See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/227693979 Perfect position/force tracking of robots with dynamical terminal sliding mode control Article in Journal of Robotic Systems · September 2001 DOI: 10.1002/rob.1041 CITATIONS READS 12 37 3 authors, including: V Parra-Vega Alejandro Rodrguez‐Ángeles Center for Research and Advanced S… Center for Research and Advanced S… 166 PUBLICATIONS 899 CITATIONS 36 PUBLICATIONS 415 CITATIONS SEE PROFILE SEE PROFILE All content following this page was uploaded by V Parra-Vega on 16 January 2015 The user has requested enhancement of the downloaded file All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately Perfect Position/Force Tracking of Robots with Dynamical Terminal Sliding Mode Control • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • V Parra-Vega∗ Sección de Mecatrónica Depto de Ing Eléctrica CINVESTAV A.P 14-740, México, D.F 07000 México e-mail: vparra@mail.cinvestav.mx A Rodríguez-Angeles Systems Signals and Control Group Faculty of Applied Mathematics University of Twente P.O Box 217 7500 AE Enschede The Netherlands G Hirzinger Institute of Robotics and Mechatronics German Aerospace Center–DLR P.O Box 1116 82230 Wessling, Germany Received 28 May 2000; accepted March 2001 According to a given performance criteria, perfect tracking is defined as the performance of zero tracking error in finite time It is evident that robotic systems, in particular those that carry out compliant task, can benefit from this performance since perfect tracking of contact forces endows one or many constrained robot manipulators to interact dexterously with the environment In this article, a dynamical terminal ∗ To whom all correspondence should be addressed Journal of Robotic Systems 18(9), 517–532 (2001) © 2001 by John Wiley & Sons, Inc 518 Journal of Robotic Systems—2001 sliding mode controller that guarantees tracking in finite-time of position and force errors is proposed The controller renders a dynamic sliding mode for all time and since the equilibrium of the dynamic sliding surface is driven by terminal attractors in the position and force controlled subspaces, robust finite-time convergence for both tracking errors arises The controller is continuous; thus chattering is not an issue and the sliding mode condition as well the invariance property are explicitly verified Surprisingly, the structure of the controller is similar with respect to the infinite-time tracking case, i.e., the asymptotic stability case, and the advantage becomes more evident because terminal stability properties are obtained with the same Lyapunov function of the asymptotic stability case by using more elaborate error manifolds instead of a more complicated control structure A simulation study shows the expected perfect tracking and a discussion is presented © 2001 John Wiley & Sons, Inc INTRODUCTION In constrained motion tasks, the end-effector moves in compliant direction so as to exert a desired profile of force in the constrained force degree of freedom (FDoF) while moving along the unconstrained position degree of freedom (PDoF) To achieve this goal, a combination of position and force control loops are required to drive simultaneously the manipulator along each DoF and to keep the end-effector in contact to the environment Over the last decade numerous contributions have proposed alternative approaches for what was considered for long time an open problem in robot control.1–4 Among all these approaches, we focus on the explicit force feedback control algorithms for rigid, fully actuated robot manipulators in contact to known (infinitely) stiff environments, and with known upper bound of physical parameters.a The system is modeled using differential algebraic equations (DAE)5 , and hence the contact force stands for the Lagragian of the constrained system The control objective is to achieve simultaneously finite-time convergence of force and position tracking errors under parametric uncertainty, with a continuous controller Now, we discuss the background and address the contribution of this article 1.1 Explicit Force Control Two basic approaches have prevailed over the years in explicit force feedback robot control research implicit equation that models the constraint to obtain a decoupled dynamics for the open-loop PDoF and the FDoF Thus, robot dynamics are explicitly obtained for each DoF in terms of a unique set of independent generalized coordinates Though this set always exists, it is not evident how to handle it in a large workspace.6 Control structure is rather involved, though simple stability arguments are used to prove the global stability and several control techniques have been proposed.7–12 A2 In the second approach,13 a passivity-based algorithm that does not use any coordinate partition of system dynamics, but introduces two orthogonal projections to construct an orthogonalized error coordinate system, leads to a local asymptotic stability result This approach is computationally more efficient since the solution of the implicit equation is not required at all; besides that this scheme exploits effectively some fundamental physical properties of robot dynamics The control objective is translated into reshaping the desired closed-loop mechanical energy of the system such that a local minimum arises on desired trajectories The control structure is simpler with respect to that in ref 5, though involved stability arguments are used to prove the local asymptotic convergence of force and position tracking errors.13 14 A1 The first one proposed5 exploits the partition coordinates of the solution of the A third alternative that appears as an efficient combination of the first and second approaches has been proposed: We study only the stage of constrained motion, leaving out the impact and transition phases A3 The partition coordinates of A1 are proposed in order to obtain globally the orthogonal projections of A2; robot dynamics are a Vega et al.: Perfect Position/Force Tracking of Robots not embedded in the solution of the implicit function, which leads to the control structure of the second approach to yield global asymptotic stability with simple stability arguments.15 16 In ref 17 closed-loop error dynamics are partially decoupled, and in refs 18 and 19 overcompensated controllers are proposed along this line In any case, the literature available up to date, not only for explicit force control but also for all the other force control strategies,1 does not assure finite-time convergence (FTC) of tracking errors (FTT) 1.2 The Paradigm of FTT and Applications FTT for physical systems such as robot manipulators has attracted little attention; therefore it is convenient now to define the FTT paradigm: “Design a control system that yields FTC of tracking errors with a real-time compliant control input; that is, the controller that renders FTT should be realizable with current software and hardware technology.” To this end, we impose some constraint on the control design, such as the controller should be: (1) continuous; (2) with no unbounded effort; (3) with no high frequency; (4) causal; and (5) robust to parametric uncertainty and initial conditions The paradigm FTT is not only fundamental to yield perfect tracking and great performance, but for many problems in robotics it is a desirable property To name a few, we review briefly the following tasks that can benefit if FTT is implemented: (a) walking robots where it is needed to assure that the state of the leg is in the given desired trajectory before other leg deattaches from ground; (b) event-based algorithms where the discrete states are assumed to belong to given compact sets at given time; (c) contact transition tasks where contact detection depends on complex algorithms to detect the exact state of the system at given instant; (d) dynamic simulation systems where complex and high order models are used to render realistic motion of articulated bodies (however the complexity of the system requires that stringent assumptions are imposed on the model and important information is neglected, and thus FTC can yield better realistic simulators by relaxing such assumptions since convergence time could be set arbitrarily); (e) closed-loop identification of robot parameters, where weaker conditions on the regressor can be imposed since real trajectories can be substituted by desired trajectories 519 at given time and the persistent excitation condition can be designed beforehand; (f) obstacle avoidance methods, wherein consider that real trajectories follow exactly the planned trajectories (otherwise overdesigned desired trajectories are given); (g) optimal path planning usually considers that the system follows exactly the optimal path, which is not always the case, and then optimality is not achieved; (h) object manipulation and multirobot coordination usually require perfect timing for all finger robots to release and grasp the object, and during (i) constrained motion, where the end-effector is in contact to the environment and thus real position and force trajectories follow the trajectories that satisfy the constraint (otherwise the system may damage the constrained object) In this article we focus on the FTT paradigm for constrained motion using second order sliding mode control with terminal attractors20 in the sliding surface 1.3 Terminal Attractors In order to design a control system that fully complies with the FTT paradigm, we exploit the littleused technique called terminal sliding mode control Although terminal attractors have been subject of intensive research in the numerical and neural networks research community, where the application is mainly bound to computer computations,21 that is not the case for physical systems The philosophy of design of terminal sliding mode control is basically a conventional static sliding mode controller with a nonlinear sliding surface of tracking errors, where the dynamics of this surface exhibits an attractor with FTC (called the terminal attractor20 ), and thus tracking errors converge in finite time The terminal attractor is modeled as a first order differential equation that violates the Lipschitz condition, and the attractive singularity located precisely in zero position error renders unbounded control in the bounded domain with internal instability of the differential equation In the second order derivative of this differential equation appears the singularity in zero position tracking error, as is the case for second order systems such as robot arms This puzzling behavior seems then disastrous, and not surprisingly, the consequence of this unusual and unconventional formulation is that few control algorithms based on terminal attractors are available for robot manipulators in free motion,22–25 though these controllers are not fully compliant with the FTT paradigm because they may 520 Journal of Robotic Systems—2001 violate (2), (3) above These control schemes need a discontinuous control input to achieve FTC, and since it is virtually impossible to reproduce the theoretically infinite bandwidth of a signum function in real time, a saturation function is proposed to realize the controller Thus, a sliding mode does not strictly arise and the singularity induces internal instability all the time, which eventually may render unbounded control input and instability, while in ref 26 a complex procedure is proposed to achieve finite-time convergence with high frequency control inputs In ref 27 an algorithm is proposed to sequentially induce sliding modes to avoid singularity; however, a discontinuous controller is needed, and again in order to realize the controller a saturation function is implemented In this condition, the sequence to avoid singularity cannot be guaranteed To induce well-posed terminal attractors, it is fundamental then to induce a sliding mode for all time, and in order to realize this controller in real time, the control action must be continuous, otherwise the sliding mode condition cannot be strictly verified 1.4 Sliding Modes with Continuous Control Chattering is a problem that arises in variable structure control systems due to the finite bandwidth of the software and hardware, and many techniques have been proposed to attenuate to some extent this phenomena.28 29 However, boundary-layer-like methods30 not strictly verify the sliding mode condition, while numerical solutions31 or second order sliding surfaces32 are not well developed yet for mechatronic systems On the other hand dynamic sliding modes33–35 seem to be a promising technique; however, backsteeping methods are really complicated in comparison to the class of controller obtained in passivity-based robot control.36 Thus, we develop further the passivity-based dynamic sliding mode control proposed in ref 16 to obtain a sliding mode regime with continuous control for DAE systems 1.5 Contribution We propose a solution for using a continuous controller tors that are subject to known and known upper bound of the FTT paradigm for robot manipulaholonomic constraint physical parameters To this end, we further elaborate on the third alternative16 outlined in A3 to introduce terminal attractors in orthogonal position and force dynamic sliding surfaces The controller compensates the parametric uncertainty, and terminal attractors show their implosive attractiveness to induce FTC of both position and force tracking errors The closed-loop system is free of singularity and preserves the passivity of the open-loop system, and thus this algorithm might be extended to other classes of mechanical systems that are passive in the open loop Computer simulation shows the performance of a DoF rigid arm The article has been organized as follows Section presents the formulation of the problem Section shows the robot dynamics in the error space In Section the controller and its stability proof is presented, while in Section some remarks are presented A simulation study is discussed in Section 6, and conclusions are presented in Section PROBLEM FORMULATION When the robot end-effector is in contact to a smooth surface, a geometric (holonomic) constraint is imposed by the forward kinematic equation X = f q , where X ∈ stands for the x y x T ∈ Cartesian coordinates and the three Euler angles, q ∈ n stands for the joint generalized coordinates, and f q is the forward kinematic mapping Usually the holonomic constraint is formulated in X coordinates as X = 0, where it is assumed that the constraint is twice differentiable Using the constraint can be expressed in q coordinates as ≡ q = f q = A classical mechanics formulation37 has been used in ref 13 to yield a model of a rigid serial n-link robot manipulator with all revolute joints as H q qă + C q q + B0 q˙ + G q = U + J T+ − =0 f (1) (2) where H q denotes the n × n symmetric positive definite inertial matrix, B0 is an n × n positive definite matrix of damping coefficients, G q models the gravity loads, C q q˙ represents the n × n Coriolis matrix, U stands for the n torque inputs, and f represents a model of the sliding friction force at the contact point For simplicity we assume a viscous model such that f is linear in terms of the ˙ where f = velocity q˙ by f = f q, X˙ JxT Jx and 0< X˙ < It is assumed that the constraint Vega et al.: Perfect Position/Force Tracking of Robots n ∈ C2 → m denotes m−smooth surfaces and it is consistent and independent in a sense that J T has full column rank m and a unique and analytic solution of (1) exists if initial conditions are chosen to satisfy (2) and its derivatives up to order two.5 38 J T+ = J T+ q = J T q J q J T q −1 models the normalized matrix that points outward and therefore the Lagragian ∈ m physically stands for the magnitude of the force applied at the contact point (2) with J ≡ J q = Xq Jx ∈ n×m , and Jx = f qq stands as the direct Jacobian In order to obtain the representation of the system in error coordinates, we use the standard linear parametrization Yr of robot dynamics in terms of the nominal reference q˙r and its derivative qăr , where arguments are omitted from now on when no confusion arises, as40 Yr = Yr q q qr qăr = H qăr + C + B0 + f q˙r + G (3) where ∈ l is composed of, and is possibly a product of, physical parameters and Yr ∈ n×l stands for the regressor If we add (3) to (1) we obtain H S˙ + C + B0 + f S = U + J T+ − Yr S = q˙ − q˙r (5) S˙ = qă qăr (6) We now have the following Statement of the problem: Design a continuous controller U which guarantees trajectory tracking in finite time of desired time-varying pose and contact force It is assumed that: (i) the upper bound of the unknown parameter vector is known; (ii) the regressor Yr is available; (iii) the kinematic constraint is twice differentiable ˙ and exactly known; (iv) the state (position q, velocity q, and contact momentum F , and thus ) is available; and T T (v) desired trajectories qdT q˙dT , qădT FdT are known d bounded analytical functions Error equation (4) and constraint (2) define a differential algebraic system whose solution is constrained to evolve in an invariant manifold defined by = q q˙ i=0 t ∈ R n × Rn × R m × R+ This manifold will be exploited in the following section to synthesize a convenient orthogonalized error coordinate system using terminal sliding modes in order to design the controller U according to the statement ERROR DYNAMICS FOR CONSTRAINED MOTION To design an appropriate error equation, we keep in mind at this stage that the regressor to be compensated must be continuous since a continuous controller must be designed Let us note also that we want to preserve the passivity in the closed loop and thus we are looking for a similar control structure and stability analysis of refs 13 and 16 As can be seen now the problem is translated into the reformulation of a new error state S in (5), which means the design of new nominal references qrT qărT T 2n in (5), (6) We present now the open-loop error dynamics using the partitioning method5 for the error manifolds13 with terminal attractors,20 and without coordinate reduction of system dynamics.15 16 (4) where 521 i d dt i q =0 (7) 3.1 An Orthogonalized Terminal Sliding Surface The m geometric constraints (2) give rise to n − m independent generalized coordinates q2 ∈ n−m , and m dependent generalized coordinates q1 ∈ m Thus, the following partition of joint space coordinate q ∈ n arises: q = q1T q2T T (8) Now according to (7), the derivative of (2), that is, d = J q˙ ≡ 0, with its corresponding partition given dt by (8), yields J q˙ = J q˙1 + J q˙2 = J J q˙1 ≡0 q˙2 where J = / q1 ∈ m×m and J m× n−m Solving (9) for q˙1 yields q˙1 = q˙2 where (9) =− J = −1 J / q2 ∈ (10) n−m and → m has full column rank m since by assumption rank = m and thus J −1 is well posed in the finite workspace imposed by the holonomic constraint (2) Taking into account the 522 Journal of Robotic Systems—2001 partition (8) and using (10), the generalized velocity q˙ = q˙1T q˙2T T can be written as q˙ = = q˙2 q˙2 q In−m and subtract QK1 (16) one obtains where qr n q2 (12) qăcont = Q˙ q˙2d − q2r + Sdp − K1 F − SdF + K2 p F (13) + J˙T +J with bounded ∈ (18) (19) n−m × n−m + , ∈ m×m , + and Zp = Sqp − sgn Sqp (20) ZF = SqF − sgn SqF (21) xk T stands The vector x = x1 for the hyperbolic tangent function of X ∈ k , and every entry zp ∈ Zp zF ∈ ZF are bounded by ±1 Substituting (13) into (5) and (17) into (6) gives rise to (15) ˙ vp + Q S˙qp + K1 S˙ = QS −JT (22) S˙qF + K2 Sqp SqF JT SvF + qădisct Svp = Sqp + K1 p (24) SvF = SqF + K2 F (25) with Sqp = Sp − Sdp Sp = q˙2 + q˙ + S˙dp − K1 sgn Sqp − S˙dF + K2 sgn SqF SqF = SF − SdF (16) where the vector sgn x = sgn x1 sgn xj T stands for the signum function of X ∈ j However, Eq (16) is discontinuous and it is not allowed because (3) would be discontinuous Then, if we add (23) where p F − SdF +JT SqF K ZF F = sgn SqF + Q qă2d r F T Sqp F S = QSvp − J T SvF q2r + Sdp − K1 p SdF + K2 T qădisct = QK1 Zp − J t + K2 F − SdF + K2 (14) q2r−1 to (17) q˙ + S˙dp − K1 q2r−1 ˙ p = sgn Sqp d , and the with q2 = q2 − q2d , F = t0 − d subscript d denotes the desired reference value n−m × n−m Diagonal feedback gains are K1 ∈ R + , K2 ∈ Rm×m , and r is a terminal attractor param+ eter.b The passivity approach13 36 40 suggests that in order to fully exploit the physical structure of robot dynamics, the nominal reference qăr must be equal to d q˙ , then (13) becomes dt r + J˙T q2r + Sdp K1 + Q qă2d r and qăr = Q q2d SqF where where Q ∈ n× n−m is well posed Since J Q ≡ 0m× n−m , the image of J lies on the null space of Q; that is, the state space is decomposed into two orthogonal subspaces such that n can be written as the direct sum Rn = R J ⊕ R Q where R ∗ stands for the range of ∗ Now, with the unique set of joint independent generalized coordinates q2T q˙2T T ∈ n−m , consider the nominal reference +JT + J T K2 qăr = qăcont + qădisct (11) Q qr = Q q2d − Sqp (26) q2r (27) where SF ≡ F (28) and Sdp SdF are to be defined yet Equation (4) in terms of Eqs (22), (23) can be written as H S˙ + C +B0 + f S = U +J T+ Ycont H qădisct (29) where Ycont = Yr q q qr qăcont = H qăcont + C + B0 + f q˙r + G b We define as a terminal attractor parameter any scalar x that powers y, that is, y x , such that x = xn /xd xn xd ∈ Z+ xn < xd 12 < x < and xn xd odd is continuous On the other hand, Eq (17) allows one to cast the discontinuous term H qădisct as bounded 523 Vega et al.: Perfect Position/Force Tracking of Robots disturbances into the right hand side of the openloop error equation (29), which in turn allows one to derive a continuous controller since the regressor Ycont is continuous In (22) we can see that due to Q ∩ J = 0, the orthogonal complements Q and J globally project the position–velocity and integral of the force tracking errors onto orthogonal subspaces, respectively These projections are instrumental in the proof of stability, as becomes clear in the following section 4.1 Boundedness of State Trajectories S Sv F Consider the Lyapunov candidate function V= V˙ = −S T B0 + ≤ −S Kd S T U = −Kd S c ∗ ≤ −S Kd S T SqF i sat ≤ −S Kd S T d + Ycont + SvF (30) ≤ −S Kd S Sj Ycontji i=1 l (31) where sat X = X/ X + stands for i > i the inputwise saturation function of vector X = x1 xl T , and Ycont = Ycontji The parameter > defines the width of the saturation function, feedback gains Kd = KdT ∈ n×n = T ∈ m×m , and c + , + d are terminal attractor parameters The closed-loop error equation between (29) and (30), (31) yields H S˙ = − C + B0 + − d f S + Kd S c + J T+ S˙vF + T − Ycont − Ycont ∗ T sat Ycont S d SvF (32) where d = H qădisct J T+ K2 ZF is considered a disturl×l bance, and ∗ = diag We are now l ∈ in a position to state the stability properties of the closed-loop system (32) in the next theorem Theorem 1: Consider robot dynamics (1) in closed loop with the controller (30), (31) Then, the global finitetime convergence of tracking errors arises with continuous control and singularity-free closed-loop dynamics Proof: The proof is organized in the following sections c c T − ST d − S Ycont ∗ sat d SvF − S T Ycont ∗ T sat Ycont S ∗ T sat Ycont S d SvF T T Ycont S d T − SvF + ST T − SvF + S T Ycont T j=1 for c d SvF T sat Ycont Sr − SrT Ycont T − SvF + S T Ycont − d − S˙dF + J T+ n =− T S − S T Kd S c − SvF − S Ycont − S + K2 i c f T Consider the controller U given by (33) T where VS = S T HS and VF = SvF SvF The total derivative of (33) along its solution (32) leads to − SrT Ycont MAIN RESULT V + VF S d SvF d − S T Ycont + S T − SvF d SvF d + 0+ ST d (34) where we have used the fact that −S T Yr ∗ sat YrT S + ≤ Now, note that S T Yr since ≥ S T d is radially unbounded only for S and for bounded signals S T d attains a unique equilibrium point at S = 0; see also that for admissiwe have that S T HQK1 Zp ≤ S T HQK1 , ble q2 T T S HJ K2 ZF ≤ S T HJ T K2 On the other hand, according to the boundedness property of the inertial matrix and the fact that Q and J are functions of bounded constant and trigonometric functions, then d is also bounded Along with the fact that are bounded feedback gains, then there K K2 exists always a positive scalar = sup S SvF limt + + , where = M H M Q M K1 , = T T H J K , and = J M M M M M M K2 + such that S T d ≤ S , where M ∗ stands as the maximum eigenvalue of ∗ Thus, Eq (34) becomes V˙ ≤ − VS − VF + 0+ S (35) where = + c/2, ≡ m Kd 2/ m H q = 2/ H q If Kd is large + d/2, and ≡ m enough, and according to refs 23 and 38 one obtains the terminal convergence of S → and SvF → , where and are bounded hyperballs with radii r0 > and r1 > 0, respectively Hence, for the region outside the union of the boundaries of the domains = ∪ centered in the equilibrium S = and SvF = 0, we can conclude the terminal ultimate 524 Journal of Robotic Systems—2001 boundedness of error dynamics within the neighborhood of such that terminal trajectories for V can be obtained as −1 ¯ V< + −1 ¯ S ≤ >0 Thus, there exist bounded scalars 5, such that SvF ≤ and S˙vF ≤ S˙ ≤ (41) since Q Q = I n−m × n−m and Q J T = n−m ×m Using (14) and (24), the derivative of Eq (41) can be written as follows: S˙qp = −K1 sgn Sqp + Q˙ S + Q S˙ (42) (36) If we multiply (22) by the pseudoinverse Q = QT Q −1 QT ∈ n−m ×n one obtains Q S = Q QSvp − Q J T SvF = Svp for some h S where ¯ = + i > 0, for i = +h S 4.3 Sliding Mode and FTC for Sqp Now, consider the following Lyapunov function T Vqp = Sqp Sqp This establishes the boundedness of S SvF and their derivatives S˙ S˙vF (43) Using (42) into the total derivative of (43) gives rise to 4.2 Sliding Mode and FTC for SqF T T V˙qp = −Sqp K1 sgn Sqp + Sqp Q˙ S + Q S˙ We now show that the properties of the dynamical system defined by Eqs (25) and (15) S˙qF = −K2 sgn SqF + S˙vF (37) yields a sliding mode at SqF = To see this, consider the following Lyapunov function: T SqF VqF = SqF (38) The total derivative of (38) along its solution (37) gives rise to T T ˙ SvF K2 sgn SqF + SqF V˙qF = −SqF To proceed, notice that Q is full column rank n − m and is composed of constant and trigonometric functions, then Q is bounded by a constant, and its derivative is also bounded by a function of q˙2 That is, there exists bounded positive scalars such that ≤ Q Q˙ ≤ q˙2 V˙qp ≤ −K1 Sqp + Sqp Q˙ S + Q S˙ Q˙ S + Q S˙ ≤ −K2 SqF + SqF S˙vF ≤ −K1 Sqp + Sqp ≤ −K2 SqF + ≤− ≤− SqF SqF (39) where we have used (36), and = K2 − Thus, in order to prove that SqF → in finite time, we can always choose K2 > (40) in such a way that a > in (40) guarantees the existence of a sliding mode since Eq (39) is the sliding mode condition.29 This indicates that a sliding mode is established in finite time tqF ≤ SqF t0 / , and since for any initial condition SqF t0 = 0, then a sliding mode in SqF t = is enforced for all time without reaching phase in the force controlled subspace, then tqF ≡ (45) Using (45) into (44) one obtains ≤ −K1 Sqp + Sqp (44) q˙2 2+ Sqp (46) where we have used (36), and = K1 − q˙2 + Thus, in order to prove that SqF → in finite time, we can always choose K1 > sup × q2 q˙2 q˙2 T q1 q2 =0 ˙ q1 q2 q˙2 =0 2+ (47) in such a way that > guarantees the existence of a sliding mode since Eq (46) is equivalent to the sliding mode condition.29 This indicates that a sliding mode is established in finite time tqp ≤ Sqp t0 / , and since for any initial condition Sqp t0 = 0, then a sliding mode in Sqp t = is enforced for all time without reaching phase in the position controlled subspace; thus tqp ≡ Vega et al.: Perfect Position/Force Tracking of Robots 4.4 FTC of Sp and SF which implies that We have shown that Sqp t = ∀t ≥ tdp > is enforced for all time; then from (26) we have the invariant system Sp = Sdp and hence Sdp plays the role of a desired trajectory for Sp and also as an input, and since Sdp t = ∀t ≥ , where is the convergence time of Sdp t , then Sp t = ∀t ≥ A similar procedure can be followed for the sliding surface SF to obtain SF t = ∀t ≥ tF where tF is the convergence time of SdF t 4.5 Terminal Sliding Mode and FTC of q T q˙ T T for ∀t (48) T q q 2 (49) The total derivative of (49) along its solution (48) gives rise to V˙tp = − =− q2T q2r n + ≤ −2 ≤ −2 m m n q2i2 + q2T Sd (51) T = 0Tn 0Tn ∀t ≥ T 2ttp F and Dynamic sliding surface SqF t = implies F = SdF ∀t (52) Since we design SdF such that it exhibits FTC at t = tF , then F t converges equally at time t = tdF Note that the derivative of (52) is d F ≡ dt ⇒ = S˙dF ∀t (53) (50) 4.7 Singularity-Free Closed-Loop Dynamics i=1 n q2i2 + q2T Sdp i=1 Vtp + q2T Sdp where = + r/2 Since Sdp t achieves FTC at time t = tdp , and according to ref 22, then Vtp t = 2ttp If we design S˙F such that it exhibits FTC at t = t > 0, then t converges equally at time t = t i=1 q˙T 4.6 Terminal Sliding Mode of FTC of q2T Sdp q2i2 + q2T Sdp = −2 ∀t ≥ regardless of system parameters Convergence of the generalized coordinates q2 t q˙2 t implies convergence of the dependent coordinates q1 t q˙1 t since establishes a diffeomorphism between q˙1 and q˙2 and the desired reference has been designed consistently with the partition of the vector q q˙ ∈ R2n and with the constraints (2) Then, we can finally conclude the global finite-time convergence of the complete set of original joint position and joint velocity error trajectories, Now consider the following Lyapunov function: Vtp = T where ttp can be fixed arbitrarily Using (26), (27) we can see that the existence of a dynamic sliding mode in Sqp t = for all time implies q2r + Sdp q˙2T t = 0Tn−m 0Tn−m q2T t qT q˙2 = − 525 We exclude the trivial case when the system is already in the singularity q t0 = at given initial conditions,24 since any terminal-attractor-based control algorithm fails at this point at t = t0 c Then, we analyze the case of q2 t0 = Note that the equation qăr in (17) violates the Lipschitz condition in the open loop However, considering that for closed-loop ∀t ≥ ttp > where ttp ≤ tdp + Vtp1− t0 m 1− c Initial position tracking error must be different from zero at any given initial conditions, as is usually the real case 526 Journal of Robotic Systems—2001 dynamics Sqp = ∀t ⇒ q˙2 = Eq (17) becomes qăr = +Q r q22r−1 − r q2r + Sdp ∀t, then q2r−1 Sdp Qr Qr 2 q22r−1 − r q 2r−1 + q r−1 Sdp + t ≤ tdp t > tdp S˙vF = −b F a (54) where = Q˙ q˙2d q2r + Sdp K1 p + Q qă2d + S˙dp − K1 Sqp + J˙T SvF + K2 F + J T S˙qF + K2 SqF + qădisct and there is not discontinuity Since r > 1/2 then there is not singularity in the term r q22r−1 , and the term r q2r−1 Sdp is singular in q2 = 0; however, since we can shape the transient response of Sdp independently of system dynamics, then we can design a Sdp with shorter convergence time than the convergence time of q2 in such a way that Sdp tends to zero faster than q2 Note that q˙2 = − q2r + Sd has at least exponential convergence and no overshoot can happen, and then q2 cannot be or cross zero before the FTC of Sdp d Thus, the term r q2r−1 Sd of (54) tends to zero before singularity occurs; that is, qăr = or SF t0 ) A particular design for SdF that complies with (52), (53) is at tdF = ⇒ F tdF = 0 F t0 1−a b 1−a where b is a terminal attractor parameter Other similar Sd s can be designed with spatial and temporal attributes For instance,27 O3 Polynomials of order three where input data are final time td and Sd t0 since initial time t0 and final connecting point Sd td are both zero Notice that Sdp t0 and SdF t0 are always available using only reliable position sensor and momentum (force sensor) readings as well as desired trajectories.e On the other hand, a design of Sd that yields terminal convergence of tracking errors with exponential decay is O4 Sd3 = S t0 exp− (55) where t−t0 > which is free of singularity Finally, note that there is not need to implement (55) instead of (17) Q.E.D T T Note that since Sqp SqF = 0Tn−m 0Tm T ∀t and according to Sections 4.5 and 4.6 we can shape arbitrarily the transient response of tracking errors for t < td (either tdp or tdF ) as long as Sd (either Sdp or SdF ) fulfills the following three conditions DISCUSSIONS Remark (terminal sliding mode for all time) The theorem states that if feedback gains are tuned properly then S SvF S˙ S˙vF are stable, and the following chain of implications is established: dynamic sliding modes at Sqp = SqF = ∀t → FTC C1 Sd ∈ C C2 Sd t0 = S t0 C3 Sd t S˙d t = 0 for ∀t = td In this way, critically damped response can be obtained For instance, consider the options O1 S˙d1 = − t Sd1 O2 S˙d2 = −bSda2 , where d = + a/2, t stands for a time base generator according to ref 42, a b are terminal attractor parameters, and Sd1 t0 = Sd2 t0 (either Sp t0 Since Sd is independent of system dynamics, then S˙dp t t→tdp → from either the positive or the negative side of the phase plane Sdp t always with critically damped response, and hence S˙dp t is continuous in the segment ≤ t ≤ td ; afterwards S˙dp t = Sdp t = d for Sp SF terminal sliding modes at q2 q˙2 F ∀t → FTC for q q˙ (56) It is interesting to note that since there are orthogonalized terminal attractors, then there could be a terminal sliding mode in the position subspace without a terminal sliding mode in the force subspace, and vice versa Remark (continuous controller) A key observation to design the continuous controller is that the high frequency component of qădisct in qăr can be cast as a bounded disturbance rejection problem, leaving plant dynamics only with continuous signal to e This is the practical case for robots that start motionless Vega et al.: Perfect Position/Force Tracking of Robots compensate, and the dynamic sliding mode yields the missing energy to withstand the disturbance It is noteworthy that a dynamical sliding mode is obtained using a simple continuous controller in comparison to the involved procedure proposed in refs 33 and 35 Remark (feedback gains) In theory, the control law is continuous for any value of positive feedback gains; however, in practice if are large enough then high frequency will appear The same effect is caused if r c d are small enough However, notice that neither large nor small r c d are required to obtain FTC of tracking errors On the other hand, passivity-based design usually allows one insight into the closed-loop energy balance which in turn may guide the tuning procedure In our controller, although it achieves terminal dissipativity and we have exploited the natural structure of constrained robot dynamics, the high nonlinear nature of the closed-loop system makes it difficult to obtain a systematic and intuitive tuning procedure for the 12 feedback gains involved in the controller, and this certainly is a drawback of this controller Careful attention is paid to the interplay of each gain to be able to tune properly the controller, and if K1 and K2 are not large enough then the dynamic sliding mode will not appear on Sqp SqF , and when q2 crosses zero, a singularity will a appear exactly at the time when q2 crosses zero To avoid this, overestimated feedback gains K1 K2 are used, and a short convergence time of Sdp SdF must be set Needless to say, a systematic tuning procedure would be very much welcomed, and it remains a challenge to design simpler feedback controllers that comply with the FTT paradigm Remark (control structure) The structure of the controller reveals three nonlinear control loops: a proportional plus derivative (PD) position control loop working in the position controlled subspace defined in the image of Q; a proportional plus integral (PI) force control loop working in the force controlled subspace defined in the image of J T ; and the sliding mode control loop for compensation of nonlinear inertial, centrifugal, centripetal, friction, and gravitational forces This loop is composed of two coupled control loops that compensate for such forces in each controlled subspace It is worth noticing that for the case when the parameters are known, the PD and the PI control loops not interfere to each other, a well-known property of the 527 orthogonalization principle13 that satisfies the invariance and static consistency principles, in joint space, neatly presented in ref Remark (local versus global terminal stability) In ref 14 the experimental verification of the locally asymptotically stable adaptive controller of ref 13 is presented, where very good performance is obtained for smooth rigid surfaces under small error at initial conditions Thus, if local terminal stability can be afforded by a particular application, then the decomposition of Eq (8) is not needed, and we can use a simpler Q matrix13 with the same control structure In this case, the stability proof is more involved, but we arrive at similar results as in Theorem since the Lyapunov function is the very same one As discussed in ref 6, the decomposition of joint coordinates always exists and nonlinear numerical errors not propagate as long as exact switching is implemented to commute to the set of independent coordinates q2 at any given time Remark (known parameters case implies control without force sensor) When is exactly known, and discontinuous control input can be implemented, then it is easy to obtain FTT without using a force sensor if we follow similar developments of refs 13 and 41 If discontinuous control is not allowed, then we can still obtain FTT for position errors and force errors remain bounded Details are omitted Remark (robustness) The invariance property29 is verified for each subspace and the system is theoretically invariant to parametric uncertainty of a class of bounded unmodeled dynamics, and since the state of the system is at least on an exponential stable regime, we can call for total stability arguments to withstand this disturbance Note that in presence of such disturbances, the scalars i , for i = will be bigger and then bigger feedback gains K1 , K2 are required to meet the sliding mode condition to guarantee the global terminal stability Remark (FTT for constrained second order systems) The extension of our control system for the general class of MIMO nonlinear second order systems described in ref 36 seems straightforward, considering the previsions of ref 36 for fully actuated rigid electromechanical systems On the other hand it is not evident how to extend our controller to higher order systems.27 528 Journal of Robotic Systems—2001 SIMULATIONS Table II Simulation of conventional (static) sliding mode control systems renders a numerically stiff set of equations since fast (the discontinuity around the sliding surface) and slow dynamics (the physical system) are combined Numerical results of such systems may produce biased results if sampling rate as well as tolerances are not set properly.31 In our case, the system itself is an infinitely stiff system;38 however, the dynamic terminal sliding mode controller is not stiff since no discontinuities are present See remark The closed-loop system is DAE and it is simulated using a commercial simulator wherein one implements the backward differentiation formula (BDF) of Petzold and Gear38 and we additionally implement an efficient finite-time convergent constraint violation algorithm41 to bound in finite time the numerical error of the constraint violation Using this simulator, a simulation study on a degrees of freedom rigid robot arm has been conducted, where robot parameters and feedback gains, including the parameters for O4, are given Tables I and II, respectively System setup and task definition are presented in such a way that the end-effector slides up and down over a infinitely stiff inclined wall modeled by X = a1 x + b1 y − c1 = 0, with a slope equal to −0 804 rad such that at y = 0m → x = 1m Then, for consistency to the DAE system, the desired Cartesian trajectories are Parameter xd t = − sin t +0 m t +0 m t = 20 + cos 5t Nm yd t = sin d where t is the independent time variable Since we not have an explicit tuning procedure then Table I Parameters of the robot arm Parameter Value Units Mass1 Link1 Linkc1 Inertia1 B01 Mass2 Link2 Linkc2 Inertia1 B02 10 10 57 02 75 37 01 Kg m m K gm2 Nm−s−1 Kg m m K gm2 Nm−s−1 Kd K1 K2 Feedback gains Value 350 ∗ I2×2 15 01 13 b 13 c ∗ I2×2 13 11 13 r d 20 10 15 feedback gains have been tuned in a trial-anderror-basis As discussed in Remark 2, attractor parameters were set far from zero to avoid high frequency, while K1 K2 were initially set large enough to finally set them down as given in Table II Parameters are tuned 10% higher than the real value of , and the sampling period is h = 001 s, which is a sampling rate of kHz, with the default tolerances of the BDF solver Figures 1, 2, and show the smooth and underdamped FTC of Cartesian, articular, and contact force tracking errors, respectively Figure shows the smooth controller, where it can be observed that a numerical spike is introduced in the very first six nonfixed samplings of the BDF numerical algorithm Figures and show the expected behavior of the dynamic sliding surface Sqp = ∀t and the FTC of terminal sliding surface Sp as well as the stable error manifolds S, respectively After the trajectories hit the zero manifold, they remain there afterward, only bounded by the numerical zero (for instance, after 20 seconds of simulation, the variable q2 moves around ±3 × 10−300 deg) Finally, when a large time-varying sinusoidal perturbation P = sin t is implemented the controller rejected them with the same control gains and without any significant effort; thus figures are omitted CONCLUSIONS Using terminal attractors as the equilibrium of a dynamic sliding surface, a new globally finite-time convergent passivity-based controller has been proposed A novel parametrization allows one to cast Vega et al.: Perfect Position/Force Tracking of Robots Figure Smooth FTC of Cartesian errors Figure Smooth FTC of both dependent and independent joint error coordinates Figure FTC of contact force tracking errors 529 530 Journal of Robotic Systems—2001 Figure Smooth control input A large numerical spike appears in the first five sampling periods, and it is therefore omitted Figure Dynamic sliding mode for Sqp for all time, and FTC of sliding mode Sp Figure The state S of the error equation is around zero (see magnitude of the vertical axis) Vega et al.: Perfect Position/Force Tracking of Robots discontinuous signals as bounded disturbances; thus control input is continuous The controller guarantees a dynamic sliding mode all the time which allows: (i) desired transient response with critically damped response; (ii) simultaneous finite-time convergence of position and force tracking errors; (iii) singularityfree closed-loop dynamics; and finally (iv) perfect tracking for constrained motion tasks for uncertain robot manipulators Implementation of the controller requires similar information of established explicit force feedback controller, which are measurements of position–velocity and contact–force, the regressor, and exact knowledge of the geometry of the constraint surface It is worth noticing that the additional synthesis and computational costs to derive the new signals for the terminal sliding surfaces are simply negligible since only the initial position and momentum are required as well as n integrators more Simulation data allow one to visualize the expected perfect performance This algorithm might be useful for several robotic tasks where FTC is a desired property, and it seems plausible to establish FTT for other passivity-based control algorithms.36 This work has been carried out under an Alexander von Humboldt Fellowship held by first author The second author acknowledges support from the CONACYT Scholarship 72368 10 11 12 13 14 15 16 17 18 REFERENCES D.E Whitney, A historical 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