PASSIVITY AND POWER BASED CONTROL OF A SCARA ROBOT MANIPULATOR

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1 PASSIVITY AND POWER BASED CONTROL OF A SCARA ROBOT MANIPULATOR Gabriel V. Paim, Lucas C. Neves, Ubirajara F. Moreno, Edson R. De Pieri Departamento de Automação e Sistemas (DAS), Universidade Federal de Santa Catarina (UFSC), Florianópolis, Brasil. s: gpaim@das.ufsc.br, lcneves@das.ufsc.br, moreno@das.ufsc.br, edson@das.ufsc.br Abstract This paper presents the design of two Passivity-based controllers (PBC) to the problem of motion control of a robot manipulator with a SCARA configuration, one called IDA-PBC (Interconnection and Damping Assignment) and the other called Power-shaping. The first controller uses the system described as a Portcontrolled Hamiltonian (PCH) to apply balancing of energy. The second controller, uses Brayton-Moser equations to implement balancing of power. Through these two methods, at the end, it is obtained the same controller known as PD with gravity compensation, which is implemented in the SCARA manipulator robot. Keywords Control. SCARA Robot, Passivity-based Control, Port-Controlled Hamiltonian Model and Power-shaping Resumo Este artigo apresenta o projeto de dois controladores baseados em passividade (PBC) para o problema de controle de movimento de um robô manipulador com uma configuração SCARA, um chamado IDA-PBC (Atribuição por Interconexão e Amortecimento) e outro chamado Power-shaping. O primeiro controlador utiliza o sistema descrito na forma do Hamiltoniano de Porta Controlada (PCH) para aplicar um balanceamento de energia. O segundo controlador utiliza equações de Brayton-Moser para implementar o balanceamento de potência. Através destes dois métodos, ao final, é obtido o mesmo controlador conhecido como PD com compensação de gravidade, que é implementado no robô manipulador SCARA. Palavras-chave Robô SCARA, Controle Baseado em Passividade, Modelo Hamiltoniano de Porta Controlada e Controle via Power-shaping 1 Introduction Control design problems have traditionally been approached from a viewpoint of signal processing. The control objectives are expressed in order to keep the error signals small and reduce the effect of certain disturbances, despite of the presence of some unmodeled dynamics (Ortega et al., 21). According to (Ortega et al., 21), the analysis and design of controllers for nonlinear systems is best suited when the plant to be controlled and the controller are viewed as devices that transform energy, breaking them down into simpler systems whose energies, after their interconnections, are added together to determine the behavior of the system as a whole. The control problem can then be redrafted in order to find the system dynamics and standard interconnection, such that the total energy function takes the desired shape. This approach, called energy-shaping, is the essence of passivity-based controllers (PBC). When it comes to fully actuated mechanical system, the use of PBC methodology provides a natural procedure to model the system only from the potential energy, resulting in controllers with a clear physical interpretation (Ortega, Spong, Gomez-Estern and Blankenstein, 22). However, unfortunately, modeling the total energy of the system destroys its physical structure, i.e., no longer represents a Lagrangian system and the storage function of the map is not a passive energy function with a physical meaning. Thus, the IDA-PBC (Interconnection and Damping Assignment) method was developed, which aims at modeling the behavior of the energy of the closed loop system through the manipulation of interconnection and damping arrays (Ortega, van der Schaft, Maschke and Escobar, 22). On the other hand, there is a new approach well suited to these types of systems, which takes as an example the concepts applied in IDA-PBC methodology, but shapes and designs controllers based on the modeling of power instead of energy. This method is called power-shaping and, recently, the power-shaping methodology has been applied to different fields as physical (García-Canseco et al., 21) and chemical systems (Favache and Dochain, 29). This article aims at drawing parallels between these two design methods of control of the system described in section 2, IDA-PBC (section 5) and Power-shaping Control (section 6), through demonstrating the equivalence of the controllers obtained under some design conditions. It is made by comparing the models (sections 3 and 4) and the controllers (section 7), followed by applications of these controllers on a SCARA robot manipulator (section 8). Finally, the section 9 presents the final considerations about the methods applied in the study case. 2 The SCARA Robot Manipulator The system used to study the application of the control techniques is the well known manipulator robot in a SCARA (Selective Compliant Ar- ISSN: Vol. X 827

2 Figure 1: Robotic manipulator. ticulated Robot for Assembly) configuration. Its physical prototype is shown in Fig. 1. The problem of modeling the dynamics of this type of robot has already been solved (Tsai, 1999). The model used in this paper takes into account three actuated joints, q 1, q 2 and d 3. Thus, the manipulator s end-effector moves in a 3-DOF trajectory along the workspace. The robot s dynamic model with a fourth joint (revolute) and its parameters are described in (Vargas, 25). Based on this model, and removing the fourth joint, the model takes the well known form: M(q) q + C(q, q) q + G(q) = τ, (1) where q = [q 1 q 2 d 3 T are the joint space coordinates, M(q) is the Inertia Matrix, C(q, q) the Coriolis vector, G(q) the gravity vector and τ the torque applied to each manipulator joint. From this point, in order to simplify the notation, the third joint coordinate d 3 will be called q 3, despite being a prismatic joint. Note that the friction forces are not taken into account in this model. 3 Port-controlled Hamiltonian Model According to (Ortega et al., 21), in order to characterize a class of stabilizable systems with energy balancing PBC and simplify the solution of PDEs, we need to incorporate more structure in the system dynamics, in particular, making explicit the damping terms and the dependence of the energy function. Therefore, the Port-controlled Hamiltonian (PCH) models, which cover a large class of nonlinear physical systems, will be used. They have been considered an alternative to the classical models and their standard form is expressed as: { ẋ = [J(x) R(x) H (x) + g(x)u Σ : y = g T (x) H (x), (2) where H(x) R is the energy function, J(x) = J T (x) R 2n 2n represents the interconnection structure, R(x) R 2n 2n (R(x) = R T (x) ) is the matrix of dissipation, u, y R n are respectively the input and output of the system, g(x) is the function that couples the control to the system dynamics. Note that, since the system is fully actuated, u (in the case of the SCARA manipulator, τ) has the dimension of q, i. e., each measured joint is actuated. For mechanical systems, the PCH model is written in terms of generalized coordinates x = [q p T, which are composed by the generalized coordinates of position q = [q 1,..., q n T and the generalized coordinates of momentum p = [p 1,..., p n T, where the relationship between them can be expressed as q = M 1 (q)p, (3) where M(q) R n n (M(q) = M T (q) > ) is the generalized mass matrix or matrix of inertia of the manipulator. The system Σ, represented in (2), is passive from u to y, because it satisfies the equation of energy balance H[x(t) H[x() = t u T (t)y(t)dt d(t), (4) where d(t) is a non-negative function that represents the phenomenon which describes the energy dissipation of the system. For the robotic manipulator in question, the energy function (Hamiltonian) chosen for the system is represented as H(q, q) = T(q, q) + V(q) = 1 2 qt M(q) q + V(q), (5) where T(q, q) R is the kinetic energy, V(q) R is the potential energy. The energy function of the system can also be written as H(q, p) = 1 2 pt M 1 (q)p + V(q). (6) Thus, the PCH model of the manipulator gets the following form: [ {[ [ } [ [ q In q H = + u ṗ I n D R n p H G, (7) y = g T (x) H where u R n is the vector of external inputs (torques applied on the motors), q H = H, ph = H p, G is the input matrix,and D Rn n a damping matrix. For our application, the system is fully actuated. Then, G = I n and D =. Notice that the system is affine in control, because g(x) = [ I n T. Thus, the output vector is precisely the generalized velocities (y = q). In order to demonstrate the equivalence of the PCH and dynamic models, the PCH model will be rearranged so that the dynamic models can be obtained. Thus, starting from the PCH system model definition (7) and making some multiplications, we get: { q = p H = M 1 (q)p ṗ = q H R( q) p H + u. (8) The introduction of the derivative of p (ṗ = Ṁ(q) q + M(q) q) in the second line of (8) and its manipulation, yield: M(q) q + Ṁ(q) q + T(q, q) + V(q) + R( q) q = u. (9) Thus, the classical dynamic model of manipulators, just as shown in equation (1), can be obtained by using the relationships: Ṁ(q) q+ T(q, q) = C(q, q), the Coriolis vector; V(q) = G(q), the gravity vector; and R( q) q = F at ( q), the friction force vector. ISSN: Vol. X 828

3 4 Brayton-Moser Model The introduction of models written in the form of Port-controlled Hamiltonian allowed the description of a large number of nonlinear physical systems using a different view, which considers systems (plants, controllers, etc) as energy transformers. Its popularity comes from its wide applicability with respect to the analysis and control systems. However, as mentioned in (Ortega et al., 21), energy-balancing control is stymied by the existence of pervasive dissipation. Several control methodologies have been developed to over-come this dissipation obstacle. One of these methodologies, called power-shaping, was originally introduced in (Ortega et al., 23) and later extended to general nonlinear systems in (García-Canseco et al., 21). The method relies on the solution of a PDE, which allows us to write the original dynamics in term of the Brayton- Moser equations (Brayton and Moser, 1964), and yields storage functions which have units of power. In general, the BM models have the following form: Q(x)ẋ = P(x) + B(x)u, (1) where x R 2n is the states vector of the system, Q(x) R 2n 2n a full rank symmetric matrix, B(x) R 2n n the input matrix and P(x) R a mixed potential function, which has the unity of power, e.g., watt in SI. It has been shown in (Jeltsema and Scherpen, 27) that the BM model of a mechanical system described by equation (1) can be obtained from the PCH model making: Q(q, p) = x = [q p T ; (11) 2 V (p T M 1 (q)p) (M 1 (q)p) [ B(q, p) = (p T M 1 (q)p) M 1 (q)g (p T M 1 (q)) M 1 (q) ; (12) ; (13) P(q, p) = VT M 1 (q)p pt M 1 (q)dm 1 (q)p ( ) T. M 1 (q)p (14) It is important to highlight that this transformation demonstrated in (Jeltsema and Scherpen, 27) is only applied to frictionless systems, i.e. systems with R( q) =. Therefore, for sake of simplicity, the frictions will be disconsidered. 5 IDA-PBC Control 5.1 Concepts Associated with the Method Consider the nonlinear system as ẋ = f (x) + g(x)u y = h(x). (15) Assume the existence of g (x), J d (x) = J T d (x), R d (x) = R T d (x) and the function H d : R 2n R which satisfy the PDE g (x) f (x) = g (x)[j d (x) R d (x) H d, (16) where g (x) is a full rank annihilator of g(x), i.e., g (x)g(x) = and H d (x) is such that x = arg min H d (x), (17) with x R 2n an equilibrium point to be stabilized. Then, the closed loop system (15), where u(x) = [g T (x)g(x) 1 g T (x){[j d (x) R d (x) H d (x) f (x)}, (18) takes the PCH form ẋ = [J d (x) R d (x) H d (x), (19) with x a stable equilibrium (locally). It will be asymptotically stable if, in addition, x is an isolated minimum of H d (x) and the largest invariant set inside the closed loop dynamics (19) contained in {x R 2n [ H d T R d (x) H d = } (2) is equal to x. An estimation of the respective domain of attraction is given by the highest level of a limited set {x R 2n H d (x) c}. 5.2 Implementation of IDA-PBC Method The method used in this work does not consider the friction forces applied at the joints of the system. Thus, the model used for the controller design is the same as (7), but without the presence of R n matrix, which represents the friction applied to the robotic system. Therefore, the system takes the form: [ [ [ [ q In q H = + u ṗ }{{} x I n p H } {{ } f (x) y = g T (x) H G }{{} g(x). (21) As shown previously, the total energy of this system is given by (6). Similarly, we can naturally define the desired energy function as: H d (q, p) = 1 2 pt M 1 d (q)p + V d(q), (22) where the index d indicates that this is the reference. Since the energy function of the system was determined, the matrix of interconnection will be also determined: [ J d (q, p) = M 1 (q)m d (q) M d (q)m 1 (q) J 2 (q, p). (23) Taking the control action given by (18), and in order to make clear the role of each term of the equation, we can divide the control action into two terms: the energy-shaping term and damping term (u(x) = u es (x) + u di (x)). In this case we have that (Ortega and Canseco, 24): u es (x) = [ g T (x)g(x) 1 g T (x) {[ J d (x) H d f (x) } u di (x) = K v g T (x) H d. (24) ISSN: Vol. X 829

4 We can now make some assumptions to simplify the control law and get a simpler law, but not necessarily less efficient. The following simplifications are made: it will only be done the energy-shaping of the potential energy, which naturally implies that M d (q) = M(q); the tuning parameter J 2 (q, p) will not be used (J 2 (q, p) = ). The above simplifications lead to J d = J, i.e., the desired interconnection matrix is not modified and is equal to the interconnection matrix of the system. Therefore, the part of the control related to the energy-shaping is: u es (x) = [ I n {[ I n I n [ q H d p H d Solving the equation (25), we obtain: [ [ } In q H. I n p H (25) u es (x) = q V q V d. (26) We can now make an appropriate choice of V d (q) in order to find a control law. A smart choice of V d (q) can be done as follows: V d (q) = 1 2 (q q d) T K p (q q d ), (27) considering a reference of position. Since we know what is the desired potential energy of the system, and that we defined the desired energy function, we can go further and rewrite equation (26): u es (x) = q V K p (q q d ). (28) In the same way, the control action responsible for the damping can be calculated: u di (x) = [ K v Hd = K v q. (29) Finally, the whole control action is: u(x) = q V K p (q q d ) K v q. (3) The controller of the equation (3) is the well known PD (proportional with derivative action) with gravity compensation (Wang and Goldsmith, 28; Ortega et al., 21; Takegaki and Arimoto, 1981), widely used in mechanical control systems. 6 Power-shaping Control The purpose of this section is to establish a control signal to stabilize the system and to adjust the mixed potential function P(x) into a function with the desired shape P d (x). Thus, it uses the same idea as in IDA-PBC control, which also adjusts the system into the desired shape. In this case, the shaping does not happen in the total energy function (Hamiltonian) of the system, but in the mixed potential function (P(x)), which also results in shaping the Q(x) matrix. These matrix changes of the system lead to the desired shape Q d (x)ẋ = P d(x), (31) where the d index refers to desired matrices. The method used in this article is based on the method described in (García-Canseco et al., 21), which takes into account the definition of the nonlinear system in state-space form as shown in equation (15). With the system described in this way, we can get the driver through the proposition 6.1 described below. Proposition 6.1 Consider the general nonlinear system (15). Given an equilibrium point x X R n, where X := { x R n g ( x) f ( x) = }, and g (x) is a full-rank left annihilator of g(x). Assume A.1 There exists a nonsingular matrix Q : R n R n n that (i) solves the partial differential equation (Q(x) f (x)) = [ (Q(x) f (x)) T, and (32) (ii) verifies Q(x) + Q T (x) in neighborhood of x. A.2 There exists a scalar function P a : R n R, (locally) positive definite in a neighborhood of x, that verifies (iii) g (x)q 1 (x) P a (x) =, (iv) P d (x ) =, 2 P d (x ) >, with P d (x) = P(x) + P a (x), and P(x) satisfies P(x) = Q(x) f (x). Then, the control law u = [g T (x)q T (x)q(x)g(x) 1 g T (x)q T (x) P a (x) (33) ensures x is a (locally) stable equilibrium with Lyapunov function P d (x). Assume, in addition, A.3 x is an isolated minimum of P d (x) and the largest invariant set contained in the set {x R n T P d (x)[q 1 (x) + Q T (x) P d (x) = } (34) equals {x }. Then, x is an asymptotically stable equilibrium and an estimate of its domain of attraction is given by the largest bounded level set {x R n P d (x) c}. 7 Linking IDA-PBC Control and Power-shaping Control Let us compare the two different ways used to describe the system model, (1) and (15). By multiplying (1) by Q 1 (x), we notice that f (x) = Q 1 (x) P(x) and g(x) = Q 1 (x)b(x). Therefore, replacing Q(x)g(x) by B(x) in the control law (33), one gets u = [B T (x)b(x) 1 B T (x) P a (x). (35) To be able to obtain the Q d (x) and P d (x) matrices, and still put the expression of the control action in a form that can be compared with the classic controls, we must perform some manipulations of the system. Thus, subtracting (1) from (31), we get Q(x)ẋ = [Q d (x) Q(x) ẋ = P a(x) B(x)u. (36) ISSN: Vol. X 83

5 By (36), we notice that to obtain the expression (35), it is necessary that Q(x) =. This can be done with the following choice of Q d (x) and P d (x) and Q d (q, p) = 2 V d (p T M 1 (q)p) (M 1 (q)p) (p T M 1 (q)) M 1 (q), (37) P d (q, p) = VT d M 1 (q)p pt M 1 (q)k v M 1 (q)p ( ) T. M 1 (q)p (p T M 1 (q)p) (38) To show the equivalence of the control obtained by this method with that obtained via IDA-PBC, we must expand (36) as: ( 2 V d 2 V 2 2 ) ẋ = P a(x) P a(x) p [ M 1 (q) u. (39) Figure 2: Desired and obtained trajectories. Using equations (39), (38), (14), and taking into account the assumption A.2 (iv), we obtain: M 1 (q)u = P(x) p M 1 (q)u = P d(x) p ( V V d M 1 (q)k v (4) ) M 1 (q). (41) Whereas the purpose of control is the same as the IDA-PBC, i.e., perform an energy-shaping of the potential energy plus a damping assignment, we must use the desired mixed potential function as V d = 1 2 (q q d) T K p (q q d ), where q d is the desired angular position. And finally, knowing that M(q) and M 1 (q) are symmetric matrices, q = M 1 (q)p and q = (q q d ), we get u = q V K p q K v q. (42) Note that the controller obtained is exactly the one obtained by the IDA-PBC method, i.e., a PD controller (proportional with derivative action) with compensation of gravity. It is important to highlight that, as demonstrated by (García- Canseco et al., 21) in proposition 6.1, we can prove the asymptotic stability of the closed loop system using the controller designed. 8 Simulations and Results This section is responsible for presenting the simulation performed, along with the subsequent results. It was simulated the PD controller with compensation of gravity in Simulink/Matlab environment using the plant as the nonlinear dynamic model, as shown in equation (1). In order to better visualize the robot s movements in its workspace, it was used a trajectory in which the end-effector moves along the plane x = y. The Fig. 2 shows the desired and obtained trajectories of the end-effector in the Cartesian space, and the Fig. 3 shows the errors curves of X, Y and Z axes separately for the same trajectory. The desired trajectory has the ranges x =, 3m, Figure 3: Error of the Final Efector s Position. y =, 3m and z =, 3m, and the time to complete the round trip is t = 1s. It can also be seen in sections 5.2 and 6 that the controller design method of energy-shaping and power-shaping provide a class of controllers whose parameters should be calibrated according to the final application and desired trajectory. Thus, for the simulation results, the control system was first calibrated with the aid of a Simulink toolbox called Design Optimization, for the trajectory defined in this section. This tool tunes parameters in the controller to meet specified constraints in the closed-loop system. The constraints include bounds on signal amplitudes and matching of reference signals. The optimized values found for K p and Kv were diagonal matrices with the elements K p 1 = 515, 5, K p 2 = 2.562, K p 3 = 87.51, Kv 1 = 9, 48, Kv 2 = 3, 82 and Kv 3 = 192, 1. Although the method presented a satisfactory result, once the speed for the trajectory was increased, it became harder to optimize the controller, even though the same optimization method as described above was performed. Therefore, it could be noticed that the robot motion is degraded when the speeds and accelerations increase or some disturbance is added. In this way, in order to improve performance, another control method should be implemented, involving an integral or adaptive action or even other control theories. 9 Conclusion It was presented in this article, the application of one controller obtained through two different ISSN: Vol. X 831

6 methods - one Passivity-based (PBC) and another power-based - to the problem of motion control of a manipulator robot with a SCARA configuration. With the first method, called Interconnection and Damping Assignment (IDA-PBC), it is possible to modify the kinetic energy of mechanical systems by modeling its potential energy. This class of controllers was used before by (Ortega, Spong, Gomez-Estern and Blankenstein, 22), seeking to reach the global stability of an inverted pendulum with an inertial disk and a ball-beam system, but the methodology extends to a wider class of systems, nonlinear systems in general. With the second method, called Power-shaping, the Brayton- Moser equations are used to modify the mixed potential energy function of the system, which has the same unity as power. Simulations of the controlled system were performed using the classical dynamic model of the robot (equation (1)) as the plant. It was imposed one single trajectory in a number eight form, moving along the x = y plane. With a calibration of the controller parameters, it was possible to obtain satisfactory results. As the speeds and accelerations increase, the robot points less precision, pointing the possibility of using other control methods or adding new dynamics in the controller to increase its performance, as friction compensation or integrative dynamics. The control methodologies IDA-PBC and Power-shaping consists of an alternative way in the analysis and project of control systems. Instead of considering the signals involved in the process, the energy of systems is manipulated. The results showed in this paper indicate that the controllers designed with these methodologies are not always unique, i.e., they can be obtained through other synthesis methods. However, energy based systems do not represent a simple controller design technique, but a wider way of understanding systems. The role of energy and the interconnections between subsystems provide the basis for various control strategies. Thus, energy can serve as a common language to facilitate communication among scientists and engineers from different fields (Jeltsema and Scherpen, 29). References Brayton, R. K. and Moser, J. K. (1964). A theory of nonlinear networks - i, Quarterly of Applied Mathematics 22(4): Favache, A. and Dochain, D. (29). Analysis and control of the exothermic continuous stirred tank reactor: the power-shaping approach, Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference. García-Canseco, E., Jeltsema, D., Ortega, R. and Scherpen, J. M. A. (21). Powerbased control of physical systems, Automatica 46(1): Jeltsema, D. and Scherpen, J. M. A. (27). A power-based description of standard mechanical systems, Systems and Control Letters 56(5): Jeltsema, D. and Scherpen, J. M. A. (29). Multidomain modeling of nonlinear networks and systems, Control Systems Magazine, IEEE 29(4): Ortega, R. and Canseco, E. G. (24). Interconnection and damping assignment passivitybased control: A survey, European Journal of Control 1(5). Ortega, R., Jeltsema, D. and Scherpen, J. M. A. (23). Power shaping: a new paradigm for stabilization of nonlinear rlc circuits, Automatic Control, IEEE Transactions on 48(1): Ortega, R., Schaft, A. J. V. D., Mareels, I. and Maschke, B. (21). Putting energy back in control, Control Systems Magazine, IEEE 21(2): Ortega, R., Spong, M. W., Gomez-Estern, F. and Blankenstein, G. (22). Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment, Automatic Control, IEEE Transactions on 47(8): Ortega, R., van der Schaft, A., Maschke, B. and Escobar, G. (22). Interconnection and damping assignment passivity-based control of port-controlled hamiltonian systems, Automatica 38(4): Takegaki, M. and Arimoto, S. (1981). A new feedback method for dynamic control of manipulators, Journal of Dynamic Systems, Measurement and Control 12: Tsai, L. W. (ed.) (1999). Robot Analysis: The Mechanics of Serial and Parallel Manipulators, John Wiley & Sons, Inc., New York, N.Y., USA. Vargas, F. J. T. (25). Análise e Síntese de Controladores de Força-Posição de Robôs Manipuladores: Aspectos Teóricos e Experimentais, PhD thesis, UFSC, PPGEEL, Florianópolis/SC, Brasil. Wang, Z. and Goldsmith, P. (28). Modified energy-balancing-based control for the tracking problem, Control Theory Applications, IET 2(4): ISSN: Vol. X 832