A High'Gain Observer-Based PD Manipulator. Control for Robot. M(q) q+c(q,q)q+g(q)+fq=t (1) x= f(x, z), z= g(x, z, (2) x= f(x, (x)) (4) Abstract

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1 Proceedings of the American Control Conference Chicago, Illinois June 2000 A High'Gain Observer-Based PD Manipulator Control for Robot Jose Antonio Heredia and Wen Yu Departamento de Control Automatico, CINVESTAV-IPN Av.IPN 2508, A.P , Mexico D.F., 07360, Mexico yuw~ctrl.cinvestav.mx, fax: Abstract In this paper, the popular PD control of robot manipulator is modified. A high-gain observer is proposed for the estimation of the velocity. The main contributions of this paper are: (1) By means of singular perturbation analysis, we prove that the closed-loop system is asymptotic stable; (2) We solve the problem of how faster the observer should be than the observer-based control. 1 Introduction It is well known that most of the industrial manipulators are equipped with the simplest PD control [8]. Various modifications and experimental tests of PD control have been published. There exist one main weakness in PD control: it requires the measurements of joint positions and velocities [11]. The joint positions measurements can be obtained by means of encoders, which gives very accurate measurements. The joint velocities are usually measured by velocity tachometers, which are expensive and often contaminated by noise [10]. server's dynamic) and a slow subsystem (PD controller and the robot's dynamic). We prove that observer error and tracking error are stable. Taking advantage of the results in [12], we give a necessary condition of the asymptotic stability for PD control. This condition corresponds to the minimum convergence speed of high-gain observer. 2 Robot Dynamic and Singularly Perturbed System The dynamics of a serial n-link rigid robot manipulator can be written as [81 M(q) q+c(q,q)q+g(q)+fq=t (1) where q E!~ denotes the links positions, qc ~ de- notes the links velocity, M(q) E!}~,~xn is the inertia matrix, C(q, q) C ~xn is the centripetal and Coriolis matrix, G(q) E!~ is the gravity vector, F C R nx~ is a positive definite diagonal matrix of frictional terms (Coulomb friction), and 7- E!}~'~ is the input control vector. One possible solution is to implement a velocity observer. Many papers have been published devoted to the theory and implementation of velocity observers for manipulators. Two kinds of observers may be used: model-based observer and model- free observer. The model-based observer assumes that the dynanfics of the robot is complete known or partial known. If the inertia matrix of the robotic's dynamic is known, sliding model observer [2] and adaptive observer [3] were proposed. Passivity-based observer was developed in [1]. The model-free observer means that no exact knowledge of robot dynamics is required. Most popular used observers are the high-gain observers, which can estimated the derivative of the output [4]. In this paper, we use singular perturbation method to analyze the PD control with high-gain observer [4], [13]. The closed-loop system may be divided into two separated sub-systems: a fast subsystem (high-gain ob $ AACC 2518 A nonlinear system is said to be singularly perturbed if it has the following form: x= f(x, z), z= g(x, z, (2) where x E R n, z E R m, ~ > 0 is a small constant parameter. It is assumed that (2) has a unique solution with a unique equilibrium point (x, z) = (0, 0). Since the second equation of (2) can be written as: z= g(~,z,,), its velocity is very fast when E ~ 0. Mak- E inge=0in (2) :~= f(x,z), 0 = g(x,z,o) (3) It is assumed that (3) has a unique root z = (x), and it is said that (2) is in the standard form. Then it is possible to express the slow subsystem as: x= f(x, (x)) (4)

2 The fast subsystem is: dz d-~ = g(x, z(r), 0) (5) where T = t and x is treated as a fixed unknown parameter. The slow subsystem is called the quasisteady-state system and the fast subsystem is called the boundary-layer system. Theorem 1 If the previous two assumptions are satis-.fled.for (2), then the solutions of (~) and (5) approach the solutions of (2) in an interval [t0, T] and [tl,t], respectively, where 0 _< to < tl < T. wheree::[z~,z~l T,A:: -Hv 0 'B:= E 0] I " Let us propose the following PD-like controller d U = M(Xl) x2 +C(Xl,X2)x2 d +Y~2 + a(x,) - K,(x, - Xl ~) - K.(~2 - x~) (12) where x~ d E!}~ is the desired position, x~ d is the desired velocity, and Kp, Kd e R n~'~ are constant positive matrices. In this paper, we assume that the desired trajectory and it's first two derivatives are bounded. The compensation terms in (12) are d M(xl) x2 +C(xl, 2)x~ + F222 + G(xl). 3 An Observer-Based PD Control. The motion equations of the serial n-link rigid robot manipulator (1) can be rewritten in the state space form [4]: Xl= X2' x2=f(xl'x2)-'i-g(xl)u (6) y=xl where xl = q E!}~ is the vector of joint positions, x2 =qe ~ is the vector of joint velocities, y E!~n is the measurable position vector. f(xl,x2) = M(xl) -l[-c(xl, x2)x2 - a(xl) - Fx2] g(xl) = M(Xl)-lu (r) The high-gain observer proposed in [4] is in following form: ~1 =. x2 + }Hv(Y - Xl) (S) ~2= ~Hv(y - ~) where 221 E ~, x2 E!}~ denote the estimated values of xl, x2 respectively; e is chosen as a small positive parameter; and Hp, Hv are positive definite matrices ch sensuchthat[ -Hp-Hv 0I]isaHurwitzmatrix'Let define the observer error as xl :-- Xl -xl, x2 := x2-~2. From (6) and (8) the dynamic of observer error can be formed as: xl = x2 - ~ Hpxl (9) Y2= -~Hv~l + f(x,, x~) + g(xl)u If we define a new pair of variables g~ := 71, 72 := ex2. (9) can be rewritten as: zl = z2 -- gp~l (10) ~2 = -Hv~l + e 2 If(x1, x2) + g(xl)u] or in matrix form: Remark 1 If the joint velocities are measurable, this PD controller is also suitable. We only need to change ~2 to x2. The next theorem shows that the PD control, with velocity measurements, can make the closed-loop system asymptotic stable. Let define the tracking error as From (6) and (12) the tracking error equation can be formed as:.d 51~_2i 1 -- X 1 = 5 2 d.d.d 2=/(Xl, x2) + g(xl)u(x~, x~, ~2, x2, x2)- x2 which becomes: xl= 52 (13) 52= H(51,52, Xl", x~, ~2) where H = M -1 [-F22 - Kp51 + Kdx2 - Kd52 - C22 - C52]. Now, it is possible to include the control (12) into (10), to get 8~1 = ~2 -gpzl The closed-loop is the combination of (6), (12) and (8), i.e., tracking error and observer error can be written in the form: x~=52, 52= H (14) with the equilibrium point (51,52, zl, z2) = (0, 0, 0, 0). Clearly (14) has the singularly perturbed form (2). Making e = 0, 0 = ~2 - Hp~l and 0 = -Hv~l imply that ~= A~+ ~2B [f(xl, x2) + g(xl)u] (11) 2519

3 which has an equilibrium point (xl, xz) = (21,22). The system (14) therefore is in the standard form. Substituting the equilibrium point into the first two equations of (14), we obtain the quasi-steady-state model, Xl=X2, ~2=H(~1,- Z2'Zl'ZZ' d d 0) (15) =M-I[-Kp5i --Kd~2 -- Cx2] On the other hand, the boundary layer system of (14) is: 77,(~) = ~2(~) - H,?i(~) (16) 5~72(7) = -H,?l(7) where 7 = ~. t (16) can be written as:?(r) = A?(7) (17) where?(~-) = [~r(w),z~(t)] w and A is defined as in (11). The following theorem will show the stability properties of the equilibrium points (xl,x2) = (0,0) and (71,72) = (0, 0) for (15) and (17), respectively. Theorem 2 The equilibrium point (~1,x2) = (0,0) o.f (15) and the equilibrium point (71,?2) = (0, 0) of (17) are asymptotic stable. Proof." For the first system, consider the following candidate Lyapunov function The advantage of this approach is that the singularly perturbed analysis may divide the original problem in two systems: the slow subsystem or quasi-steady state system and the fast subsystem or boundary layer system. Then both systems can be studied independently with the boundary layer system faster enough compared with the slow subsystem. These two subsystems explain why the dynamic of the high-gain observer is faster than the dynamic of the robot and the PD control. If the slow subsystem is considered as static, the high-gain observer in the fast time scale 7- can make the estimated states (xl, x2) converge to the real states (xl, x2). In the slow time scale t, the slow subsystem composed by the robot and the PD control can use then the estimated states (~,, x2). Remark 2 From the point of the singular perturbation analysis, one can see that high-gain observer (8) has a.faster dynamic than the robot (6) and the PD control (12). Under the assumption o fe = O, the observer error and the tracking error of PD control are asymptotic stable if the joint velocities are measurable. Since it is impossible for the high-gain observer (8) to have e = 0, it is necessary to find a positive value of e for which the stability properties are valid. For this purpose we proposed a modified version of [12]. The following theorem is an extend version of [12]. V,(2,, 2) =!52TM~2 + -istkp~, (18) 2 2 with its derivative with respect to time and along (15), V~: ~ [-gp~, - Kj~ - C~] + ½~ M ~ + ~K~ : M -c] - : < 0 Applying Lasalle's theorem, the only solutions of (15) evolving in the set f~v, = { (!21, 22)12.2 = 0} are #,=o, o=-[m-'kp]~, which implies that 31 = 0 (using property 2 of the robot's dynamic). For the second system, since A is a Hurwitz matrix, there exist a positive definite matrix P such that ATp + PA = -Q where Q is a positive definite matrix. candidate Lyapunov function V2(71,?2) = 7P? Consider the with its derivative with respect to time and along (17), V2 = Ev 'ATp + PA) ~ = _frq~ < Theorem 3 If there exist a continuous interval F = (0, ~) such that.for all e E F satisfies: ( ~ ) e4 + ((1- d)dfllk2) e2+ ((1 - d)alk1) e + (1-d)~Z~ -- (1 -- d)doqct2 < 0 2 (19) where oq, o~2, 13 a, IK,, K2 are nonnegative constants, 0 < d < 1, g is the positive root of (19), then the origin (~, z~ = (0, O) of (1~) is asymptotically stable Proof: Let us select Lyapunov function for (14) as vc~(x, z) = (1 - d)v,(~l,~2) + dr2(?1, ;2) (20) From theorem 2 we have Vc,=-(1-d)x [ 0 --T 0 0 _ Kd ] x +(1 - d) [52TM(~z + xl d) [H(e) - H(0)]] (21) _dz--'rq?+ d [22Tpe2Bg(e)] At this point we need to solve, [52TM( 1 + xf)[h(e) - H(0)]] r 0 0 = ~2 v [-P~2 + Kd~2 - C~2] = ~T! L 0 -F+Kd-C (22)

4 d)alk1) Notice that this matrix is bounded because F and Kd are constant matrices and property 6 of the robot's dynamic. Using the robot's dynamic properties we can concluded that: [2~Tp~2BH(e)] 0 0 ] - 0-2M -1 (F - Kd + C) "z (23) +yrpd[ 0 o ] -M-1Kp -M -1 (Kd +C) Applying (22) and (23) to (21), Vd-< -(1 - d)al~b2(~) + (1 - d)}fll (?2) (z ~ (24) -d [~a _ K1] 2(z~ + dek2 (~) (z~ [oo] where c~ = o Ka, ~2 = IIQII, Kx K2 /31 =,(z-) = as: 0 (?- Kd + ' = P[0-2M-: C) ] 0 1 = -M-1Kp -t~1-1 (Kd + C) ' [ F + Kd - C ' = II~tl, I1~11 (24) can be written in a matrix form -- (z) ] TT (z)] (1 - d)al _ (l-d)01 _ edna 2e 2 where T = [ -O-d)~'-EdK~ d[~-k1] ]" 2e 2 Now, T has to be a positive definite matrix. And T is positive definite if there exist a continuous interval F = (0, ~) such that for all e E F satisfies: ((1 - + ((1- d)d,k2) e2+ e + 0-d)~ 2 -- (1 -- d)dalct2 < 0 Then ~ will be the upper bound of e. Remark 3 Since (19) has 4 possible solutions, the theorem will be valid if there exist a positive real root of (19) such that in the interval [0,7] (19) is negative. The condition (I9) is only necessary (. epsilon = ~ / Figure 1: Polinomial of epsilon. The condition for ~ found in [12] is a simple formula depending on tile constants al, a2, fil, K1, K2 and not a more complex polynomial as in our case. 4 Simulation Results. To develope the simulations, a two-link planar robot manipulator is considered. It is assumed that each link has its mass concentrated as a point at the end. The manipulator is in vertical position, with gravity and friction. The robot parameters are: ml = m2 = 1, l 1 = 1, 12 = 2. The two friction coefficients are 0.3, and gravity is 9.8. So the real ma- [ 4cos(q2) +6 2cos(q2) + 4 ] trices of (1): M = 2cos(q2)+4 4 ' [ -4cos/42)sin(q2) \~] " " 2qlsin(q2) ] 0 G = C = -2 cos [d2~ sin (q2) ' 19.6 cos(q1 + q2) cos ql F = [ 19.6cos(ql + q2) ] ' ' q = [ql, q2] T. All data are given with the appropriate unities. The following PD coefficients are chosen: =[31 o] d=[oo o] 0 45 ' 0 80 The main differences between [12] and our paper are: We neglect assumption 3-a of the original theorem, because our Lyapunov function does not depend on 5. In assumption 3-b of the original theorem, the right side of the inequality does not depend on e. In assumption 3-c of the original theorem, the term which includes the constant K2 is multiplied by e and not by e 2 as in the previous theorem Let us calculate the constants in Theorem3, cq = [00 KdOJ = 80, a2 = [,Qii = 45, K1 = ' K2 = ,/31 = Then (19 / becomes: f (e) = d2e (1 - d)de (1 - d)e (1 - d) (1 - d)d (25) With d = 0.5 the polynomial (25) is shown in Figure1. One can see that for 0 < ~ < (25) is negative. So, ~ = 0.056, i. e., F = (0, 0.056). The highgain observer (8) is determined as e = Figure 2

5 shows the simulation results of the performance of the high-gain observer when a perturbation at t = 3. The perturbation is a 20% increment on the second link's mass. This is intended to see the robustness of the observer-based PD control. Figure 3 shows the tracking done by both links of the robot with the same perturbation at t = 3. Since the observer and controller are independence of the from robot's dynamics, the influence of perturbations are very small link 1..~../v:-:..!!.n.k e.s.t.!~.a.t..ed :... 5 Conclusions. In this paper, singularly perturbation method is used to prove the stability of PD control with [13]. Based on [12], an upper bound of e is given. The asymptotic stability of the equilibrium point of the closed-loop system is reached. The simulations show that the robot can follow the desired trajectories accurately. i L link 2 estim,ted Figure 2: High-gain observer. _~_L_._ link ::1,~f~-_ link ::1 desired / References [1] H.Berghuis and H.Nijmeijer, A Passivity Approach to Controller-Observer Design for Robots, IEEE Tran. on Robot. Automat., Vol. 9, , [2] C.Canudas de Wit and J.J.E.Slotine, "Sliding Observers for Robot Manipulator", Automatica, Vol.27, No.5, , [3] C.Canudas de Wit and N.Fixot, Adaptive Control of Robot Manipulators via Velocity Estimated Feedback, IEEE Tran. on Automatic Control, Vol. 37, , [4] S. Nicosia, A. Tornambe, P. Valigi, "Experimental Results in State Estimation of Industrial Robots", Proceedings of the 29th Conference on Decision and Control, [5] S. Nicosia, P. Tomei, "Robot control by using only joint position measurements", IEEE Tran. on Automatic Control, Vol. 35, No. 9, [6] P.Tomei, Adaptive PD Controller for Robot Manipulator, IEEE Tran. on Automatic Control, Vol. 36, , [7] M.Takegaki and S.Arimoto, A New Feedback Method for Dynamic control of Manipulator, ASME J. Dynamic Syst. Measurement, and Contr., Vo1.103, , 1981 [8] M. Spong, M. Vidyasagar, Robot Dynamics and Control, New York, Wiley, [9] P.A.Ioannou and J.Sun, Robust Adaptive Control, Prentice-Hall, NJ:07458, [10] R.Kelly, Global Positioning on Robot Manipulators via PD control plus a Classs of Nonlinear Integral Actions, IEEE Trans. Automat. Contr., vol.43, No.7, , Figure 3: PD control with high gain observer [11] R.Ortega and M.W.Spong, Adaptive Motion Control of Rigid Robot: A Tutorial, Automatica, Vol.25, no.6, , systems", Proc. IEEE Conf. Decision and Control, pp. 7-12, Tucson, Dec [12] Ali Saberi, Hassan Khalil, "Quadratic-Type Lyapunov Functions for Singularly Perturbed Systems", IEEE Transactions on Automatic Control, Vol. AC-29, No. 6, June [13] S.Nicosia and A.Tomambe, "High-Gain Observers in the State and Parameter Estimation of Robots Having Elastic Joins", System ~4 Control Letter, Vol.13, , 1989