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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Introduction Implicit Runge-Kutta
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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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Introduction Implicit Runge-Kutta
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MathsExample
Conclusion
PolynomialsReformulationNonLinear Programming
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.
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IntroductionM h
Implicit Runge-KuttaP l i l
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MathsExample
Conclusion
PolynomialsReformulationNonLinear Programming
Polynomial Representation
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.
Gilbert Gede The Direct Collocation Method for Optimal Control
IntroductionM ths
Implicit Runge-KuttaP l i ls
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MathsExample
Conclusion
PolynomialsReformulationNonLinear Programming
Polynomial Representation
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
PolynomialsReformulationNonLinear Programming
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
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IntroductionMaths
Implicit Runge-KuttaPolynomials
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MathsExample
Conclusion
PolynomialsReformulationNonLinear Programming
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.
Gilbert Gede The Direct Collocation Method for Optimal Control
IntroductionMaths
Implicit Runge-KuttaPolynomials
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ExampleConclusion
ReformulationNonLinear Programming
NonLinear Programming
NLP is the most general case of local
optimization.
Gilbert Gede The Direct Collocation Method for Optimal Control
IntroductionMaths
Implicit Runge-KuttaPolynomials
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ExampleConclusion
ReformulationNonLinear Programming
NonLinear Programming
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.
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IntroductionMaths
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Implicit Runge-KuttaPolynomialsR f l i
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ExampleConclusion
ReformulationNonLinear Programming
Using NLP
Now it is clear what the previously defined defectswill be used for: inputs into the NLP problem asequality constraints.
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ExampleConclusion
ReformulationNonLinear Programming
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.
Gilbert Gede The Direct Collocation Method for Optimal Control
IntroductionMaths
Example
Implicit Runge-KuttaPolynomialsReformulation
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ExampleConclusion
ReformulationNonLinear Programming
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).
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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
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pConclusion
Results
What is This System?
This system is analogous to a mass with velocity in onedirection.
Gilbert Gede The Direct Collocation Method for Optimal Control
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DescriptionImplementationResults
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pConclusion
Results
What is This System?
This system is analogous to a mass with velocity in onedirection.
We want to switch the sign of that velocity within the timeinterval, and end at the starting position.
Gilbert Gede The Direct Collocation Method for Optimal Control
IntroductionMaths
Example
DescriptionImplementationResults
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ConclusionResults
What is This System?
This system is analogous to a mass with velocity in onedirection.
We want to switch the sign of that velocity within the timeinterval, and end at the starting position.
The mass isnt allowed to move further than laway from thestarting position in the positive direction.
Gilbert Gede The Direct Collocation Method for Optimal Control
IntroductionMathsExample
C l i
DescriptionImplementationResults
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ConclusionResults
What is This System?
This system is analogous to a mass with velocity in onedirection.
We want to switch the sign of that velocity within the timeinterval, and end at the starting position.
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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IntroductionMathsExample
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.
Gilbert Gede The Direct Collocation Method for Optimal Control
IntroductionMathsExample
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.
While it has shortcomings, it is very good at
problems which do not encounter a lot of inequalityconstraints.
Gilbert Gede The Direct Collocation Method for Optimal Control
IntroductionMathsExample
Conclusion
ConclusionReferences
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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.
Gilbert Gede The Direct Collocation Method for Optimal Control
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