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Convex MPC and Thrust Allocation with Deadband for Spacecraft Rendezvous

Pedro Taborda, Hugo Matias, Daniel Silvestre, Pedro Lourenço

TL;DR

The paper addresses the challenge of real-time rendezvous control for a chaser spacecraft using Model Predictive Control with deadband-induced mixed-integer constraints. It linearizes actuator dynamics and derives an affine discretization to enable convexification, then introduces two fast solvers, Projected and Relaxed, that approximate the optimal mixed-integer MPC by relaxing or partially handling the integer constraints. Simulations show that both solvers achieve trajectories nearly indistinguishable from the true optimum while dramatically reducing computation time, even for large prediction horizons, making real-time implementation feasible. The results demonstrate that convex relaxation combined with projection provides a practical path to efficient thrust allocation in rendezvous missions, with future work targeting elliptical orbits and safety considerations.

Abstract

This paper delves into a rendezvous scenario involving a chaser and a target spacecraft, focusing on the application of Model Predictive Control (MPC) to design a controller capable of guiding the chaser toward the target. The operational principle of spacecraft thrusters, requiring a minimum activation time that leads to the existence of a control deadband, introduces mixed-integer constraints into the optimization, posing a considerable computational challenge due to the exponential complexity on the number of integer constraints. We address this complexity by presenting two solver algorithms that efficiently approximate the optimal solution in significantly less time than standard solvers, making them well-suited for real-time applications.

Convex MPC and Thrust Allocation with Deadband for Spacecraft Rendezvous

TL;DR

The paper addresses the challenge of real-time rendezvous control for a chaser spacecraft using Model Predictive Control with deadband-induced mixed-integer constraints. It linearizes actuator dynamics and derives an affine discretization to enable convexification, then introduces two fast solvers, Projected and Relaxed, that approximate the optimal mixed-integer MPC by relaxing or partially handling the integer constraints. Simulations show that both solvers achieve trajectories nearly indistinguishable from the true optimum while dramatically reducing computation time, even for large prediction horizons, making real-time implementation feasible. The results demonstrate that convex relaxation combined with projection provides a practical path to efficient thrust allocation in rendezvous missions, with future work targeting elliptical orbits and safety considerations.

Abstract

This paper delves into a rendezvous scenario involving a chaser and a target spacecraft, focusing on the application of Model Predictive Control (MPC) to design a controller capable of guiding the chaser toward the target. The operational principle of spacecraft thrusters, requiring a minimum activation time that leads to the existence of a control deadband, introduces mixed-integer constraints into the optimization, posing a considerable computational challenge due to the exponential complexity on the number of integer constraints. We address this complexity by presenting two solver algorithms that efficiently approximate the optimal solution in significantly less time than standard solvers, making them well-suited for real-time applications.
Paper Structure (14 sections, 14 equations, 4 figures, 4 tables, 2 algorithms)

This paper contains 14 sections, 14 equations, 4 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries & Problem Statement
  3. Full Dynamics Model
  4. Simplified Model
  5. Actuation and Model Discretization
  6. Linearization of the Actuator Dynamics
  7. Model Predictive Control with Mixed-Integer Constraints
  8. Projected Algorithm
  9. Relaxed Algorithm
  10. Simulation Results
  11. Variation of the Minimum Activation Time
  12. Variation of the Horizon Length
  13. Computation Times
  14. Conclusion & Future Research

Figures (4)

  • Figure 1: and reference frames.
  • Figure 2: Resulting trajectories and activation profiles obtained using the Standard algorithm for different values of the minimum activation time. Chaser's initial position , target position .
  • Figure 3: Resulting trajectories and activation profiles obtained using each algorithm for different horizon lengths. Chaser's initial position , target position .
  • Figure 4: Computation times obtained using each algorithm for different horizons - Standard (blue), Projected (red), Relaxed (yellow). Mean solve time (dash-dotted, histograms), mission time (dash-dotted, profiles), maximum solve time (dotted, profiles).