joint 2 0.0075 -0.0067 0.0060 0.0076 0.0141 -0.0052
Table 6.5: Peformance comparison benveen RDC itnd STC Iregtriur irujecrot? i
)jxI YI
Percentage improvement in performance Joint No. o Joint 1 Joint 2
o
v
Z k&cJ c e,k . r'
e
-GDCJ
66% 46%
Table 6.6: Percenrage improvemenr in peformance of RDCover !TC (regiilarr r u j e c r o ~ ~
Chapter 6: Experimental Validation
-99-
STC Contraller 1 ROC Controller Error Measurements M ~ X 1 Max 1 Surnof 1 Max 1 Max 1 Sum of lrregular Negative squared Positive Negative squaredl Positive Trajectories error (rad) error (rad) enor (rad2) enor (rad) error (rad) error (rad ) -0.0042 0.0074 -0,0019 0.0038 Joint 1 0.0089 0.0066 Joint 2 -0.0086 -0.0105 0.0076 0.0039 0.0126 0.0048
Table 6.7: Performance cornparison benveen RDC utid STC (irregltlur rrcijecror?)
1
1
Joint 1 Joint 2
Percentage improvement in performance
49% do0/,
Table 6.8: Percerirage i~nprovrnrettriit pe$orniance of RDCover STC (irrqrilur rrujrcror~,
Chaoter 6: Exoerimental Validation
Figure 6.12: Desired rrajectories of the nvo joints (reguinr r m j e c r o ~ )
Figure 6.13: Position errors ofjoinr I (top: STC;botrom: RDCl
-
lnn -
Chapter 6: Expenmental Validation
Figure 6.14: Velocity errors of joint 1 (top: STC: botrom: RDC)
Figure 6.15: Driving rorqim ofjoinr 1 (top: STC: bortorn: RDC)
- 101 -
Cha~ter6: Exaerîmental Validation
Figure 6.16: Posiriori rrrors oj'joirir 2 (rop: STC: bottom: RDCI
Figure 6.17: Velaci- errors of joinr 2 (top: STC: borrom: RDC)
- ln' -
Cha~ter6: Exoerimental Validation
Figure 6.18: Driving rorqries ofjoint 2 (top: STC; bottom: RDC)
Figure 6.19: Desired rrajecories ofthe w o joints (irregnlar rrajectoy)
- 10; -
Chapter 6: Experimental Validation
w 16' 10,
Joint 1: Position Enors (rad) b
I
Figure 6.20: Position errors ofjoint 1 (top: STC: borrom: RDC) Joint 1: Vcloaty Ermrs(nus) 1
Figure 6.21: Velocity errors of joint 1 (top: STC;bottom: RDC)
- lm-
Chaoter 6: Exoerirnental Validation
Jânt 1: Toques (Nm) I
1
Figure 6.22: Driving rorqrtes ofjoint I (top: STC;botrom: RDC)
Figure 6.23: Position errors ofjoint 2 (top: STC; bartom: RDC)
- in5 -
Chaoter 6: Exoerimental Validation
Figure 6.24: Veloci~errors of joinr 2 (top: STC;bonom: RDC) Joint 2: Toques (Nm)
2
Figure 6.25: Driving torqries of joinr 2 (top: STC;bononr: RDC)
- 1015-
- 107 -
Chapter 6: Experimental Validation
6.5 Observer-based Controller From the experimentai resuIts presented in the previous sections. we c m see that significant noise occurring in the closed-loop substantially degraded the control performance, as a result of estimating the velocity signals using the inaccurate nurnçrical differentiation procedure. In this section, we present the experimental results to show how the velocity observer proposed in Chapter 5 may irnprove the control performtinct.. In this set of experiments, two joints are used for tracking a sinusoidal curve iFiprt'
6.16) using the PID and OBC controiier, respectively. A simple integral control term w r ~ ! added ont0 the independent joint PD control in order to compensate for gravity an3 nominal dynamics that is generally required (Lewis et ni. 1993): and as such, it was also added onto the OBC controlier for comparison purposes. Both controllers have sirnilar control structures and exactly the same proportional, derivative and integral pins. Thc only difference between the two is the acquiring methods of velocity signal?;.
L!
illustrated in Chapcer 5. Figures 6.27 to 6.3 1 shows that the high-t'requency noise in thc controller loop caused by velocity inaccuracy has been substantially reduced by usin2 t tic proposed observer. The experiment parameters of this set of experiments are given in Table 6.9. #
Robot Parameters
ib
i4
kd
ma
Joint 1
3.0
1.5
0.03
10.0
0.6
1.0
10.0
10.0
Joint 2
2.5
2.0
0.02
30.0
0.8
5.0
500.0
100.0
PI0 Gains
Obsenrer Based Controller a0 A2 = A15 Al
13
Table 6.9: Erperimental robot parumeters Table 6.10 presents the comparison of error measurements between PID and OBC. Table 6.1 1 gives the percentage irnprovements of the sum of the square errors for rtich joint that OBC provides compared with that of PID,which proves that OBC is much more robust than P D , By using the proposed veIocity observer to estimate the velocity signais for P D ,the magnitudes of both maximum positive and negative tracking m o r s of the two joints are comparably smdier than those using numerical differentiation. and the sums of squared errors of the joints decreased about 77% a d i5%. respectively. while the required controI torques of the two are almost the same. It should be noted that the
Chapter 6: Experimental Validation
- IQX -
comparably less improvement of the cracking performance in Joint 2 is mainiy due to the significant backlash found to occur in the joint of RoboTwiri, which significantly degrride the control performance.
In the experiment, the proposed velocity observer-based linear controller achirved better performance in terms of position and velocity tracking errors. This can be mainly attribuced
CO:
(i) the observer generating a better velocity measurement. which
significantly reduced the noise contained in the velocity signal (see Figures 6.25 and 6.3 1) and resulted in a much smoother joint control torque (see Figures 6.19 and 6.32):
md (ii) the nonlinear dynamic coupling (Le., the centripetal and Coriolis vectori. as iswell as the gravity and disturbances, have k e n largely compensated for by OBC becriuse of
its capability of dealing with dynamic uncertainties, while PID exhibits much w:ikr.r ability in this situation. In conclusion, the proposed observer can provide a better estirnition on the velocity signal than using the differentiation of joint position measurement. Error PID Controller Observer Based Controller Measurements Max Max Sum of Sum of Max Max Sinurioidal squared Negative squared Positive Negative Positive Tralectories error (rad) error (rad) enor (rad2) error (rad) error (rad) error (rad2) 0.001 7 -0.0033 0.0071 -0.0030 0.0058 0.0037 Joint 1 0.01O1 -0.0068 0.01 20 0.0098 -0.0093 0.0075 Joint 2
Table 6.10: Peformunce cornpurison benveen PfD und OBC Percentage improvement in performance Joint No.
Joint 1 Joint 2
77% 15%
Table 6.11: Percenrage improvenlenr in peformance of PID over OBC
Chaoter 6: Ex~erimentaiValidation
Figure 6.26: Desired rrajectories o f t h nvo joints Joint 1: Posaon Enon (md)
0.01
Figure 6.27: Position errors ofjoint 1 (top: PID; bottom: OBC)
- 109 -
Cha~ter6: Experimental Validation
Joint 1:Vaiacity Ermn (mdls) 0.1,
I
Figure 6.28: Velocis, errors of joint 1 (top: PID: bortom: OBC)
Figure 6.29: Dtfving torqrtes of joint 1 (top: PID; botrom: OBCl
- 110-
Chapter 6: Experimental Validation
Figure 6.30: Position errors ofjoinr 2 (top: PID: borrorn: OBC)
Figure 6.31: Velociq errors ofjoinr 2 (top: PID; botrom: OBC)
-
111 -
Cha~ter6: Ex~erimentalValidation
- 112-
Joid 2: Toques (ml
Figure 6.32: Driving rorqiies ofjoint 2 (top: PID; bortonl: OBC)
6.6 Summary [n this chapter, experimental results have k e n presented to illustrate the effectiveness of the controiiers proposed in the previous chapters. It is shown that al1 proposcd controiiers have a significant control performance in cornparison with the convrnriontii ones. This further validates the theory developed in this thesis. [t should be pointed out, however, that the experirnental effectiveness does not
auromticaiiy provide a guarantee of the appiicability in real applications of the proposed control strategies since many facts may hinder them from king utiiized in reality. For thc NNC, for instance, we found that the number of the hidden-layer neurons hris u
significant influence on the leacnîng effect of the neural network: tao few may greatly
degrade the WC's performance, Unfortunately, a large number of hidden-Iayer ncurons requires a large on-line computational cost and a large memory to store the data: this mriy be unredistic for most industriai robotic applications because of the iimited and relatively
Chapter 6: Experirnental Validation
- 113 -
low computing capability of most embedded systems. Unfortunarely, we are not able to make improvement in this regard. ~ S O aii , experiments
presented in this chapter were perfomed on a manipulator a m
with a fuced base. Thus, the dynamic interactions between the manipulator arm and the vehicle base were not present. Such coupled dynamics has k e n shown to have a sipnificant Uifluence on the control performance in certain circumsrances. cg.. when the mobile manipulator moves at a high speed (Dubowsky and Vance. 1989). Theretorr. further work to verify the effectiveness of the proposed controllers when they are applictl to the control of a mobile manipulator is necessary. Unlike NNC,both RDC and OBC have very simple conrro1 structures. and they only require a comparably Iower extra-computational cost in comparison to NNC. This is ont. of their signZcant advamages over NNC, which may facilitate their utilizrition in industrial applications.
Conclusions and Future Research 7.1 Conclusions This thesis addressed the position control of mobile manipulator subject to kinemaric constraints in the presence of unstructured dynamics and unstructured disturbances. in the previous chapters. the main results achieved in the thesis have been presented in detail. In summary, this research has provided the following main contributions: 1) A neural network-based hierarchical intelligent architecture was proposed in
Chapter 3 for the control of a constrained mobile manipulator.
The controllet-
consists of four-levels: (i) decision level; (ii) learning level: (iii) adaptation Ievel: and (iv) execution level. In the learning and adaptation layers, two controlters ivere deveioped to independently control the vehicle and the manipulator. taking intu account the dynamic interaction between the two. Each controller output consists of LI linear controuer and an NN compensator. The NN compensation cerm is ussd
tu
compensate for dynamic uncertainties and unmodeled disturbance. To mrike t hc proposed theory practical, new NN on-line weight-updating mles were derived in the learning levei. No off-line learning phase for NN weights is required. thus t h q mi! simply be initialized at zero. The NN controUer provides good learnin_o ancl adaptation capabilitities. Furthermore, it is capable of compensating h r the çoupld d y n d c s between the vehicle and mmîpulator, as weii as rejecting unmodeled disturbances. As a resuIt, the proposed NN control architecture is especidly suitable for applications where the mobile tnanipulator moves in unknown and changins environments in which leaming and adaptation capabiiities are crucial, Morever. the cIosed-loop conuol stabiiity, convergence of NN learning processes and boundedness
Chapter 7: Summary
-
115-
of NN weight estimation errors are ail garanteed. The above can be viewed ris improvements that the proposed NN controiier can offer over most published NNbased control schemes, which typicaliy require an off-line learning stage of NX weights andior rnay not guarantee the closed-loop stability. 2) A new robust control technique, Robust Damping Control, was introduced in Chapter 4. A RDC-based hierarchical robust control archicecture was further developed for a mobile manipulator.
The controller consists of three-levelb: i i l
decision level; (ii) robustness level; and (E) execution level. The detailed design of the robustness level was presented, The RDC controiier is capable of dealin: with dynamic uncenainties and unmodeled disturbances. Compared with NNC and mosi conventional robust controllers. RDC has a sllnpler controlier structure and dsmancls much less on-line computation. Moreover, it is system-independent (Le. the dynarnics properties of mobile manipulators are not required). This is especially interesting 6-om a realistic viewpoint since we cm design a RDC controller h r universal use with mobile manipulators so that. when the dynamic properties of the robotic systerns change, the controLier can stiII provide a guarantee of achieving good performance without modifyig the structure of the controller. The RDC controlltir is especially suitable for applications where the availabie systern resources of rhr mobile manipulaior, such as sue of embedded memory and computation capability o f on-board computer, are restricted, or when the mobile robot moves in pürtiriily or completely known environrnents where learning and adaptation capabilities are lrss crucial.
3) A reduced-order adaptive velocity observer was proposed in Chapter 5. A velocity observer-based linear feedback controller was further developed.
The
proposed observer targets the position control of robotic manipulators in t h presencr of dynamic uncertainties and unmodeted disturbances. requiring only the measurement of position signai, The main advantages of the proposed velocity observer-based controiier over m s t conventionai observer-based control schemes are: (ij it has a reduced-order observer dynamics, thus it provides a rnuch h t e r dynamic response; (hi it does not require any knowledge of the robot dynamics: moreover, the bounds of uncertainties are not required except for some knowIedgt of
- 116-
Chapter 7: Sumrnary
the lower bound of inertia m s which c m be easily detennined; and (üi) it is capable of compensiiting for unknown friction, gravity and unmodeled disturbances:
1) Extensive experiments were conducted on a manipulator arm (IRIS RoboTwin), A
RDC controuer, an RBF-based NN controiier and an observer-bosed controllsr
were irnplemented experimentdiy for verifkation purposes. Each of the concrolkr
has k e n tested for trajectory tracking joint-space control. The experimentd rcsiilr~ illustrate the advantages of the proposed theory.
7.2 Future Research The following topics are suggested as ri continuation of this research: 1) Extend the
NNC
to the multilayer networks instead of
RBF
networks.
Tiit
proposed RBF-net based controuer (3.36) displayed gwd performnce without requiring much detriiled knowiedge of the system dynamics. However. it does rcqiiirt. the bounds of the parameters (see (A.13) in Appendir Il, which is generilty dit'ticult CO
acquire in realistic applications. The RBF net is a hnctional-link nrurril net
( F L i i ' . Using a F L N requires one io select the activation hncrions ocsi
corresponding to ri bais set ofsmooth functionsAx) (see (2.20) md (7.22))-whilt. for
a multiiayer neural net, such pre-seiection is generdty not required. Also, the RBF ncr is Iinear in the tunable weights W (see (2.22)), which is undesirabLe. iilthough a le>> severe restriction than hearity in the system parameters is required in the standarc! adaptive robotic control applications (Slotine et al.. 1991). By using a rnultiltiyer neural net, rhis probiem may be overcome (Lewis et aL, 1999): 2 ) Enhance the RDC controller with adaptation capabilities. The RDC cont ro ller
proposed in Chapter 4 has a very simple control structure and c m be easily irnplemented in reai systems. However, in order to achieve the desired trxkinz performance, sufficientlylarge control gains in (4.45)and (4.46).or (4.57)and i 3.58i. may be required. It is weU knowvn that such high-gain feedback rnay cause undesirable chattering in reai applications. This is due mainly to the Ioss of adaptation in detemiining the conuol gains of the proposed RDC conuoiier. Therefore. hrther
Chapter 7: Surnmary
- 117-
research is expected to modiîy the RDC control laws so that the adaptation capabilities may be provided;
3) Combine the OBC with NNCJRDC. The velocity observer-based linear feedbrick controller developed in Chapter 5 has a rather simple Linear structure. However. the linear structure of the PD-like controiler (5.18) rnay not be optimal when highly nonlinear dynamics of robotic systems and large disturbances are present. In t his crise. unnecessary large torque inputs may be generated in (5.18) when nr, is chosen to bc too iarge to satisfy condition (A.25) (see AppendLr II). Thus, it is desirable to provide
an adaptation method of determining m, in (5.8). Furthemore, it would also be ver! interesting to combine the proposed Linear velocity observer (5.6) and (5.8) with the NNC (e.g., (3.36)) and RDC (e-g., (4.46), (4.47) and (4.57). (4.58)) so thrit the
requirements of the velocity measurements in the NNC and RDC can be eliminated.
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[2] Akpan, U. and M. Kujath, "Nonstationary random dynamics of a mobile robotic: manipulator", ASME Journal of Dynamic Systems, Mraairemrnr, id Corirroi. vol. 119. 135-139, 1997
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[ID] Bloch A. and N, McClamroch, "Control and stabiiization of nonho tunomic dynamical systems", IEEE Trans. Automaric Control, vol. 37, 1746- 1757. 1991
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IlIIIli Appendix
A.l Proof of Theorem 3.1 The proof of Theorem 3.L starts from the Lyapunov design result represrnted by (3.261. The unknown dynamic terms Y, and Yz respectively described by (3.27) and i3-281 are noniinear continuous functions of position and velocity signals. Thus. thry crin bt. approximated by RBF networks with the estimation errors sufficiently small. in other words, the NN approximation property (3.30) hold for the nonlinear functions (3.77) and (3.28) with given accuracy E , , and E : , for allx in the compact set
where b, > q, (Lewis et al., 1999). Let us further de fine a compact set
with P(O)E S, (Lewis et al. 1999). Frorn (3.33) we m y show that the YX approximation property also holds for aii P in the compact set S, . Substituthg the torque control law (3.36) into (3.26) yields
where
=w,-w,,
@?
=wL-wL.
Appendix
]
Let us d e h e
and r D=
[
W
-.-lI+.%
h(r) =( hl (-VI h&)
-
)
W =[ \VI O
-IE
O
131 -
~ i l ! l - * l : ~ ~ ~ l l ~ ~1 - ~ : - f
\v,
r Then, it is straightforward to verify that the folloring
The Lyapunov hnction V consists of the Lyapunov îünction V4 (proposed in (3.17)) t'or the mobile manipulator, and two additional terms that are used to count for the Sli learning dynamics. Differentiating (A.6) and substituthg (k3)into it yields
- 132 -
Appendix
which is guaranteed negative as long as
Furthemore, to ensure that the approximation property of the two NN on-line estimators (3.30) holds throughout, the tracking error-vector P should be always kcpt in
-
the compact set S,. This may be achieved by selecting the minimum control gain k to satisfy
6, - q 6 Therefore, the compact set defmed by
is contained within S,; as a result, the approxhiïiion pmperty of the two NN estimators holds throughout. Thus,
v
is negative outside a compact set. According to the standard Lyapi~nuv
theory and the extension of LaSulle theory, this demonstrates the ultimate unitortn boundedness of the u a c b g erron z and r, and the neural net weights subsequently, the weight estimates
@, and
c.and
and W~ (notirtg chat W Land W- are constants) 3
(Lewis et al. 1999). Therefore, the conml torques are &O yaranteed to be bounded.
Moreover. the n o m of the tracking enors increasing the minimum gain
=/[:]l
11~1
c m be kcpt arbitrarily srnrll hy
k in (A. 11).
[7
A.2 Proof of Theorem 5.1 Substituting the control law (5.18) into che observer error dynrunics and trrickin, rrror dynamics (5.14) and (5. 15), respeccively, we obtain
- AK,é2 - T,P2 + P .
ii= -PK,e,
(,A.ij~
ê2 =-K,e, -K2ê2+ï3i1-x:.
where A = (M-'- M i l ) M O It. is easy to show that A is bounded.
(A161
~. E . I A I
Sd,, . with
6, > O some positive constant. Equations (5.9), (5.1 l),(A. 15) and (A. 16)represent the error dynamics of the combined controuer-observer closed-Ioop system. Let us choose the Lyapunov hnction as
The Lyapunov function V in (A. 17) consists of four terms. The first term accounts t'or chs actud position tracking error, and the other three terms are introduced to account for the observer error dynarnics and the actual veiocity tracking error. Diiierentiating (A. 17) dong the system trajectories yields
L
where Â, is chosen to satisfy Al 2 -, and 2
In (A. 191, a, is a positive number that should be chosen to satisQ O < a,, < 1 .
Fust, let us consider (A. 19). From (A.19) we obtain
From (A.21) we can see that when 1 -
we may guarantee that
iv,
(6,
+,1)'
4a;
10.Le., when
IO.
Let m, in the control law (5.18) (notice that M, = n r , l ) be chosen such thrit the folIowing inequality holds: 1
6, 5 [ ~ M - 1 ~ ~ + $ ] ~ 5 [ ~ II+, + $ ] - 5-2l < ~ 1.l o where m - is the Iower bound of the inertia rnatrix defined in (2.50). From (A.23) we obtain
Hence, (A.22) c m be satisfied when
Now let us consider (A.20). Using the property (5,17), (A.20) t u m out to be
i X.73 I
To apply Lemma A.l to the previous proof, we first choose a controller gain
k, = k,> O and make it fuied. Then the contmiier gain kZ and the observer gain chosen to satisfy
&, = S,k,, where 6,> O
À,
are
is a positive constant. Now (A.28) turns out
tci
be
Pl Pl and 1 are al1 bounded by some positive constants Notice that 1-a, > O ; -, k2
and dirninish when k, and
S
&
4 increase. Therefore, from Lemma 5.1 it is shown thcit thc
terrns in the two brackets in the right-hand side of (A.3 1) can be made negrit ive by choosing sufficiently large k, and
4.
Thus, from (A.18). (A.21) and (A.3 1 ) and
following the proof in (Qu 19981, we conclude that the system is ultimnrely bot inded.
riri$)rrrr&
En
