IROS2019国际学术会议论文集0777

Tracking Control of Fully-Constrained Cable-Driven Parallel Robots using Adaptive Dynamic Programming Shuai Li, Student Member, IEEE, Damiano Zanotto, Member, IEEE AbstractIn this paper, a new adaptive tracking controller with learning ability is proposed for fully-constrained cable- driven parallel robots (CDPRs). For these systems, the necessity of maintaining positive and bounded tensions in all cables while coping with disturbances represents a critical control requirement. To achieve this goal, we propose a control law based on adaptive dynamic programming (ADP), with an actor- critic structure. In the critic part, an artifi cial neural network (NN) approximates the value function which is to evaluate the system performance; in the action part, the controllers parameters are tuned online to achieve optimal control per- formance. Additionally, the anti-windup (AW) technique is combined with the adaptive controller to cope with the input saturation problem. The stability of the closed-loop system with the proposed control algorithm is proved using the Lyapunov method. Numerical simulations show the eff ectiveness of the proposed controller. Index Termscable-driven parallel robots, adaptive dynamic programming, neural networks, anti-windup, tracking control. I. Introduction CDPRs are parallel manipulators in which rigid links are replaced by cables. Each cable is wound around an actuated pulley that extends or retracts the cable to control position and orientation of an end-eff ector (EE). In a fully-constrained CDPR, the number of cables is greater than the degrees of freedom (DOF) of the EE. Compared with conventional rigid-link robots, CDPRs have several advantages, which may include a large workspace 1, 2, high operating speed 3 and high load capacity 4. Replacing rigid links with cables brings some disad- vantages. Unlike rigid links, cables can only apply tensile forces on the EE and therefore they must be constantly kept tensioned. Additionally, the maximum torques attainable by the actuated pulleys defi ne upper bounds on the cable tensions 5. Because the confi guration of the cables changes with the position and orientation of the EE, these constraints on cable tensions cause the Available Wrench Set (i.e., the set of generalized forces that can be exerted by cables on the EE) to be pose-dependent, unless special designs are devised 6, 7. Given a Required Wrench Set, a pose is said to be wrench-feasible if any external wrench within the required set can be counteracted by feasible (i.e., positive and bounded) tensile forces in the cables. Consequently, the Wrench-Feasible Workspace (WFW) is defi ned as the set of wrench-feasible poses 8, 9. Computing a devices WFW usually requires the use of numerical methods, either discrete Shuai Li and Damiano Zanotto () are with the Dept. of Mechanical Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, USA. or continuous 10. Due to these specifi c features, conventional control meth- ods for trajectory tracking are typically not suitable for CDPRs. To date, researchers have proposed control laws designed in either cable length coordinates or task-space coordinates. The H-infi nite control investigated in 11 and the nonlinear feedforward control proposed in 12, for example, belong to the fi rst group. The computed torque method presented in 13, 14 and the sliding mode control described in 15 belong to the second group. More recently, modern control methods have also been applied to fully- constrained CDPRs. H. J. Asl etal. 16 proposed an adaptive neural trajectory tracking control law for CDPRs, which includes input saturation. F. Tajdari et al. 17 implemented an adaptive neuro-fuzzy controller to cope with the noise and disturbance of the system, and proved the stability of the closed-loop system using the Lyapunov method. Most of these control strategies, however, assume that all the states are measurable, which might be unpractical or impossible in real-life applications. Further, most controllers do not use explicit indexes (e.g., the long-run cost) to measure and optimize tracking performance. Additionally, most of these controllers do not account for the devices WFW. In recent years, a subclass of reinforcement learning techniques known as adaptive dynamic programming (ADP) has become more and more popular in optimal control. Due to the universal approximation property of artifi cial neural networks (NNs), many applications of ADP employ NNs to estimate the solution of the Hamilton-Jacobi-Bell (HJB) equation, thus alleviating the computational burden 18, 19. The actor-critic structure is one of the most widely used techniques in the optimal control of nonlinear systems 20 26. The actor-critic approach allows an actor component to generate the control action, while the critic component is used to approximate a cost function that represents the long-term system performance. Thus, the critic part helps to improve the quality of the action output. In this paper, the vast capability of ADP methods is leveraged to develop a novel adaptive controller for fully- constrained CDPRs, which takes input saturation into ac- count. A critic NN is employed to approximate the long-term cost function, while the control law is designed in parametric form, wherein the parameters are tuned online based on the feedback information from the critic part. Input saturation may degrade the tracking performance of the closed-loop system. Previous works have addressed this issue in diff erent ways. In 18, for example, an auxiliary linear system was proposed to embed the input constraints 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Macau, China, November 4-8, 2019 978-1-7281-4003-2/19/$31.00 2019 IEEE6781 as saturation nonlinearity in the controller structure, and the stability of the closed-loop system was demonstrated. However, in general the eff ects of input saturation on the system cannot be reliably modeled by a linear auxiliary system. In 2730, a non-quadratic reward function was used to formulate the long-term cost, the optimal control law took the form of a hyperbolic tangent, and NNs were employed to approximate the HJB equation. In this case, the control structure is fi xed and only applies to certain types of value function. Conversely, anti-windup (AW) techniques can be applied to linear and nonlinear systems to avoid instability and reduce the performance loss when saturation occurs 31, 32, More importantly, AW techniques can be designed independently of the controller. In this work, AW techniques are combined with the aforementioned adaptive controller to cope with the input saturation problem and guarantee the stability of the closed-loop system. Further, a supervisor term is used to guarantee boundedness of the states. This paper is organized as follows: Section II summarizes the dynamic model of a generic fully-constrained CDPR. Section III illustrates the controller design, followed by the anti-windup design and a stability proof. The proposed controller is then validated through numerical simulations which are reported in Section IV. II. CDPR Model The layout of a CDPR with m cables controlling n DOF is shown in Fig. 1. Cables are assumed massless and infi nitely stiff . The ith cable is connected to the ith winch at point Aiand has the other extremity attached to point Bion the EE. Vector bi, whose coordinates are known in the EE frame O0, connects the center of mass of the EE (which coincides with O0) to the attachment point Bi. uidenotes the unit vector along the ith cable, directed from the EE to the base. R R33 represents the rotation matrix from the fi xed coordinate frame O to O0 and is a function of three independent Euler angles: , and . Based on these defi nitions and given Fig. 1, Bican be written as Bi= r + Rbi,(1) where r = x,y,zTis the position of the EE center of mass. The length of the ith cable Lican be obtained from: L2 i = (Ai r Rbi)T(Ai r Rbi)(2) By diff erentiating (2) with respect to time and combining all m equations in matrix form, the following expression is obtained ? Li=J? rT,TT,(3) where = x,y,zTindicates the angular velocity of the EE, L = L1.LmTand the Jacobian matrix J Rmn is defi ned as: J = ? u1u2.um b1 u1b2 u2.bm um ?T (4) Further, the vector of angular velocities is related to the time derivatives of the Euler angles through the following expression = E? ,?, ? T,(5) O x y z i b i B r i u i A x z y O 1i A m A Fig. 1.Kinematic model of a generic CDPR. where matrix E is given by: E = 10s 0ccs 0scc (6) Thus, (3) can be rewritten as ? L =J ?I 33 033 033E ? ? x = J? x,(7) where x = x,y,z,Tis the vector of the generalized coordinates. The equation of motion of a generic CDPR with m cables and n DOF can be written as M(x)? x + C(x, ? x)? x + G(x) + d= JTT = ,(8) where M(x) Rnnis the inertia matrix, C Rnnis the centrifugal and Coriolis matrix, G Rnis the gravity vector, d Rnis the disturbance vector, T Rmis the vector of the cable tensions and Rnis the wrench generated by the m cables on the EE. Equation (8) holds as long as the pose is wrench-feasible. For the dynamic system (8), the following properties are valid 33: Property 1. The inertia matrix M(x) is positive defi nite and bounded (i.e., there exist positive scalars m and m such that m 6k M(x) k6 m). Property 2. The Coriolis and centrifugal matrix and the gravity vector are also bounded, such that kC(x, ? x)k 6 ck? xk, kG(x)k 6 g, with c,g 0. Further, we make the following assumptions: Assumption 1. The time-varying disturbance dis un- known but bounded, with k dk6 d, d 0. Assumption 2. The lower and upper bounds m and m are known. III. Adaptive tracking Control In this section, a novel adaptive control law is built for fully-constrained CDPRs based on the ADP framework without considering the saturation. First, (8) is rewritten in discrete-time form and the tracking error is defi ned. Then, the implicit function theorem is used to show the relationship between states and control input, and this transformation is used to design the controller. Let the state vector q be defi ned as q = x, ? xT, and let tbe the time step. The discrete-time 6782 Anti- windup Action PlantESO Tension distribution Critic T aw x p + aw xq aw qx + - q + - + + d x Fig. 2.Block diagram of the proposed adaptive controller form of (8) can be written as qi(k + 1) = qi(k) + t fi(q(k),(k),d(k) y(k) = x(k), ? y(k) =? x(k), i = 1.2n (9) where qiis the i-th state, fi() is the i-th system function (which is continuous and continuously diff erentiable with respect to the states), y is the output vector and ? y is the estimated Cartesian velocity vector. We assume that the desired trajectory xd Rnis an arbitrary smooth function of time which lies in the devices WFW. The WFW corresponds to a required wrench set min,max and a set of feasible cable tensions Tmin,Tmax . Starting from the defi nition of the tracking error e = xd x, the following error equations are introduced to be used in the action part: = ? e + e ? q = ? xd+ e ? q = ? xd+ ? e (10) In the previous equations, Rnis the new error, q Rn is the error state and Rnn is a