Giáo trình robot - Phần 4

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Giáo trình robot - Phần 4 Part IIIMotion ControlIntroduction to Part IIIConsider the dynamic model of a robot manipulator with n degrees of freedom,rigid links, no friction at the joints and with ideal actuators, (3.18), which werepeat here for ease of reference: M (q )q + C (q , q )q + g (q ) = τ . ¨ ˙˙ (III.1) TIn terms of the state vector q T q T ˙ these equations are rewritten as ⎡⎤ ⎡ ⎤ q q˙ d⎣ ⎦ ⎣ ⎦ = dt ˙ M (q )−1 [τ (t) − C (q , q )q − g (q )] q ˙˙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 torquesand τ ∈ IRn is a vector of external forces and torques applied at the joints.The vectors q , q , q ∈ IRn denote the position, velocity and joint acceleration ˙¨respectively. The problem of motion control, tracking control, for robot manipulatorsmay be formulated in the following terms. Consider the dynamic model of ann-DOF robot (III.1). Given a set of vectorial bounded functions q d , q d and q d ˙ ¨referred to as desired joint positions, velocities and accelerations we wish tofind a vectorial function τ such that the positions q , associated to the robot’sjoint coordinates follow q d accurately. In more formal terms, the objective of motion control consists in finding τsuch that lim q (t) = 0 ˜ t→∞where q ∈ IR stands for the joint position errors vector or is simply called n ˜position error, and is defined by q (t) := q d (t) − q (t) . ˜224 Part III ˙ Considering the previous definition, the vector q (t) = q d (t) − q (t) stands ˜ ˙ ˙for the velocity error. The control objective is achieved if the manipulator’sjoint variables follow asymptotically the trajectory of the desired motion. The computation of the vector τ involves in general, a vectorial nonlinearfunction of q , q and q . This function is called “control law” or simply, “con- ˙ ¨troller”. It is important to recall that robot manipulators are equipped withsensors to measure position and velocity at each joint henceforth, the vectorsq and q are measurable and may be used by the controllers. In some robots, ˙only measurement of joint position is available and joint velocities may beestimated. In general, a motion control law may be expressed as τ = τ (q , q , q , q d , q d , q d , M (q ), C (q , q ), g (q )) . ˙¨ ˙¨ ˙ However, for practical purposes it is desirable that the controller does notdepend on the joint acceleration q since accelerometers are usually highly ¨sensitive to noise. Figure III.1 presents the block-diagram of a robot in closed loop with amotion controller. qd τ q qd CONTROLLER ˙ ROBOT q ˙ qd ¨ Figure III.1. Motion control: closed-loop system In this third part of the textbook we carry out the stability analysis of agroup of motion controllers for robot manipulators. As for the position controlproblem, the methodology to analyze the stability may be summarized in thefollowing steps. 1. Derivation of the closed-loop dynamic equation. Such an equation is ob- tained by replacing the control action control τ in the dynamic model of the manipulator. In general, the closed-loop equation is a nonautonomous nonlinear ordinary differential equation since q d = q d (t). 2. Representation of the closed-loop equation in the state-space form, d qd − q = f (q , q , q d , q d , q d , M (q ), C (q ...