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Impulsive Relative Motion Control with Continuous-Time Constraint Satisfaction for Cislunar Space Missions

Fabio Spada, Purnanand Elango, Behçet Açıkmeşe

TL;DR

This work addresses constrained impulsive relative-motion planning in nonlinear cislunar dynamics by formulating an optimal control problem with free impulse timings and continuous-time path constraints. It introduces generalized time dilation to fix timing variables, exterior penalty-based constraint augmentation, and an isoperimetric reformulation within a sequential convex programming framework, enabling exact penalization and fast convergence. A NRHO-centered case study demonstrates two- and three-impulse strategies that achieve long residence times with millisecond-level per-iteration computation and superior efficiency over mesh-refinement approaches, while preserving safety constraints. The results highlight a practical, fast, and robust method for safety-critical cislunar maneuvers, with direct applicability to MPC-based mission planning. The approach improves computational tractability for continuous-time constraint satisfaction in highly nonlinear relative-motion problems.

Abstract

Recent investments in cislunar applications open new frontiers for space missions within highly nonlinear dynamical regimes. In this paper, we propose a method based on Sequential Convex Programming (SCP) to loiter around a given target with impulsive actuation while satisfying path constraints continuously over the finite time-horizon, i.e., independently of the number of nodes in which domain is discretized. Location, timing, magnitude, and direction of a fixed number of impulses are optimized in a model predictive framework, exploiting the exact nonlinear dynamics of non-stationary orbital regimes. The proposed approach is first validated on a relative orbiting problem with respect to a selenocentric near rectilinear halo orbit. The approach is then compared to a formulation with path constraints imposed only at nodes and with mesh refined to ensure complete satisfaction of path constraints over the continuous-time horizon. CPU time per iteration of 400 ms for the refined-mesh approach reduce to 5.5 ms for the proposed approach.

Impulsive Relative Motion Control with Continuous-Time Constraint Satisfaction for Cislunar Space Missions

TL;DR

This work addresses constrained impulsive relative-motion planning in nonlinear cislunar dynamics by formulating an optimal control problem with free impulse timings and continuous-time path constraints. It introduces generalized time dilation to fix timing variables, exterior penalty-based constraint augmentation, and an isoperimetric reformulation within a sequential convex programming framework, enabling exact penalization and fast convergence. A NRHO-centered case study demonstrates two- and three-impulse strategies that achieve long residence times with millisecond-level per-iteration computation and superior efficiency over mesh-refinement approaches, while preserving safety constraints. The results highlight a practical, fast, and robust method for safety-critical cislunar maneuvers, with direct applicability to MPC-based mission planning. The approach improves computational tractability for continuous-time constraint satisfaction in highly nonlinear relative-motion problems.

Abstract

Recent investments in cislunar applications open new frontiers for space missions within highly nonlinear dynamical regimes. In this paper, we propose a method based on Sequential Convex Programming (SCP) to loiter around a given target with impulsive actuation while satisfying path constraints continuously over the finite time-horizon, i.e., independently of the number of nodes in which domain is discretized. Location, timing, magnitude, and direction of a fixed number of impulses are optimized in a model predictive framework, exploiting the exact nonlinear dynamics of non-stationary orbital regimes. The proposed approach is first validated on a relative orbiting problem with respect to a selenocentric near rectilinear halo orbit. The approach is then compared to a formulation with path constraints imposed only at nodes and with mesh refined to ensure complete satisfaction of path constraints over the continuous-time horizon. CPU time per iteration of 400 ms for the refined-mesh approach reduce to 5.5 ms for the proposed approach.

Paper Structure

This paper contains 12 sections, 1 theorem, 28 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Notation
  3. PROBLEM DEFINITION
  4. TIME DILATION, CONSTRAINT AUGMENTATION, AND SCP FRAMEWORK
  5. Generalized Time Dilation
  6. Constraint reformulation
  7. Discretization, penalization and SCP
  8. NUMERICAL EXAMPLE
  9. Problem definition
  10. Numerical results
  11. Comparison with node-only constraints
  12. CONCLUSIONS

Key Result

Theorem 1

Path constraints are satisfied continuously over the full control horizon, i.e. for $\tau \in \bm{H}_i \;\forall \, i \in \mathcal{I}^{{n_a}}$, if and only if there exists $y : [0,{n_a}]\rightarrow \mathbb{R}_+$ solving the following set of initial value problems

Figures (5)

  • Figure 1: One-dimensional representation of the considered dynamics with impulsive controls.
  • Figure 2: Earth, Moon, target and chaser spacecrafts (in black and red) in the normalized Earth-Moon synodic reference frame.
  • Figure 3: Relative trajectories in the synodic reference frame
  • Figure 4: Relative-to-target distance with isoperimetric constraint reformulation
  • Figure 5: Relative-to-target distance for node-only constraints - 3-impulse strategy

Theorems & Definitions (1)

  • Theorem 1