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Convex Block-Cholesky Approach to Risk-Constrained Low-thrust Trajectory Design under Operational Uncertainty

Kenshiro Oguri, Gregory Lantoine

Abstract

Designing robust trajectories under uncertainties is an emerging technology that may represent a key paradigm shift in space mission design. As we pursue more ambitious scientific goals (e.g., multi-moon tours, missions with extensive components of autonomy), it becomes more crucial that missions are designed with navigation (Nav) processes in mind. The effect of Nav processes is statistical by nature, as they consist of orbit determination (OD) and flight-path control (FPC). Thus, this mission design paradigm calls for techniques that appropriately quantify statistical effects of Nav, evaluate associated risks, and design missions that ensure sufficiently low risk while minimizing a statistical performance metric; a common metric is Delta-V99: worst-case (99%-quantile) Delta-V expenditure including statistical FPC efforts. In response to the need, this paper develops an algorithm for risk-constrained trajectory optimization under operational uncertainties due to initial state dispersion, navigation error, maneuver execution error, and imperfect dynamics modeling. We formulate it as a nonlinear stochastic optimal control problem and develop a computationally tractable algorithm that combines optimal covariance steering and sequential convex programming (SCP). Specifically, the proposed algorithm takes a block-Cholesky approach for convex formulation of optimal covariance steering, and leverages a recent SCP algorithm, SCvx*, for reliable numerical convergence. We apply the developed algorithm to risk-constrained, statistical trajectory optimization for exploration of dwarf planet Ceres with a Mars gravity assist, and demonstrate the robustness of the statistically-optimal trajectory and FPC policies via nonlinear Monte Carlo simulation.

Convex Block-Cholesky Approach to Risk-Constrained Low-thrust Trajectory Design under Operational Uncertainty

Abstract

Designing robust trajectories under uncertainties is an emerging technology that may represent a key paradigm shift in space mission design. As we pursue more ambitious scientific goals (e.g., multi-moon tours, missions with extensive components of autonomy), it becomes more crucial that missions are designed with navigation (Nav) processes in mind. The effect of Nav processes is statistical by nature, as they consist of orbit determination (OD) and flight-path control (FPC). Thus, this mission design paradigm calls for techniques that appropriately quantify statistical effects of Nav, evaluate associated risks, and design missions that ensure sufficiently low risk while minimizing a statistical performance metric; a common metric is Delta-V99: worst-case (99%-quantile) Delta-V expenditure including statistical FPC efforts. In response to the need, this paper develops an algorithm for risk-constrained trajectory optimization under operational uncertainties due to initial state dispersion, navigation error, maneuver execution error, and imperfect dynamics modeling. We formulate it as a nonlinear stochastic optimal control problem and develop a computationally tractable algorithm that combines optimal covariance steering and sequential convex programming (SCP). Specifically, the proposed algorithm takes a block-Cholesky approach for convex formulation of optimal covariance steering, and leverages a recent SCP algorithm, SCvx*, for reliable numerical convergence. We apply the developed algorithm to risk-constrained, statistical trajectory optimization for exploration of dwarf planet Ceres with a Mars gravity assist, and demonstrate the robustness of the statistically-optimal trajectory and FPC policies via nonlinear Monte Carlo simulation.
Paper Structure (49 sections, 2 theorems, 69 equations, 8 figures, 5 tables, 1 algorithm)

This paper contains 49 sections, 2 theorems, 69 equations, 8 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Equations of Motion
  3. Risk-Constrained Nonlinear Stochastic Optimal Control Problem
  4. Nonlinear Stochastic System
  5. Initial State Dispersion
  6. Maneuver Execution Error
  7. Stochastic Differential Equations
  8. Nonlinear Stochastic System with Navigation Uncertainty
  9. Orbit Determination Model
  10. Flight-path Control Model
  11. Original Problem
  12. Statistical Cost Metric
  13. Risk Constraints
  14. Tractable Formulation via block-Cholesky Covariance Steering
  15. Linear State Statistics Dynamics
  16. ...and 34 more sections

Key Result

Lemma 1

Using Cayley transformation, the mapping of orbital states $\bm{x}_{k}$ to $\bm{x}_{k+1}$ due to a gravity assist is expressed as a nonlinear function of $\bm{x}_k, \bm{u}_k, t_k$, as follows: where $[\cdot]_{\times}$ denotes the matrix cross product operator, and $(I_3 + [\bm{u}_k]_{\times})^{-1}$ always exists. Also, its linearized equation can be expressed as $\bm{x}_{k+1} = A_k\bm{x}_k + B_k

Figures (8)

  • Figure 1: Fundamental idea of statistical trajectory optimization under uncertainty
  • Figure 2: $l_1$-like smooth penalty function $\phi(z)$ given in \ref{['eq:penalty-l1approx']} and its gradient for different $\tau$ for $z\in[-2,2]$, compared with $l_1$ exact penalty function $\left\lvert z\right\rvert$ and quadratic penalty $z^2$
  • Figure 3: Nonlinear Monte Carlo flow chart, illustrating a simulation flow from $t_k$ to $t_{k+1}$.
  • Figure 4: Deterministic optimal Earth-Mars-Ceres low-thrust transfer
  • Figure 5: Statistically-optimal Earth-Mars-Ceres low-thrust transfer solution
  • ...and 3 more figures

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Remark 4
  • Lemma 2
  • proof
  • proof
  • proof