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Low-thrust Interplanetary Trajectories with Missed Thrust Events: a Numerical Approach

Jean-Philippe Chancelier, Pierre Carpentier, Guy Cohen, Thierry Dargent, Richard Epenoy

TL;DR

This work addresses robust design of low-thrust interplanetary trajectories in the presence of missed thrust events by formulating a stochastic optimal control problem with a probability constraint on reaching a rendezvous by final time $t_f$. Starting from a deterministic fuel-minimization formulation, the authors model engine failures as a stochastic event, reformulate the problem in terms of conditional expectations and a smoothed indicator for the constraint, and solve it using a stochastic Arrow-Hurwicz gradient algorithm. A key contribution is the inner problem approximation and the use of a no-failure scenario to reduce variance, enabling a tractable approach to balance safety (probability of success) against fuel consumption. Numerical results on a real interplanetary mission show how increasing the required success probability raises the multiplier and the fuel cost, with a noticeable breakpoint near $p\approx0.96$, highlighting the trade-off between robustness and economic efficiency. The study demonstrates a viable quantitative framework for resilient mission planning, while acknowledging substantial computational demands and avenues for theoretical improvements.

Abstract

The problem under consideration is to drive a spatial vehicle to a target at a given final time while minimizing fuel consumption. This is a classical optimal control problem in a deterministic setting. However temporary stochastic failures of the engine may prevent reaching the target after the engine usage is recovered. Therefore, a stochastic optimal control problem is formulated under the constraint of ensuring a minimal probability of hitting the target. This problem is modeled, improved and finally solved by dualizing the probability constraint and using an Arrow-Hurwicz stochastic algorithm. Numerical results concerning an interplanetary mission are presented.

Low-thrust Interplanetary Trajectories with Missed Thrust Events: a Numerical Approach

TL;DR

This work addresses robust design of low-thrust interplanetary trajectories in the presence of missed thrust events by formulating a stochastic optimal control problem with a probability constraint on reaching a rendezvous by final time . Starting from a deterministic fuel-minimization formulation, the authors model engine failures as a stochastic event, reformulate the problem in terms of conditional expectations and a smoothed indicator for the constraint, and solve it using a stochastic Arrow-Hurwicz gradient algorithm. A key contribution is the inner problem approximation and the use of a no-failure scenario to reduce variance, enabling a tractable approach to balance safety (probability of success) against fuel consumption. Numerical results on a real interplanetary mission show how increasing the required success probability raises the multiplier and the fuel cost, with a noticeable breakpoint near , highlighting the trade-off between robustness and economic efficiency. The study demonstrates a viable quantitative framework for resilient mission planning, while acknowledging substantial computational demands and avenues for theoretical improvements.

Abstract

The problem under consideration is to drive a spatial vehicle to a target at a given final time while minimizing fuel consumption. This is a classical optimal control problem in a deterministic setting. However temporary stochastic failures of the engine may prevent reaching the target after the engine usage is recovered. Therefore, a stochastic optimal control problem is formulated under the constraint of ensuring a minimal probability of hitting the target. This problem is modeled, improved and finally solved by dualizing the probability constraint and using an Arrow-Hurwicz stochastic algorithm. Numerical results concerning an interplanetary mission are presented.

Paper Structure

This paper contains 37 sections, 3 theorems, 63 equations, 7 figures.

Table of Contents

  1. Introduction and motivation
  2. Problem statement
  3. Literature review
  4. Paper's content
  5. Problem formulation
  6. Satellite model and deterministic problem
  7. Satellite model with engine failure during a time interval
  8. Engine failures and stochastic problem
  9. Initial stochastic formulation
  10. Formulation improvement
  11. Expectation formulation
  12. Missing the target should not contribute to the cost function
  13. Probability as an expectation
  14. Expectation based reformulation
  15. Introducing the no-failure scenario
  16. ...and 22 more sections

Key Result

Lemma 2

Under Assumption as:probalevel, Problem pb:stochexpect is equivalent to the following Problem pb:stochexpect-nofail

Figures (7)

  • Figure 1: Convergence and solution of the deterministic problem
  • Figure 2: Results for $p=0.550$ (top), $p=0.750$ (middle) and $p=0.925$ (bottom)
  • Figure 3: Results for $p=0.950$ (top), $p=0.960$ (middle) and $p=0.970$ (bottom)
  • Figure 4: Results for $p=0.980$ (top), $p=0.985$ (middle) and $p=0.990$ (bottom)
  • Figure 5: Results for $p=0.995$ (top) and $p=0.999$ (bottom)
  • ...and 2 more figures

Theorems & Definitions (5)

  • Remark 1
  • Lemma 2
  • Remark 3
  • Lemma 4
  • Lemma 5