Giáo trình robot - Phần 3

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Giáo trình robot - Phần 3 Part IIPosition ControlIntroduction to Part IIDepending on their application, industrial robot manipulators may be classi-fied into two categories: the first is that of robots which move freely in theirworkspace (i.e. the physical space reachable by the end-effector) thereby un-dergoing movements without physical contact with their environment; taskssuch as spray-painting, laser-cutting and welding may be performed by thistype of manipulator. The second category encompasses robots which are de-signed to interact with their environment, for instance, by applying a comply-ing force; tasks in this category include polishing and precision assembling. In this textbook we study exclusively motion controllers for robot manip-ulators that move about freely in their workspace. For clarity of exposition, we shall consider robot manipulators providedwith ideal actuators, that is, actuators with negligible dynamics or in otherwords, that deliver torques and forces which are proportional to their inputs.This idealization is common in many theoretical works on robot control as wellas in most textbooks on robotics. On the other hand, the recent technologicaldevelopments in the construction of electromechanical actuators allow one torely on direct-drive servomotors, which may be considered as ideal torquesources over a wide range of operating points. Finally, it is important tomention that even though in this textbook we assume that the actuators areideal, most studies of controllers that we present in the sequel may be easilyextended, by carrying out minor modifications, to the case of linear actuatorsof the second order; such is the case of DC motors. Motion controllers that we study are classified into two main parts based onthe control goal. In this second part of the book we study position controllers(set-point controllers) and in Part III we study motion controllers (trackingcontrollers). Consider the dynamic model of a robot manipulator with n DOF, rigidlinks, no friction at the joints and with ideal actuators, (3.18), and which werecall below for convenience:136 Part II M (q )q + C (q , q )q + g (q ) = τ . ¨ ˙˙ (II.1)where M (q ) ∈ IRn×n is the inertia matrix, C (q , q )q ∈ IRn is the vector of ˙˙centrifugal and Coriolis forces, g (q ) ∈ IRn is the vector of gravitational forcesand torques and τ ∈ IRn is a vector of external forces and torques appliedat the joints. The vectors q , q , q ∈ IRn denote the position, velocity and joint ˙¨acceleration respectively. TIn terms of the state vector q T q T ˙ these equations take the form ⎡⎤ ⎡ ⎤ q q˙ d⎣ ⎦ ⎣ ⎦. = dt ˙ −1 q M (q ) [τ (t) − C (q , q )q − g (q )] ˙˙ The problem of position control of robot manipulators may be formulatedin the following terms. Consider the dynamic equation of an n-DOF robot,(II.1). Given a desired constant position (set-point reference) q d , we wish tofind a vectorial function τ such that the positions q associated with the robot’sjoint coordinates tend to q d accurately. In more formal terms, the objective of position control consists in findingτ such that lim q (t) = q d t→∞where q d ∈ IRn is a given constant vector which represents the desired jointpositions. The way that we evaluate whether a controller achieves the control ob-jective is by studying the asymptotic stability of the origin of the closed-loopsystem in the sense of Lyapunov (cf. Chapter 2). For such purposes, it appearsconvenient to rewrite the position control objective as lim q (t) = 0 ˜ t→∞where q ∈ IRn stands for the joint position errors vector or is simply called ˜position error, and is defined by q (t) := q d − q (t) . ˜ Then, we say that the control objective is achieved, if for instance theorigin of the closed-loop system (also referred to as position error dynamics)in terms of the state, i.e. [q T q T ]T = 0 ∈ IR2n , is asymptotically stable. ˜˙ The computation of the vector τ involves, in general, a vectorial nonlinearfunction of q , q and q . This function is called the “control law” or simply, ...