Direct Collocation Presentation

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IntroductionMaths

ExampleConclusion

The Direct Collocation Method for OptimalControl

Gilbert Gede

May 26, 2011

Gilbert Gede The Direct Collocation Method for Optimal Control
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IntroductionMaths

ExampleConclusion

Outline1 Introduction

Why?Background

2 MathsImplicit Runge-Kutta

PolynomialsReformulationNonLinear Programming

3 Example

DescriptionImplementationResults

4 ConclusionConclusion

ReferencesGilbert Gede The Direct Collocation Method for Optimal Control
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IntroductionMaths

ExampleConclusion

Why?Background

What is This?

A method to solve optimal controlproblems.

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IntroductionMaths

ExampleConclusion

Why?Background

Which Optimal Control

Problems?

Usually, trajectory optimization, parameteroptization, or a combination thereof.

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I t d ti


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IntroductionMaths

ExampleConclusion

Why?Background

Why This Method?

From what I understand, the current optimization

softwares are better as you add constraints, even ifthe dimensionality increases.

This method does that, in addition to better

relating the states to the augmented cost function.

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Introduction


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IntroductionMaths

ExampleConclusion

Why?Background

History

From what I can tell, Hargraves 1987 paper in

J.G.C.D. seems to be the first major publication ofthis method.

I think use of the collocation method for optimalcontrol goes back to the 1970s, and for general useto the 1960s.

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Introduction Implicit Runge-Kutta


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IntroductionMaths

ExampleConclusion

Implicit Runge KuttaPolynomialsReformulationNonLinear Programming

ODEs

The collocation method is a way of solving

ODEs numerically.

This is actually an implicit Runge-Kutta

method.



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MathsExample

Conclusion

p gPolynomialsReformulationNonLinear Programming

ODEs

With the collocation method the states aredefined as functions of states and derivatives

(they are implicit).

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MathsExample

Conclusion


Polynomial Representation

With the prior knowledge of x (an x0 an x1) and f(x)(derivatives at those points), at s = 0 and s = 1, anddifferentiation of the polynomial, we can calculate thecoefficients.


IntroductionM h

Implicit Runge-KuttaP l i l
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MathsExample

Conclusion



We can make things simpler by examing midpoint of thisinterval. (s = 0.5)

This simplifies to:

xc=1

2(x1+ x2) +

t

8 (f1 f2) xc=

3

2t(x1 x2)

1

4(f1+ f2)

where f is the derivative of x, evaluated at x1 and x2.


IntroductionM ths

Implicit Runge-KuttaP l i ls
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MathsExample

Conclusion



Now we have an expression for the states and their

derivatives at the interval midpoint, as representedby a cubic polynomial.

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IntroductionMaths

Implicit Runge-KuttaPolynomials


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MathsExample

Conclusion


Polynomial Defects

We now are going to add equality constraints to theoptimization problem: the error in the polynomialrepresentation of the states.

This error is described by defects:

=f(xc) xc


IntroductionMaths

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MathsExample

Conclusion


Polynomial Defects

This defect is the difference between the derivative evaluated atthe approximated midpoint state, and the differentiation of theapproxiation.

Hargraves1987

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IntroductionMaths



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ExampleConclusion

yReformulationNonLinear Programming

NonLinear Programming

Nonlinear programming (NLP) is optimization of

nonlinear objective and constraint functions.

This is typically done by linearizing the functions ata point, then taking steps in the appropriatedirections.


IntroductionMaths

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ExampleConclusion

ReformulationNonLinear Programming


NLP is the most general case of local

optimization.


IntroductionMaths



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ExampleConclusion



NLP is the most general case of local

optimization.All functions can be nonlinear, and theconstraints can be inequality and/or

inequality constraints, allowing bounds to beplaced on variables.


IntroductionMaths

E l

Implicit Runge-KuttaPolynomialsR f l i
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ExampleConclusion


Using NLP

Now it is clear what the previously defined defectswill be used for: inputs into the NLP problem asequality constraints.


IntroductionMaths

E l

Implicit Runge-KuttaPolynomialsR f l ti
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ExampleConclusion


Using NLP

Now it is clear what the previously defined defectswill be used for: inputs into the NLP problem asequality constraints.

These will be driven to 0 and this must bemaintained, as the solver is free to explore different

areas in the input vector space in an attempt tominimize the objective function.


IntroductionMaths

Example

Implicit Runge-KuttaPolynomialsReformulation


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ExampleConclusion


Using NLP

I wont discuss too much mroe detail about usingNLP for these problems; it is simply the solver whichyou give your objective and constraint functions toand it returns an optimal result (hopefully).


IntroductionMaths

Example

DescriptionImplementation
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ExampleConclusion

Results

Simple Example

This is example is from vonStryk1993, which

was from Bryson before that.

We have a double integrator with specifiedboundary conditions.

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IntroductionMaths

Example

DescriptionImplementationR l


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ExampleConclusion

Results

Simple Example Cost and

Constraint

Cost function:min w (1)

Additional constraints:

l x(t) 0


IntroductionMaths

Example

DescriptionImplementationR lt
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pConclusion

Results

What is This System?

This system is analogous to a mass with velocity in onedirection.


IntroductionMaths

Example

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pConclusion

Results



We want to switch the sign of that velocity within the timeinterval, and end at the starting position.


IntroductionMaths

Example

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ConclusionResults




The mass isnt allowed to move further than laway from thestarting position in the positive direction.


IntroductionMathsExample

C l i

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ConclusionResults




The mass isnt allowed to move further than laway from thestarting position in the positive direction.

We want to minizie the integral of force over this time interval.

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Conclusion

ConclusionReferences


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Conclusion

Conclusion

The Direct Collocation Method has considerableadvantages over simpler shooting methods due to

the additionaly constraints (even when adding morevariables.



Conclusion

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Conclusion

Conclusion

The Direct Collocation Method has considerableadvantages over simpler shooting methods due to

the additionaly constraints (even when adding morevariables.

While it has shortcomings, it is very good at

problems which do not encounter a lot of inequalityconstraints.



Conclusion

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Conclusion

References

Hargraves, C. R., & Paris, S. W. (1987). Direct trajectory optimizationusing nonlinear programming and collocation. Journal of Guidance,Control, and Dynamics, 10(4), 338-342. doi: 10.2514/3.20223.

Von Stryk, O. (1993). Numerical solution of optimal control problems bydirect collocation. International Series of Numerical Mathematics,111(4), 1-13.

Runge-Kutta Methods Wikipedia, the Free Encyclopedia. 25 May2011. http://en.wikipedia.org/wiki/Runge-Kutta_methods.

http://en.wikipedia.org/wiki/Runge-Kutta_methodshttp://en.wikipedia.org/wiki/Runge-Kutta_methodshttp://en.wikipedia.org/wiki/Runge-Kutta_methodshttp://find/

Direct Collocation Presentation

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