Nonlinear Model Based Coordinated Adaptive Robust Control .
Proceedings of the 2001 IEEEInternational Conference on Robotics & AutomationSeoul, Korea May 21-26, 2001Nonlinear Model Based Coordinated Adaptive Robust Control ofElectro-hydraulic Robotic Arms via Overparametrizing Methodand Bin Yao Fanping BuSchool of Mechanical EngineeringPurdue UniversityWest Lafayette, IN 47907 Email: byao@ecn.purdue.eduAbstractThis paper studies the coordinated motion control of roboticmanipulators driven by single-rod hydraulic actuators. Compared to conventional robot manipulators driven by electricalmotors, hydraulic robot arms have a richer nonlinear dynamics and stronger couplings among various joints (or hydrauliccylinders). This paper presents a physical model based adaptive robust control (ARC) strategy to explicitly take into account the strong coupling among various hydraulic cylinders(or joints). To avoid the need of acceleration feedback forARC backstepping design, the property that the adjoint matrix and the determinant of the inertial matrix could be linearly parametrized by certain suitably selected parameters isfully exploited and overparametrizing method is used. Theoretically, the resulting controller is able to take into accountnot only the effect of parametric uncertainties coming fromthe payload and various hydraulic parameters but also the effect of uncertain nonlinearities. Furthermore, the proposedARC controller guarantees a prescribed output tracking transient performance and final tracking accuracy while achieving asymptotic output tracking in the presence of parametric uncertainties only. Simulation and experimental resultson a three degree-of-freedom (DOF) hydraulic robot arm (ascaled down version of an industrial backhoe/excavator arm)are presented to illustrate the proposed control algorithm.Robotic manipulators driven by hydraulic actuators havebeen widely used in the industry for the tasks such as material handling and earth moving due to its high power density.These types of tasks typically require that the end-effectorsof the manipulators follow certain prescribed desired trajectories in the working space. In order to meet the increasing requirement of productivity and performance of modernindustry, the development of high speed and high accuracytrajectory tracking controllers for the coordinated motion ofhydraulic robot manipulators is of practical importance.First of all, unlike the electrical motors, the hydraulic cylinders are linear actuators and complicated mechanical mechanisms are needed to drive revolute joints. Such a configuration results in additional nonlinearites and stronger couplingsamong the dynamics of various joints. Secondly, in addition to the coupled MIMO nonlinear dynamics of the rigidrobot arm, the dynamics of the hydraulic actuators must beconsidered in the control of a hydraulic arm, which substantially increases the controller design difficulties. It is wellknown that a robot arm including actuator dynamics [1] hasa ”relative degree” more than three. Synthesizing a controller for such a system usually requires joint accelerationfeedback for a complete state feedback, which may not be apractical solution. Furthermore, the single-rod hydraulic actuator studied here has a much more complicated dynamicsthan electrical motors. The dynamics of a hydraulic cylinder is highly nonlinear [2] and may be subjected to nonsmooth and discontinuous nonlinearities due to directionalchange of valve opening and friction. The dynamic equations describing the pressure changes in the two chambersof a single-rod hydraulic actuator cannot be combined into asingle load pressure equation, which not only increases thedimension of the system to be dealt with but also brings inthe stability issue of the added internal dynamics. Finally,a hydraulic arm normally experiences large extent of modeluncertainties including the large changes in load seen by thesystem in industrial use, the large variations in the hydraulicparameters (e.g., bulk modulus), leakages, friction, and external disturbances. Partly due to these difficulties, so far,the model-based coordinated robust control of a hydraulicarm has not been well studied and fewer results are available. In [3], the singular perturbation was used to synthesizea controller for a 6 axis hydraulic robot. In [4], a variablestructure controller was developed to control a Caterpillarexcavator without considering parametric uncertainties anduncertain nonlinearities associated with the system simultaneously. Theoretically, none of above schemes could addressall the difficulties mentioned above well.The controller design for the hydraulic robotic manipulatoris much more difficult both theoretically and experimentallythan those for the conventional robotic manipulators drivenby electrical motors, due to the following several reasons.In [5, 6], the ARC approach proposed by Yao and Tomizukain [7, 8, 9, 10] was generalized to provide a rigorous theoretical framework for the high performance robust motioncontrol of a one DOF single-rod hydraulic actuator. The sta-1 Introduction0-7803-6475-9/01/ 10.00 2001 IEEE3459
y2bility of zero output tracking error dynamics of single-rodhydraulic actuator was also addressed in [5, 6]. In [11], aphysical model based ARC controller, which explicitly takesinto account the strong coupling among various hydrauliccylinders (or joints), is proposed for a multi-DOF hydraulicarm. An observer motivated by the design in [1] is proposedto avoid the need of acceleration feedback for ARC backstepping design.Experimental results show that the observer in [11] may bequite sensitive to measurement noises, which may limit theachievable performance in implementation. To by-pass thispractical problem, a different method will be adopted in thispaper to avoid the need for joint acceleration feedback. Themethod makes full use of the property that the adjoint matrix and the determinant of the inertial matrix can be linearly parametrized by certain suitably selected parameters,and employs certain overparametrizing technique to by-passthe need for acceleration feedback. The design is motivatedby the researches in the electrical motor driven robot manipulators as summarized in [12] where two types of controllers are synthesized for a rigid-link flexible-joint robot.The first controller is a robust design which could ensurethe Globally Uniform Ultimate Bounded (GUUB) stabilityin the presence of parametric uncertainties and uncertainnonlinearities. The second controller is an adaptive designwhich could guarantee Globally Asymptotic Stability (GAS)in the presence of parametric uncertainties only. The difference between the proposed ARC approach and the workdone in [12] is that our approach effectively combines thedesign techniques of adaptive control (AC) and those of deterministic robust control (DRC). The basic idea is that: byusing the robust feedback technique as in DRC [13, 14], theARC will attenuate the effects of model uncertainties comingfrom both parametric uncertainties and uncertain nonlinearities as much as possible. In addition, certain parameter adaptation technique will be used to achieve a better model compensation for an improved performance. Theoretically, theproposed ARC approach achieves a guaranteed transient andfinal tracking accuracy for output trajectory tracking, whichgets rid of the drawbacks of adaptive designs. At the sametime, asymptotic output tracking is achieved in the presenceof parametric uncertainties only, which overcomes the performance limitation of robust designs. The simulation andexperimental results on a 3 DOF hydraulic arm will be presented to illustrate the proposed control algorithm.y3xstx2o2q3z1o3Stick Cylinderq2o1x3Boom Armx1Stick ArmxbSide ViewBoom CylinderSwing CylinderxsTop ViewSwing Armq1y1o2x3o3x2x1l1o1z3z2Figure 1: A Hydraulic Robot Armsired motion trajectories as closely as possible for precisionmaneuver of the inertia load of the hydraulic robot arm. Therigid-body dynamics of the hydraulic arm can be describedby:M (q)q̈ C(q; q̇)q̇ G(q) x q (A1 P1A2 P2 ) T (t ; q; q̇)(1)where P1 [P11 ; P12 ; ; P1n ]T and P1i (i 1; 2; ; n) is theforward chamber pressure of the ith cylinder.P2 [P21 ; P22 ; ; P2n ]T and P2i (i 1; 2; ; n) is the return chamberpressure of the ith cylinder. A 1 diag[A11 ; A12 ; ; A1n ] andA2 diag[A21 ; A22 ; ; A2n ] are the ram areas of the two chambers of the driving cylinders and T (t ; q; q̇) 2 R n represents thelumped disturbance torque including external disturbancesand terms like the joint friction torque.Let mL be the unknown payload mounted at the end of the ntharm, which is treated as a point mass for simplicity. Then, theinertial matrix M (q), coriolis terms C(q; q̇) and gravity termsG(q) in (1) can be linearly parametrized with respect to theunknown mass m L asM (q) Mc (q) ML (q)mL ; G(q) Gc (q) GL (q)mLC(q; q̇) Cc (q; q̇) CL (q; q̇)mL(2)where Mc (q), ML (q), Cc (q; q̇), CL (q; q̇), Gc (q), GL (q) are knownnonlinear functions of q and q̇. One of the properties of theinertia matrix M (q) is that its inverse can be written as:MjM(q)j1 (q) M̄ (q) (3)where jM (q)j represents the determinant of M (q), M̄ (q) represents the adjoint matrix of M (q). Furthermore, both M̄ (q) andjM(q)j can be written asjM(q)j I Ic ni 1 Isi miL 2 Problem Formulation and Dynamic Modelsl3l2;M̄ (q) M̄c ni 11 M̄i miL(4)where Ic , Isi , M̄c and M̄i are scalars and matrices of the knownfunctions of joint position q with I c , Isi , and I being scalars.The system under consideration is depicted in Fig.1, whichrepresents a 3 DOF robot arm driven by three single-rodhydraulic cylinders. To make the results general, let usconsider a n DOF robot arm driven by n hydraulic cylinders. The joint angles are represented by q [q 1 ; q2 ; : : : ; qn ]T .x [x1 ; x2 ; : : : ; xn ]T is the displacement vector of the hydrauliccylinders; each cylinder’s displacement is uniquely relatedto the corresponding joint angle, i.e., x 1 (q1 ), x2 (q2 ), and soon. The goal is to have joint angles q track any feasible de-Assuming no cylinder leakages, the actuator (or the cylinder)dynamics can be written as [2],V1 (x) xβe Ṗ1 A1 ẋ Q1 A1 q q̇ Q1V2 (x) xβe Ṗ2 A2 ẋ Q2 A2 q q̇ Q2(5)where V1 (x) Vh1 A1 diag[x] 2 Rn n and V2 (x) Vh2A2 diag[x] are the diagonal total control volume matrices3460
of the two chambers of hydraulic cylinders respectively,which include the hose volume between the two chambers and the valves, Vh1 diag[Vh11 ; Vh12 ; ; Vh1n ] and Vh2 diag[Vh21 ; Vh22 ; ; Vh2n ] are the control volumes of the twochambers when x 0, diag[x] diag[x1 ; x2 ; ; xn ] , βe 2 R isthe effective bulk modulus, Q 1 [Q11 ; Q12 ; ; Q1n ]T is thevector of the supplied flow rates to the forward chambersof the driving cylinders, and Q 2 [Q21 ; Q22 ; ; Q2n ]T is thevector of the return flow rates from the return chambers ofthe cylinders.Let xv [xv1 ; xv2 ; ; xvn ] denotes the spool displacements ofthe valves in the hydraulic loops. Define the square roots ofthe pressure drops across the two ports of the first controlvalve as: ppPs pP11pPP21g41 (P21 sign(xv1 )) g31 (P11 ; sign(xv1 )) ;sP11PrPrP21forforxv1 0xv1 0xv1 0xv1 0(6)where Ps is the supply pressure of the pump, and Pr is the tankreference pressure. Similarly, let g 3i and g4i be the squareroots of the pressure drops for the ith hydraulic loop. Forsimplicity of notation, define the diagonal square root matrices of the pressure drops as:g3 (P1 ; sign(xv )) diag[g31 (P11 ; sign(xv1 )); : : : ; g3n (P1n ; sign(xvn ))]g4 (P2 ; sign(xv )) diag[g41 (P21 ; sign(xv1 )); : : : ; g4n (P2n ; sign(xvn ))](7)Then, Q1 and Q2 in (5) are related to the spool displacementsof the valves xv by [2],Q1 kq1 g3 (P1 ; sign(xv ))xv ; Q2 kq2 g4 (P2 ; sign(xv ))xv(8)where kq1 diag[kq11 ; : : : ;kq1n ] and kq2 diag[kq21 ; : : : ;kq2n ] arethe constant flow gain coefficients matrices of the forwardand return loops respectively. Thus, neglecting the valve dynamics, the control objective can be stated as:Given the desired motion trajectory q d (t ), the objective is tosynthesize a control input u x v such that the output y qtracks qd (t ) as closely as possible in spite of various modeluncertainties.3 Adaptive Robust Controller Design3.1 Design Model and Issues to be AddressedIn this paper, for simplicity, we consider the parametric uncertainties due to the unknown payload m L , and the nominalvalue of the lumped disturbance T , Tn only. Other parametricucnertainties can be dealt with in the same way if necessary.In order to use parameter adaptation to reduce parametric uncertainties to improve performance, it is necessary to linearlyparametrize the system dynamics equation in terms of a setof unknown parameters. To achieve this, define the unknownparameter set as θ [θ1 ; θT2 ]T where θ1 mL and θ2 Tn . Thesystem dynamic equations can thus be linearly parametrizedin terms of θ as x (A PM (q)q̈ C(q; q̇)q̇ G(q) q1 1 A2 P2 ) θ2 T̃ (t ; q; q̇); T̃ T (t ; q; q̇) TnhṖ1 βeV11(q)Ṗ2 βeV21(q)hi xA1 qq̇ Q1 (u; g3 (P1 ; sign(u)) x q̇A2 qiQ2 (u; g4 (P2 ; sign(u))(9)Since the extent of the parametric uncertainties and uncertainnonlinearities are normally known, the following practicalassumption is made. Parametric uncertainties and uncertainnonlinearities satisfyθjT̃ (t ; q; q̇)j2 Ωθ fθ : θmin θ θmaxδT (q; q̇; t ) where θmin [θ1min ; ;θ2min ]T ,δT (t ; q; q̇) are known.gθmax [θ1max ;θ2max ]T ,(10)andAt this stage, it can be seen that the main difficulties in controlling (9) are: (i) The system dynamics are highly nonlinear and coupled, due to either the nonlinear robot dynamicsor the dependence of the effective driving torque on joint angle (terms like x q(q) ) and the nonlinearities in the hydraulicdynamics; (ii) The system has large extent of parametric uncertainties due to the large variations of inertial load m L ; (iii)The system may have large extent of lumped uncertain nonlinearities T̃ including external disturbances and unmodeledfriction forces; (iv) The added nonlinear hydraulic dynamics are more complex than the electrical motor dynamics; (v)The model uncertainties are mismatched, i.e. both parametric uncertainties and uncertain nonlinearities appear in thedynamic equations which are not directly related to the control input u xv .To address the challenges mentioned above, following general strategies will be adopted in the controller design.Firstly, the nonlinear physical model based analysis and synthesis will be employed to deal with the nonlinearities andcoupling of the system dynamics. Secondly, the ARC approach [7, 10] will be used to handle the effect of both parametric uncertainties and uncertain nonlinearities; fast robustfeedback will be used to attenuate the effect of various modeluncertainties as much as possible while parameter adaptationwill be introduced to reduce model uncertainties for highperformance. Thirdly, backstepping design via ARC Lyapunov function will be used to overcome the design difficulties caused by the unmatched model uncertainties. Finally,the property that the adjoint matrix and the determinant ofthe inertial matrix could be linearly parametrized by certainsuitably selected parameters is fully exploited so that certain overparametrizing method can be employed to avoid theneed for joint acceleration feedback. The details are outlinedbelow.3.2 Controller Design using OverparametrizingThe design parallels the recursive backstepping design procedure via ARC Lyapunov functions in [10, 5] as follows.Step 1Define a switching-function-like quantity as z 2 ż1 k1 z1 q̇ q̇r , where q̇r q̇d k1 z1 , and z1 q qd (t ), in which qd (t )is the reference trajectory and k 1 is a positive feedback gain.The design in this step is to make z 2 as small as possible witha guaranteed transient performance. The design is the sameas in [11] and is briefly outlined below.Define a positive semi-definite (p.s.d) function as V 23461
1 T2 z2 Mz2 .From(9) and the property that Ṁ (q) 2C(q; q̇) is askew symmetric matrix [15], its derivative is given byV̇2 zT2 (M ż2 12 Ṁz2 ) zT2 [ M q̈r C( q̇; q)q̇rwith a guaranteed transient performance. Since (11) has bothparametric uncertainties θ 1 and θ2 and uncertain nonlinearityT̃ , the ARC approach proposed in [10] will be generalized toaccomplish the objective. The control function PLd consistsof two parts given byPLd (q; q̇; θ̂1 ; θ̂2 ; t ) PLda PLds x 1PLda ( q) [M̂ q̈r Ĉ (q̇; q)q̇r Ĝ(q)discrepancy, we will haveφ2 θ̃ T̃ ]where φ2 [ ML q̈r CL (q̇; q)q̇rbe chosen to satisfy:condition i x zzT2 K2 (t )z2 zT2 q3condition iiT ;;βn 2φ2 θ̃ T̃ ] ε2 x PzT2 qLds 0(14)θ̂ Pro j(Γθ τθ )jMjq̈ M̄C(q̇ q)q̇ M̄G M̄ q x PL M̄θ2 M̄T̃;1 M̄ ,we θ21 ;z 3 is;θn1 ;θT2 ; θ1 θT2 ;given by ; ; βnθn1 1 θT2 ].;;βTn 1From1 x11 A22V2 ) q q̇ (A1V1 Q1 A2V2 Q2 )] PLd PLd PLdLdṖLd P q q̇ q̇ q̈ θ̂ θ̂ t1(18)Define a p.s.d function as V3 V2 12 IzT3 z3 . The derivative ofV3 is given byV̇3 V̇2 jz3 0 zT21 T xT q z3 Iz3 ż3 2 Iz3 z3 x1 V̇2 z3 0 zT3 ( q z2 I ṖL I ṖLd 2 Iz3 )j(19)where V̇2 jz3 0 represents the derivative of V2 when z3 0 andI ṖLd can be expressed byI ṖLdd g(20) I ṖLd I ṖLdwhereIdṖLd P Pˆ x P dˆ ) PLd Iˆθ̂ PLd Iˆ qLd Iˆq̇ q̇Ld ( Ĉt q̇ Ĝt M̄n q L t θ̂ PLd PLd PLdnLdθ̂ t )( i 1 Isi β̃i ) PI ṖLd ( q q̇ [ n(Cti q̇ Gti )β̃i 1i q̇ θ̂n 1 xPL β̃i M̄c β̃n 1 ni 2 M̄i 1 β̃n i d ] i 1 M̄i qIˆ Ic ni 1 Isi β̂i ; Ĉt Ctc ni 1 Cti β̂iĜt Gtc ni 1 Gti β̂i ; dˆn M̄c β̂n 1 ni 2 M̄i 1 β̂n iˆ M̄c n 1 M̄ β̂M̄i ii 1gwhere Pro j( ) denote the discontinuous projection defined in[16, 17, 7], and Γ θ denotes the adaptive gain matrix.Multiply both side of first equation of (9) by jM jMwill have;ż3 ṖL ṖLdṖL βe [ (A21V1(15)Step 2 In this step, an actual control law will be synthesizedso that z3 converges to zero or a small value with a guaranteed transient performance and accuracy. If we were to usethe backstepping design strategy via ARC Lyapunov function as in [5, 10], then, the resulting ARC law would requirethe feedback of the joint acceleration q̈ since q̈ is neededin computing Ṗˆ Ld , the calculable part of the derivative of thedesired virtual control function PLd , for adaptive model compensation. In order to avoid the need for joint accelerationfeedback, in the following, the property of the inertia matrixin (4) will be used as follows. βT2n ] [ θ1 ;(9), the derivative of(13)where ε2 is a design parameter which can be arbitrarily small.Essentially, condition i of (14) shows that PLds is synthesizedto dominate the model uncertainties coming from both parametric uncertainties θ̃ and uncertain nonlinearities T̃ , andcondition ii is to make sure that PLds is dissipating in nature so that it does not interfere with the functionality of theadaptive control part PLda . How to choose PLds to satisfy constraints like (14) can be worked out in the same way as in[8, 9]. The adaptive function τ θ and the adaptation law aregiven byτθ φ2 z2Redefine the unknown parameters as:[β 1 ; β2GL (q); In n ]. Then PLds can x PzT2 [ qLds(17)where I is a scalar. Similar to (2), Ct , Gt and dn can be expressed by Ct (q̇; q) Ctc ni 1 Cti θi1 , Gt (q) Gtc ni 1 Gti θi1 ,dn M̄c θ2 ni 11 M̄i θi1 θ2 . Where Ctc and Gtc are of the knownnonlinear functions of q and q̇.θ̂2K2 (t )z2 ](12)where K2 (t ) is a positive feedback gain matrix, M̂ (q) Mc ML θ̂1 , Ĉ(q̇; q) Cc CL θ̂1 , and Ĝ(q) Gc GL θ̂1 . Substituting (12) into (11) and let z 3 PL PLd represent the input x M̄ q PL dn d I q̈ Ct q̇ Gt x (A PG(q) q1 1A2 P2 ) θ2 T̃ ](11)Define the load pressure as PL A1 P1 A2 P2 . If we treat PLas the virtual control input to (11), a virtual control law P Ldfor PL will be synthesized such that z 2
design techniques of adaptive control (AC) and those of de-terministic robust control (DRC). The basic idea is that: by using the robust feedback technique as in DRC [13, 14], the ARC will attenuatethe effects ofmodeluncertaintiescoming from both parametric uncertainties and uncertain nonlineari-ties as much as possible.
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