Impedance Control

1
Impedance Control

• Jake Glower
• Lu Gan (MS Student),
• Jayant Singh (MS Student)
• March 7, 2006

2
Type of Control

• Position Control
• Pre 1930
• Force Control
• 1974
• Hybrid Control
• 1979
• Impedance Control
• 1985

3
Position Control

• Position Control
• Systems transfer function is strictly proper
  (more poles than zeros)
• Position, Velocity, Temperature, Water Level,
  etc.
• Large number of solutions
• Root locus, Nyquist, Lead/Lag, PID, Variable
  Structures, Saturating Control

4
Position Control Examples

• Angle of a motor
• Speed of a motor
• Temperature in a room
• Attitude of an airplane
• Humidity in a room
• Water level in a tank
• p.H. of a solution

5
Force Control

• Systems transfer function has same number of
  zeros as poles
• D matrix dominates the transfer function
• Ex Current to torque for a motor (TKtIa)

6
Force Control Applications

• Control the force applied by the tip of a robotic
  arm (Paul 1979)
• Control the force exerted on a pin during
  assembly (Inoue 1974, Shimano 1977)

7
Hybrid Control

• Define two controllers position and force
• Switch between the two controllers
• Position control when the robot is in free space
• Force control when inserting a peg

8
Hybrid Control Applications

• Allow a robot to insert a pin
• Khatib Burdick, 1986
• Anderson and Spong 1987
• Goldenberg, 1988
• Control of robotic hands (Hanafusa and Asanda,
  1982)

9
Impedance Control

• Circuit Analogy
• Voltage Position
• Current Force
• V/I Impedance
• The impedance of a circuit is the differential
  equation which related voltage to current
• V Z I

10
Impedance Control

• Controls Solutions
• Hybrid Control Position Force Control
• Position Control with a prefilter
• Force Control with a prefilter

11
Hybrid Control as Impedance Control

• Regulate both position and force
• Does not work With one input you can only do
  one thing.
• Think DC At steady state, the input is a
  constant
• T k1 V
• Angle k2 V
• Find V to satisfy both
• When you regulate both position and force you are
  regulating one thing a weighted average.

12
Position Control as Impedance Control

• Hogan (1985, 1987)
• Force position (output) to track a set point
• Define the set point as the desired impedance
  times the measured external force

13
Position Control Block Diagram
14
Position Control Equations
15
Problems with Position-Based Impedance Control

• Problems (Hogan 1987)
• The enviroment creates a feedback loop around the
  system
• Stability depends upon the environment (more
  stable for compiant surfaces less gain
• The impedance is only correct if the position
  controller results in a gain of 1.000 (infinite
  bandwidth)

16
Variations of Position-Based Impedance Control

• Position Feedback
• U k(Xref X)
• PD Feedback ()
• U k1(dXref dX) k2(Xref X)
• Measure both position and velocity
• The transfer function from Xref to X has
  dynamics. These multiply times the desired
  impedance

17

• PD Feedforward Control
• U k1(d2Xref) k2(dXref dX) k3(Xref X)
• Impedance correct
• Creates an algebraic loop

18
Variations (contd)

• Feedforward control and an inverse plant to
  cancel robot dynamics
• Liu and Goldenberg, 1991
• Variable Structures to compensate for stiction,
  other nonlienarities
• Lu and Goldenberg, 1991,
• Zu and Goldenberg, 1995
• Saturating control to eliminate chatter of
  Variable Structures controls

19
Force-Control Based Impedance Controllers

• Build upon a force controller
• This also creates a feedback loop with the
  environment
• Frequency decoupling helps assure stability

20
Force Control Impedance
21
Force Control Impedance Equations
22
Problems with Force-Control based Impedance
Control

• Frequency decoupling is a huge plus
• Current can respond very quickly (microseconds)
• The position of the robot responds very slowly
• T KtIa ZdX
• Dependence on the environment does not make sense
• A 100 Ohm resistor does not depend upon the
  circuit to be a 100 Ohm resistor
• High-gain feedback and a need for a
  second-derivative will create noise issues
• X (spring),
• dX/dt (friction), and
• d2X/dt2 (inertia)

23
Four ideas for a better solution

• Assume a DC Servo motor serves as the impedance
  control
• Control the mechanical impedance with an
  electrical circuit
• Va G(s)Ia
• Va G(s) Angle
• Ia G(s) Angle
• Va (periodic basis) constant (repetitive
  control)

24
Va G(s)Ia Equations
25
Va G(s)Ia Comments

• Works very well for G(s) 0 or infinity
• Short the motor or leave it open
• Capacitors dont act as springs
• You need to differentiate current
• You need an unstable controller.
• Open-loop unstable
• Closed-loop stable (in theory)

26
Va G(s) Angle Equations
27
Va G(s)Angle Comments

• If J and L are small, you only need to measure
  angle and velocity (good!)
• If speed is slow, you only need to measure angle
  and velocity (good!)
• In general you need to measure the third
  derivative of angle (bad)

28
Ia G(s)Angle Equations
29
Ia G(s)Angle Comments

• Very simple controller
• PID angular velocity
• Does not depend upon the environment
• If PID gains positive, stability is assured by
  passivity
• Requires a current amplifier
• Most motor controllers have a current mode
• When in current mode, dont power up the
  controller if the motor isnt attached

30
Repetitive Control

• For a heart application, the input, output, and
  control are all periodic
• Using adaptive control techniques, you can use a
  gradient search to learn and compute this
  periodic function on the fly

31
Repetitive Control
32
Repetitive Control
33
ApplicationDesign by Malshitha Kankanamge