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Variable-Horizon Guidance for Autonomous Rendezvous and Docking to a Tumbling Target

Mirko Leomanni, Renato Quartullo, Gianni Bianchini, Andrea Garulli, Antonio Giannitrapani

TL;DR

The paper tackles autonomous rendezvous and docking to tumbling targets by introducing a discrete-time, variable-horizon guidance framework that transforms the inherently nonconvex problem into a sequence of linear programs. A horizon-search algorithm coupled with a rotating-hyperplane constraint approximation enables convexification of keep-out and docking constraints, while providing an onboard-friendly, real-time capable solution. The key contribution is a practical, LP-based method that discovers favorable docking configurations as local minima and achieves substantial computational savings versus nonlinear solvers, demonstrated on EnviSat docking scenarios. This work offers a scalable approach that can operate as a standalone guidance module or as a baseline for variable-horizon MPC in uncooperative mission contexts.

Abstract

In this paper, the trajectory planning problem for autonomous rendezvous and docking between a controlled spacecraft and a tumbling target is addressed. The use of a variable planning horizon is proposed in order to construct an appropriate maneuver plan, within an optimization-based framework. The involved optimization problem is nonconvex and features nonlinear constraints. The main contribution is to show that such problem can be tackled effectively by solving a finite number of linear programs. To this aim, a specifically conceived horizon search algorithm is employed in combination with a polytopic constraint approximation technique. The resulting guidance scheme provides the ability to identify favourable docking configurations, by exploiting the time-varying nature of the optimization problem endpoint. Simulation results involving the capture of the nonoperational EnviSat spacecraft indicate that the method is able to generate optimal trajectories at a fraction of the computational cost incurred by a state-of-the-art nonlinear solver.

Variable-Horizon Guidance for Autonomous Rendezvous and Docking to a Tumbling Target

TL;DR

The paper tackles autonomous rendezvous and docking to tumbling targets by introducing a discrete-time, variable-horizon guidance framework that transforms the inherently nonconvex problem into a sequence of linear programs. A horizon-search algorithm coupled with a rotating-hyperplane constraint approximation enables convexification of keep-out and docking constraints, while providing an onboard-friendly, real-time capable solution. The key contribution is a practical, LP-based method that discovers favorable docking configurations as local minima and achieves substantial computational savings versus nonlinear solvers, demonstrated on EnviSat docking scenarios. This work offers a scalable approach that can operate as a standalone guidance module or as a baseline for variable-horizon MPC in uncooperative mission contexts.

Abstract

In this paper, the trajectory planning problem for autonomous rendezvous and docking between a controlled spacecraft and a tumbling target is addressed. The use of a variable planning horizon is proposed in order to construct an appropriate maneuver plan, within an optimization-based framework. The involved optimization problem is nonconvex and features nonlinear constraints. The main contribution is to show that such problem can be tackled effectively by solving a finite number of linear programs. To this aim, a specifically conceived horizon search algorithm is employed in combination with a polytopic constraint approximation technique. The resulting guidance scheme provides the ability to identify favourable docking configurations, by exploiting the time-varying nature of the optimization problem endpoint. Simulation results involving the capture of the nonoperational EnviSat spacecraft indicate that the method is able to generate optimal trajectories at a fraction of the computational cost incurred by a state-of-the-art nonlinear solver.

Paper Structure

This paper contains 10 sections, 1 theorem, 23 equations, 16 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation:
  3. Problem Formulation
  4. Rendezvous and Docking Constraints
  5. Rendezvous Constraints
  6. Docking Constraints
  7. Solution Strategy
  8. Search Algorithm Validation
  9. Rendezvous and Docking to the EnviSat Platform
  10. Conclusions

Key Result

Proposition 1

Consider the linear system $\mathbf{x}(k+1)=\mathbf{A} \mathbf{x}(k)+ \mathbf{B} \mathbf{\mathbf{u}}(k)$, where $\mathbf{x}(k)\in\mathbb{R}^n$ and $\mathbf{u}(k)\in\mathbb{R}^m$, and let $\mathbf{x}^d(k_0+N)\in\mathbb{R}^n$ be a target state to be reached from the initial state $\mathbf{x}(k_0)=\mat Then, the feasibility problem has no solution for $N\in\{\mathcal{I} \setminus \mathcal{F}\}$.

Figures (16)

  • Figure 1: Definition of the docking point.
  • Figure 2: Illustration of a typical rendezvous and docking maneuver. For the sake of illustration, the docking point is assumed to be static.
  • Figure 3: Illustration of the keep-out-zone approximation scheme on a two-dimensional example, for different values of $\lambda_N$.
  • Figure 4: Illustration of the polyhedral set $\mathcal{D}(k)$ used to approximate the docking cone $\mathcal{C}(k)$.
  • Figure 5: Profile of ${J}^*_{N}$ versus $N$ for the RVD scenario detailed in Section \ref{['SAp']}, with $\gamma=4$. The presence of multiple local minima is due to the target rotation. The global optimum $N^*$ is marked by an asterisk. The problem is infeasible for $N\leq 25$.
  • ...and 11 more figures

Theorems & Definitions (3)

  • Proposition 1
  • proof
  • Remark 1