INTRODUCTION TO ROBOTICS CPSC 460 Lecture 5 B

INTRODUCTION TO ROBOTICS CPSC - 460 Lecture 5 B – Control

CONTROL PROBLEM Determine the time history of joint inputs required to cause the end-effector to execute a command motion. The joint inputs may be joint forces or torques.

CONTROL PROBLEM Given: A vector of desired position, velocity and acceleration. Required: A vector of joint actuator signals using the control law.

ROBOT MOTION CONTROL (I) Joint level PID control � each joint is a servo-mechanism � adopted widely in industrial robot � neglect dynamic behavior of whole arm � degraded control performance especially in high speed � performance depends on configuration 4

ROBOT MOTION CONTROL (II) – COMPUTED TORQUE The dynamic model of the robot has the form: is the torque about zk , if joint k is revolute joint and is a force if joint k is prismatic joint Where: M(Θ) is n x n inertia matrix, is n x 1 vector of centrifugal terms G(Θ) is a n x 1 vector of gravity terms

PD CONTROL The control law takes the form Where:

PD CONTROL

MODEL BASED CONTROL The control law takes the form: Kp and KD are diagonal matrices.

CONTROL PROBLEM

STABLE RESPONSE

Evaluating the response overshoot settling time steady-state error ss error -- difference from the system’s desired value overshoot -- % of final value exceeded at first oscillation rise time -- time to span from 10% to 90% of the final value settling time -- time to reach within 2% of the final value rise time

PROJECT The equations of motion: (x , y) 2 l 2 1 l 1

PROJECT SIMULATION AND DYNAMIC CONTROL OF A 2 DOF PLANAR ROBOT Problem - - - statement: The task is to take the end point of the RR robot from (0. 5, 0. 0) to (0. 5, 0. 3, 0. 0) in the in a period of 5 seconds. Assume the robot is at rest at the starting point and should come to a complete stop at the final point. The other required system parameters are: L 1 = L 2 = 0. 4 m, m 1 = 10 kg, m 2 = 7 kg, g = 9. 82 m/s 2.

PROJECT 1. 2. Planning Perform inverse position kinematic analysis of the serial chain at initial and final positions to obtain ( 1 i, 2 i) and ( 1 f, 2 f). Then, obtain fifth order polynomial functions for 1 and 2 as functions of time such that the velocity and acceleration of the joints is zero at the beginning and at the end. These fifth order polynomials can be differentiated twice to get the desired velocity and acceleration time histories for the joints.

PROJECT Use a PD control law where Kp and Kv are 2 x 2 diagonal matrices, and s is the current(sensed) value of the joint angle as obtained from the simulation. Tune the control gains to obtain good performance

BLOCK DIAGRAM

2 DOF ROBOT The forward kinematic equations: (x , y) The inverse kinematic equations: 2 l 2 The Jacobian matrix 1 l 1