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Output Feedback Nonlinear

Output Feedback Nonlinear Hindawi Publishing Corporation Journal of Robotics Volume 2009, Article ID 718728, 10 pages doi:10.1155/2009/718728 Research Article Output Feedback Nonlinear H -Tracking Control of a Nonminimum-Phase 2-DOF Underactuated Mechanical System Luis T. Aguilar Centro de Investigacio ´n yDesarrollodeTecnolog´ıa Digital, Instituto Polit´ecnico Nacional, Avenida del Parque 1310 Mesa de Otay, 22510 Tijuana, BC, Mexico Correspondence should be addressed to Luis T. Aguilar, luis.aguilar@ieee.org Received 18 September 2008; Revised 22 January 2009; Accepted 1 April 2009 Recommended by Warren Dixon Nonlinear H synthesis is developed to solve the tracking control problem restricted to a two degrees-of-freedom (DOF) underactuated mechanical manipulator where position measurements are the only available information for feedback. A local H controller is derived by means of a certain perturbation of the differential Riccati equations, appearing in solving the H ∞ ∞ control problem for the linearized system. Stabilizability and detectability properties of the control system are thus ensured by the existence of the proper solutions of the unperturbed differential Riccati equations, and hence the proposed synthesis procedure obviates an extra verification work of these properties. Due to the nature of the approach, the resulting controller additionally yields the desired robustness properties against unknown but bounded external disturbances. The desired trajectory is centered at the upright position where the manipulator becomes a nonminimum-phase system. Simulation results made for a double pendulum show the effectiveness of the proposed controller. Copyright © 2009 Luis T. Aguilar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction ated mechanical system. Representative works in this topic include orbital stabilization of underactuated systems by The focus of this paper is to solve the tracking control means of reference models as generator of limit cycles (see, problem for a 2-DOF underactuated mechanism via non- e.g., [7–10]). In particular, this paper is devoted to the linear H -control for time-varying systems [1] where joint solution of a periodic balancing problem for a two-link position measurements are the only available information underactuated mechanical manipulator introduced in [7], for feedback. Further research applications in the control whose first link is not actuated whereas the second joint is of underactuated systems have gone in many directions, actuated. for example, fully actuated robots where it is required that motion continues in spite of a failure of any of its Contribution. For nonlinear mechanical systems, tracking actuators. Other typical examples are the systems where its control problem is known to be more difficult than sta- desired operation mode is oscillatory such as biped walking bilization mainly for underactuated systems whose initial robots where a periodic trajectory is required to produce a conditions are close to an unstable equilibrium point. coordinated motion (see, e.g., [2]); hopping robots where The central problem in nonminimum-phase underactuated thrust, decompression, flight, and compression phases are systems, solved here, is the specification and design of output also governed by a periodic motion (see, e.g., [3]); tracking feedback inner-tracking controllers to drive the output (joint control in drive systems with backlash where usually the position) to a nontrivial reference trajectory in spite of position sensor is placed on the side of the motor instead of external disturbances. the side of the load (see, e.g., [4]and [5, page 456]); juggling The prior work on the tracking control of nonmin- systems [6]; among others. iminum-phase systems includes, among others, the results of Consolini and Tosques [11] and Berkemeier and Fearing Objective. In the present paper, we address the output [7] who developed an exact tracking control via state- tracking control problem in nonminimum-phase underactu- feedback. Wang [12] partially addressed the above problem 2 Journal of Robotics by considering the regulation problem in linear systems. A procedure is also discussed. A nonlinear H -output control unified treatment of the control of such systems via output for time varying systems is also constructed. Performance feedback can be found in [13]. In the present paper, the issues of this controller are illustrated in a simulation study nonlinear H control approach is extended for the time- in Section 4. Finally, Section 5 presents conclusions. varying nonlinear nonminimum-phase systems applied to tracking control problems for underactuated mechanical 2. Background Material on Nonlinear systems. H -Control of Time-Varying Systems 2.1. Basic Assumptions and Problem Statement. Consider a Methodology. The method we use for defining a desired nonlinear system of the form trajectory for underactuated system is based on the work of Berkemeier and Fearing [7]. The method was success- x ˙ = f (x, t) + g (x, t)w + g (x, t)u, 1 2 fully applied to derive a set of exact trajectories for the nonlinear equation which involve inverted periodic motion. z = h (x, t) + k (x, t)u, (1) 1 12 This method was selected because the desired trajectories are at least twice-differentiable satisfying the smoothness ( ) ( ) y = h x, t + k x, t w, 2 21 assumption imposed on the system for the development of H control theory (see, e.g., [14]). ∞ where x ∈ R is the state space vector, t ∈ R is the time, u ∈ m r The above problem is locally resolved within the frame- R is the control input, w ∈ R is the unknown disturbance, l p work of nonlinear H -control methods from [1, 14–16]. ∞ z ∈ R is the unknown output to be controlled, and y ∈ R is Those methods do not admit a straightforward application the only available measurement on the system. The following to the problem in question because in contrast to the assumptions are assumed to hold. standard case, a partial state stabilization (i.e., asymptotic (A1) The functions f (x, t), g (x, t), g (x, t), h (x, t), stabilization of the output of the system) is only required 1 2 1 h (x, t), k (x, t), and k (x, t) are piecewise contin- 2 12 21 provided that the complementary variables remain bounded. uous in t for all x and locally Lipschitz continuous in Their modification developed in the present paper is of x for all t. the same level of simplicity, and it follows the common practice of proper solution to corresponding differential (A2) f (0, t) = 0, h (0, t), and h (0, t) = 0for all t. 1 2 Riccati equations which is performed numerically. (A3) h (x, t)k (x, t) = 0, k (x, t) k (x, t) = I , k (x, 12 12 12 21 The aforementioned H synthesis took its origins from t)g (x, t) = 0, k (x, t)k (x, t) = I . 1 21 game-theoretic approach from Basar and Bernhard [15], and the L -gain analysis from Isidori and Astolfi [14]. It fol- These assumptions are made for technical reasons. Assump- lowed the line of reasoning, used in Orlov et al. [17], where tion (A1) guarantees the well-posedness of the above the corresponding Hamilton-Jacobi-Isaacs expressions were dynamic system, while being enforced by integrable exoge- required to be negative definite rather than semidefinite. nous inputs. Assumption (A2) ensures that the origin is an In contrast to the standard L -gain analysis from equilibrium point of the nondriven (u = 0) disturbance- Isidori and Astolfi [14]and Vander Shaft[18] the result- free (w = 0) dynamic system (1). Assumption (A3) is a ing H design procedure imposed the nonstabilizability- simplifying assumption inherited from the standard H - detectability conditions on the control systems. Under control problem. appropriate assumptions the existence of suitable solutions A causal dynamic feedback compensator of Riccati differential equations, appearing in solving the H control problem for the linearized system, was shown u = K (ξ , t),(2) to be necessary and sufficient condition for a local solution ξ = η ξ , y, t (3) of the H control problem to exist. This mean that the verification of stabilizability and detectability conditions with internal state ξ ∈ R ,issaidtobeglobally(locally) will be not required. A local solution was then derived by admissible controller if the closed-loop systems (1)-(2)are means of a certain perturbation of the Riccati equations globally (uniformly) asymptotically stable when w = 0. when these unperturbed equations had bounded positive- Given a real number γ> 0, it is said that systems (1), (2) semidefinite solutions. Thus, the local stabilizability and have L -gain less than γ if the response z, resulting from w detectability properties of the control system were ensured for initial state x(t ) = 0, ξ (t ) = 0, satisfies 0 0 by the existence of the proper solutions of the unperturbed Riccati equations, and hence the H synthesis obviated any t t 1 1 2 2 extra work on verification of these properties. z(t) dt < γ w(t) dt (4) t t 0 0 Organization of the Paper. The paper is organized as follows. for all t >t and all piecewise continuous functions w(t). 1 0 Background materials on time-varying H -control synthesis The time-varying H -control problem is to find a ∞ ∞ are presented in Section 2. The tracking control problem of globally admissible controller (2)-(3) such that L -gain a 2-DOF underactuated system and its state equations are of the closed-loop systems (1), (2), (3) is less than γ.In introduced in Section 3 while desired trajectory synthesis turn, a locally admissible controller (2), (3)issaidtobe Journal of Robotics 3 −2 T a local solution of the H -control problem if there exists specified with A(t) = A(t)+ γ B (t)B (t)P(t), such ∞ 1 a neighborhood U of the equilibrium such that inequality that the system (4) is satisfied for all t >t and all piecewise continuous 1 0 T −2 T functions w(t) for which the state trajectory of the closed- x = A − Z C C − γ PB B P (t)x(t) (10) 2 2 2 2 loop system starting from the initial point (x(t ), ξ (t )) = 0 0 is exponentially stable. (0, 0) remains in U for all t ∈ [t , t ]. 0 1 According to the time-varying bounded real lemma [17], 2.2. Local State-Space Solution. Assumptions (A1)–(A3) conditions (C1) and (C2) ensure that there exists a positive allow one to linearize the corresponding Hamilton-Jacobi- constant ε such that the system of the perturbed differential Isaacs inequalities from [1] that arise in the state feedback Riccati equations and output-injection design thereby yielding a local solution T T of the time-varying H -control problem. The subsequent −P = P (t)A(t) + A (t)P (t) + C (t)C (t) ε ε ε 1 local analysis involves the linear time-varying H -control problem for the system T T ( ) ( ) ( ) ( ) ( ) ( ) + P t B t B t − B t B t P t + εI , ε 1 2 ε 1 2 x ˙ = A(t)x + B (t)w + B (t)u, 1 2 (11) z = C (t)x + D (t)u, (5) 1 12 T T Z = A(t)Z (t) + ZA (t) + B (t)B (t) + Z (t) ε ε 1 1 ε y = C (t)x + D (t)w, 2 21 T T × P(t)B (t)B (t)P(t) − C (t)C (t) Z (t) + εI , 2 2 ε where 2 2 ∂f (12) A(t) = (0, t), B (t) = g (0, t), B (t) = g (0, t), 1 1 2 2 ∂x has a unique positive definite symmetric solution (P (t), ∂h C (t) = (0, t), D (t) = k (0, t), 1 12 12 Z (t)) for each ε ∈ (0, ε )where A(t) = A(t)+ ε 0 ∂x −2 γ B (t)B (t)P (t). 1 ε ∂h Differential equations (11)and (12) are subsequently C (t) = (0, t), D (t) = k (0, t). 2 21 21 ∂x utilized to derive a local solution of the nonlinear H -control (6) problem for (1). The following resuls is extracted from [1]. Such a problem is now well understood if the linear system Theorem 1. Let conditions (C1) and (C2) be satisfied, and (5) is stabilizable and detectable from u and y,respectively. let (P (t), Z (t)) be the corresponding positive solution of (11), ε ε Under these assumptions, the following conditions are (12) under some ε> 0. Then the output feedback necessary and sufficient for a solution to exist (see, e.g., [16]). T T ξ = f (ξ , t) + g (ξ , t)g (ξ , t) − g (ξ , t)g (ξ , t) 1 2 1 2 (C1) There exists a bounded positive semidefinite sym- metric solution of the equation (13) × P (t)ξ + Z (t)C (t) y − h (ξ , t) , ε ε 2 T T 2 −P(t) = P(t)A(t) + A (t)P(t) + C (t)C (t) 1 1 u =−g (ξ , t)P (t)ξ (7) 2 T T + P(t) B (t)B (t) − B (t)B (t) P(t), 1 2 1 2 is a local solution of the H -control problem. such that the system In what follows, Theorem 1 is used to design an H T −2 T tracking controller for the underactuated system. x ˙ = A − B B − γ B B P (t)x(t) (8) 2 1 2 1 is exponentially stable. ( Throughout, a time- 3. H -Control of Underactuated System dependent n×n-matrix P(t) is positive semidefinite if and only if x P(t)x ≥ 0for all n-vectors x and all time 3.1. Problem Statement. Consider the equation of motion of instants t whereas P(t) is positive definite if and only an underactuated mechanical system given by the Lagrange T T if x P(t)x ≥ mx x for all x and t, and some constant equation m> 0. Respectively, P(t) is bounded if and only if P(t)≤ m for all t and some constant m > 0. ) 0 0 M q q ¨ + N q, q ˙ = Bτ + w (t), (14) (C2) There exists a bounded positive semidefinite sym- where q = [q , q ] ∈ R is a vector of generalized 1 2 metric solution to the equation coordinates where q and q are the unactuated and actuated 1 2 T T Z (t) = A(t)Z (t) + Z (t)A (t) + B (t)B (t) + Z (t) 1 joints, respectively; τ ∈ R is the vector of applied joint torques; B = [0, 1] is the input matrix that maps the (9) T T torque input τ to the joint of coordinates space; w (t) ∈ ( ) ( ) ( ) ( ) ( ) ( ) ( ) x × P t B t B t P t − C t C t Z t 2 2 2 2 γ 2 R is the unknown disturbance vector to account for 4 Journal of Robotics 0.5 destabilizing model discrepancies due to hard-to-model nonlinear phenomena such as friction and backlash, t ∈ R 2×2 0 is the time; M(q) ∈ R is the symmetric positive-definite inertia matrix; N (q, q ˙ ) = [N (q, q ˙ ), N (q, q ˙ )] ∈ R is the 1 2 −0.5 vector that contains the Coriolis, centrifugal, and gravity torques. Appendix A presents the dynamic model of the −1 double pendulum. 0 5 10 15 20 The control objective is to design a nonlinear H tracking ∞ Time (s) controller that ensures (a) lim q(t) − q (t) = 0 (15) t→∞ to be achieved asymptotically, while also attenuates the influence of external disturbances. Here, q (t) ∈ R is a continuously differentiable desired trajectory. −1 0 5 10 15 20 3.2. The Desired Trajectory. We point out that the present Time (s) formulation is different from typical formulation of output tracking and regulation [1, 19], where the set point or the (b) reference trajectory is a priori given because underactuated Figure 1: Plot of desired trajectories for Acrobot by selecting several systems are not feedback or input-state linearizables due to values of φ. its complexity. Therefore, special attention is required in the selection of the desired trajectory for the system under study. There are a few procedures to find desired trajectories that can be interpreted as the zero dynamics of the system for underactuated systems in literature [7, 8, 10, 20, 21], (16) with respect to the output y (t). Time evolution of the and under reasonable hypotheses all of them can be used desired trajectory is illustrated in Figure 1 where the value of to obtain a desired trajectory. The methodology from [7] φ is modified along the time: is used here, where a set of exact trajectories is derived for the nonlinear equation of motion which involves inverted 0, if t< 5, periodic motion. To this end, let us consider the desired ⎪ trajectory which is solution of φ = 0.025, if 5 ≤ t< 12, (20) ⎡ ⎤ ⎡ ⎤ ⎪ 0.1, if 12 ≤ t ≤ 20, q ˙ d d ⎣ ⎦ ⎣ ⎦ =       , (16) −1 dt q ˙ M q Bτ q , q ˙ − N q , q ˙ d d d d d d where t ∈ R is given in seconds. Notice that frequency and amplitude of oscillations change according to variations in φ. 2 2 where q (t) ∈ R , q ˙ (t) ∈ R are the desired joint positions d d Figure 2 shows the profile of the frequency and amplitude of and velocities, respectively, and oscillations for several values of φ. 2M q − M q 22 d 12 d ˙     ˙ τ = N q , q − N q , q (17) 3.3. The Task. Our objective is to design a controller of the 2 d d 1 d d 2M q − M q 12 d 11 d form is the control input that makes the desired virtual output τ = τ q , q + u (21) d d y (t) = 2q (t) + q (t) − φ (18) d d d 1 2 with internal state ξ (t) ∈ R , that ensures (15). Thus, the controller to be constructed consists of the trajectory remains at zero for all t ≥ 0 when y (t) starts at y (0) = d d compensator (17) and a disturbance attenuator u given y ˙ (0) = 0, φ is a constant parameter that parameter- in (2), (3) internally stabilizing the closed-loop system izes the equilibrium manifold of the pendulum, and the around the desired trajectory. In the sequel, we confine our oscillations given by (16)–(18) are around this manifold. investigation to the H tracking problem, where Throughout, we confine our research interest in desired (1) the output to be controlled is given by oscillations around the upright position of the pendulum ⎡ ⎤ ⎡ ⎤ which correspond to the more difficult case due that the 0 1 open-loop system has an unstable zero dynamics. Toward ⎣ ⎦ ⎣ ⎦ z = ρ + u(t) (22) this end, we choose φ = π for all t ≥ 0in(18). It was shown 2q + q − π 0 1 2 in [7] that (16)and (17) generate a set of exact periodic with a positive weight coefficient ρ; trajectories given by (2) the joint position vector q(t) ∈ R is the only c sin q + c sin φ − q 4 d 5 d 1 1 available measurement, and this measurement is q ¨ = , (19) d1 c − c 1 2 corrupted by the error vector w (t) ∈ R , that is, q (rad) d1 q (rad) d2 Journal of Robotics 5 1 4 φ = 0.75 0.9 φ = 0.5 0.8 φ = 0.25 0.7 0.6 0.5 φ = 0.1 0.4 −1 φ = 0.075 0.3 φ = 0.05 −2 0.2 φ = 0.025 −3 φ = 0 0.1 −4 −0.1 −0.05 0 0.05 0.10.15 3.54 4.55 5.56 6.5 q (rad) Frequency ω (rad/s) 1 (a) Figure 2: Profile of the frequency and amplitude of oscillations for several values of φ. 8 −2 τ 1 −4 l q 1 1 −6 −8 33.13.23.33.4 q (rad) (b) Figure 4: Phase portrait of the first joint trajectory and desired Figure 3: Schematic diagram of the acrobot where l and l denote 1 2 trajectory (+) for the unperturbed case (a) and perturbed case (b). the length of the nonactuated and actuated links, respectively; and m and m are the masses of each link. 1 2 y = q(t) + w (t). (23) that let us rewrite the system (14), the output to be controlled (22), and the output (23) in terms of the state vector The H control problem in question is thus stated as x: follows. Given the system representation (14)–(23), the desired trajectory q (t) ∈ R ,and arealnumber γ> 0, it is required to find (if any) a causal dynamic feedback ⎡ ⎤ ⎡ ⎤ x x 1 2 controller (2), (3) such that the undisturbed closed-loop d ⎣ ⎦ ⎣ ⎦ system is uniformly asymptotically stable around the origin, −1 dt x M x + q N x + q , x + q ˙ − q ¨ 2 1 d 1 d 2 d d and its L -gain is locally less than γ, that is, inequality (4)is ⎡ ⎤ ⎡ ⎤ satisfied for all t >t and all piecewise continuous functions 1 0 0 0 T ⎣ ⎦ ⎣ ⎦ +   w +   τ , w(t) = [w (t), w (t)] for which the corresponding state x o −1 −1 M x + q M x + q B 1 d 1 d trajectory of the closed-loop system, initialized at the origin, (24) remains in some neighborhood of this point. ⎡ ⎤ ⎡ ⎤ 0 1 ⎣ ⎦ ⎣ ⎦ z = ρ + u(t), 3.4. H Synthesis. To begin with, let us introduce the 2x + x +2q + q − π 0 1 1 d1 d2 1 2 (25) state deviation vector x = (x , x ) ∈ R where x = 1 2 1 T T (q − q , q − q ) and x = (q ˙ − q ˙ , q ˙ − q ˙ ) .After ( ) ( ) 1 d1 2 d2 2 1 d1 2 d2 y = x + q t + w t . 1 d o Amplitude A (rad) dq /dt (rad/s) dq /dt (rad/s) 2 6 Journal of Robotics 0.6 obviates an extra work (formidable in the nonlinear case) on verification of these properties. 0.4 0.2 4. Simulation Results The controller performance was studied in simulation by −0.2 02 4 6 8 10 applying the exposed ideas to the Acrobot, depicted in Time (s) Figure 3, which is a two-link planar robot with no actuator at the shoulder (link 1) and actuator at the elbow (link 2). In (a) the simulation, performed with MATLAB, the Acrobot was required to move from [q (0), q (0)] = [−0.07, 3.3] to the 1 2 desired trajectory q (t) ∈ R and φ = π . The initial velocity 2 4 q(0) ∈ R and the initial compensator state ξ (0) ∈ R were 3.5 set to zero for all the simulations. The matrices M(q)and N (q, q ˙ ) for the Acrobot are given in Appendix A.Weseek for orbital stabilization of the unactuated link q around the 2.5 equilibrium point q = (0, π ). 02 4 6 8 10 The control goal was achieved by implementing the Time (s) nonlinear H controller with a weight parameter ρ = 1 (b) on the Acrobot. By iterating on γ, we found the infimal achievable level γ 250. However, in the subsequent Figure 5: Time evolution of the output (continuous line) following simulations γ = 2000 was selected to avoid an undesirable the desired trajectory (dashed line) under perturbed torques. high-gain controller design that would appear for a value of γ close to the optimum. With γ = 2000 we obtained that for ε = 0.1 the corresponding differential Riccati equations (11)-(12)with Clearly, the above H tracking control problem is nothing ⎡ ⎤ else than a standard nonlinear H control problem from [1] ∞ 0010 ⎢ ⎥ stated for a time-varying nonlinear system (1)specifiedas ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A(t) = , ⎢ ⎥ ⎢ A (t) A (t) A (t) A (t)⎥ 31 32 33 34 ⎡ ⎤ ⎣ ⎦ A (t) A (t) A (t) A (t) ⎣ ⎦ 41 42 43 44 f (x, t) = , −1 M x + q N x + q , x + q ˙ − q ¨ 1 d 1 d 2 d d ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 0 2×2 2×2 2×1 ⎣ ⎦ ⎣ ⎦ 0 0 B (t) =   , B (t) =   , 2×2 2×2 1 2 −1 −1 ⎣ ⎦ g (x, t) =   , M q 0 M q 1 d 2×2 d −1 M x + q 0 1 d 2×2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0000 1000 ⎣ ⎦ ⎣ ⎦ 0 ( ) ( ) 2×1 C t = ρ , C t = , 1 2 ⎣ ⎦ g (x, t) = , 2100 0100 −1 M x + q B 1 d (26) ⎡ ⎤ ⎡ ⎤ 1 ⎣ ⎦ 0 0 I D (t) = , D (t) = 12 22 2×2 2 ⎣ ⎦ h (x, t) = ρ , 0 2x + x +2q + q − π 1 1 d1 d2 1 2 (27) ⎡ ⎤ have positive-definite solutions. These solutions can be ⎣ ⎦ h (x, t) = x + q , k (x, t) = , 2 1 d 12 numerically found with MATLAB. Matrix A(t)isgiven in Appendix B. It should be pointed out that the constant k (x, t) = 0 I . 2×2 2 φ = π does not appear in C (t) due to straightforward 21 1 calculation of (6), but it is definitely required in (22)to improve the selection of γ which affects inequality (4)thus Now by applying Theorem 1 to system (1)thusspecified, we avoiding the synthesis of a high-gain controllers. Resulting derive a local solution of the H tracking control problem. trajectories is depicted in Figure 4. This figure demonstrates Thus, the output feedback controller (13), specified that the H controller does asymptotically stabilize the according to (26), locally solves the H tracking control system motion around the desired trajectory. In addition, the problem (4)–(24). Stabilizability and detectability properties H controller was successfully applied to the Acrobot under of the control systems are ensured by the existence of external disturbances the proper solutions of the unperturbed differential Riccati equations, and hence the corresponding synthesis procedure w (t) = b cos(0.1t) exp(−2t), i = 1, 2, (28) x i q (rad) q (rad) 1 2 Journal of Robotics 7 150 200 0 0 05 10 05 10 Time (s) Time (s) (a) (b) 150 150 100 100 50 50 0 0 05 10 05 10 Time (s) Time (s) (c) (d) 4×4 Figure 6: Time evolution of the determinants of the principal minors of the matrix P ∈ R P , P , P , P . m1 m2 m3 m4 −3 −3 where b = 1 × 10 [N · m] and b = 2 × 10 [N · m] are These Figures highlight that matrices P(t)and Z (t), which 1 2 the disturbance levels at first and second joints, respectively. are solution of (11)and (12), respectively, are bounded and Resulting trajectories are depicted in Figure 5. Figures 6 and positive definite for all t ≥ 0. 7 show the time evolution of the determinant of minors of 4×4 4×4 matrices P(t) ∈ R and Z (t) ∈ R denoted as 5. Conclusions ⎛ ⎞ P (t) P (t) 11 12 The output feedback Nonlinear H tracking control problem ⎝ ⎠ P (t) = P (t), P (t) = , m1 11 m2 is locally solved for an underactuaded mechanical system. P (t) P (t) 12 22 The desired periodic orbit is centered at the upright position ⎛ ⎞ where the open-loop plant becomes a nonminimum-phase P (t) P (t) P (t) 11 12 13 ⎜ ⎟ system. The developed controller drives the trajectories of ⎜ ⎟ P (t) = P (t) P (t) P (t) , 12 22 23 m3 ⎝ ⎠ the robot into a set of inverted exact desired trajectories P (t) P (t) P (t) governed by its zero dynamics. Simulation studies, made 31 32 33 (29) for the Acrobot, showed the effectiveness of the controller. ⎛ ⎞ Z (t) Z (t) 11 12 The design of methods to generate reference trajectories ⎝ ⎠ Z (t) = Z (t), Z (t) = , m1 11 m2 evolving more frequencies and amplitudes in the upright Z (t) Z (t) 12 22 position is in progress, and few results have been published ⎛ ⎞ for double-pendulums in [20, 22]. In future work there are Z (t) Z (t) Z (t) 11 12 13 ⎜ ⎟ two extensions of the result of the paper. First, one would like ⎜ ⎟ Z (t) Z (t) Z (t) Z (t) = . m3 12 22 23 ⎝ ⎠ to synthesize the H control taking into account reference ( ) ( ) ( ) trajectories derived from alternative methods. The other Z t Z t Z t 31 32 33 Det. P Det. P m3 m1 Det. P Det. P m2 8 Journal of Robotics 30 800 0 0 05 10 05 10 Time (s) Time (s) (a) (b) 30 30 20 20 10 10 0 0 05 10 05 10 Time (s) Time (s) (c) (d) 4×4 Figure 7: Time evolution of the determinants of the principal minors of the matrix Z ∈ R Z , Z , Z , Z . m1 m2 m3 m4 Table 1: Parametervaluesfor theAcrobot. with c c c c c 1 2 3 4 5 0.0043 0.0051 0.0034 0.0494 0.038 M q = c + c − 2c cos q , 11 1 2 3 2 M q = c − c cos q , 12 2 3 2 M q = c , would also like to extend the result of the paper for the 22 2 (A.2) nonsmooth case. N q, q ˙ = c sin q q ˙ q ˙ + c q ˙ q ˙ + q ˙ sin q 1 3 2 1 2 3 2 1 2 2 − c sin q + c sin q + q , 4 1 5 1 2 Appendices N q, q ˙ =−c sin q q ˙ + c sin q + q , 2 3 2 5 1 2 A. Dynamic Model of Acrobot The equation motion of Acrobot, described by (14), was where the values of c (i = 1,... ,5), given in Table 1,were specified by applying the Euler-Lagrange formulation [23] taken from the experimental Acrobot provided in [7]. where ⎡ ⎤ ⎡ ⎤ M q M q N q, q ˙ 11 12   1 B. Matrix A(t) ⎣ ⎦ ⎣ ⎦ M q =     , N q, q = M q M q N q, q ˙ 12 22 2 In this appendix we provide the computed matrix A(t)for (A.1) the Acrobot which was used in the solution of differential Det. Z m1 Det. Z m3 Det. Z m2 Det. Z Journal of Robotics 9 Riccati equations (11)and (12): References ⎡ ⎤ [1] L. T. Aguilar, Y. Orlov, and L. Acho, “Nonlinear H -control of nonsmooth time-varying systems with application to friction ⎢ ⎥ ⎢ ⎥ mechanical manipulators,” Automatica,vol. 39, no.9,pp. ⎢ ⎥ ⎢ ⎥ ( ) 1531–1542, 2003. A t = ,(B.3) ⎢ ⎥ ⎢ A (t) A (t) A (t) A (t)⎥ 31 32 33 34 [2] E. R. Westervelt, J. Grizzle, C. Chevallereau, J. Cho, and B. ⎣ ⎦ Morris, Feedback Control of Dynamic Bipedal Robot Locomo- ( ) ( ) ( ) ( ) A t A t A t A t 41 42 43 44 tion, Taylor & Francis/CRC, London, UK, 2007. [3] R. T. M’Closkey, J. W. Burdick, and A. F. Vakakis, “On the where periodic motions of simple hopping robots,” in Proceedings of the IEEE International Conference on Systems, Man and −1 Cybernetics (ICSMC ’90), pp. 771–777, Los Angeles, Calif, A (t) = c ΔM(q ) c cos q + q − c cos q 31 2 d 5 d1 d2 4 d1 USA, November 1990. −1 [4] L. T. Aguilar, Y. Orlov, J. C. Cadiou, and R. Merzouki, − ΔM(q ) c cos q + q M q , d 5 d1 d2 12 d “Nonlinear H -output regulation of a nonminimum phase −1 servomechanism with backlash,” Journal of Dynamic Systems, A (t) = c ΔM(q ) c cos q q ˙ q ˙ 32 2 d 3 d2 d1 d2 Measurement and Control, vol. 129, no. 4, pp. 544–549, 2007. [5] B. Brogliato, Nonsmooth Mechanics: Models, Dynamics and ˙ ˙ ˙ + c q q + q cos q + c cos q + q 3 d2 d1 d2 d2 5 d1 d2 Control, Springer, London, UK, 2nd edition, 1999. −2 [6] B.Brogliato andA.Z.Rio,“On thecontrol of complementary- + −2c c ΔM(q ) sin q cos q N q , q ˙ 2 3 d d2 d2 1 d d slackness juggling mechanical systems,” IEEE Transactions on −1 Automatic Control, vol. 45, no. 2, pp. 235–246, 2000. − c ΔM(q ) N q , q ˙ sin q 3 d 2 d d d2 [7] M. D. Berkemeier and R. S. Fearing, “Tracking fast inverted −2 trajectories of the underactuated Acrobot,” IEEE Transactions +2c ΔM(q ) sin q cos q M q N q , q ˙ 3 d d2 d2 12 d 2 d d on Robotics and Automation, vol. 15, no. 4, pp. 740–750, 1999. −1 [8] J. W. Grizzle, C. H. Moog, and C. Chevallereau, “Nonlinear − ΔM(q ) −c cos q q ˙ + c cos q + q d 3 d2 d1 5 d1 d2 control of mechanical systems with an unactuated cyclic × M q , variable,” IEEE Transactions on Automatic Control, vol. 50, no. 12 d 5, pp. 559–576, 2005. −1 [9] Y.Orlov,L.T.Aguilar,and L. Acho,“Modelorbit robust A (t) = 2c c ΔM(q ) sin q q ˙ 33 2 3 d d2 d2 stabilization (MORS) of pendubot with application to swing −1 up control,” in Proceedings of the 44th IEEE Conference on +2c ΔM(q ) M q sin q q ˙ , 3 d 12 d d2 d1 Decision and Control and European Control Conference (CDC- −1 ECC ’05), vol. 6, pp. 6164–6169, Seville, Spain, December ( ) ˙ ˙ A t = 2c c ΔM(q ) sin q q +sin q q , 34 2 3 d d2 d1 d2 d2 −1 [10] A. Shiriaev, J. W. Perram, and C. Canudas-de-Wit, “Construc- A (t) = ΔM(q ) c cos q + q M q 41 d 5 d1 d2 11 d tive tool for orbital stabilization of underactuated nonlinear −1 systems: virtual constraints approach,” IEEE Transactions on − ΔM(q ) c cos q + q − c cos q d 5 d1 d2 4 d1 Automatic Control, vol. 50, no. 8, pp. 1164–1176, 2005. [11] L. Consolini and M. Tosques, “A morphing method for exact × M q , 12 d tracking control of nonminimum phase non linear systems,” −1 −1 in Proceedings of the 46th IEEE Conference on Decision and A (t) = 2c ΔM q N q , q sin q + ΔM(q ) 42 3 d 2 d d d2 d Control (CDC ’07), pp. 5240–5245, New Orleans, La, USA, 2 December 2007. × c cos q + q − c cos q q ˙ M q 5 d1 d2 3 d2 11 d d1 [12] W. Wang, “Output feedback control of unstable non- −2 minimum phase systems,” in Proceedings of the 25th Chinese −2c ΔM q M q N q , q ˙ sin q cos q 3 d 11 d 2 d d d2 d2 Control Conference (CHICC ’06), pp. 781–784, Harbin, China, August 2006. −1 −1 − c ΔM(q ) N q , q ˙ sin q − ΔM(q ) 3 d 1 d d d2 d [13] A. Isidori, “A tool for semiglobal stabilization of uncertain non-minimum-phase nonlinear systems via output feedback,” × 2c cos q q ˙ q ˙ + c cos q q ˙ 3 d2 d1 d2 3 d2 d2 IEEE Transactions on Automatic Control, vol. 45, no. 10, pp. 1817–1827, 2000. + c cos q + q M q [14] A. Isidori and A. Astolfi, “Disturbance attenuation and H - 5 d1 d2 12 d ∞ control via measurement feedback in nonlinear systems,” IEEE −2 Transactions on Automatic Control, vol. 37, no. 9, pp. 1283– +2c ΔM(q ) M q N q , q sin q cos q , 3 d 12 d 1 d d d2 d2 1293, 1992. −1 [15] T. Basar and P. Bernhard, H -Optimal Control and Related A (t) =−2c ΔM(q ) M q + M q sin q q ˙ , 43 3 d 11 d 12 d d2 d1 Minimax Design Problems: A Dynamic Game Approach, −1 Birkhauser ¨ , Boston, Mass, USA, 1990. A (t) =−2c ΔM(q ) q ˙ + q ˙ sin q M q , 44 3 d d1 d2 d2 12 d [16] R. Ravi,K.M.Nagpal, andP.P.Khargonekar,“H control (B.4) of linear time-varying systems: a state-space approach,” SIAM Journal on Control and Optimization, vol. 29, no. 6, pp. 1394– where ΔM(q ) = M (q )M (q ) − M (q ). 1413, 1991. d 11 d 22 d d 12 10 Journal of Robotics [17] Y. Orlov, L. Acho, and V. Solis, “Nonlinear H -control of time- varying systems,” in Proceedings of the 38th IEEE Conference on Decision and Control (CDC ’99), vol. 4, pp. 3764–3769, Phoenix, Ariz, USA, December 1999. [18] A. J. van der Shaft, “L -gain analysis of nonlinear systems and nonlinear state feedback H control,” IEEE Transactions on Automatic Control, vol. 37, no. 6, pp. 770–784, 1992. [19] Y. Orlov and L. T. Aguilar, “Non-smooth H -position control of mechanical manipulators with frictional joints,” Interna- tional Journal of Control, vol. 77, no. 11, pp. 1062–1069, 2004. [20] L. T. Aguilar, I. Boiko, L. Fridman, and R. Iriarte, “Output excitation via continuous sliding-modes to generate periodic motion in underactuated systems,” in Proceedings of the 45th IEEE Conference on Decision and Control (CDC ’06), pp. 1629– 1634, San Diego, Calif, USA, December 2006. [21] L. Freidovich, A. Robertsson, A. Shiriaev, and R. Johansson, “Periodic motions of the Pendubot via virtual holonomic constraints: theory and experiments,” Automatica, vol. 44, no. 3, pp. 785–791, 2008. [22] L. T. Aguilar, I. Boiko, L. Fridman, and R. Iriarte, “Periodic motion of underactuated mechanical systems self-generated by variable structure controllers: design and experiments,” in Proceedings of the European Control Conference (ECC ’07),pp. 3796–3801, Kos, Greece, July 2007. [23] J. Craig, Introduction to Robotics: Mechanics and Control, Addison-Wesley, Reading, Mass, USA, 1989. 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Output Feedback Nonlinear

Aguilar, Luis T.
Journal of Robotics , Volume 2009 – Jun 15, 2009
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Hindawi Publishing Corporation
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Copyright © 2009 Luis T. Aguilar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1687-9600
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10.1155/2009/718728
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Hindawi Publishing Corporation Journal of Robotics Volume 2009, Article ID 718728, 10 pages doi:10.1155/2009/718728 Research Article Output Feedback Nonlinear H -Tracking Control of a Nonminimum-Phase 2-DOF Underactuated Mechanical System Luis T. Aguilar Centro de Investigacio ´n yDesarrollodeTecnolog´ıa Digital, Instituto Polit´ecnico Nacional, Avenida del Parque 1310 Mesa de Otay, 22510 Tijuana, BC, Mexico Correspondence should be addressed to Luis T. Aguilar, luis.aguilar@ieee.org Received 18 September 2008; Revised 22 January 2009; Accepted 1 April 2009 Recommended by Warren Dixon Nonlinear H synthesis is developed to solve the tracking control problem restricted to a two degrees-of-freedom (DOF) underactuated mechanical manipulator where position measurements are the only available information for feedback. A local H controller is derived by means of a certain perturbation of the differential Riccati equations, appearing in solving the H ∞ ∞ control problem for the linearized system. Stabilizability and detectability properties of the control system are thus ensured by the existence of the proper solutions of the unperturbed differential Riccati equations, and hence the proposed synthesis procedure obviates an extra verification work of these properties. Due to the nature of the approach, the resulting controller additionally yields the desired robustness properties against unknown but bounded external disturbances. The desired trajectory is centered at the upright position where the manipulator becomes a nonminimum-phase system. Simulation results made for a double pendulum show the effectiveness of the proposed controller. Copyright © 2009 Luis T. Aguilar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction ated mechanical system. Representative works in this topic include orbital stabilization of underactuated systems by The focus of this paper is to solve the tracking control means of reference models as generator of limit cycles (see, problem for a 2-DOF underactuated mechanism via non- e.g., [7–10]). In particular, this paper is devoted to the linear H -control for time-varying systems [1] where joint solution of a periodic balancing problem for a two-link position measurements are the only available information underactuated mechanical manipulator introduced in [7], for feedback. Further research applications in the control whose first link is not actuated whereas the second joint is of underactuated systems have gone in many directions, actuated. for example, fully actuated robots where it is required that motion continues in spite of a failure of any of its Contribution. For nonlinear mechanical systems, tracking actuators. Other typical examples are the systems where its control problem is known to be more difficult than sta- desired operation mode is oscillatory such as biped walking bilization mainly for underactuated systems whose initial robots where a periodic trajectory is required to produce a conditions are close to an unstable equilibrium point. coordinated motion (see, e.g., [2]); hopping robots where The central problem in nonminimum-phase underactuated thrust, decompression, flight, and compression phases are systems, solved here, is the specification and design of output also governed by a periodic motion (see, e.g., [3]); tracking feedback inner-tracking controllers to drive the output (joint control in drive systems with backlash where usually the position) to a nontrivial reference trajectory in spite of position sensor is placed on the side of the motor instead of external disturbances. the side of the load (see, e.g., [4]and [5, page 456]); juggling The prior work on the tracking control of nonmin- systems [6]; among others. iminum-phase systems includes, among others, the results of Consolini and Tosques [11] and Berkemeier and Fearing Objective. In the present paper, we address the output [7] who developed an exact tracking control via state- tracking control problem in nonminimum-phase underactu- feedback. Wang [12] partially addressed the above problem 2 Journal of Robotics by considering the regulation problem in linear systems. A procedure is also discussed. A nonlinear H -output control unified treatment of the control of such systems via output for time varying systems is also constructed. Performance feedback can be found in [13]. In the present paper, the issues of this controller are illustrated in a simulation study nonlinear H control approach is extended for the time- in Section 4. Finally, Section 5 presents conclusions. varying nonlinear nonminimum-phase systems applied to tracking control problems for underactuated mechanical 2. Background Material on Nonlinear systems. H -Control of Time-Varying Systems 2.1. Basic Assumptions and Problem Statement. Consider a Methodology. The method we use for defining a desired nonlinear system of the form trajectory for underactuated system is based on the work of Berkemeier and Fearing [7]. The method was success- x ˙ = f (x, t) + g (x, t)w + g (x, t)u, 1 2 fully applied to derive a set of exact trajectories for the nonlinear equation which involve inverted periodic motion. z = h (x, t) + k (x, t)u, (1) 1 12 This method was selected because the desired trajectories are at least twice-differentiable satisfying the smoothness ( ) ( ) y = h x, t + k x, t w, 2 21 assumption imposed on the system for the development of H control theory (see, e.g., [14]). ∞ where x ∈ R is the state space vector, t ∈ R is the time, u ∈ m r The above problem is locally resolved within the frame- R is the control input, w ∈ R is the unknown disturbance, l p work of nonlinear H -control methods from [1, 14–16]. ∞ z ∈ R is the unknown output to be controlled, and y ∈ R is Those methods do not admit a straightforward application the only available measurement on the system. The following to the problem in question because in contrast to the assumptions are assumed to hold. standard case, a partial state stabilization (i.e., asymptotic (A1) The functions f (x, t), g (x, t), g (x, t), h (x, t), stabilization of the output of the system) is only required 1 2 1 h (x, t), k (x, t), and k (x, t) are piecewise contin- 2 12 21 provided that the complementary variables remain bounded. uous in t for all x and locally Lipschitz continuous in Their modification developed in the present paper is of x for all t. the same level of simplicity, and it follows the common practice of proper solution to corresponding differential (A2) f (0, t) = 0, h (0, t), and h (0, t) = 0for all t. 1 2 Riccati equations which is performed numerically. (A3) h (x, t)k (x, t) = 0, k (x, t) k (x, t) = I , k (x, 12 12 12 21 The aforementioned H synthesis took its origins from t)g (x, t) = 0, k (x, t)k (x, t) = I . 1 21 game-theoretic approach from Basar and Bernhard [15], and the L -gain analysis from Isidori and Astolfi [14]. It fol- These assumptions are made for technical reasons. Assump- lowed the line of reasoning, used in Orlov et al. [17], where tion (A1) guarantees the well-posedness of the above the corresponding Hamilton-Jacobi-Isaacs expressions were dynamic system, while being enforced by integrable exoge- required to be negative definite rather than semidefinite. nous inputs. Assumption (A2) ensures that the origin is an In contrast to the standard L -gain analysis from equilibrium point of the nondriven (u = 0) disturbance- Isidori and Astolfi [14]and Vander Shaft[18] the result- free (w = 0) dynamic system (1). Assumption (A3) is a ing H design procedure imposed the nonstabilizability- simplifying assumption inherited from the standard H - detectability conditions on the control systems. Under control problem. appropriate assumptions the existence of suitable solutions A causal dynamic feedback compensator of Riccati differential equations, appearing in solving the H control problem for the linearized system, was shown u = K (ξ , t),(2) to be necessary and sufficient condition for a local solution ξ = η ξ , y, t (3) of the H control problem to exist. This mean that the verification of stabilizability and detectability conditions with internal state ξ ∈ R ,issaidtobeglobally(locally) will be not required. A local solution was then derived by admissible controller if the closed-loop systems (1)-(2)are means of a certain perturbation of the Riccati equations globally (uniformly) asymptotically stable when w = 0. when these unperturbed equations had bounded positive- Given a real number γ> 0, it is said that systems (1), (2) semidefinite solutions. Thus, the local stabilizability and have L -gain less than γ if the response z, resulting from w detectability properties of the control system were ensured for initial state x(t ) = 0, ξ (t ) = 0, satisfies 0 0 by the existence of the proper solutions of the unperturbed Riccati equations, and hence the H synthesis obviated any t t 1 1 2 2 extra work on verification of these properties. z(t) dt < γ w(t) dt (4) t t 0 0 Organization of the Paper. The paper is organized as follows. for all t >t and all piecewise continuous functions w(t). 1 0 Background materials on time-varying H -control synthesis The time-varying H -control problem is to find a ∞ ∞ are presented in Section 2. The tracking control problem of globally admissible controller (2)-(3) such that L -gain a 2-DOF underactuated system and its state equations are of the closed-loop systems (1), (2), (3) is less than γ.In introduced in Section 3 while desired trajectory synthesis turn, a locally admissible controller (2), (3)issaidtobe Journal of Robotics 3 −2 T a local solution of the H -control problem if there exists specified with A(t) = A(t)+ γ B (t)B (t)P(t), such ∞ 1 a neighborhood U of the equilibrium such that inequality that the system (4) is satisfied for all t >t and all piecewise continuous 1 0 T −2 T functions w(t) for which the state trajectory of the closed- x = A − Z C C − γ PB B P (t)x(t) (10) 2 2 2 2 loop system starting from the initial point (x(t ), ξ (t )) = 0 0 is exponentially stable. (0, 0) remains in U for all t ∈ [t , t ]. 0 1 According to the time-varying bounded real lemma [17], 2.2. Local State-Space Solution. Assumptions (A1)–(A3) conditions (C1) and (C2) ensure that there exists a positive allow one to linearize the corresponding Hamilton-Jacobi- constant ε such that the system of the perturbed differential Isaacs inequalities from [1] that arise in the state feedback Riccati equations and output-injection design thereby yielding a local solution T T of the time-varying H -control problem. The subsequent −P = P (t)A(t) + A (t)P (t) + C (t)C (t) ε ε ε 1 local analysis involves the linear time-varying H -control problem for the system T T ( ) ( ) ( ) ( ) ( ) ( ) + P t B t B t − B t B t P t + εI , ε 1 2 ε 1 2 x ˙ = A(t)x + B (t)w + B (t)u, 1 2 (11) z = C (t)x + D (t)u, (5) 1 12 T T Z = A(t)Z (t) + ZA (t) + B (t)B (t) + Z (t) ε ε 1 1 ε y = C (t)x + D (t)w, 2 21 T T × P(t)B (t)B (t)P(t) − C (t)C (t) Z (t) + εI , 2 2 ε where 2 2 ∂f (12) A(t) = (0, t), B (t) = g (0, t), B (t) = g (0, t), 1 1 2 2 ∂x has a unique positive definite symmetric solution (P (t), ∂h C (t) = (0, t), D (t) = k (0, t), 1 12 12 Z (t)) for each ε ∈ (0, ε )where A(t) = A(t)+ ε 0 ∂x −2 γ B (t)B (t)P (t). 1 ε ∂h Differential equations (11)and (12) are subsequently C (t) = (0, t), D (t) = k (0, t). 2 21 21 ∂x utilized to derive a local solution of the nonlinear H -control (6) problem for (1). The following resuls is extracted from [1]. Such a problem is now well understood if the linear system Theorem 1. Let conditions (C1) and (C2) be satisfied, and (5) is stabilizable and detectable from u and y,respectively. let (P (t), Z (t)) be the corresponding positive solution of (11), ε ε Under these assumptions, the following conditions are (12) under some ε> 0. Then the output feedback necessary and sufficient for a solution to exist (see, e.g., [16]). T T ξ = f (ξ , t) + g (ξ , t)g (ξ , t) − g (ξ , t)g (ξ , t) 1 2 1 2 (C1) There exists a bounded positive semidefinite sym- metric solution of the equation (13) × P (t)ξ + Z (t)C (t) y − h (ξ , t) , ε ε 2 T T 2 −P(t) = P(t)A(t) + A (t)P(t) + C (t)C (t) 1 1 u =−g (ξ , t)P (t)ξ (7) 2 T T + P(t) B (t)B (t) − B (t)B (t) P(t), 1 2 1 2 is a local solution of the H -control problem. such that the system In what follows, Theorem 1 is used to design an H T −2 T tracking controller for the underactuated system. x ˙ = A − B B − γ B B P (t)x(t) (8) 2 1 2 1 is exponentially stable. ( Throughout, a time- 3. H -Control of Underactuated System dependent n×n-matrix P(t) is positive semidefinite if and only if x P(t)x ≥ 0for all n-vectors x and all time 3.1. Problem Statement. Consider the equation of motion of instants t whereas P(t) is positive definite if and only an underactuated mechanical system given by the Lagrange T T if x P(t)x ≥ mx x for all x and t, and some constant equation m> 0. Respectively, P(t) is bounded if and only if P(t)≤ m for all t and some constant m > 0. ) 0 0 M q q ¨ + N q, q ˙ = Bτ + w (t), (14) (C2) There exists a bounded positive semidefinite sym- where q = [q , q ] ∈ R is a vector of generalized 1 2 metric solution to the equation coordinates where q and q are the unactuated and actuated 1 2 T T Z (t) = A(t)Z (t) + Z (t)A (t) + B (t)B (t) + Z (t) 1 joints, respectively; τ ∈ R is the vector of applied joint torques; B = [0, 1] is the input matrix that maps the (9) T T torque input τ to the joint of coordinates space; w (t) ∈ ( ) ( ) ( ) ( ) ( ) ( ) ( ) x × P t B t B t P t − C t C t Z t 2 2 2 2 γ 2 R is the unknown disturbance vector to account for 4 Journal of Robotics 0.5 destabilizing model discrepancies due to hard-to-model nonlinear phenomena such as friction and backlash, t ∈ R 2×2 0 is the time; M(q) ∈ R is the symmetric positive-definite inertia matrix; N (q, q ˙ ) = [N (q, q ˙ ), N (q, q ˙ )] ∈ R is the 1 2 −0.5 vector that contains the Coriolis, centrifugal, and gravity torques. Appendix A presents the dynamic model of the −1 double pendulum. 0 5 10 15 20 The control objective is to design a nonlinear H tracking ∞ Time (s) controller that ensures (a) lim q(t) − q (t) = 0 (15) t→∞ to be achieved asymptotically, while also attenuates the influence of external disturbances. Here, q (t) ∈ R is a continuously differentiable desired trajectory. −1 0 5 10 15 20 3.2. The Desired Trajectory. We point out that the present Time (s) formulation is different from typical formulation of output tracking and regulation [1, 19], where the set point or the (b) reference trajectory is a priori given because underactuated Figure 1: Plot of desired trajectories for Acrobot by selecting several systems are not feedback or input-state linearizables due to values of φ. its complexity. Therefore, special attention is required in the selection of the desired trajectory for the system under study. There are a few procedures to find desired trajectories that can be interpreted as the zero dynamics of the system for underactuated systems in literature [7, 8, 10, 20, 21], (16) with respect to the output y (t). Time evolution of the and under reasonable hypotheses all of them can be used desired trajectory is illustrated in Figure 1 where the value of to obtain a desired trajectory. The methodology from [7] φ is modified along the time: is used here, where a set of exact trajectories is derived for the nonlinear equation of motion which involves inverted 0, if t< 5, periodic motion. To this end, let us consider the desired ⎪ trajectory which is solution of φ = 0.025, if 5 ≤ t< 12, (20) ⎡ ⎤ ⎡ ⎤ ⎪ 0.1, if 12 ≤ t ≤ 20, q ˙ d d ⎣ ⎦ ⎣ ⎦ =       , (16) −1 dt q ˙ M q Bτ q , q ˙ − N q , q ˙ d d d d d d where t ∈ R is given in seconds. Notice that frequency and amplitude of oscillations change according to variations in φ. 2 2 where q (t) ∈ R , q ˙ (t) ∈ R are the desired joint positions d d Figure 2 shows the profile of the frequency and amplitude of and velocities, respectively, and oscillations for several values of φ. 2M q − M q 22 d 12 d ˙     ˙ τ = N q , q − N q , q (17) 3.3. The Task. Our objective is to design a controller of the 2 d d 1 d d 2M q − M q 12 d 11 d form is the control input that makes the desired virtual output τ = τ q , q + u (21) d d y (t) = 2q (t) + q (t) − φ (18) d d d 1 2 with internal state ξ (t) ∈ R , that ensures (15). Thus, the controller to be constructed consists of the trajectory remains at zero for all t ≥ 0 when y (t) starts at y (0) = d d compensator (17) and a disturbance attenuator u given y ˙ (0) = 0, φ is a constant parameter that parameter- in (2), (3) internally stabilizing the closed-loop system izes the equilibrium manifold of the pendulum, and the around the desired trajectory. In the sequel, we confine our oscillations given by (16)–(18) are around this manifold. investigation to the H tracking problem, where Throughout, we confine our research interest in desired (1) the output to be controlled is given by oscillations around the upright position of the pendulum ⎡ ⎤ ⎡ ⎤ which correspond to the more difficult case due that the 0 1 open-loop system has an unstable zero dynamics. Toward ⎣ ⎦ ⎣ ⎦ z = ρ + u(t) (22) this end, we choose φ = π for all t ≥ 0in(18). It was shown 2q + q − π 0 1 2 in [7] that (16)and (17) generate a set of exact periodic with a positive weight coefficient ρ; trajectories given by (2) the joint position vector q(t) ∈ R is the only c sin q + c sin φ − q 4 d 5 d 1 1 available measurement, and this measurement is q ¨ = , (19) d1 c − c 1 2 corrupted by the error vector w (t) ∈ R , that is, q (rad) d1 q (rad) d2 Journal of Robotics 5 1 4 φ = 0.75 0.9 φ = 0.5 0.8 φ = 0.25 0.7 0.6 0.5 φ = 0.1 0.4 −1 φ = 0.075 0.3 φ = 0.05 −2 0.2 φ = 0.025 −3 φ = 0 0.1 −4 −0.1 −0.05 0 0.05 0.10.15 3.54 4.55 5.56 6.5 q (rad) Frequency ω (rad/s) 1 (a) Figure 2: Profile of the frequency and amplitude of oscillations for several values of φ. 8 −2 τ 1 −4 l q 1 1 −6 −8 33.13.23.33.4 q (rad) (b) Figure 4: Phase portrait of the first joint trajectory and desired Figure 3: Schematic diagram of the acrobot where l and l denote 1 2 trajectory (+) for the unperturbed case (a) and perturbed case (b). the length of the nonactuated and actuated links, respectively; and m and m are the masses of each link. 1 2 y = q(t) + w (t). (23) that let us rewrite the system (14), the output to be controlled (22), and the output (23) in terms of the state vector The H control problem in question is thus stated as x: follows. Given the system representation (14)–(23), the desired trajectory q (t) ∈ R ,and arealnumber γ> 0, it is required to find (if any) a causal dynamic feedback ⎡ ⎤ ⎡ ⎤ x x 1 2 controller (2), (3) such that the undisturbed closed-loop d ⎣ ⎦ ⎣ ⎦ system is uniformly asymptotically stable around the origin, −1 dt x M x + q N x + q , x + q ˙ − q ¨ 2 1 d 1 d 2 d d and its L -gain is locally less than γ, that is, inequality (4)is ⎡ ⎤ ⎡ ⎤ satisfied for all t >t and all piecewise continuous functions 1 0 0 0 T ⎣ ⎦ ⎣ ⎦ +   w +   τ , w(t) = [w (t), w (t)] for which the corresponding state x o −1 −1 M x + q M x + q B 1 d 1 d trajectory of the closed-loop system, initialized at the origin, (24) remains in some neighborhood of this point. ⎡ ⎤ ⎡ ⎤ 0 1 ⎣ ⎦ ⎣ ⎦ z = ρ + u(t), 3.4. H Synthesis. To begin with, let us introduce the 2x + x +2q + q − π 0 1 1 d1 d2 1 2 (25) state deviation vector x = (x , x ) ∈ R where x = 1 2 1 T T (q − q , q − q ) and x = (q ˙ − q ˙ , q ˙ − q ˙ ) .After ( ) ( ) 1 d1 2 d2 2 1 d1 2 d2 y = x + q t + w t . 1 d o Amplitude A (rad) dq /dt (rad/s) dq /dt (rad/s) 2 6 Journal of Robotics 0.6 obviates an extra work (formidable in the nonlinear case) on verification of these properties. 0.4 0.2 4. Simulation Results The controller performance was studied in simulation by −0.2 02 4 6 8 10 applying the exposed ideas to the Acrobot, depicted in Time (s) Figure 3, which is a two-link planar robot with no actuator at the shoulder (link 1) and actuator at the elbow (link 2). In (a) the simulation, performed with MATLAB, the Acrobot was required to move from [q (0), q (0)] = [−0.07, 3.3] to the 1 2 desired trajectory q (t) ∈ R and φ = π . The initial velocity 2 4 q(0) ∈ R and the initial compensator state ξ (0) ∈ R were 3.5 set to zero for all the simulations. The matrices M(q)and N (q, q ˙ ) for the Acrobot are given in Appendix A.Weseek for orbital stabilization of the unactuated link q around the 2.5 equilibrium point q = (0, π ). 02 4 6 8 10 The control goal was achieved by implementing the Time (s) nonlinear H controller with a weight parameter ρ = 1 (b) on the Acrobot. By iterating on γ, we found the infimal achievable level γ 250. However, in the subsequent Figure 5: Time evolution of the output (continuous line) following simulations γ = 2000 was selected to avoid an undesirable the desired trajectory (dashed line) under perturbed torques. high-gain controller design that would appear for a value of γ close to the optimum. With γ = 2000 we obtained that for ε = 0.1 the corresponding differential Riccati equations (11)-(12)with Clearly, the above H tracking control problem is nothing ⎡ ⎤ else than a standard nonlinear H control problem from [1] ∞ 0010 ⎢ ⎥ stated for a time-varying nonlinear system (1)specifiedas ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A(t) = , ⎢ ⎥ ⎢ A (t) A (t) A (t) A (t)⎥ 31 32 33 34 ⎡ ⎤ ⎣ ⎦ A (t) A (t) A (t) A (t) ⎣ ⎦ 41 42 43 44 f (x, t) = , −1 M x + q N x + q , x + q ˙ − q ¨ 1 d 1 d 2 d d ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 0 2×2 2×2 2×1 ⎣ ⎦ ⎣ ⎦ 0 0 B (t) =   , B (t) =   , 2×2 2×2 1 2 −1 −1 ⎣ ⎦ g (x, t) =   , M q 0 M q 1 d 2×2 d −1 M x + q 0 1 d 2×2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0000 1000 ⎣ ⎦ ⎣ ⎦ 0 ( ) ( ) 2×1 C t = ρ , C t = , 1 2 ⎣ ⎦ g (x, t) = , 2100 0100 −1 M x + q B 1 d (26) ⎡ ⎤ ⎡ ⎤ 1 ⎣ ⎦ 0 0 I D (t) = , D (t) = 12 22 2×2 2 ⎣ ⎦ h (x, t) = ρ , 0 2x + x +2q + q − π 1 1 d1 d2 1 2 (27) ⎡ ⎤ have positive-definite solutions. These solutions can be ⎣ ⎦ h (x, t) = x + q , k (x, t) = , 2 1 d 12 numerically found with MATLAB. Matrix A(t)isgiven in Appendix B. It should be pointed out that the constant k (x, t) = 0 I . 2×2 2 φ = π does not appear in C (t) due to straightforward 21 1 calculation of (6), but it is definitely required in (22)to improve the selection of γ which affects inequality (4)thus Now by applying Theorem 1 to system (1)thusspecified, we avoiding the synthesis of a high-gain controllers. Resulting derive a local solution of the H tracking control problem. trajectories is depicted in Figure 4. This figure demonstrates Thus, the output feedback controller (13), specified that the H controller does asymptotically stabilize the according to (26), locally solves the H tracking control system motion around the desired trajectory. In addition, the problem (4)–(24). Stabilizability and detectability properties H controller was successfully applied to the Acrobot under of the control systems are ensured by the existence of external disturbances the proper solutions of the unperturbed differential Riccati equations, and hence the corresponding synthesis procedure w (t) = b cos(0.1t) exp(−2t), i = 1, 2, (28) x i q (rad) q (rad) 1 2 Journal of Robotics 7 150 200 0 0 05 10 05 10 Time (s) Time (s) (a) (b) 150 150 100 100 50 50 0 0 05 10 05 10 Time (s) Time (s) (c) (d) 4×4 Figure 6: Time evolution of the determinants of the principal minors of the matrix P ∈ R P , P , P , P . m1 m2 m3 m4 −3 −3 where b = 1 × 10 [N · m] and b = 2 × 10 [N · m] are These Figures highlight that matrices P(t)and Z (t), which 1 2 the disturbance levels at first and second joints, respectively. are solution of (11)and (12), respectively, are bounded and Resulting trajectories are depicted in Figure 5. Figures 6 and positive definite for all t ≥ 0. 7 show the time evolution of the determinant of minors of 4×4 4×4 matrices P(t) ∈ R and Z (t) ∈ R denoted as 5. Conclusions ⎛ ⎞ P (t) P (t) 11 12 The output feedback Nonlinear H tracking control problem ⎝ ⎠ P (t) = P (t), P (t) = , m1 11 m2 is locally solved for an underactuaded mechanical system. P (t) P (t) 12 22 The desired periodic orbit is centered at the upright position ⎛ ⎞ where the open-loop plant becomes a nonminimum-phase P (t) P (t) P (t) 11 12 13 ⎜ ⎟ system. The developed controller drives the trajectories of ⎜ ⎟ P (t) = P (t) P (t) P (t) , 12 22 23 m3 ⎝ ⎠ the robot into a set of inverted exact desired trajectories P (t) P (t) P (t) governed by its zero dynamics. Simulation studies, made 31 32 33 (29) for the Acrobot, showed the effectiveness of the controller. ⎛ ⎞ Z (t) Z (t) 11 12 The design of methods to generate reference trajectories ⎝ ⎠ Z (t) = Z (t), Z (t) = , m1 11 m2 evolving more frequencies and amplitudes in the upright Z (t) Z (t) 12 22 position is in progress, and few results have been published ⎛ ⎞ for double-pendulums in [20, 22]. In future work there are Z (t) Z (t) Z (t) 11 12 13 ⎜ ⎟ two extensions of the result of the paper. First, one would like ⎜ ⎟ Z (t) Z (t) Z (t) Z (t) = . m3 12 22 23 ⎝ ⎠ to synthesize the H control taking into account reference ( ) ( ) ( ) trajectories derived from alternative methods. The other Z t Z t Z t 31 32 33 Det. P Det. P m3 m1 Det. P Det. P m2 8 Journal of Robotics 30 800 0 0 05 10 05 10 Time (s) Time (s) (a) (b) 30 30 20 20 10 10 0 0 05 10 05 10 Time (s) Time (s) (c) (d) 4×4 Figure 7: Time evolution of the determinants of the principal minors of the matrix Z ∈ R Z , Z , Z , Z . m1 m2 m3 m4 Table 1: Parametervaluesfor theAcrobot. with c c c c c 1 2 3 4 5 0.0043 0.0051 0.0034 0.0494 0.038 M q = c + c − 2c cos q , 11 1 2 3 2 M q = c − c cos q , 12 2 3 2 M q = c , would also like to extend the result of the paper for the 22 2 (A.2) nonsmooth case. N q, q ˙ = c sin q q ˙ q ˙ + c q ˙ q ˙ + q ˙ sin q 1 3 2 1 2 3 2 1 2 2 − c sin q + c sin q + q , 4 1 5 1 2 Appendices N q, q ˙ =−c sin q q ˙ + c sin q + q , 2 3 2 5 1 2 A. Dynamic Model of Acrobot The equation motion of Acrobot, described by (14), was where the values of c (i = 1,... ,5), given in Table 1,were specified by applying the Euler-Lagrange formulation [23] taken from the experimental Acrobot provided in [7]. where ⎡ ⎤ ⎡ ⎤ M q M q N q, q ˙ 11 12   1 B. Matrix A(t) ⎣ ⎦ ⎣ ⎦ M q =     , N q, q = M q M q N q, q ˙ 12 22 2 In this appendix we provide the computed matrix A(t)for (A.1) the Acrobot which was used in the solution of differential Det. Z m1 Det. Z m3 Det. Z m2 Det. Z Journal of Robotics 9 Riccati equations (11)and (12): References ⎡ ⎤ [1] L. T. Aguilar, Y. Orlov, and L. Acho, “Nonlinear H -control of nonsmooth time-varying systems with application to friction ⎢ ⎥ ⎢ ⎥ mechanical manipulators,” Automatica,vol. 39, no.9,pp. ⎢ ⎥ ⎢ ⎥ ( ) 1531–1542, 2003. A t = ,(B.3) ⎢ ⎥ ⎢ A (t) A (t) A (t) A (t)⎥ 31 32 33 34 [2] E. R. Westervelt, J. Grizzle, C. Chevallereau, J. Cho, and B. ⎣ ⎦ Morris, Feedback Control of Dynamic Bipedal Robot Locomo- ( ) ( ) ( ) ( ) A t A t A t A t 41 42 43 44 tion, Taylor & Francis/CRC, London, UK, 2007. [3] R. T. M’Closkey, J. W. Burdick, and A. F. 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