University of Bridgeport Introduction to ROBOTICS Control University of Bridgeport 1 1

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.

Dynamic model 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 and G(Θ) is a n x 1 vector of gravity terms.

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

PD control The control law takes the form Where:

PD control

Model based control Kp and KD are diagonal matrices. The control law takes the form: Kp and KD are diagonal matrices.

Control Problem

Stable Response

Project The equations of motion: 2 1 (x , y) l2 l1

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, 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 come to a complete stop at the final point. The other required system parameters are: L1 = L2 = 0.4m, m1 = 10kg, m2 = 7kg, g = 9.82m/s2.

Project Planning Perform inverse position kinematic analysis of the serial chain at initial and final positions to obtain (1i, 2i) and (1f, 2f). 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 2x2 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

2DOF robot The forward kinematic equations: The inverse kinematic equations: The Jacobian matrix 2 1 (x , y) l2 l1