A new Control Synthesis Approach of P-Time Petri Nets
Patrice Bonhomme · InTech eBooks · 2010
Will-be-set-by-IN-TECHdynamics of rigid robots describe a passivity mapping from torque input to velocity output.This property is know as the passive structure of rigid robots (Ortega & Spong, 1989).The controller design methodology for robot manipulators introduced by (Takegaki & Arimoto, 1981), also called energy shaping plus damping injection technique, allows to naturally split the controller tasks into potential energy shaping for stabilization at the desired equilibrium, and damping injection, to make this equilibrium attractive (Ortega et al., 1995b).As a feature of this kind of controllers, it can be shown that the passivity property, from a new input torque to output velocity, is preserved for robots in closed-loop with the energy shaping term and damping injection term of the controller.Furthermore, considering that corresponding feedback to the integral action define a passive mapping, then it is possible to use a passivity theorem of interconnected systems to explain the stability of a class of nonlinear PID global regulators for robots.The theorem used allows to conclude global asymptotic stability of the origin of an unforced feedback system, which is composed by the feedback interconnection of state strictly passive dynamic systems with a passive and zero state observable system.The objective of this chapter is to present in a simple framework the global asymptotic stability analysis, by using passivity theory for a class of nonlinear PID regulators for robot manipulators.The remainder of this chapter is organized as follows: Section 2 summarizes the dynamics for rigid robots and also recalls some of their important properties.The rationale behind the energy shaping plus damping injection technique for rigid robots are given in Section 3. The class of nonlinear PID regulators is given in Section 4. In Section 5 we recall the definition of the passivity concepts for dynamical systems and we present the passivity theorem useful for asymptotic stability analysis of interconnected systems.In Section 6 we present a passivity analysis and application of passivity theorem to conclude global asymptotic stability.An Evaluation in simulation to verify the theoretical results is presented in Section 7. Finally, our conclusions are shown in Section 8. Throughout this chapter, the norm of a vector x is defined as x = √ x T x and that of a matrix A is defined as the corresponding induced norm A = λ M {A T A}.L n 2 and L n 2e denote the space of n-dimensional square integrable functions and its extension, respectively. Robot dynamicsFor control design purposes, it is necessary to have a mathematical model that reveals the dynamical behavior of a system.Robots manipulators are articulated mechanical systems composed of links connected by joints.Links and joints are usually made as rigid as possible so as to achieve high precision in robot positioning.The joints are mainly of two types: revolute and prismatic.Its dynamic model is characterized by nonlinear coupled secondorder differential equations, which describe the temporal interactions of the joint motions in response to the inertial, centrifugal and Coriolis, gravitational and actuating torques or forces.The most commonly used equations to model the dynamics of a robot are the Euler-Lagrange and Newton-Euler formulations.Here we use the Euler -Lagrange formulation.In this section we consider robot manipulators formed by an open kinematic chain.We assume that all the links are joined together by revolute joints.In the absence of friction and other disturbances, the Lagrangian L(q,q) of a mechanical system is defined by L(q,q)=K(q,q) -U(q), 44 Advances in PID Control www.intechopen.com