1. IVR 30/10/2009 Herrmann1 IVR: Control Theory Overview: 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. IVR 30/10/2009 Herrmann2 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. IVR 30/10/2009 Herrmann3 Motor command Robot in environment Control Law Disturbances Measurement Proportional error control 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 Desired output Actual output

4. IVR 30/10/2009 Herrmann4 Steady state error Servo control law: New process: Battery voltage V B Vehicle speed s ? ss goal gain=K With steady state (ds/dt=0): Steady state error :

5. IVR 30/10/2009 Herrmann5 Steady state error k 1 is determined by the motor physics : e= k 1 s Can choose K p : to reach target want But we can’t put an infinite voltage into our motor! Battery voltage V B Vehicle speed s ? ss goal gain=K

6. IVR 30/10/2009 Herrmann6 Steady state error For any sensible K p the system will undershoot the target velocity. t s s goal

7. IVR 30/10/2009 Herrmann7 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 K p and K i this can eliminate the steady state error.

8. IVR 30/10/2009 Herrmann8 Motor command Robot in environment Control Law Disturbances Measurement Feedback with time delays Imagine trying to move a robot to some zero position with the simple control law: –If x t < -δ move forward at 1cm/second, –If x t > δ move backward at 1cm/second, –If -δ < x t < δ stop What happens if there is a 1 second delay in feedback? Desired output Actual output

9. IVR 30/10/2009 Herrmann9 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. IVR 30/10/2009 Herrmann10 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. AB 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

11. IVR 30/10/2009 Herrmann11 www.sarcos.com Auke Ijspeert: Bastian

12. IVR 30/10/2009 Herrmann12 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. IVR 30/10/2009 Herrmann13 Second order system Want to move to desired postion θ goal For a DC motor on a robot joint where J = joint inertia, F = joint friction Proportional control : θ goal Amplification K of current I Output torque T θ error θ Need more sophisticated model for oscillations

14. IVR 30/10/2009 Herrmann14 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, K p as follows: where [May be complex]

15. IVR 30/10/2009 Herrmann15 Over and Under Damping The system returns to the goal (critical damping) The system returns to the goal with over-damping The system returns to the goal with under-damped, sinusoidal behaviour

16. IVR 30/10/2009 Herrmann16 Steady state error – Load Droop If the system has to hold a load against gravity, it requires a constant torque But P 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. IVR 30/10/2009 Herrmann17 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. IVR 30/10/2009 Herrmann18 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. IVR 30/10/2009 Herrmann19 Combine as PID control KpKp θ error θgθg KiKi K d d/dt Torque

20. IVR 30/10/2009 Herrmann20 PID control demo Rearranging becomes (almost) solvable as: where

21. IVR 30/10/2009 Herrmann21 PID control demo II Pure proportional control with friction & inertia Overdamped, overshoot, oscillate Choosing K p, K d, K i a bit tricky

22. IVR 30/10/2009 Herrmann22 PID control demo III Use K p =0.003 Add integral control Less overdamped, overshoot, oscillate

23. IVR 30/10/2009 Herrmann23 PID control demo IV Add derivative control Less overdamped, overshoot, oscillate

24. IVR 30/10/2009 Herrmann24 PID control demo V Goal: track oscillating goal position (red) Original K p =0.003 gain lags, undershoots K p =0.006 ok, but lags. K p =0.03 tracks better but overshoots

25. IVR 30/10/2009 Herrmann25 Limitations of PID control Tuning: largely empirical e.g. by Ziegler– Nichols method Delays: Combine with Feedforward Non-linearities: gain scheduling, multi-point controller, fuzzy logic Noise: D-term particularly sensitive: use low-pass filter (May cancel out D → PI) Often PI or PD is sufficient

26. IVR 30/10/2009 Herrmann26 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 2935-2940 S. Schaal and C.G. Atkeson. Open Loop Stable Control Strategies for Robot Juggling. In Proc. IEEE Conf. Robotics and Automation, pages 913 -- 918, Atlanta, Georgia, 1993. CPGs are used in many more recent examples of walking robots.

27. IVR 30/10/2009 Herrmann27 Bose suspension

28. IVR 30/10/2009 Herrmann28 Next time: Bob Fisher