1. Control 3 Keypoints: PID control
Steady-state error and the integral method
Overshoot and ringing in system with time delay
Overshoot and ringing in second order systems
The derivative method

2. PID control P = Proportional error I = Integral D = Derivative
A standard and very useful method for augmenting simple feedback control
E.g. used in wheel position/speed control in the Khepera

3. Proportional error control
Desired
output
Control Law
Motor command
Robot in environment
Disturbances
Actual
output
Measurement
Same as servo, i.e. using a control signal proportional to the difference between the actual (measured) output and the desired (target) output.
I.e. want to make large change when error is large, small change if error is small

4. Steady state error Servo control law: New process:
Battery voltage VB
Vehicle speed s ?
gain=K
sgoal s
With steady state (ds/dt=0):
Steady state error:

5. Steady state error k1 is determined by the motor physics: e= k1 s
Battery voltage VB
Vehicle speed s ?
gain=K
sgoal s
k1 is determined by the motor physics: e= k1 s
Can choose Kp: to reach target want
But we can’t put an infinite voltage into our motor!

6. Steady state error
For any sensible Kp the system will undershoot the target velocity. s
sgoal t

7. Proportional Integral (PI) Control
If we could estimate this error we could add it to the control signal:
The best way to estimate it is to integrate the error over time:
Obtain new control law:
Basically, this sums some fraction of the error until the error is reduced to zero.
With careful choice of Kp and Ki this can eliminate the steady state error.

8. Feedback with time delays
Desired
output
Control Law
Motor command
Robot in environment
Disturbances
Actual
output
Measurement
Imagine trying to move a robot to some zero position with the simple control law:
If xt < -δ move forward at 1cm/second,
If xt > δ move backward at 1cm/second,
If -δ < xt < δ stop
What happens if there is a 1 second delay in feedback?

9. Feedback with time delays
In general, a time-lag in a feedback loop will result in overshoot and oscillation.
Depending on the dynamics, the oscillation could fade out, continue or increase.
Sometimes we want oscillation, e.g. CPG

10. Central Pattern Generators
Many movements in animals, and robots, are rhythmic, e.g. walking
Rather than explicitly controlling position, can exploit an oscillatory process, e.g.
If A is tonically active, it will excite B
When B becomes active it inhibits A
When A is inhibited, it stops exciting B
When B is inactive it stops inhibiting A A B

12. Feedback with time delays
In general, a time-lag in a feedback loop will result in overshoot and oscillation.
Depending on the dynamics, the oscillation could fade out, continue or increase.
Sometimes we want oscillation, e.g. CPG
Note that integration introduces a time delay.
Time delays are equivalent to energy storage e.g. inertia will cause similar effects.

13. AmplificationK of current I
Second order system
Need more sophisticated model for oscillations
AmplificationK of current I
Output torque T θ
error
θgoal
Want to move to desired postion θgoal
For a dc motor on a robot joint
where J = joint inertia, F = joint friction
Proportional control:

14. Second order system solution
For simplicity let
Then the system process is
The solution to this equation has the form
The system behaviour depends on J,F,Kp as follows:
where
[May be complex]

15. Over and Under Damping The system returns to the goal
The system returns to the goal with over-damping
The system returns to the goal with under-damped, sinusoidal behaviour

16. Steady state error – Load Droop
If the system has to hold a load against gravity, it requires a constant torque
But PE controller cannot do this without error, as torque is proportional to error (so, needs some error)
If we knew the load L could use
But in practice this is not often possible

17. Proportional integral (PI) control
As before, we integrate the error over time, i.e.
Effectively this gradually increases L until it produces enough torque to compensate for the load
PI copes with Load Droop

18. Proportional derivative (PD) control
Commonly, inertia is large and friction small
Consequently the system overshoots, reverses the error and control signal, overshoots again…
To actively brake the motion, we want to apply negative torque when error is small and velocity high
Make
Artificial friction:

19. Combine as PID control Kp θ
error θg Ki
Kdd/dt
Torque

20. PID control demo
Rearranging becomes (almost) solveable as:

21. PID control demo II Pure proportional control with friction & inertia
Overdamped, overshoot, oscillate
Choosing Kp, Kd, Ki a bit tricky

22. PID control demo III Use Kp=0.003 Add integral control
Less overdamped, overshoot, oscillate

23. PID control demo IV Add derivative control
Less overdamped, overshoot, oscillate

24. PID control demo V Goal: track oscillating goal position (red)
Original Kp=0.003 gain lags, undershoots
Kp=0.006 ok, but lags.
Kp=0.03 tracks better but overshoots

25. Further reading:
Most standard control textbooks discuss PID control.
For the research on robot juggling see:
Rizzi, A.A. & Koditschek, D.E. (1994) Further progress in robot juggling: Solvable mirror laws. IEEE International Conference on Robotics and Automation
S. Schaal and C.G. Atkeson. Open Loop Stable Control Strategies for Robot Juggling. In Proc. IEEE Conf. Robotics and Automation, pages , Atlanta, Georgia, 1993.
CPGs are used in many more recent examples of walking robots.