Adaptive Control for Robot Manipulators Under Ellipsoidal Task Space Constraints

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1 22 IEEE/RSJ International Conference on Intelligent Robots and Systems October 7-2, 22. Vilamoura, Algarve, Portugal Adaptive Control for Robot Manipulators Under Ellipsoidal Task Space Constraints Keng Peng Tee, Shuzhi Sam Ge, Rui Yan, and Haizhou Li Abstract Motivated by applications in robot-assisted physical rehabilitation, this paper presents an adaptive control design for robot manipulators operating in an ellipsoidal constrained region. The ellipsoidal constraint problem is more challenging than the box constraint problem tackled in previous works, since the nonlinear constraint boundary cannot be handled in a decoupled manner along the dimensions of the task space. We introduce a novel Barrier Lyapunov Function (BLF) which contains a quotient of the squared norm of the tracking error over the ellipsoidal task space constraint. This function allows the task space constraint to be handled directly without requiring an intermediate mapping to the error space. We show that, under the proposed BLF-based adaptive control, the end-effector always remains in the constrained region despite the perturbing effects of online parameter adaptation and also the presence of bounded external disturbances. A simulation example illustrates the performance of the proposed control. Fig.. robot constrained task space Ellipsoidal constrained region for robot end-effector. I. INTRODUCTION Robotics applications are moving from factory floors to unstructured human environments. As such, research on the safe control of robot manipulators, in close proximity or direct physical interactions with human users, is gaining importance. In particular, the topic of adaptive interaction control, which aims to reduce dependency on a precise knowledge of the dynamics of the robot and the environment, has seen substantial advancement through works such as adaptive impedance control [], [2], [3], adaptive admittance control [4], adaptive Jacobian tracking [5], and approximation-based impedance control [6], [7]. While adaptive interaction control has received much attention, position constraints in the operating environment are usually neglected in the control design. During physical interactions with humans, the interactive forces can be unpredictable. Therefore, constraints need to be enforced to prevent endangerment to human safety as well as collisions with itself or the surroundings. Consider the example of robot-assisted physical rehabilitation [8] for patients who suffer from motor disabilities (Figure ). The motion of the rehabilitation robot needs to be constrained strictly within a subset of the task space to minimize the risk of overstretching and injuring the patient s limb. Depending on the extent of muscle spasticity, abnormal muscle tone, and other conditions, a patient s limb can The corresponding author is K.P. Tee, K.P. Tee, R. Yan, and H. Li are with the Institute for Infocomm Research, A*STAR, Singapore S. S. Ge is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 7576, and also with the The Robotics Institute, and School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 683, China. exhibit varying movement resistance in different regions and along different directions in the task space. Thus, the motor rehabilitation program needs to customize the constrained region, in terms of size, shape or orientation, according to patient s motor condition and progressive outcome. Given the constrained region, the robot controller needs to guarantee that the end-effector, which is the point of interaction, always stays inside. Furthermore, the controller is required to adaptively compensate for uncertainty in the combined inertia of the coupled system between the patient s limb and the robot manipulator. Based on the practical scenario described above, we see a need for a rigorous guarantee of constrained region satisfaction in adaptive interaction control. Such a constrainedregion control problem, which is concerned with keeping the manipulator inside a region described by inequality constraints, is different from existing works that deal with keeping the manipulator on a surface represented by equality constraints [5]. A promising approach to ensure constrained region satisfaction is to employ Barrier Lyapunov Functions (BLF) when designing the control [9], [], []. The method uses barrier functions that grow to infinity at some finite limits, and designs a control to keep the candidate BLF bounded in the closed loop system, thus guaranteeing that the barriers are not transgressed. Our previous work on BLF-based control of robot manipulator in constrained task space [2] focused on box constraints of a fixed orientation, and was unable to handle more general shapes of constrained regions, such as ellipsoids, in arbitrary orientations. In this paper, we propose a new method of adaptive constrained control design for robot manipulators, based on a novel barrier function that is a quotient of the squared /2/S3. 22 IEEE 67

2 norm of the tracking error over the task space constraint. This function is different from the logarithmic type barrier function of [2], which was formulated in terms of the tracking error only, and necessitated a mapping of the task space constraint to a dynamic error space constraint. Another difference is that [2] only handles box constraints, while the BLF proposed here allows the handling of more general constraints such as ellipsoids. Compared with box constraints, ellipsoidal constraints are harder to deal with, since the constraint boundary is a nonlinear curve or surface in task space, and cannot be handled in a decoupled manner along each dimension of the task space as in the case of box constraints. Our new barrier function simplifies the control design and enables the direct handling of ellipsoidal task space constraints without prior mapping to the error space, which can be a complicated process and can lead to a conservative design. Under a unified framework of BLFbased Lyapunov control synthesis and stability analysis, we show that the end-effector is able to track a desired trajectory while never leaving the ellipsoidal constrained region. II. PROBLEM FORMULATION Throughout this paper, we denote by R the set of real numbers, and the Euclidean vector norm in R m. A. Robot Dynamics Consider a robot manipulator described by: M(q) q + C(q, q) q + G(q) + F( q) = τ + τ e (t) () where M(q) R n n is a symmetric positive definite matrix, C(q, q) q R n the Coriolis and centrifugal forces, G(q) R n the gravitational forces, F( q) R n the frictional forces, q R n the robot joint position, τ R n the input torque, and τ e R n a bounded interaction torque from the environment (or human) satisfying τ e (t) τ e, t, with τ e a constant. The terms M(q), C(q, q), F( q) and G(q) contain uncertain dynamic parameters. To track a desired trajectory in task space, the joint space dynamics () are transformed into task space dynamics [3]: M x ẍ + C x ẋ + G x + F x = f + f e (t) (2) via the forward kinematics and the Jacobian: x = Ω(q), ẋ = Ω q =: J(q) q (3) q where x = [x,,...,x m ] T is a vector of task variables, and the coefficient matrices are defined as M x = J T MJ, G x = J T G, F x = J T F C x = J T (C MJ J)J, f = J T τ, f e = J T τ e (4) Note that the external force f e is bounded by a constant f e, i.e. f e (t) f e, t. For simplicity, we consider only non-redundant (m = n) non-singular manipulators with known Jacobian J in this paper. Henceforth, we consider x R n. Furthermore, we consider only positioning of the end-effector (i.e. n 3), although our method can be easily modified to control end-effector orientation too. The following properties hold [3]. Property : The inertia matrix M x is symmetric positive definite. Property 2: The matrix Ṁx 2C x is skew symmetric. Property 3: The left-hand-side expression of (2) can be linearly parameterized in terms of the robot system parameters as follows: M x φ + C x φ 2 + G x + F x = ψ(φ,φ 2,x,ẋ)θ (5) for any φ,φ 2 R m, where θ R l are constant parameters and ψ R l is a known regressor function. B. Control Objective The control objective is to ensure that the end-effector position x = [x,...,x n ] T tracks a desired trajectory x d = [x d,...,x dn ] T while always satisfying position constraints imposed on the end-effector. Specifically, the end-effector is required to satisfy position constraints, which involve always staying in the interior of an ellipsoidal region x T Sx =, where S is a positive definite symmetric matrix. Thus, the position constraint set can be described by x Ω x, where Ω x = {x R n : x T Sx < } (6) In other words, the controller needs to ensure that x(t) Ω x t >, given that x() Ω x. Assumption : The desired trajectory x d (t) Ω x t, and there exist positive constants Y and Y 2 such that ẋ d (t) Y and ẍ d (t) Y 2, t. III. CONTROL SYNTHESIS USING A BARRIER LYAPUNOV FUNCTION Denote the error variables as: z = [z,...,z n ] T = x x d w = [w,...,w n ] T = ẋ α (7) where α = [α,...,α n ] T is a stabilizing function to be designed shortly. Then, we define the following constraint functions: (x) = x T Sx (8) For system (2), consider the Barrier Lyapunov Function (BLF) candidate V (z,x,w,q, θ) = V z (z,x) + V w (w,q) + V θ ( θ) (9) z T z V z (z,x) = () 2 (x) V w (w,q) = 2 wt M x w () V θ ( θ) = 2 θ T Γ θ (2) where σ = [σ,...,σ n ] T, Γ is a positive definite matrix, θ = ˆθ θ is the error between θ and its estimate ˆθ, and () is 68

3 a quotient of mixed variables z and x. It can be shown that V is positive definite and continuously differentiable in the set x Ω x. We employ adaptive backstepping procedure [4] to design the controller. For the robot manipulator (), the procedure consists of 2 steps. Step First, we design the stabilizing function. To this end, we obtain the partial derivatives of V z (z,x) as follows: V z (z,x) z V z (z,x) = z T (x) = zt z 2g 2 s(x) (x) Thus, the time-derivative of V z in () is given by V z = where = z T zt z 2g 2 s(x) (3) (x) (w + α) (x) (w + α ẋ d) z T (x) [h(z,x)(w + α) ẋ d] (4) h(z, x) = ( (x) = xt Sx d x T Sx ) z 2 (x) We design the stabilizing function α as follows: α(z,x,ẋ d ) = (5) h(z,x) ( K zz + ẋ d ) (6) where K z is a positive definite matrix. By substituting (6) into (4), we obtain: V z = zt K z z + h(z,x)zt w (7) The following lemma shows that h(z,x) is bounded away from in the set x,x d Ω x, which is important for ensuring that α(z,x,ẋ d ) is bounded for x,x d Ω x and bounded ẋ d. Lemma : The term h(z, x), defined in (5), satisfies h(z,x) > in the set x,x d Ω x. Proof: By completion of squares, we obtain the inequality: 2x T Sx d x T Sx + x T d Sx d (8) Furthermore, in the set x,x d Ω x, we have x T Sx < and x T d Sx d <. Then, it is clear, from (8), that x T Sx d < (9) Finally, from (5), it follows that h(z,x) > in the set x,x d Ω x. Step 2 In this step, we design the adaptive control torque and the adaptation law. Similar to Step, we obtain the timederivative of V w from (2) and () as: V w = w T ( C x ẋ G x F x + f + f e M x α) + 2 wt Ṁw (2) Substituting ẋ = w + α and using Property 2, we eliminate the last term on the right hand side and obtain V w = w T ( C x α G x F x + f + f e M x α) (2) Then, based on Property 3, we can rewrite V w such that the robot dynamics are linearly parameterized: V w = w T ( ψ( α,α,x,ẋ)θ + f + f e ) (22) where ψ is a known regressor. Then, we obtain the timederivative of the Lyapunov function candidate (9) as V = zt K z z + w T ( ψ( α,α,x,ẋ)θ + f + f e ) + θ T Γ ˆθ (23) Now, we design the control as f = ψ( α,α,x,ẋ)ˆθ K w w h(z,x)z (24) τ = J T (q)f (25) where K w > I/2 is a positive definite matrix. For robust adaptation of the parameter estimate ˆθ, we employ the following adaptation law: ˆθ = Γ( ψ( α,α,x,ẋ)w µˆθ) (26) where µ is a positive constant. The leakage term provides robustness against unmodelled disturbances and noisy force measurements. Substituting (24) and (26) into (23), it can be shown that V zt K z z w T (K w 2 I)w + 2 f 2 e µ 2 θ 2 + µ 2 θ 2 ρv + c (27) in the set x Ω x, where ρ = { min 2λ min (K z ), 2λ min(k w 2 I) } µ, λ max (M x ) λ max (Γ ) (28) c = 2 f 2 e µ 2 θ 2 + µ 2 θ 2 (29) The closed loop error system is given by: ż = h(z,x) ( K zz + ẋ d ) + w ẋ d ( ẇ = Mx ψ( α,α,x,ẋ) θ K w w h(z,x)z ) θ = Γ( ψ( α,α,x,ẋ)w µ( θ + θ)) (3) which can be rewritten in the form ξ = h(t,ξ) (3) where ξ = [z,w, θ] T, h(t,ξ) is piecewise continuous in t and locally Lipschitz in ξ, uniformly in t, in the set ξ Ψ = {ξ R 2n+l : > } (32) 69

4 Thus, (27) and (3) allow us to establish the existence and uniqueness of the solution ξ(t) t [, ), according to [5]. In the following result, we establish the fact that x(t) remains in the constrained set Ω x t >, as well as the convergence properties of z(t). Theorem : Consider the robot manipulator () under Assumption, control torque (25), and parameter adaptation law (26). For initial condition x() Ω x, the following properties hold: i) The tracking error z(t) is bounded by: ( z(t) 2 V t= + c ), t > (33) ρ ii) The state x(t) remains, for all t >, in the constrained set Ω x. iii) The stabilizing function α(t) and control torque τ(t) are bounded t >. Proof: i) From (27), we obtain that V t V t= + c/ρ in the set x Ω x. Since initial condition x() Ω x, we have V t V t= + c/ρ t >. Then, using the fact that V t z(t) 2 /2, we thus arrive at z(t) 2 2(V t= + c/ρ), which leads to (33). ii) Using proof by contradiction, we first assume that there exists some t = T such that x(t) T Sx(T) = (34) starting from the initial condition x() Ω x. Then, using (), we rewrite the result V t=t V t= + c/ρ t > as z(t) 2 (x(t)) V t= + c ρ (35) Now, substituting (34), the left hand side becomes infinite, contradicting the boundedness result V t=t V t= + c/ρ t >. As such, x(t) T Sx(t) (36) for all t >. Then, since x() Ω x, it is clear that x(t) Ω x t >. iii) Since V t V t= + c/ρ, we know that ˆθ(t) and w(t) are bounded t >. Based on Lemma, we know that h(z,x) > for all x,x d Ω x. Thus, we see, from (6), that the stabilizing function α(t) is bounded too. This leads to the boundedness of ẋ(t), since ẋ = w + α. We can also show that α is bounded by noting the boundedness of the partial derivatives of α(z,x,ẋ d ) in the set x Ω x. Since f is a continuous function of bounded signals in the set x Ω x, we know that f(t) is bounded t >. Since the Jacobian J is nonsingular in the constrained work space, τ(t) = J T f(t) is bounded t >. Corollary : Consider the robot manipulator () in the special case of f e. Then the control law (6), (24)- (25), and adaptation law ˆθ = Γψ T v, with initial condition x() Ω x, ensure that the tracking error z(t) converges to zero asymptotically, i.e., x(t) x d (t) as t. Proof: Substituting (24), f e, and ˆθ = Γψ T v into (23) yields V = zt K z z =: (z,x) (37) (z,x) Since V, we can show that (z(t),x(t)) > t >, using a similar approach as the proof of Theorem (ii). t Thus, it can be shown that lim t (τ) dτ and (t) are bounded. Then, we invoke Barbalat s Lemma [6] to show that (z(t),x(t)) as t. Since (z,x) = only if z =, we conclude that z(t) as t. IV. SIMULATION To illustrate the performance of the proposed controller, we consider a simple example of a two-link frictionless robot moving in a horizontal plane subject to a ellipsoidal constraint in the task space. All units are S.I. The robot dynamics are modeled by [7] τ = m 2 l 2 2( q + q 2 ) + m 2 l l 2 c 2 (2 q + q 2 ) τ e +(m + m 2 )l 2 q m 2 l l 2 s 2 q 2 2 2m 2 l l 2 s 2 q q 2 τ 2 = m 2 l2( q 2 + q 2 ) + m 2 l l 2 c 2 q + m 2 l l 2 s 2 q 2 τ e2 (38) where c i = cos(q i ), s i = sin(q i ), s ij = sin(q i + q j ), and c ij = cos(q i + q j ). The uncertain parameters are m and m 2, whose true values are.5kg and.kg respectively. The lengths of the links are l = l 2 =.35m. The ellipsoidal constraint region is described by x T Sx = where S =.224, S 2 = S 2 = , and S = This ellipse has major length.35m, minor length.25m, and is rotated anticlockwise by 3deg. The origin of the task space x = [x, ] T is at [,.32] T m with respect to the position of the base joint. Initially, the end-effector is at rest at the origin. We apply the proposed BLF-based adaptive controller (24)-(26), with design parameters K z =.5I, K w = 8I, Γ = 5I, and µ = 4. The parameter estimates are initialized as ˆθ () = ˆθ () =.75. To test the ability of the controller to maintain the endeffector within the constrained region in the presence of disturbances, we apply, on the end-effector, an interaction force (in N) that is proportional to the desired setpoint/trajectory: f e (t) = kx d (t) (39) where k is a positive constant. These are indicated by the arrows emanating from the center in Figure 2. Then, the interaction torque τ e = [τ e,τ e2 ] T is given by τ e = J T f e. For comparison purpose, we also simulate a conventional adaptive controller without BLF-based constraint satisfaction, described by: α = K z z + ẏ d f = K w w + ψˆθ z ˆθ = Γ( ψw µˆθ) (4) 7

5 A. Setpoint Reaching The setpoint reaching task requires that the robot endeffector, starting from the center of the elliptic constrained region, reach a set of 8 target points near the boundary. The interaction force (39) is set with k = 2.5Nm. z with BLF z time [s] Fig. 3. Trajectories of the end-effector under BLF-based control for the setpoint reaching task. N x without BLF.3.2 parameter estimate ˆθ ˆθ x Fig. 2. Traversed paths of the end-effector with (top) and without (bottom) BLF-based control for the setpoint reaching task. The grey ellipse represents the boundary of the constrained region. Figure 2 shows that the proposed BLF-based adaptive controller succeeds in preventing the end-effector from transgressing the boundary of the ellipse, despite the fact that the target points are very close to the boundary and a constant outward force is trying to pushing the end-effector out of the constrained region. Online parameter adaptation contribute an additional source of transient perturbation, but the proposed control is able to ensure that the targets are reached accurately without ever leaving the constrained region. By examining Figure 3, it is clear that the BLFbased control provides good error convergence properties. The parameter estimates ˆθ and ˆθ 2,, as well as the control torque τ, are always bounded and converge quickly to steady values, as shown in Figure 4. However, for the conventional adaptive controller without BLF-based constraint satisfaction, the end-effector escapes the constrained region and is unable to converge to any of the target points due to the bias force disturbance, as shown in Figure 2. N control torque [Nm] time [s] Fig. 4. Evolution of parameter estimates (top) and control torques (bottom) for a representative movement in the setpoint reaching task. B. Trajectory Tracking The trajectory tracking task requires the end-effector to track a desired trajectory lying on an ellipse within the constrained region. The trajectory is described by: x d (t) = Acos ( π 4 + ωt ), x d2 (t) = A 2 sin( π 4 + ωt) (4) The interaction force (39) is set with k =.Nm. As observed in Figure 5, the end-effector starts from the origin and tracks the desired trajectory in the presence of model uncertainty and interaction force (39), progressively reducing its tracking error with the help of the adaptive controller. With our proposed BLF-based constraint satisfaction mechanism, the end-effector never leaves the constrained region despite the fact that the desired trajectory almost touches the boundary of the constrained region at some points. In contrast, the traversed path corresponding to the conventional adaptive controller (4) transgresses the boundary of the constrained region substantially. Figure 6 shows that the endeffector trajectory under our proposed BLF-based control eventually converges to the desired trajectory. τ τ 2 7

6 with BLF x without BLF x Fig. 5. Traversed paths of the end-effector with (top) and without (bottom) BLF-based control for the trajectory tracking task. The grey ellipse represents the boundary of the constrained region. x time [s] Fig. 6. After parameter adaptation, the end-effector trajectory (solid) under BLF-based control tracks the desired trajectory (dotted) closely. V. CONCLUSIONS In this paper, we have presented the design of an adaptive controller for non-redundant robot manipulators operating in a virtually constrained task space. Specifically, we deal with the class of ellipsoidal constrained regions, which constitute a more difficult problem compared with rectangular constrained regions that have been handled in previous works. We have used a barrier function that explicitly contains the ellipsoidal task space constraint. This has allowed the simultaneous satisfaction of tracking and constraint requirements without the need to, first, map the ellipsoidal constraint from the task space to the error space. This mapping process, though successfully applied for the box constraint problem in [2], cannot be easily extended to the ellipsoidal constraint problem, because the nonlinear ellipsoidal surface cannot be treated in a decoupled manner along each dimension of the task space, as had been done in [2]. Hence, the barrier function proposed in this paper provides a relatively simple way to circumvent the difficulties associated with the existing approach. Using Lyapunov-based control synthesis and stability analysis, we have proven theoretically that the tracking error and control signals are bounded, and that the ellipsoidal task space constraints are always met. ACKNOWLEDGMENTS This work is supported by the A*STAR Science and Engineering Research Council (SERC) Grant No REFERENCES [] R. Kelly, R. Carelli, M. Amestegui, and R. Ortega, On adaptive impedance control of robot manipulators, in Proc. IEEE Conf. Robotics & Automation, (Scottsdale,Arizona), pp , 989. [2] R. Colbaugh, H. Seraji, and K. Glass, Direct adaptive impedance control for robot manipulators, J. Robotic Systems, vol., no. 2, pp , 993. [3] H. Park and J. Lee, Adaptive impedance control of a haptic interface, Mechatronics, vol. 4, no. 3, pp , 24. [4] H. Seraji, Adaptive admittance control: an approach to explicit force control in compliant motion, in Proc. IEEE Conf. Robotics & Automation, (San Diego, CA), pp , 994. [5] C. C. Cheah, Y. Zhao, and J. J. E. Slotine, Adaptive Jacobian motion and force tracking control for constrained robots with uncertainties, in Proc. IEEE International Conference on Robotics and Automation, (Orlando, Florida), pp , May 26. [6] L. Huang, S. S. Ge, and T. H. Lee, Neural network based adaptive impedance control of constrained robots, in Proc. IEEE Symposium on Intelligent Control, pp , 22. [7] M. C. Chien and A. C. Huang, Adaptive impedance control of robot manipulators based on function approximation technique, Robotica, vol. 22, no. 4, 24. [8] H. I. Krebs, N. Hogan, M. L. Aisen, and B. T. Volpe, Robot-aided neurorehabilitation., IEEE Trans Rehabil Eng, vol. 6, no., pp , 998. [9] K. P. Tee, S. S. Ge, and E. H. Tay, Adaptive control of electrostatic microactuators with bidirectional drive, IEEE Trans. Control Systems Technology, vol. 7, no. 2, pp , 29. [] K. P. Tee, S. S. Ge, and E. H. Tay, Barrier Lyapunov functions for the control of output-constrained nonlinear systems, Automatica, vol. 45, no. 4, pp , 29. [] K. P. Tee, B. Ren, and S. S. Ge, Control of nonlinear systems with time-varying output constraints, Automatica, vol. 47, no., pp , 2. [2] K. P. Tee, R. Yan, and H. Li, Adaptive admittance control of a robot manipulator under task space constraint, in IEEE Conference on Robotics & Automation, pp , 2. [3] M. W. Spong and M. Vidyasagar, Robot Dynamics and Control. New York: Wiley, 989. [4] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley and Sons, 995. [5] E. D. Sontag, Mathematical Control Theory. Deterministic Finite- Dimensional Systems, Volume 6 of Texts in Applied Mathematics, Second edition. New York: Springer-Verlag, 998. [6] J. E. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliff, NJ: Prentice-Hall, 99. [7] J. J. Craig, P. Hsu, and S. S. Sastry, Adaptive control of mechanical manipulators, in Proc. IEEE International Conf. Robotics & Automation, pp. 9 95,

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