20/10/2009 IVR Herrmann IVR: Introduction to Control OVERVIEW Control systems Transformations Simple control algorithms.

20/10/2009 IVR Herrmann

IVR: Introduction to Control

OVERVIEW

Control systems

Transformations

Simple control algorithms


History of control Boulton and Watt’s governor

J. C. Maxwell (1868) On Governors.

Wright Brothers (1899)pilot control rather than “inherent stability”

flight = wings + engines + control


Examples of Control

Pilot control Cruise control Robot control Electronics Power control Thermostat Fire control Process control Space craft control Homeostasis &

biological motor control Control in economy


Control Example

• Dynamical system(plant)

• Continuous states

• Physical inputs and outputs (to the system)

• Control actuators

• Controller

• A room(containing air)

• Temperature at certain points in the room

• Heater and measurement device

• A way of switching the heater on or off

• Thermostat


Questions & Problems

What control strategy? Stability

Does the system really behave as desired? Controlability and observability

Are the critical variables accessible and measurable Delays

Is the measurement up to date, when does the control take effect?

Efficiency Can the same effect be achieved with less effort?

Adaptivity Is the control strategy appropriate for changing conditions?


Controller ‘A device which monitors

and affects the operational conditions of a given dynamical system’

The controller receives “outputs” and adjusts “input” variables.

It may also receive signals from a (human) operator or from another controller

Controllers often affect the “outputs” to stay close to a desired setpoint (homeostasis)

“Input” from the controlled system can serve as feedback telling the controller to what extend the control goal was achieved

Controller

System

input

output


Dynamical systems

• Differ from standard computational view of systems:– Perception-Action loop rather than

input → processing → output– Analog vs. digital, thus set of states describe a

state-space, and behaviour is a trajectory• On-going debate whether human cognition is better

described as computation or as a dynamical system (e.g. van Gelder, 1998)


The control problem

How to make a physical system (such as a robot) function in a specified manner?

Particularly when:• The function would not happen naturally• The system is subject to a large class of

changes

e.g. get the mobile robot to a goal, get the end-effector to a position, move a camera…


“Bang-bang” control

• Simple control method is to have physical end-stop…

• Stepper motor is similar in principal:

Moff


The control problem

• For given motor commands, what is the outcome?

• For a desired outcome, what are the motor commands?

• From observing the outcome, how should we adjust the motor commands to achieve a goal?

Motor command

Robot in environment

OutcomeGoal

= Forward model

= Inverse model

= Feedback control

Action


Want to move robot hand through set of positions in task space: X(t)

X(t) depends on the joint angles in the arm A(t)A(t) depends on the coupling forces C(t)delivered by transmission from motor torques T(t)T(t) produced by the input voltages V(t)

A less-than-perfectrobotic arm


The control problem

Depends on:• Kinematics and geometry: Mathematical

description of the relationship between motions of motors and end effector as transformation of coordinates

• Dynamics: Actual motion also depends on forces, such as inertia, friction, etc…

V(t) T(t) C(t) A(t) X(t)command voltage torque force angle position camera


The control problem

• 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!

V(t) T(t) C(t) A(t) X(t)


Inverse model

(V given X)Solution might not existNon-linearity of the forward transformIll-posed problems in redundant systems Robustness, stability, efficiency, ... Partial solution and their composition

V(t) T(t) C(t) A(t) X(t)


Beyond Inverse Models

Feed-back controlDynamical systemsAdaptive controlLearning control

1788 by James Watt following a suggestion from Matthew Boulton


Problem: Non-linearity

• In general, we have good formal methods for linear systems

• Reminder:

Linear function:

• In general, most robot systems are non-linear

bxaxF )(

)()()( 2121 xFxFxxF

x

F(x)


Kinematic (motion) models

• Differentiating the geometric model provides a motion model (hence sometimes these terms are used interchangeably)

y = F(x)

• This may sometimes be a method for obtaining linearity (i.e. by looking at position change in the limit of very small changes)

)()( ygxfdt

dyy if x=x(y)


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:

Battery voltage

VB

Vehicle speed

s? VB

IR

e


A fairly simple control algorithm

Compensator

High-frequencyoscillator

Compensator in orderto determine the effector

characteristicsEffector

High passfilter

Control of the compensator characteristics

Addition

N. Wiener: Cybernetics, 1948


K = Σ ci xi

e.g. xpred

= xold

System: dx/dt = f(x)

System + Controller

What if system description is not analytically given?

Stabilizing controller for box pushing or wall-followingmore complex behaviors for more complex predictors

A Simple Controller


How to find better parameters ci in K = Σ ci xi ?

Perform “test actions” at both sides of the trajectory works best in 1D (e.g. for steering)

A Simple Controller

cexpl

= c + a sin(w t)

Δc = < E(t) sin (w t) >

short-term average


Summary

Control systemsCalculating control is hardControlling by probing

Standard control schemes (next time)

20/10/2009 IVR Herrmann IVR: Introduction to Control OVERVIEW Control systems Transformations Simple control algorithms.

Documents

control goal

control problemhow

motor commands

control strategy appropriate

physical system

moffthe control problemfor

motor commandrobot

physical endstopstepper