IVR: Control Theory

1
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
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
Motor command
Robot in environment
Control Law
Desired output
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

gainK
s
sgoal
With steady state (ds/dt0) Steady state error
5
Steady state error
gainK
s
sgoal

• k1 is determined by the motor physics e k1 s
• Can choose Kp to reach target want
• But we cant 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
Motor command
Robot in environment
Control Law
Desired output
Disturbances
Actual output
Measurement

• Imagine trying to move a robot to some zero
  position with the simple control law
• If xt lt -d move forward at 1cm/second,
• If xt gt d move backward at 1cm/second,
• If -d lt xt lt d 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
11
www.sarcos.com
Auke Ijspeert Bastian
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
Second order system
Need more sophisticated model for oscillations
?goal

• Want to move to desired postion ?goal
• 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

May be complex
where
15
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
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
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
20
PID control demo
Rearranging becomes (almost) solvable as
where
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 Kp0.003 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 Kp0.003 gain lags,
undershoots Kp0.006 ok, but lags. Kp0.03 tracks
better but overshoots
25
Limitations of PID control

• Tuning largely empirical e.g. by ZieglerNichols
  method
• Delays Combine with Feedforward
• Non-linearities gain scheduling, multi-point
  controller, fuzzy logic
• Noise D-term particularly sensitiveuse
  low-pass filter (May cancel out D ? PI)
• Often PI or PD is sufficient

26
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
Bose suspension
28

• Next time Bob Fisher