I NTRODUCTION TO R OBOTICS CPSC Lecture 5B – Control

C ONTROL P ROBLEM 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.

C ONTROL P ROBLEM Given: A vector of desired position, velocity and acceleration. Required: A vector of joint actuator signals using the control law.

4 R OBOT M OTION C ONTROL (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

R OBOT M OTION C ONTROL (II) – C OMPUTED T ORQUE The dynamic model of the robot has the form: is the torque about z k,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

M ODEL BASED CONTROL The control law takes the form: K p and K D are diagonal matrices.

C ONTROL P ROBLEM

S TABLE R ESPONSE

Evaluating the response steady-state error settling time rise time overshoot 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 ss error -- difference from the system’s desired value

P ROJECT The equations of motion: 22 11 (x, y) l2l2 l1l1

P ROJECT S IMULATION AND D YNAMIC C ONTROL OF A 2 DOF P LANAR R OBOT 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: L 1 = L 2 = 0.4m, m 1 = 10kg, m 2 = 7kg, g = 9.82m/s2.

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

P ROJECT Use a PD control law where K p and K v 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

B LOCK DIAGRAM

2DOF R OBOT The forward kinematic equations: The inverse kinematic equations: The Jacobian matrix 22 11 (x, y) l2l2 l1l1