1. Control of Robot Manipulators
Professor Nicola Ferrier
ME Room 2246,

2. Control Tasks Robot Level task Pe(t) Trajectory Planning (IK, J, etc)
controller
Power
electronics
Current to motors

3. Control Tasks Robot Level task Pe(t) Trajectory Planning (IK, J, etc)
controller
Power
electronics
Current to motors

4. Independent Joint Control (chapter 6*)
Use computed reference points (setpoints) for each joint
Control each joint “independently”
Ignore dynamic effects
Treat each joint as a stand alone “motor”
Dynamics Based control (Chapter 8*)
Use dynamics model to facilitate control
Compute torque feedforward
Inverse Dynamic Control
Operation Space control
And Compliance, Impedance, Force….
*Spong, Hutchinson, Vidyasagar, “Robot Modeling and Control”, Wiley, 2006

5. Jointed system components

6. Independent Joint Control
Use computed reference points (setpoints) for each joint
Control each joint “independently”
Ignore dynamic effects
Treat each joint as a stand alone “motor”
Simplifies control
Block Diagram (next slide)

7. Block Diagram of PE controller for a single joint
Energy source (current)
Reference angle
Error signal
control signal
torque u
PE controller
u = Kpe e
Motor
+reduction
+transmission
Joint
measured joint position, qm
Actual joint position, q
Joint
position
sensor

8. Independent Joint Control
Control each joint independently without “communication” between actuators
Basic Steps:
Model actuator
Use kinematics to obtain set-points for each joint
Develop a controller for each joint
Error for joint i:

9. Need to model relationships:
Actuator Model
Need to model relationships:
between actuator input (current) and output (torque)
Section 6.1: Permanent magnet DC-motor
Torque is approximately linear with applied current
(equation 6.4)
Applied current, amp
Motor torque, Nm
Motor constant, Nm/amp

10. actuator current vs torque

11. Need to model relationships:
Actuator Model
Need to model relationships:
between actuator torque and motor angle (q)
Section 6.1: Permanent magnet DC-motor
Second order ode
(equation 6.8 and 6.16)
disturbance
control input
Rotational inertia of joint, kg m^2
Effective damping (friction, back emf), Nm/amp

12. Independent Joint Control
Control each joint independently without “communication” between actuators
Basic Steps:
Model actuator
Use kinematics to obtain setpoints for each joint (recall trajectory planning – chapter 5)
Develop a controller for each joint
Error for joint i:

13. Proportional control for each joint
Input proportional to position error:
Neglect disturbance, wlog set reference position to zero or

14. Proportional control for each joint
Second order linear differential equation:
has general form solution:
where

15. Block Diagram of PE controller
Kpe 1 + -
s(Js+F)
Sensor
transfer
function

16. Three solutions
What does this term do?

17. Three solutions
Over-damped (w2 > 0)
Critically damped (w2 = 0)

18. If B is small and KPE is large: unstable!
Three solutions
Under-damped (w2 < 0)
w has complex roots
Oscillates with frequency
If B is small and KPE is large: unstable!

19. Example Step Responses (1 radian)

20. PE controllers can lead to
PI, PID controllers
PE controllers can lead to
Steady state error
Unstable behavior
Add Integral Term:
….but now we can have overshoot
Add derivative term (PID Controller)

21. Block Diagram of PE controller
Kpe 1 + -
s(Js+F)
Ki(1/s)
Kd(s)
Sensor
transfer
function

22. Set Gains for PID Controller
wlog set (we already have )
Convert to third order equation
Solution will be of the form
where

23. Set Gains for PID Controller
Critically damped when w = 0 or
An equation in 3 unknowns
Need two more constraints:
Minimum energy
Minimum error
Minimum jerk
And we need the solution to double minimization
Beyond the scope of this class – topic of optimal control class
More on gains in Advanced Robotics Course ….

24. Problems with Independent Joint Control
Synchronization ?
If one joint does not follow the trajectory, where is the end-effector???
Ignores dynamic effects
Links are connected
Motion of links affects other links
Could be in-efficient use of energy

25. Dynamics: Equations of Motion
Dynamic Model
forces/torques motion of manipulator+load
Equations of Motion
Ideally we can use
Modeling errors
Friction
synchronization