a multiple-shooting differential dynamic programming algorithm
TRANSCRIPT
A Multiple-Shooting Differential Dynamic Programming Algorithm
Etienne PellegriniRyan P. Russell
27th Spaceflight Mechanics Meeting, San Antonio, TX02/06/2017
Summary
• Introduction and Background
• Multiple-Shooting Problem Formulation
• Solving the multiple-shooting problem- Augmented Lagrangian methods- Single-leg expansions- Multi-leg expansions
• Numerical results- Validation of the quadratic expansions and updates- Van Der Pol Oscillator- 2D Spacecraft Orbit Transfer
• Conclusions and future work
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Introduction and Background
• Modern spacecraft trajectories are increasingly complex (flight times, multiple fly-bys, tour design)
• Need high-fidelity solvers (combined to global search tools for preliminary design)
• Goals:- Improve robustness- Improve computational efficiency
GTOC 1 trajectory, www.esa.int
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Motivations
• Multi-phase capabilities: - Separation at flybys, interception, rendez-vous, etc… - Each phase can have different dynamics, constraints, etc…
• Multi-shooting framework:- Allows for the decoupling of the legs and phases- Reduces sensitivities and improves robustness- Parallel implementation (solving each leg independently).
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Hybrid Differential Dynamic Programming
Classic NLP Solvers DDP Methods
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Summary
• Introduction and Background
• Multiple-Shooting Problem Formulation
• Solving the multiple-shooting problem- Augmented Lagrangian methods- Single-leg expansions- Multi-leg expansions
• Numerical results- Validation of the quadratic expansions and updates- Van Der Pol Oscillator- 2D Spacecraft Orbit Transfer
• Conclusions and future work
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Problem Statement
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The Multi-Shooting Subinterval: the Legs
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Discretization
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Multi-Shooting Problem Statement
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Summary
• Introduction and Background
• Multiple-Shooting Problem Formulation
• Solving the multiple-shooting problem- Augmented Lagrangian methods- Single-leg expansions- Multi-leg expansions
• Numerical results- Validation of the quadratic expansions and updates- Van Der Pol Oscillator- 2D Spacecraft Orbit Transfer
• Conclusions and future work
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• Mixed approach:- Box constraints on the state or controls- “Hard’’ constraints, can not be violated to first order by the
update laws- Accounted for using constrained quadratic programming- In MDDP as it is, using a null-space method in the TR algorithm
- Terminal constraints (intra- and inter-phase) and path constraints
- Accounted for using an augmented Lagrangian technique
Handling the constraints
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The Augmented Lagrangian Algorithm
• Penalization method for constrained optimization• Transforms a constrained problem into an unconstrained
one by adding a penalty and Lagrange multipliers to the cost function:
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Single-Leg Problem
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Single-Leg Quadratic Expansions
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Multi-leg Quadratic Expansion
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Multi-Leg Quadratic Expansion
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Inner Loop of the MS Algorithm
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Lagrange Multipliers Update
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The MDDP Algorithm with AL
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Summary
• Introduction and Background
• Multiple-Shooting Problem Formulation
• Solving the multiple-shooting problem- Augmented Lagrangian methods- Single-leg expansions- Multi-leg expansions
• Numerical results- Validation of the quadratic expansions and updates- Van Der Pol Oscillator- 2D Spacecraft Orbit Transfer
• Conclusions and future work
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Validation and Numerical Results
• Validation of the quadratic expansions and updates is done using a quadratic problem with linear constraints:
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Linear Quadratic Problem
Initial States Final States
Controls
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Van Der Pol Oscillator
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Classical VDP
Final State Controls
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Minimum-Time VDP
Final State Controls
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20-Leg Solution of Min-Time VDP
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2D Spacecraft Orbit Transfer
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2DSpacecraft SolutionInitial State
Final State
Controls
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Single Shooting
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Multiple-Shooting
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Conclusions & Future work
• The theoretical developments necessary to the formulation of a multiple-shooting differential dynamic programming algorithm are presented for the first time.
• An algorithm based on augmented Lagrangian methods and multiple-shooting DDP is described and tested, allowing to confirm:- The validity of the quadratic expansions and update equations- The applicability of the multiple-shooting principles to DDP- The resulting reduction in sensitivity for the subproblems
• Future work:- Parallel implementation- Robustness and performance analysis on more complex test problems
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