Copyright c 9 by ABCM January 4-8, 1, Foz do Iguaçu, PR, Brazil LPV MODELING AND CONTROL OF A -DOF ROBOTIC MANIPULATOR BASED ON DESCRIPTOR REPRESENTATION Houssem Halalchi, houssem.halalchi@unistra.fr Edouard Laroche, laroche@unistra.fr Gabriela Iuliana Bara, iuliana.bara@lsiit.u-strasbg.fr LSIIT Laboratory, University of Strasbourg and CNRS. Pôle API, Boulevard Sébastien Brant, 6741 Illkirch, France Abstract. This paper deals with the practical applicability of the control design methods that are based on the descriptor representation for robotic manipulators. The nonlinear model of a manipulator can be reformulated as a linear parameter varying (LPV) model with rational parametric dependence. The statespace LPV model can then be converted into a LPV descriptor model with affine parametric dependence by using a linear fractional representation (LFR). In this paper, a two degrees of freedom (-DOF) robotic planar manipulator is considered. The affine dependence allows to derive analysis and design procedures that consist in a finite set of LMI conditions. Such an approach is used for the design of an output-feedback LPV controller which guarantees the stability and the H performance of the closed-loop system. The validity of the H control scheme is evaluated through simulations. Keywords: robotic manipulator, LPV systems, descriptor representation, H control. 1. INTRODUCTION The control of robotic manipulators is a broad area of research in which several theoretical and practical problems, such as nonlinearites, flexibility and some difficulties posed by non-colocated control, are encountered. Classical control methods in the joint space include simple PD control, PID control, inverse dynamics control, Lyapunov-based control, and passivity-based control (Canudas de Wit, Siciliano and Bastin, 1996). Although control schemes based on a nominal model of the robot have shown some effectiveness in practice, they generally do not guarantee performance and robustness over the full operating range. The basic idea of our study is to take advantage of the recent theoritical advances in linear parameter varying (LPV) control theory in order to propose generic methods allowing to address the control problems cited above. Descriptor models are an appropriate mathematical representation for numerous real-world dynamic systems. Indeed, the use of both algebraic and dynamical equations involving system state variables allows to obtain a generalized state-space representation called descriptor representation. Several types of applications are concerned such as the ones encountered in the fields of Mechanics and Robotics. Developing the Euler-Lagrange equations of a robotic manipulator leads to the derivation of the dynamic model of the robot. This dynamic model takes the form of a set of ordinary differential equations whose coefficients may depend on one or more parameters of the system, typically the joint angular positions and velocities of the robotic manipulator. In this paper, we focus on the dependence of the inertia matrix on the cosine of the second joint angle (in an affine manner). Taking the state vector as the joint positions and velocities of the robot, the state-space representation involves the inverse of the inertia matrix and yields a rational LPV state-space model with respect to the scheduling parameter. Due to the compactness of the admissible parameters set, robust stability and performance analysis of LPV systems generally leads to an infinite number of Linear Matrix Inequality (LMI) constraints (Apkarian and Adams, 1998). In the particular case of an affine or polytopic dependence of the state-space LPV matrices, these LMI contraints can be evaluated only on the vertices of a convex polytope, leading to a finite number of conditions (Gahinet, Apkarian and Chilali, 1996). This makes the problem of stability and performance analysis more easily solvable in practice for this class of LPV systems. This simplification does not hold for LPV systems of higher parametric complexity such as rational or more complex parameter dependence. In order to obtain a finite number of LMI conditions used for the analysis of stability and performance of higher complexity LPV systems, several techniques have been developed such as: parameter gridding (Apkarian and Adams, 1998) and minimal convex polytope finding (Anstett, Millerioux and Bloch, 9). In the case of rational LPV systems, an equivalent affine LPV descriptor representation has been proposed in (Masubuchi, Akiyama and Saeki, 3). This technique consists in deriving an affine decriptor realization of a rational LPV system using an augmented state vector composed of the original state vector and some additional variables satisfying algebraic relations with the original ones. A way to choose the additional variables, based on LFR (linear fractional representation, see (Magni, 4) for an overview) representation of the system, was proposed in (Bouali, Chevrel and Yagoubi, 6). However, it is not always necessary to establish this LFR model. In many cases, a simple change of variables is sufficient for the derivation of the affine descriptor model.
Copyright c 9 by ABCM January 4-8, 1, Foz do Iguaçu, PR, Brazil The paper is organized as follows. In section, we introduce the LPV modeling of robotic manipulators and the possibility of reformulating a rational state-space LPV model into an affine descriptor LPV one. In the third section, we adress the application of a -DOF planar manipulator. The design of an output-feedback LPV controller under H constraint is discussed in section 4. The notations used in the paper are defined as follows: If M 1 and M are symmetric matrices, M 1 > (resp. ) denotes a positive [ definite ] (resp. positive semi-definite) matrix, I n denotes the identity matrix of M1 dimension n and diag(m 1, M ) =. M. ROBOT MANIPULATORS AND LPV SYSTEMS.1 Context and motivation In the general case including friction and joint flexibility, the dynamic model of a robotic manipulator can be expressed as: M(q) q + C(q, q) q + D(q) q + K(q)q(t) + g(q) = Γ(t) (1) where q is the vector of generalized coordinates that consists of the joint positions and some additional variables (fictitious joints) describing flexibility if it appears. M(q), D(q) and K(q) are the inertia, damping and stiffness matrices respectively. C(q, q) is the Coriolis and centripetal matrix and g(q) is the gravitational torques vector. Γ(t) is the external torque applied to the manipulator. For a large class of manipulators, this complete nonlinear model can be reformulated as a LPV model by defining a vector of scheduling parameters that are functions of the joint positions and velocities. These state-space models are also referred to as "quasi-lpv" models since the scheduling parameter vector is a function of the state vector of the system (See (Siqueira and Terra, 4) for a design example in the field of robotics). When considering a partially linearized model where the Coriolis and gravitational forces are supposed to be compensated in the control torque, for instance by using the feedback linearization techniques (Spong, Hutchinson and Vidyasagar, 5), the dynamic equation (1) leads to a rational LPV system due to the inversion of the inertia matrix M(q). This rational LPV model causes some implementation issues in the resolution of LMIs for the controller sythesis with reduced conservatism. Thus, an affine dependence is suitable. In the following, we define the class of rational LPV sytems and present a method for the elimination of the rational parametric dependence via the use of the descriptor systems representation.. The class of rational LPV systems The rational LPV systems are described by the following state-space equations: { ẋ r (t) = A r (ρ(t))x r (t) + B r (ρ(t))u(t) y(t) = C r (ρ(t))x r (t) () where the state and observation matrices A r (ρ(t)), B r (ρ(t)) and C r (ρ(t)) are rational functions of the time-varying parameter vector ρ(t). A natural representation for this class of systems (Apkarian and Gahinet, 1995) is the Linear fractional representation (LFR): ẋ r (t) M 11 M 1 M 13 x r (t) y(t) = M 1 M M 3 u(t) and v(t) = (ρ)z(t) (3) z(t) M 31 M 3 M 33 v(t) where x r (t) R r is the state, v(t) R nv is the external input, u(t) R nu is the control input, z(t) R nz is the controlled output and y(t) R ny is the measured output. (ρ) is the uncertainty matrix. The problem of LPV control based on the LFR of gain-scheduling systems, where the controller has a LFR structure, has been adressed in (Apkarian and Gahinet, 1995). Furthermore, the LFR of a rational LPV system can be exploited in order to obtain an equivalent affine descriptor realization of the system..3 Descriptor realization of LPV systems Any rational LPV system can be reformulated as an affine descriptor LPV system of higher order (Masubuchi, Akiyama and Saeki, 3). New states that verify some algebraic relations with the original ones, are added to obtain an affine descriptor model. The adressed LPV descriptor systems, also referred to as singular or generalized state-space systems, are systems of the form: { Eẋ(t) = A(ρ(t))x(t) + B(ρ(t))u(t) (4) y(t) = C(ρ(t))x(t)
Copyright c 9 by ABCM January 4-8, 1, Foz do Iguaçu, PR, Brazil where x(t) R n is the vector of descriptor variables. It is built by concatenating the rational state vector x r (t) and a vector of additional variables called x (t): x(t) = [x T r (t) x T (t)] T. The matrix E may be singular. In (Bouali, Chevrel and Yagoubi, 6), it has been shown that rational LPV systems () can be equivalenty rewritten in the descriptor form (4) with E = diag(i r, n r ) and the state and observation matrices A(ρ), B(ρ) and C(ρ) being affine functions of the parameter vector ρ(t). Derivation of an affine LPV descriptor model based on LFR: The transition between a rational LPV model described by () and an affine descriptor model of the form (4) can be done in different ways. One can distinguish the so-called "ad-hoc" methods from the "systematic" methods. The former consist in observing the rational functions of the state equations and then making some changes of variables to eliminate this rational dependence, while the latter uses a generic formulation to define the new generalized state variables. In the following, we describe a systematic method that uses the LFR representation introduced above. From equation (3) we have: v(t) = (ρ)m 31 x r (t) + (ρ)m 3 u(t) + (ρ)m 33 v(t). (5) Defining the additional state variables x as: x = [v T (t) u T (t)] T, we obtain the affine descriptor LPV representation of the system (4) in which only the state matrix A(ρ) depends on the scheduling parameter: ẋ r M 11 M 13 M 1 M 1 nv = (ρ)m 31 (ρ)m 33 I nv (ρ)m 3 x + nv u(t) (6) nu r nv I nu I nu where x = [x T r x T ] T = [x T r v T (t) u T (t)] T. The size of the generalized state vector x(t) obtained by this method is r + n v + n u (r initial and (n v + n u ) additional variables). We have adopted this method for the application discussed in the paper because it has led to a more compact descriptor representation than the one obtained with an "ad-hoc" change of variables method. 3. LPV MODEL OF A -DOF HORIZONTAL ROBOTIC MANIPULATOR The LPV models discussed in the previous section may be valid for a broad class of dynamical systems including articulated mechanical systems. As our field of interest is the control of robotic manipulators, we adress the simple case of a -DOF horizontal and completely rigid robotic manipulator as a first step towards dealing with more complex configurations. Assuming that the Coriolis and centripetal forces are compensated by the control torque (Adams, Apkarian and Chrétien, 1996), the dynamic model of a -DOF planar manipulator can be described by the equation: M(q ) q + D q + Kq(t) = Γ(t) (7) where M(q ) is the inertia matrix, D and K are the constant damping and stiffness matrices and Γ(t) is the vector of control torques applied to the manipulator. [ ] m11 m The inertia matrix is a symmetric positive definite matrix expressed as: M(q ) = 1. It depends on the m 1 m geometric configuration of the robot. More precisely, M(q ) varies in an affine manner with respect to the cosine of the second joint angle, which will be considered as the scheduling parameter ρ = cos(q ): [ ] [ ] m11 m M(q ) = M + M c cos(q ) = M + M c ρ with M = 1 m11c m and M m 1 m c = 1c. (8) m 1c The admissible set for the scheduling parameter ρ is the interval [ρ min, ρ max ]. Let us now define a state vector of the robot that consists of the joint angles q 1, q and their time-derivatives q 1 and q : x r (t) = [q 1, q, q 1, q ] T = [q T, q T ] T. The system has a state-space LPV representation of the form () where the parameter ρ(t) can be available from measurements: [ ] [ ] A r (ρ) = I M (ρ)k M, B (ρ)d r (ρ) = M and C (ρ) r (ρ) = [ ] I. (9) Note that the damping and stiffness matrices are respectively: D = diag(f 1, f ) and K = diag(k 1, K ). The control input is u(t) = Γ(t) = [Γ 1 (t) Γ (t)] T. [ ] [ ] I The rational LPV system (9) can be rewritten in the LFR form (3) with: M 11 = M K M D, M 1 = M,
Copyright c 9 by ABCM January 4-8, 1, Foz do Iguaçu, PR, Brazil [ ] M 13 =, M I 1 = [ ] I, M = M 3 =, M 31 = [ M M cm K M M cm D], M 3 = M M cm and M 33 = M M c. The uncertainty matrix is (ρ) = ρ I. The LPV descriptor model of the -DOF robotic manipulator is now fully described by (6), where r = 4 and n v = n u =. In the next section, this model is used for the design of an output-feedback LPV controller under H constraint. 4. H CONTROL SCHEME FOR THE ROBOTIC MANIPULATOR 4.1 Augmented system for H control In order to ensure both the closed-loop stability and the robustness of the control system against the parameter variations, we have implemented the H control scheme illustrated in Figure 1. It is possible to use more sophisticated design schemes. However, a one block scheme was sufficient for our application. w(t) W 1 (s) r(t) + z(t) = e(t) G(s, ρ) y(t) u(t) G 1 (s, ρ) K(s, ρ) e(t) Figure 1. H control of the robot In Figure 1, G(s, ρ) = C(sE A(ρ)) B represents the transfer function of the LPV descriptor system obtained by the systematic method previously described and K(s, ρ) is the state-space LPV controller to be synthesized. Through a convenient tuning of the structure and parameters of the weighting function W 1 (s), it is possible to impose the following desired requirements on the closed-loop transfer T r e (Skogestad and Postlethwaite, 5): the bandwidth ω c, the steady-state position error E p and the modulus margin M mod, for any frozen value of the parameter ρ. We have chosen here the following weighting function structure: W 1 (s) = s+a K I (s+b). The augmented LPV descriptor system G 1 (s, ρ) can be determined under the following generalized state-space representation, where A 1 (ρ) is affine: { E 1 ẋ(t) = A 1 (ρ)x(t) + B 1 w(t) + B u(t) (1) z(t) = C 1 x(t) + D 1 w(t) The LMI-based controller synthesis method discussed in the next paragraph requires that the system is strictly proper (ie D 1 = ). It is possible to satisfy this condition by defining the new generalized state vector x = [x(t) T w(t) T ] T. Equation (1) then becomes (with n w = length(w)): {Ẽ1 x(t) = Ã1(ρ) x(t) + B 1 w(t) + B u(t) z(t) = C x(t) (11) where Ẽ1 = diag(e 1, nw ), Ã 1 = diag(a 1, I nw ), B1 = [B T 1 I nw ] T, B = [B T nw ] T and C = [C 1 D 1 ]. Note that another way to satisfy this design requirement is to add a high-frequency pole (1 + εs) in W 1 (s). However, this method is more restrictive than the above because it releases the closed-loop constraints in the high frequency range. 4. LMI conditions for the controller synthesis The synthesis problem consists in the design of a LPV controller K(s, ρ) that minimizes the L induced gain of the performance channel T w z of the closed-loop system described by Figure 1. In (Masubuchi et. al, 4), a synthesis method in the LPV descriptor case based on the resolution of LMI constraints has been proposed. When choosing constant Lyapunov matrices, a LPV controller ensuring that the L gain is less than γ does exist if the following LMIs have a solution p = {X, Y, F, G, H} with appropriate dimensions (See (Masubuchi et. al, 4) for more detail): [ ] [ ] Y Ẽ1 T Ẽ 1 Y Ẽ Ẽ1 T Ẽ1 T 1 = T T M Ẽ A + MA T M B MC T 1 X Ẽ1 T Ẽ1 T and MB T γi < (1) X M C γi
Copyright c 9 by ABCM January 4-8, 1, Foz do Iguaçu, PR, Brazil where [Ã1 (ρ)y M A = T + B ] [ ] F T Ã 1 (ρ) B1 H T X T Ã 1 (ρ) + G T, M B = C X T B1 and M C = [ CY T C]. Due to the affine dependence of Ã1(ρ) with respect to the scheduling parameter ρ, the parameter-dependent LMIs (1) wich must hold for any ρ belonging to the admissible parameters set (the interval [ρ min, ρ max ]) can be equivalently applied only on the vertices of this set: {ρ min, ρ max }. Note that the gain scheduled controller obtained by this method is given in a state-space form: (A cs (ρ), B cs, C cs ) and is applied to the rational state-space model of the robot: G r (s, ρ) = C r (si r A r (ρ)) B r (ρ). The LMIs (1) imply: S y (s, ρ).w 1 (s) < γ, ρ [ρ min, ρ max ], where S y (s, ρ) = (I + G r (s, ρ).k(s, ρ)) is the sensitivity function and. denotes the H norm. Therefore, given the particular structure of W 1 (s), this condition is equivalent to: S y (s, ρ) < γ W 1 (s), ρ [ρ min, ρ max ]. Assuming the following requirements for our control application: M mod >.991, ω c > 1 rad.s and E p < 1 3, the obtained performance index is γ = 1.783. Figure illustrates the singular values of the sensitivity function S y (s, ρ), γ compared to the frequential template W for three particular values of the parameter ρ: ρ 1(s) min =, ρ nominal = and ρ max = 1. Note that the LMIs (1) have been solved by using the semidefinite programming (SDP) solver SeDuMi associated with the YALMIP toolbox within the MATLAB environment. The resolution on an Intel Core Duo processor takes 1.51 seconds. sensitivity frequential template sensitivity frequential template sensitivity frequential template singular values (db) 4 4 4 6 6 6 1 4 1 1 1 1 4 frequency (rad/sec) 1 4 1 1 1 1 4 frequency (rad/sec) 1 4 1 1 1 1 4 frequency (rad/sec) (a) (b) (c) Figure. The realized sensitivity function when (a): ρ =, (b): ρ =, (c): ρ = 1 and the template γ W 1(s) joint positions (rad) 3.5 1.5 1.5.5 5 1 15 time (sec) (a) joint positions (rad) 3.5 1.5 1.5.5 5 1 15 time (sec) (b) 4.3 Simulation results Figure 3. Nominal PD controller (a) and the proposed LPV controller (b) The simulations we present in this section have been carried out using a dynamic model of a horizontal -DOF robotic manipulator which is the rigid part of the laboratory structure SECAFLEX presented in (Adams, Apkarian and Chrétien, 1996). The interested reader will find in (Chrétien, 1989) a full description of the device. In order to assess the system response over the operating range, especially the admissible scheduling parameters set given by q [, π] (we do not address the negative part for symmetry reasons), the desired trajectory we have chosen for the second joint is a series of progressive step signals allowing to browse the whole interval. In order to evaluate the coupling effects, we have chosen a square signal in phase quadrature as a reference trajectory for the first joint. We have made a comparison between the temporal response of the closed-loop system obtained with the LPV controller described in the previous section (Figure 3.b), and the one obtained with a simple PD (proportional and derivative) LTI controller. The PD controller has been designed on the nominal system (ρ = ie q = π ) in order to obtain, for each joint considered separately, a second order behavior with a damping ratio ξ = 1 and a natural frequency ω n = 1 rad/sec (Figure 3.a).
Copyright c 9 by ABCM January 4-8, 1, Foz do Iguaçu, PR, Brazil Due to the simplicity of the configuration, the PD controller stabilizes the simulation system and annuls the steadystate position error. However, unlike the LPV controller, this PD LTI controller does not allow an input-output decoupling of the system. The coupling effects are important near q = (rad.) which corresponds to a maximal inertia of the robot, and are low near q = π (rad.) corresponding to the minimal inertia. It would be possible to synthesize a LTI decoupling controller but its decoupling effect would not cover the full opertaing range since the system is parameter-varying. We want to emphasize that the simulations presented here were made on a simple system for which a simple controller is quite satisfactory. In order to assess the overall benefits of the LPV H control scheme presented here compared to a classical LTI approach, simulations may be performed on more complex systems such as: robots with a larger number of degrees of freedom and/or with flexible elements, taking into account of a higher number of non-linearities in the LPV modeling stage, etc. 5. CONCLUSION In this paper, we have explored the possibility of reformulating a nonlinear dynamic model of a robotic manipulator into a LPV state-space model. This modeling process is also referred to as quasi-lpv modeling since the time-varying parameter depends on the joint positions and velocities of the robot. Then, we have presented some equivalent models, namely a LFR model and an affine LPV descriptor model for this mechanical device. The latter is useful for the design of a dynamic output-feedback LPV controller under H constraint, synthesized by resolving a finite number of LMI conditions. The simulations have been done on a -DOF horizontal and completely rigid robotic manipulator. The results of these simulations have demonstrated the validity of the proposed control approach for this dynamical system. We have observed that, for this relatively simple system, the contribution of the LPV control method adressed here (compared to a classical PID control) is input-output decoupling of the two joints of the robot over the full operating range. In a future work, our goal is to use these approaches for the controller design of more complex articulated systems including nonlinearity and flexibility phenomena. 6. REFERENCES Adams R.J., Apkarian P. and Chrétien J.-P., 1996. "Robust control approaches for a two-link flexible manipulator", In Proceedings of the 3rd Int. Conference on Dynamics and Control Structures in Space, pp. 11-116. Anstett F., Millerioux G. and Bloch G., 9. "Polytopic observer design for LPV systems based on minimal convex polytope finding", Journal of Algorithms and Computational Technology, Vol.3, No 1, pp. 1-9. Apkarian P. and Adams R.J., 1998. "Advanced gain-scheduling techniques for uncertain systems", IEEE Transactions on Control Systems Technology, Vol.6, No. 1, pp. 1-3. Apkarian P. and Gahinet P., 1995. "A convex characterization of gain-scheduled H controllers", IEEE Transactions on Automatic Control, Vol.4, No. 5, pp. 853-864. Bouali A., Chevrel P. and Yagoubi M., 6. "About gain-scheduled state feedback controllers for rational LPV systems", In Proceedings of the 9th International Conference on Control, Automation, Robotics and Vision, pp. 71-77. Canudas de Wit C., Siciliano B. and Bastin G., 1996. "Theory of robot control", Springer, London, 39 p. Chrétien J.-P., 1989. "SECAFLEX: An experimental set-up for the study of active control of flexible structures", In Proceedings of the 1989 American Control Conference, pp. 1397-144. Gahinet P., Apkarian P. and Chilali M., 1996. "Affine parameter-dependant Lyapunov functions and real parameter uncertainty", IEEE Transactions on Automatic Control, Vol.41, No. 3, pp. 436-44. Magni J.-F., 4. "Linear Fractional Representation Toolbox: Modelling, Order Reduction, Gain Scheduling", Technical Report, ONERA, 193 p. Masubuchi I., Akiyama T. and Saeki M., 3. "Synthesis of output feedback gain-scheduling controllers based on descriptor LPV system representation", In Proceedings of the 4nd IEEE Conference on Decision and Control, pp. 6115-61. Masubuchi I., Kato J., Saeki M. and Ohara A., 4. "Gain-scheduled controller design based on descriptor representation of LPV systems: Application to flight vehicle control", In Proceedings of the 43rd IEEE Conference on Decision and Control, pp. 815-8. Siqueira A.A.G. and Terra M.H., 4. "Nonlinear and markovian H controls of underactuated manipulators", IEEE Transactions on Control Systems Technology, Vol.1, No. 6, pp. 811-86. Skogestad S. and Postlethwaite I., 5. "Multivariable feedback control - Analysis and design", nd Edition, Wiley, 5, 59 p. Spong M.W., Hutchinson, S. and Vidyasagar, M., 5. "Robot Modeling and Control", Wiley and Sons, New York, 47 p. 7. RESPONSIBILITY NOTICE The authors are the only responsible for the printed material included in this paper.