Structure and Synthesis of Robot Motion

Topics in Control

Subramanian RamamoorthySchool of Informatics

19 February, 2009

In this Lecture

• We will survey some issues that occur in many realistic robotic systems and discuss control methods for dealing with them

• We will look at three ideas: predictive, adaptive and nonlinear control

• We will only try to get an understanding of what the essential issue is – detailed techniques beyond the scope of this lecture

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Start from the Basics: A Simple System

• Consider the mechanical system shown• From rest, if you apply force u(t), the position

y(t) changes in response• In fact, the response is given by an ODE:

• Define the state as:• You get equations of motion:

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m

b

ku(t)

y(t)

ukyybym

)()(),()( 21 tytxtytx

2

1

12

1

2

1

01

010

x

xy

ux

x

x

x

mmb

mk

How to Control the Simple System?

• For any system of the general form (very often D = 0),

• One could use many standard linear control techniques• For instance, linear quadratic regulator, u = - Kx

• So, in general, properties of such a system are essentially that of the canonical system,

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DuCxy

BuAxx

Cxy

xAxBKABKxAxx

ˆ)(

Axx

Properties of a Linear System

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Properties of a Linear System, contd.

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Makes design

convenient!t!

Is that It? Where are the Complications?

In real robotics problems, we face many difficulties that are not captured in the simple model.

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m

b

ku(t)

y(t)

Workspace may change (e.g., Obstacles and constraints)at “run-time”- No longer possible tofind and use global solutions

System dynamics oroperating conditionsmight change over time

Finally, most robot systems’ dynamics are more complex and non-linear

Ways to Deal with these Complications

• Model Predictive ControlGiven a basic system dynamics model, pose long-term control problem as a sequence of finite-horizon optimal control problems – constraints can then be easily accommodated

• Adaptive ControlIf system dynamics is going to change over time, estimate the changes continually and change controller accordingly

• Nonlinear ControlFinally, if the system dynamics is clearly nonlinear then one needs to do things differently – we will discuss linearization methods (many other possibilities exist!)

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Model Predictive Control – Basic Idea

Use a combination of system (sometimes called plant) model and state/control history to anticipate significant changes in control actions

Example: Cruise control system in a car• Fuel sent to engine depends on desired cruise velocity and

actual velocity of the car• Actual velocity depends on slope of the road (varies…)• An MPC would try to predict significant changes in slope and

correct for it

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Factors Influencing the MPC Problem

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How does MPC differ from Simple Feedback?

• What happens as the slope of the road increases?– Car velocity decreases– Fuel input to engine must be increased by

the controller

• In a traditional feedback controller (u = -Kx), car must first slow down for corresponding fuel increase – Controller catches up, but you have

oscillations• A predictive controller uses knowledge of

a finite-horizon forward in time to make corrections much earlier

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How does Predictive Control work?

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Use measurements to decide on set-points, e.g., is there a bump ahead?

Use model and set-point to solve an optimal control

problem over some horizon

Apply control action over a small horizon – no

change in control during remaining interval

Sometimes, map frommeasurements to outputs is not direct- Need to estimate intermediate states

Moving the Prediction Horizon

After each iteration, the prediction horizon is moved forward and a new finite-horizon optimization problem is started

An essential tradeoff:• If you use a long prediction horizon, you have a much larger

problem on each iteration• Short control horizons (w.r.t. prediction horizon) are desirable -

keeps the actions tame (aggressive changes get penalized in the ‘uncontrolled’ part of prediction horizon)

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MPC – Problem Formulation

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[Source: ZK Nagy, ECMI 2004]

Important Feature of MPC - Constraints

• Once the basic problem is setup as one of (numerical) optimal control, it becomes possible to incorporate constraints on the state and control equations– Inequality constraints, e.g., actuator bounds– Equality constraints, e.g., end-effector stays on a surface

• The fact that the problem is repeatedly solved over a finite horizon means that these constraints can come and go

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Adaptive Control

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A Core Idea in Adaptive Control

• Adaptive control is a well-established area with numerous possible architectures – we will not survey them all

• Instead we will only look at one core idea: on-line determination of process parameters

• If you can use a batch of data from a moving window to ‘identify’ the currently valid system model then we could adapt controller parameters accordingly

• Ideally, all this happens on-line – one data point at a time!

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Least Squares Estimation

Known since Gauss (late 18th century):

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Least Squares Example: Cubic Polynomial

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Least Squares can be Solved Recursively

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I am not going to show how these equations and quantities like K(t) are derived.If you are curious, please refer to:- K. Astrom, B. Wittenmark, Adaptive Control, Addison-Wesley 1995 (Ch 2)- S. Satry, M. Bodson, Adaptive Control (on-line version), Ch 2

Nonlinear Control

• Very few robotic systems actually exhibit linear dynamics across the entire state space– Variety of nonlinearity: limit stops, hysteresis, static friction, complex

dynamics (e.g., trigonometric terms in equations of motion)

• There are a number of different ways to cope with this:– One could deal with a specific nonlinearity in a “one-off” sense,

e.g., friction compensation– Many techniques (including MPC and Adaptive control) can

cope with nonlinearities in a ‘slowly varying’ sense• Another alternative is to use systematic linearization tools

from nonlinear control theory

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Simple Idea: Jacobian Linearization

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0 at equilibrium pointSimilar to a linear system,

But, this is only true in that small neighbourhood!

Better Approach: Feedback Linearization

• Algebraically transform nonlinear system dynamics into linear ones, so that linear control techniques can be applied

• Different from using the Jacobian – feedback linearization is achieved by exact state transformation and feedback.

• Used in many demanding applications ranging from mobile robotics and aircrafts to complex chemical processes

• Biggest disadvantage is reliance on a good model – need significant sophistication to ensure robustness in practice

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Feedback Linearization: Basic Procedure

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Summary

• This lecture surveyed three major techniques for controlling systems when they deviate from the assumptions of simple linear systems

• We looked at– Model Predictive Control– Adaptive control– Nonlinear control

• They all attack closely related issues but the focus of attention of each method is different

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