GMV's Algorithms & Techniques for Low-Thrust Trajectory - ESA

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GMV’s Algorithms & Techniques for Low-Thrust Trajectory Design

3rd ESA Workshop on Astrodynamics Tools and Techniques

Jesús Gil-Fernández

Carlos Corral van Damme

10/05/06 Page 2GMV’s Algorithms & Techniques for Low-Thrust Trajectory Design © GMV, 2006

1. Introduction

2. Direct Methods

3. Indirect Methods

4. Hybrid Methods

5. Conclusions



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Introduction



Maturity of EP as main propulsion in interplanetary missions

Deep Space 1 (1998) SMART-1 (2003) Hayabusa/MUSES-C (2003)

Trade off at early stage of mission design

Chemical (impulsive) vs. electrical (low thrust)

Conversion of impulsive big delta-V into finite burns

Optimization of thrust direction & burns duration

Motivation

Deep Space 1 (credits JPL)


Flexible configuration for different problems Different objective functions

– Minimum time– Maximum mass– Maximum target deflection

Different boundary conditions– Orbit transfer– Rendezvous– Impact/Fly-by

Multiple swing-by (gravity assists)– Multiple fly-by

Different launchers– Ariane5, Soyuz/Fregat, Dnper, constant m0

Different thruster models Different thrust/gravity acceleration

– Low-thrust vs. very low-thrust

Mission types

Hayabusa (© JAXA)


Different approaches to cope with the different mission types Different optimization methods

– Direct, Indirect, Hybrid Different types of initial guess Different NLP methods

– Gradient-based vs. Non-gradient– Separate objective&constraints vs.

Penalty function All combinations of free/fixed initial &

final times– Trajectory decomposed into sub-arcs

Dynamics – Cartesian coor. and orbital elements– Forward and backward propagation– Switching function vs. fixed burns

duration

Methodology

Bepi Colombo


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Direct Methods



Find the extreme of objective function fulfilling constraints Finite set of variables and constraints Variety of methods for discretization

– Transcription– Collocation– Pseudo-spectral

Solve the resulting NLP problem– Sequential Quadratic Programming

• Line search– Simplex– Genetic Algorithms

Main advantages (vs. indirect)– Higher convergence radius

• Less number of equations• Lower sensitivity to initial guess

– Less problem dependent formulation

Overview

© Mathworks

© AIAA (1990)


Solves two-point BVP Maximum final mass Optimal thrust direction & magnitude

Methodology Global search

– Shape-based technique: • Exponential sinusoid (Petropoulos)

– Genetic Algorithms– Initial guess for local search

Local optimization– Direct transcription– Available NLP solvers

• Simplex (Nelder-Mead)• SNOPT• Matlab optimization toolbox algorithms

– Solves thrust level constraint

inilow (1/2)

Earth – Temple1


Optimization scheme

inilow (2/2)

Global Search: Shape Based Technique

Genetic Algoritms

Out-Of -PlaneIn-Plane

Trajectory Refinement

Direct Transcription Local Optimiser

Simplex

Snopt

SOLUTION

Problem Definition


Optimization of EP closing maneuver Problem statement

– Closing from 35km to –1km in V-bar– Burning time ~25% of transfer time– Constant thrust direction ( V-approach) not

valid• High propellant expenditure due to TCM• Final residual velocity (drift during OD)

Fast & robust optimization required for operational purposes– Cost: propellant mass– Optimization parameters: burn times & thrust

direction (angles expansion)– Dynamics: Hill equations with perturbations

• Final state obtained by quadrature (low numerical sensitivity)

– Gradient method• Good initial guess (impulsive approximation)• Fast computation of derivatives

Validation in realistic simulations– Final error due to navigation uncertainty

Rendezvous maneuvers

Constant thrust direction (optimal impulse)

Starting Point

Starting Point

Optimal thrust direction (finite burn)


Autonomous low-thrust guidance Compensate deviations from nominal

trajectory– On-board algorithm

Discretization– Thrust arc segmentation– Piece-wise linear approximation of control

• Thrust magnitude, spherical angles Problem statement

– Minimize control variations (times+angles)– Constrained control– Newton method + weighted line search

• Fast convergence• Avoid wandering of Newton method

Open-loop validation– Missions: SMART-1 & BepiColombo– Monte-Carlo

Closed-loop validation– AutoNav simulator

Low-thrust guidance


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Indirect Methods



Find extremal functions (optimal control): fulfill the optimality conditions Based on variational calculus

(Pontryagin, Lawden) New features appear

– Adjoints (or co-states)– Multi-Point BVP– Switching points (due to constraints)

Methods to solve the MPBVP– Multiple shooting– Gradient-Restoration (Miele)– Discretize into a NLP problem

Main advantages (vs. direct)– Reduced set of parameters for control

representation– Reduced # of optimization variables– May give idea of solution structure

Overview

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Optimal control formulation of simple problem


ApplicationDirect transfers to asteroids

Objective function– Impact: max. target major-axis variation– RendezVous: max. final mass

Optimization variables:– Launch and arrival dates– Launcher delta-V– Initial adjoints

• Adjoint-control transformation reduce sensitivity and search space

Thrust structure defined by switching function– High sensitivity of equations & solution

Solver– Global search with genetic algorithm– Local search with simplex


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Hybrid Methods



Try to get the advantages of each method

Direct methods– Direct evaluation of objective function– Less sensitivity of solution– Physical meaning of optimization variables

Indirect methods– Optimal control analytical representation– Reduced set of optimization variables– Optimality conditions assure extremality of solution (at least locally)

Final remark

A suitable formulation allows easy set-up of different problems– Similar physics and numerical issues

Overview


Transform impulsive mission into low-thrust mitrades provides feasible impulsive trajectory

– Minimum delta-V– Impact with an asteroid– Maximum number of visited asteroids (2 additional flyby)

Applications (1/4)

Hybrid algorithm provides feasible low-thrust trajectory– Impulsive trajectory is initial guess

• A priori knowledge of thrusting structure

– Direct maximization of final mass– Discretization by arc decomposition– NLP solver

• SQP provides fast convergence• Simplex is slower

– Negligible difference (smooth function)


Pure low-thrust missions

Mission design Reconstruction and/or refinement

Example: Rendezvous with Nereus

Soyuz/Fregat launcher Low-thrust EGA Assumed thrust structure Free dates

– Launch, EGA, arrival

Applications (2/4)


merpro

Application to optimal abort and re-entry trajectories

Constrained optimization– Optimize mass – Assumed thrust structure

• Multiple arc decomposition

Reduced set of optimization variables for each arc– Optimal thrust direction and burn

duration– Optimal lift direction (re-entry)

NLP solver– Global search: genetic algorithms– Local optimizer: simplex

Applications (3/4)

Abort 2 Atlantic & abort 2 orbit trajectories

Abort 2 Woomera trajectories


Very low thrust

Specific problems– Numerical stability– Computational time & load– Fidelity of control representation– Initial guess construction

Application of optimization methods requires tailoring– New techniques appear: averaging

GMV solution: geoexpress– Take advantage of physics of problem– Problem decomposition

• Optimal control for fast-evolution sub-problem• Direct method for outer-loop (secular trajectory)

Applications (4/4)


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Conclusions



Preferred optimization method depends on:

Problem Expertise Resources …

Human intervention required for each new problem

Different mission types usually require formulation tailoring Initial guess is usually crucial Solution analysis required

– Usually no a priori existence theorems– Feasibility– Optimality– Convergence criteria

Conclusions


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Thank you

Jesús Gil-Fernández

Mission Analysis and Advanced Studies

Email: [email protected]

www.gmv.com

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GMV's Algorithms & Techniques for Low-Thrust Trajectory - ESA