IVR:Control Theory

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IVRControl Theory

• OVERVIEW
• Control problems
• Kinematics
• Examples of control in a physical system
• A simple approach to kinematic control

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The control problem
Outcome
Action
Goal
Motor command
Robot in environment

• For given motor commands, what is the outcome?
  ? Forward model
• For a desired outcome, what are the motor
  commands? ? Inverse model
• From observing the outcome, how should we adjust
  the motor commands to achieve a goal?
  ? Feedback control

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The control problem
command voltage torque
force angle position
camera

• Forward kinematics is not trivial but usually
  possible
• Forward dynamics is hard and at best will be
  approximate
• But what we actually need is backwards kinematics
  and dynamics
• 
  Difficult!

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Inverse model

• Find motor command given desired outcome
• Solution might not exist
• Non-linearity of the forward transform
• Ill-posed problems in redundant systems
• Robustness, stability, efficiency, ...
• Partial solution and their composition

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Problem Non-linearity

• In general, we have good formal methods for
  linear systems
• ReminderLinear function
• In general, most robot systems are non-linear

F(x)
x
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Kinematic (motion) models
Example A simple arm model

• Differentiating the geometric model provides a
  motion model (hence sometimes these terms are
  used interchangeably)
• This may sometimes be a method for obtaining
  linearity (i.e. by looking at position change in
  the limit of very small changes)

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Differential Equations
Using known relations between quantities and
their rate of change in order to find out how
these quantities change

• Mathematics Equation that is to be solved for
  an unknown function
• PhysicsDescription of processes in nature
• EngineeringRealizability of a goal by a plant
  by including control terms
• Informatics Tool for realistic modeling

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Differential Equations
fast growth starting from initial value x(t0)x0
decay with time scale -1/at
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Dynamic models

• Kinematic models neglect forces motor torques,
  inertia, friction, gravity
• To control a system, we need to understand the
  continuous process
• Now An Example for control of a physically
  realistic model
• Next A simple example of control

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Dynamic models

• Kinematic models neglect forces motor torques,
  inertia, friction, gravity
• To control a system, we need to understand the
  continuous process
• Start with simple linear example

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Example Electric motor

• Ohms law Kirchhoff's law
• Motor generates voltage
• proportional to speed
• Vehicle acceleration
• (M is a motor constant)
• Torque t is proportional to current
• Putting together

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General form

• VB Control variable input
• s State variable output
• ABd/dt Process dynamics
• Dynamics determines the process, given an initial
  state s(t0)s0.
• State variable s(t) separates past and future
• Continuous process models are often differential
  equations!

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Process Characteristics

• Given the process, how to describe the behaviour?
• Concise, complete,
  implicit, obscure

• Characteristics
• Steady-state What happens if we wait for the
  system to settle, given a fixed input?
• Transient behaviour What happens if we suddenly
  change the input?
• Frequency response What if we smoothly/regularly
  change the inputs?

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Control theory

• Control theory provides tools
• Steady-state ds/dt 0,
• Transient behaviour (e.g. change in voltage from
  0 to 7V) exponential decay
  towards steady state
• Half-life of decay

(Solve for
using )
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Example

• Suppose
  M vehicle mass
  R setti
  ng
• If robot starts at rest, and apply 7 volts
• Steady state speed
• Half-life
• Time taken to cover half the gap between current
  and steady-state speed

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Motor with gears
Battery voltage VB
Gear ratio g where more gear-teeth near output
means g gt 1
?
smotor
sout

smotor g sout for g gt 1, output velocity is
slower torquemotor g -1 torqueout for g gt 1,
output torque is higher
Thus Same form, different steady-state,
time-constant etc.
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Motor with gears

• Steady-state
• Half-life
• i.e. for ? gt 1, reach lower speed in faster time,
  robot is more responsive, though slower.
• N.B. we have modified the dynamics by altering
  the robot morphology.

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Electric Motor Over Time

• Simple dynamic example
• We have a process model
• Solve to get forward model
• Derivation of this andmore general cases using
  e.g. Laplace transformation

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A fairly simple control algorithm
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A Simple Controller
System
System Controller
a simple choice for the controller
a simple choice for prediction xpred
xold
What if there is no analytical description of the
system? Stabilizing controller for box pushing
or wall-following more complex behaviors for
more complex predictors
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A Simple Controller
How to find better parameters ci in K S ci xi
?
cexpl c a sin(w t)
?c lt E(t) sin (w t) gt
short-term average
Perform test actions at both sides of the
trajectory works best in 1D (e.g. for steering)
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Summary

• Forward and inverse models
• Calculating control is hard but not impossible
  for many control problems
• Controlling by probing
• Feed back control (next time)

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Beyond Inverse Models

• Feed-back control
• Dynamical systems
• Adaptive control
• Learning control

1788 by James Watt following a suggestion from
Matthew Boulton