Software Engineering, CPSC-4360-01, CPSC-5360-01, Lecture 11
Introduction to Robotics cpsc - 460
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INTRODUCTION TO ROBOTICS CPSC - 460 Lecture 5B – Control
description
Introduction to Robotics cpsc - 460. Lecture 5B – 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. - PowerPoint PPT Presentation
Transcript of Introduction to Robotics cpsc - 460
INTRODUCTION TO ROBOTICSCPSC - 460
Lecture 5B – 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.
4
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
RobotContro ller_
tor qqdq dq
dqTrajectoryP lanner
qqe d e e
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
( ) ( , ) ( )M V G
( , )V
PD CONTROL
The control law takes the form
Where:EKEK DP
dE
dE
PD CONTROL
Robot
KP
+
KD
Torque
+
-
e
e
+-
d
d
RobotContro ller_
tor qqdq dq
dqTrajectoryP lanner
qqe d e e
MODEL BASED CONTROL
The control law takes the form:
Kp and KD are diagonal matrices.
( )( ) ( , ) ( )d D pM K E K E V G
dE
dE
RobotContro ller_
tor qqdq dq
dqTrajectoryP lanner
qqe d e e
CONTROL PROBLEM
STABLE RESPONSE
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
PROJECT
The equations of motion:
2
1
(x , y)
l2
l1
( ) ( , ) ( )M v g
PROJECTSIMULATION 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
Planning1. 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.
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
( ) ( )p d D dK K
BLOCK DIAGRAM
2DOF ROBOT
The forward kinematic equations:
The inverse kinematic equations:
The Jacobian matrix
)cos(cos 21211 llx
)sin(sin 21211 lly
)2
(cos21
22
21
22
2 ll
llyx
)cos
sin(tan)(tan
221
22111
ll
l
x
y
)cos()cos(cos
)sin()sin(sin
21221211
21221211
lll
lllJ
2
1
(x , y)
l2
l1
