Rigid motions

1
Controlling Manipulators

• Lets consider the problem of controlling a
  manipulator
• The inertial dynamics result in non-linear
  torques experienced at the joints.
• We will extend approaches from SISO control to
  this problem.

2
Electric dynamics of a DC motor

• The motor is essentially an inductor in series w/
  a resistor
• The dynamics of this circuit are

net voltage in circuit
3
Mechanical dynamics of the joint

• Suppose theres an r-to-1 gear box on the motor
  output
• The velocity on the output of the gearbox is
  times the velocity of the motor

Therefore, the gearbox reduces the effect of any
disturbances applied to the gearbox output by a
factor of
4
Modeling a DC motor
The torque applied by the motor is
Motor torque constant

• Because the motor is turning in a magnetic field,
  it experiences a back EMF in the rotor
• The motion of the motor causes an opposing
  voltage proportional to angular velocity

5
Combined dynamics of the joint
Mechanical dynamics
Electrical dynamics
6
Combined dynamics of the joint
disturbance torque
7
Dynamics of the joint manipulator
In reality, that disturbance torque is partly
due to the (as yet) un-compensated inertial
dynamics of the manipulator.
8
Motion Control

• Motion control problem consists of
• Achieve desired position, velocity, acceleration
  in either joint space or operation space
• Manipulator inertial dynamics can make this
  difficult.

9
Independent joint control of the manipulator
For starters, lets just do PI, PD, PID control
around each joint independently of the others
Simplify plant model
where
10
Independent joint PI control
Whats the root locus of this look like?
Unstable
Stable
Best
11
Independent joint PID control

• Use one of these zeros to cancel the off-origin
  pole
• whats the root locus of this look like?

12
Independent joint rate feedback PID control
13
Disturbance torques are reduced by the gear ratio

• Increasing K_p reduces the magnitude of
  disturbances.

14
Combining motor and manipulator dynamics
Motor transfer fn from before
In the time domain
15
Combining motor and manipulator dynamics
These are diagonal matrices containing the
parameters for each joint.
16
Compensating for gravity
PD rate feedback control (notice no integrator)
where
At equilibrium
17
Compensating for gravity
Resulting equation of motion
At equilibrium
18
Inverse dynamics control
19
Inverse dynamics control
20
Control law partitioning
Non-linear compensation
Linear controller
Non-linear compensation
Linear controller
21
Operational space (Cartesian) control
Jacobian (pseudo)inverse control
Jacobian transpose control
22
Operational space PD control
With gravity compensation

• requires fast computation of Jacobian

23
Operational space inverse dynamics control
Desired dynamics behavior
where
Passive manipulator dynamics
First, partition control law
Then, solve
for
24
Operational space inverse dynamics control
Therefore
25
Operational space inverse dynamics control
26
Interaction Control

• Interaction control problem consists of
• Achieve desired impedance, force characteristics
  w.r.t. environment.

27
When motion control comes into contact
What happens when the manipulator contacts the
environment under operational space inverse
dynamics control?

• What happens when the manipulator contacts the
  environment under operational space PD control?
• The environment will be modeled as a stiff spring
  (elastic environment)

28
When motion control comes into contact
Assume that the environment is modeled as a stiff
spring (elastic contact)
elastic model of contact
Operational space PD control
At steady state
steady state relationship
29
When motion control comes into contact
If were using joint space PD control instead
steady state relationship
joint space compliance
equivalent operational space compliance
30
Impedance control
Design a controller with the following equation
of motion at the end effector

• You must be able to measure the wrench to
  implement this
• Otherwise, you get unavoidable coupling among
  joints

Use the following partitioning scheme
measured wrench
31
Impedance control
Manipulator equation of motion
this wrench displaces end effector
Substituting
You get
Solving for
32
Impedance control
Solving for
Of course, your force measurements must be
accurate
33
Impedance control equivalent block diagram
Solving for
34
Force control inner velocity loop

• Feedback loop on interaction force
• One possibility PD loop on force a force
  derivative term is noisy

• Instead, modify the impedance control loop
• Impedance controller
• Set to zero

force error gain
What does steady state look like?
In steady state, position depends on environment
position and reference wrench
35
Force control equivalent block diagram
Solve for acceleration
36
Hybrid force/position control
Used when you want to apply forces in some
directions and control motion in other directions.

• Consider a diagonal selection matrix, ,
  with elements either one or zero
• The two matrices, and are
  orthogonal
• One of these matrices identifies the force
  directions
• The other identifies the motion directions

37
Hybrid force/position control