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Proc. of CDC’2000 - 39th IEEE Conference on Decision and Control, Sydney, Australia, December 2000Regulation of a Nonholonomic Dynamic Wheeled MobileRobot with Parametric Modeling Uncertainty usingLyapunov FunctionsAntónio Pedro Aguiar†1†Ahmad N. Atassi‡António M. Pascoal†ISR/IST - Institute for Systems and Robotics, Instituto Superior Técnico,Torre Norte 8, Av. Rovisco Pais, 1049-001 Lisboa, PortugalE-mail: {antonio.aguiar,antonio}@isr.ist.utl.pt‡UCSB - University of California Santa Barbara, Center forControl Engineering and Computation, CA 93106-9560, USAE-mail: atassiah@seidel.ece.ucsb.eduAbstractattractive are discontinuous control laws, which in somecases can overcome the complexity and lack of goodperformance (e.g., low rates of convergence and oscillating trajectories) that are often associated with timevarying control strategies. The reader is referred to(Astolfi, 1999) for a discussion of this interesting circleof ideas.This paper addresses the problem of regulating the dynamic model of a nonholonomic wheeled robot of theunicycle type to a point with a desired orientation. Asimple controller is derived that yields global convergence of the trajectories of the closed loop system inthe presence of parametric modeling uncertainty. Controller design relies on a non smooth coordinate transformation in the original state space, followed by thederivation of a Lyapunov-based, adaptive, smooth control law in the new coordinates. Convergence to theorigin is analyzed and simulation results are presented.Despite the vast amount of papers published on the stabilization of nonholonomic systems, the majority hasconcentrated on kinematic models of mechanical systems controlled directly by velocity inputs, while lessattention has been paid to the control of nonholonomicdynamical mechanical systems where forces and torquesare the true inputs. See for example (M’Closkey andMurray, 1994) where the authors extended time-varyingexponential stabilizers to dynamic nonholonomic systems.Another less studied problem is that of controlling uncertain nonholonomic systems. See (Jiang and Pomet,1996) where a backstepping based time-varying adaptive control scheme for a special class of uncertain nonholonomic chained systems was proposed.1 IntroductionThe control of nonholonomic systems has been the subject of considerable research effort over the last fewyears. The reason for this trend is threefold: i) there area large number of mechanical systems such as robot manipulators, mobile robots, wheeled vehicles, and spaceand underwater robots that have non integrable constraints; ii) there is considerable challenge in the synthesis of control laws for systems that are not transformable into linear control problems in any meaningful way and, iii) as pointed out in a famous paper ofBrockett (Brockett, 1983), nonholonomic systems cannot be stabilized by continuously differentiable, timeinvariant, state feedback control laws. To overcome thelimitations imposed by the celebrated Brockett’s result,a number of approaches have been proposed for stabilization of nonholonomic control systems to equilibriumpoints. See (Kolmanovsky and McClamroch, 1995) andthe references therein for a comprehensive survey of thefield. Among the proposed solutions are smooth timevarying controllers (Samson, 1995; Godhavn and Egeland, 1997), discontinuous or piecewise smooth controllaws (Canudas de Wit and Sørdalen, 1992; Bloch andDrakunov, 1994; Astolfi, 1999), and hybrid controllers(Hespanha, 1996; Aguiar and Pascoal, 2000). SpeciallyMotivated by the above considerations, this paper addresses the problem of regulating the dynamic modelof a nonholonomic wheeled robot of the unicycle typeto a point with a desired orientation. A simple, discontinuous, adaptive state feedback controller is derivedthat yields global convergence of the trajectories of theclosed loop system in the presence of parametric modeling uncertainty. This is achieved by resorting to a polar representation of the kinematic model of the mobilerobot that is a non smooth transformation in the original state space, followed by the derivation of a smooth,time-invariant control law in the new coordinates. Thenew control algorithm proposed as well as the analysisof its convergence build on Lyapunov stability theoryand LaSalle’s invariance principle. For an introductionto the polar representation and how it can be exploitedto overcome the basic limitations imposed by Brockett’sresult, see (Aicardi et al., 1995) and (Astolfi, 1999).1 The work of António Aguiar was support by a Graduate Student Fellowship from the Portuguese PRAXIS XXI Programmeof FCT.The paper is organized as follows: Section 2 describes1

the model of a wheeled mobile robot of the unicycle typeand formulates the problem of regulating its motion toa point with a desired orientation in the presence ofmodel uncertainties. A non-smooth polar representation of its kinematics is also presented. Section 3 introduces a simple Lyapunov-based adaptive strategy forvehicle control that builds on a series of candidate Lyapunov functions related to vehicle heading regulation,target distance regulation, and parameter adaptation.Section 4 offers a formal proof of convergence of the resulting adaptive regulation system. Section 5 containssimulation results that illustrate the performance of theproposed control strategy and show how it yields natural vehicle’s behaviour. The paper concludes with asummary of results and recommendations for furtherresearch.where v and ω denote the linear and angular velocityof {B} with respect to {U }, respectively. The controlinputs are the force F along the vehicle axis xB and thetorque N about its vertical axes zB . It is easy to seethatyU yGxU xGyBbxBv2 The wheeled mobile robot. Control problemformulationq2LThis section describes the kinematic and dynamic equations of the wheeled mobile robot of the unicycle typeshown in Figure 1 and formulates the problem of controlling it to a point with a desired orientation. Thevehicle has two identical parallel, nondeformable rearwheels which are controlled by two independent motors, and a steering front wheel. It is assumed that theplane of each wheel is perpendicular to the ground andthe contact between the wheels and the ground is purerolling and nonslipping, i.e., the velocity of the centerof mass of the robot is orthogonal to the rear wheelsaxis1 . It is further assumed that the masses and inertias of the wheels are negligible and that the centerof mass of the mobile robot is located in the middle ofthe axis connecting the rear wheels. Each rear wheel ispowered by a motor which generates a control torqueτi , i 1, 2.2RFigure 1: A wheeled mobile robot of the unicycle-type.1(τ1 τ2 )RLN (τ1 τ2 )RF (2.2)where R is the radius of the rear wheels and 2L is thelength of the axis between them. The symbols m andI are the mass and the moment of inertia of the mobilerobot, respectively.With the above notation, the problem considered in thispaper can be formulated as follows:Derive a feedback control for τ1 and τ2 to regulate {B}to {G} {U } in the presence of uncertainty in theparameters m, I, R, and L.2.1 Robot Model. Problem FormulationThe following notation will be used in the sequel. Thesymbol {A} : {xA , yA } denotes a reference frame withorigin at OA and unit vectors xA , yA . Let {U }, {G},and {B} be inertial, goal, and body reference frames,respectively. Assume, for simplicity of presentation,that {U } {G} and that the origin OB of {B} is coincident with the center of the rear wheels axle. Let [x, y]Tspecify the position of OB in {U } and let θ be the parameter that describes the orientation of {B} with respect to {U } (i.e., the robot orientation with respect tothe inertial x-axis). The kinematics and dynamics ofthe mobile robot are modeled by the equations2.2 Polar representationThis section introduces a change of coordinates thatplays a crucial role in the development that follows.See (Astolfi, 1999) and the references therein for anintroduction to this transformation, its rationale, andimportance. Consider the coordinate transformation(see Figure 1)pe x2 y 2x e cos(θ β)y e sin(θ β)¶µ y 1θ β tan xẋ v cos θẏ v sin θθ̇ ωmv̇ FI ω̇ Ndw(2.1)(2.3a)(2.3b)(2.3c)(2.3d)where d is the vector from OB to OU , e is the lengthof d, and β denotes the angle measured from xB to d.Notice that in equation (2.3d) care must be taken toselect the proper quadrant for β. Differentiating (2.3)with respect to time, the dynamics of the wheeled robot1 By assuming that the wheels do not slide, a nonholonomicconstraint on the motion of the mobile robot of the form ẋ sin θ ẏ cos θ 0 is imposed.2

in the new coordinate system can be written asė v cos βsin βv wβ̇ eθ̇ wmv̇ FI ω̇ Nand compute its time derivative along trajectories of(3.1) to obtain ·sin βV̇1 β k1 kσ σ k1 sin β ω .β(2.4)Following the nomenclature in (Krstić et al., 1995) letω be a virtual control input andα1 (σ, β) k1 kσ σRemark 1 Notice that the coordinate transformation(2.3) is only valid for non zero values of the distanceerror e, since for e 0 the angle β is undefined. Thiswill introduce a discontinuity in the control law thatwill be derived later, which will obviate the basic limitations imposed by the celebrated result of Brockett.The creation of this singular point is at the core of thedesign methodology adopted in this paper, which wasinspired by the work of (Astolfi, 1999).(3.3)k2 0, a virtual control law. Introduce the error variablez1 ω α1 ,(3.4)and compute V̇1 to obtainV̇1 k2 β 2 βz1 .Step 2. (Backstepping) The function V1 is now augmented with a quadratic term in z1 to obtain the newcandidate Lyapunov function3 Nonlinear Controller Design1V2 V1 z12 .2This section proposes a nonlinear adaptive control lawto regulate the motion of the mobile robot described byequations (2.1) and (2.2) to a point with a desired orientation, in the presence of parametric modeling uncertainty. Only the rationale for the control law proposedis introduced, a formal proof of convergence being deferred to Section 4. For the sake of clarity, candidateLyapunov functions are introduced recursively in a sequence of logical steps directly related to vehicle heading regulation, target distance regulation, and parameter adaptation. This methodology borrows heavily fromthe techniques of backstepping (Krstić et al., 1995). Aswitching term is introduced in the control law at thelast stage in order to solve the indeterminacy at e 0caused by the polar representation adopted.The time derivative of V2 can be written as· N2V̇2 k2 β z1 f1 (·) ,Iwhere α ασ̇ β̇ β σ ββ cos β sin βsin β k1 kσ σ β̇ k1 kσ σ̇ββ2f1 (σ, β, z1 , ρ) k1 β̇ cos β k2 β̇ β.βNotice that the terms sinβ β and β cos β sinare well deβ2fined and continuous at zero. Using L’Hopital’s rule itis easy to see that when β 0 the first and secondterms are equal to 1 and 0, respectively.Step 1. (Heading regulation) Define the variablesvρ , σ β θ,eand rewrite the equations of motion (2.4) asLet the control law for N be chosen asσ̇ ρ sin β(3.1a)β̇ ρ sin β ωNω̇ I(3.1b)ė ρ cos βeFρ̇ ρ2 cos βme(3.2a)N If1 (·) k3 z1 ,k3 0.Thenk3 2z 0,I 1that is, V̇2 is negative semidefinite.(3.1c)V̇2 k2 β 2 andStep 3. (Distance regulation) Consider now the distancesubsystem (3.2). A new error variable z2 ρ k1is defined and a third candidate Lyapunov function isintroduced as1V3 V2 z22 .2(3.2b)where system (2.4) has been divided in two subsystemsthat will henceforth be referred to as the heading anddistance subsystems, respectively. Consider the heading subsystem (3.1) and suppose (only at this stage)that ρ k1 0. Define the control Lyapunov functionV1 sin β k1 sin β k2 β,βComputing its time derivative gives· Fk3 f2 (·) ,V̇3 k2 β 2 z12 z2Ime11kσ σ 2 β 2 ,223

whereto yieldf2 (σ, β, ρ) ρ2 cos β kσ σ sin β β sin β.V̇4 k2 β 2 The last two terms of f2 are due to the fact that ρ isnot constant, but ρ k1 z2 instead. They are simplycomputed by replacing ρ by k1 z1 in the expressionfor V1 and propagating the corresponding terms downto V̇3 . Now, by choosing the control inputthe time derivative of V3 becomesk3 2 k4 2z z ,I 1 m 2that is, V̇3 is negative semidefinite.when e 0, where kθ̇ and kθ are positive constants.The motivation for this control law can be simply understood by noticing that it aims at rotating the vehicle in place under the action of the proportional andderivative terms kθ θ and kθ̇ θ̇, respectively.Step 4. (Parameter adaptation) Suppose that the values of the physical constants m, I, R, and L are notknown precisely. Define the control inputs ui , i 1, 2as u1 τ1 τ2 and u2 τ1 τ2 . Then, from (2.2) thedynamic equations for the mobile robot can be writtenasu2u1, v̇ ,ω̇ c1c2The complete control law is thus given by· ½u1(3.5), (3.6) e 6 0u u2(3.8)e 0where c1 IRL and c2 mR are positive unknownparameters.Consider the augmented candidate Lyapunov function(3.9)4 Convergence Analysis11 c2 c2 ,V4 V3 2c1 γ1 1 2c2 γ2 2This section proves convergence to zero of the trajectories of the closed-loop system consisting of equations(2.1) and (3.9). The following theorem establishes themain result.where ĉi ; i 1, 2 are nominal value of the parameters ci , ci ci ĉi are parameter estimation errors,and γi 0; i 1, 2 are adaptation gains. The timederivative of V4 can be computed to yield· · u1u22V̇4 k2 β z1 f1 (·) z2 f2 (·)c1c2 e c1 c2 ĉ1 ĉ2 .c1 γ1c2 γ2Theorem 1 Consider the closed-loop nonlinear invariant system Σ described by (2.1) and (3.9). Let X :[t0 , ) R7 , X (t) (x, y, θ, v, w, c1 , c2 )0 ; t0 0denote any solution of Σ. The following properties hold.1. X (t) exists, is unique and defined for all t t0and all X (t0 ) X0 .Motivated by the choices in steps 2 and 3, choose thecontrol lawsu1 ĉ1 f1 (·) k3 z1 ,u2 ĉ2 f2 (·)e k4 z2 e,(3.7)Step 5. (Switching control law) So far, it has beenassumed that the mobile robot will never start at orreach2 the position x y 0 in finite time, becausethe polar representation (2.3) and consequently the control law described above are not defined at e 0. Todeal with this situation, a switching control law mustbe introduced at this stage. A possible solution is tomakeu1 kθ̇ θ̇ kθ θ(3.8)u2 0F mf2 (·)e k4 z2 e,V̇3 k2 β 2 k3 2 k4 2z z 0.c1 1 c2 22. X (t) is bounded for any X0 .3. For any initial condition X0 , the state variablesq (x, y, θ, v, ω)0 converge to zero as t .(3.5)to obtainProof: The existence and uniqueness of X (t) is provenby showing first that for the closed-loop system Σ theonly switching scenario is when e 0 and v 6 0. Itcan be easily checked that the manifold F {X : x 0 y 0 v 0} is positively invariant. Also fore 6 0, if solutions to Σ exist then (3.7) and LaSalle’sinvariance principle (La Salle and Lefschetz, 1961) showthat ρ(t) k1 which means that e and v must be eitherzero or non-zero at the same time. The only case wherethis is violated is when e(t0 ) 0 and v(t0 ) 6 0. Ink3k4V̇4 k2 β 2 z12 z22c1c2"#"# c1ĉ 1 c2ĉ 2 z1 f1 (·) z2 f2 (·) .c1γ1c2γ2Notice in this equation how the terms containing cihave been grouped together. To eliminate them, choosethe parameter adaptation law asĉ 1 γ1 z1 f1 (·),ĉ 2 γ2 z2 f2 (·),2 In fact, it will be shown later that if the initial condition ise(t0 ) 6 0 than this situation will never arise.(3.6)4

this case, from (2.1) it can be seen that for t t0 δwith δ 0, e(t0 δ) 6 0, i.e., the control system willswitch to the case e 6 0. It can thus be concludedthat for any initial condition X0 there occurs at mostone switching, and the closed-loop system Σ has a finitenumber of discontinuities. Using (Hale, 1980, Theorem5.3, Section I.5), a unique solution to Σ exists over amaximal interval [0, tf ). Now, it remains to prove thatthe maximum interval of existence is infinite. For e 6 0,solutions of (3.1) and (3.2) exist. Also, from (3.7) onecan conclude that σ(t), β(t), z1 (t), ρ(t), z2 (t), ĉ1 (t),and ĉ2 (t) are bounded when X0 / F . Thus, the abovevariables are well defined on the infinite interval. Noticethat β(t) 0 and ρ(t) k1 as t . Thus, thereexists a finite time T t0 0 such that for all t T ,ρ cos β 0. From (3.2a)µ Z t¶e(t) e(t0 ) exp ρ(τ ) cos β(τ )dτ e(t0 )e RTt0t0ρ(τ ) cos β(τ )dτ eRtTρ(τ ) cos β(τ )dτkθ̇ 0.8, and kθ 0.32. The initial estimates for thevehicle parameters were m̂ 20.0 Kg, Iˆ 1.6 Kg m2 ,L̂ 0.4 m, and R̂ 0.15 m.Figure 2 shows the vehicle trajectories in the xy-planefor different initial conditions in θ. Figures 3-5 displaythe time responses of the relevant state space variablesfor the initial condition q(t0 ) (x0 , y0 , θ0 , v0 , ω0 ) ( 10, 2, 0, 0, 0). Notice how, in spite of parameter uncertainty, the mobile robot converges asymptotically tothe origin with a ”natural”, smooth trajectory. Notice also that the estimated parameters ci , i 1, 2 arebounded, as expected. However, as Figure 5 shows, theestimation error c1 does not converge to zero. This isdue to the particular structure of the adaptive controlsystem adopted that allows for asymptotic convergenceof q to zero with values of the estimated parameter ĉ1different from the ”true” one.108,64for all t t0 . Therefore, it can be immediately seenthat e(t) is bounded and e(t) 0 as t . Sinceσ(t) and β(t) are bounded, then θ(t) is bounded. Thus,the trajectory X (t) is bounded for all X0 / F . WhenX0 F the same conclusions can be immediatelydrawn. Since the trajectory X (t) is bounded, it existsover the infinite interval, that is, tf .y [m]20 2 4 6 8 10The rest of the proof shows that q converge to zero. IfX0 F then θ 0 and ω 0 as t . If X0 / F,e(t) 0 which implies that x(t) 0 and y(t) 0as t . Moreover, since ρ(t) k1 , then v(t) 0as t . It remains to prove convergence of θ andω. This is done by resorting to the LaSalle’s invariance4principle. Define Ω {X : V4 (X ) V4 (X0 ) c} whichis a positively invariant set since V̇4 0. Let E bethe set of all points in Ω such that V̇4 (X ) 0, thatis, E {X Ω : β 0 z1 0 z2 0}. Let Mbe the largest invariant set contained in E. LaSalle’stheorem assures that every bounded solution starting inΩ converges to M as t . To characterize the set M ,observe that in the set E the variables β and β̇ are zero.Therefore, from (3.1b) ω 0. Notice also from (3.3)and (3.4), that if X E then z1 α1 k1 kσ σ, andsince z1 0 in E it follows that σ 0. Consequently,one can conclude that ω(t) 0, σ(t) 0 and thereforeθ(t) 0 as t . Thus, X (t) 0 as t . Thisconcludes the proof of Theorem 1. 15 10 505x [m]Figure 2: Trajectories in the xy-plane.Initial conditions: e 10, β v ω 0, and θ π2 , π3 , π6 , 0, π6 , π3 , π2 .x [m]0 5 1005101520253035402530354025303540time [s]y [m]10 1 205101520time [s]0.4θ [rad]0.30.20.1005101520time [s]Figure 3: Time evolution of position variables x(t), y(t),and orientation variable θ(t).5 Simulation results6 ConclusionsThis section illustrates the performance of the proposedcontrol scheme (in the presence of parametric uncertainty) using computer simulations. The objective is toregulate the position and attitude of the robot to zero.The following parameters were adopted: m 10.0 Kg,I 1.25 Kg m2 , L 0.5 m, and R 0.1 m. The control parameters were selected as k1 0.47, k2 0.1,k3 1.0, k4 0.5, γ1 0.1, γ2 5.0, kσ 2.0,This paper proposed a new solution to the problemof regulating the dynamic model of a nonholonomicwheeled robot of the unicycle type to a point with adesired orientation. A discontinuous, bounded, timeinvariant, nonlinear adaptive control law that yieldsglobal convergence of the trajectories of the closed loopsystem in the presence of parametric modeling uncertainty was derived. Controller design relied on a non5

via Lyapunov techniques. IEEE Robotics & Automation Magazine 2(1), 27–35.e [m]15105005101520Astolfi, A. (1999). Exponential stabilization of awheeled mobile robot via discontinuous control. Journal of Dynam. Syst. Measur. and Contr. 121, 121–126.2530354025303540Bloch, A. and S. Drakunov (1994). Stabilization of anonholonomic system via sliding modes. In: Proc. 33rdIEEE CDC. Orlando, Florida, USA.40Brockett, R. W. (1983). Asymptotic stability and feedback stabilization. In: Differential Geometric ControlTheory (R. W. Brockett, R. S. Millman and H. J. Sussman, Eds.). Birkhäuser, Boston, USA. pp. 181–191.time [s]β [degree]20100 1005101520time [s]θ [degree]2015105005101520253035time [s]Figure 4: Time evolution of variables e(t), β(t), and θ(t).Canudas de Wit, C. and O.J. Sørdalen (1992). Exponential stabilization of mobile robots with nonholonomic constraints. IEEE Transactions on AutomaticControl 37(11), 1791–1797. 0.34 0.342 0.344 c1Godhavn, J. M. and O. Egeland (1997). A Lyapunovapproach to exponential stabilization of nonholonomicsystems in power form. IEEE Transactions on Automatic Control 42(7), 1028–1032. 0.346 0.348 0.35 0.3520510152025303540time [s]0.5Hale, J. K. (1980). Ordinary differential equations. 2nded. Krieger Publishing Company. New York.0 c2 0.5 1Hespanha, J. P. (1996). Stabilization of nonholonomicintegrators via logic-based switching. In: Proc. 13thWorld Congress of IFAC. Vol. E. S. Francisco, CA,USA. pp. 467–472. 1.5 20510152025303540time [s]Figure 5: Time evolution of parameter estimation errorsJiang, Z. P. and J. B. Pomet (1996). Global stabilization of parametric chained-form systems by timevarying dynamic feedback. International Journal ofAdaptive Control and Signal Processing 10, 47–59. c1 and c2 .smooth coordinate transformation in the original statespace, followed by the derivation of a Lyapunov-based,smooth control law in the new coordinates. Convergence to the origin was analyzed and simulations wereperformed to illustrate the behaviour of the proposedcontrol scheme. Simulation results show that the control objectives were achieved successfully. Future research will address the extension of the method proposed to underwater vehicles. This poses considerablechallenges to control system analysis and design, sincethe models of those vehicles typically include a driftvector field that is not in the span of the input vectorfields, thus precluding the use of input transformationsto bring them to driftless form. Another open problemthat warrants further research is the control and analysis of mechanical nonholonomic systems in the presenceof noisy measurements, actuator saturation constraintsand observer dynamics.Kolmanovsky, I. and N. H. McClamroch (1995). Developments in nonholonomic control problems. IEEEControl Systems Magazine 15, 20–36.Krstić, M., I. Kanellakopoulos and P. Kokotovic (1995).Nonlinear and Adaptive Control Design. John Wiley &Sons, Inc. New York.La Salle, J. and S. Lefschetz (1961). Stability by Liapunov’s Direct Method With Applications. AcademicPress Inc. London.M’Closkey, R. T. and R. M. Murray (1994). Extendingexponential stabilizers for nonholonomic systems fromkinematic controllers to dynamic controllers. In: Proc.4th IFAC Symposium on Robot Control. Capri, Italy.Samson, C. (1995). Control of chained systems: Application to path following and time-varying pointstabilization of mobile robots. IEEE Transactions onAutomatic Control 40(1), 64–77.ReferencesAguiar, A. P. and A. Pascoal (2000). Stabilization ofthe extended nonholonomic double integrator via logicbased hybrid control: An application to point stabilization of mobile robots. In: SYROCO’00 - 6th International IFAC Symposium on Robot Control. Vienna,Austria.Aicardi, M., G. Casalino, A. Bicchi and A. Balestino(1995). Closed loop steering of unicycle-like vehicles6

the presence of parametric modeling uncertainty. Con-troller design relies on a non smooth coordinate trans-formation in the original state space, followed by the derivation of a Lyapunov-based, adaptive, smooth con-trol law in the new coordinates. Convergence to the origin is analyzed and simulation results are presented. 1 Introduction

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