Transcript Document

Introduction to ROBOTICS
Control
University of Bridgeport
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:
  M ()  V (, )  G()
•  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,
• V (, ) 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
  K P E  K D E
• Where:
E  d  
 

E  
d
PD control
d
e
+
KP

+


d
e
+
-
KD
Torque
Robot


Model based control
• The control law takes the form:
  M ()(d  KD E  K p E)  V (, )  G()
E  d  
 

E  
d
• Kp and KD are diagonal matrices.
Control Problem
•
Stable Response
Project
• The equations of motion:
(x , y)
M ( )  v( , )  g ( )  
2
l2
1 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
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.
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
  K p (d  )  KD (d  )
Block diagram
2DOF robot
• The forward kinematic equations:
x  l1 cos1  l2 cos(1  2 )
y  l1 sin 1  l2 sin(1   2 )
• The inverse kinematic equations:
y
x
1  tan1 ( )  tan1 (
(x , y)
2
l2 sin  2
)
l1  l2 cos 2
x 2  y 2  l12  l22
cos 2  (
)
2l1l2
• The Jacobian matrix
 l sin 1  l2 sin(1   2 )  l2 sin(1   2 )
J  1

 l1 cos1  l2 cos(1   2 ) l2 cos(1   2 ) 
l2
1 l1