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Expected Time-Optimal Control: a Particle Model Predictive Control-based Approach via Sequential Convex Programming

Kazuya Echigo, Abhishek Cauligi, Behçet Açıkmeşe

Abstract

In this paper, we consider the problem of minimum-time optimal control for a dynamical system with initial state uncertainties and propose a sequential convex programming (SCP) solution framework. We seek to minimize the expected terminal (mission) time, which is an essential capability for planetary exploration missions where ground rovers have to carry out scientific tasks efficiently within the mission timelines in uncertain environments. Our main contribution is to convert the underlying stochastic optimal control problem into a deterministic, numerically tractable, optimal control problem. To this end, the proposed solution framework combines two strategies from previous methods: i) a partial model predictive control with consensus horizon approach and ii) a sum-of-norm cost, a temporally strictly increasing weighted-norm, promoting minimum-time trajectories. Our contribution is to adopt these formulations into an SCP solution framework and obtain a numerically tractable stochastic control algorithm. We then demonstrate the resulting control method in multiple applications: i) a closed-loop linear system as a representative result (a spacecraft double integrator model), ii) an open-loop linear system (the same model), and then iii) a nonlinear system (Dubin's car).

Expected Time-Optimal Control: a Particle Model Predictive Control-based Approach via Sequential Convex Programming

Abstract

In this paper, we consider the problem of minimum-time optimal control for a dynamical system with initial state uncertainties and propose a sequential convex programming (SCP) solution framework. We seek to minimize the expected terminal (mission) time, which is an essential capability for planetary exploration missions where ground rovers have to carry out scientific tasks efficiently within the mission timelines in uncertain environments. Our main contribution is to convert the underlying stochastic optimal control problem into a deterministic, numerically tractable, optimal control problem. To this end, the proposed solution framework combines two strategies from previous methods: i) a partial model predictive control with consensus horizon approach and ii) a sum-of-norm cost, a temporally strictly increasing weighted-norm, promoting minimum-time trajectories. Our contribution is to adopt these formulations into an SCP solution framework and obtain a numerically tractable stochastic control algorithm. We then demonstrate the resulting control method in multiple applications: i) a closed-loop linear system as a representative result (a spacecraft double integrator model), ii) an open-loop linear system (the same model), and then iii) a nonlinear system (Dubin's car).
Paper Structure (8 sections, 1 theorem, 17 equations, 6 figures, 4 tables)

This paper contains 8 sections, 1 theorem, 17 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation and Particle Model Predictive Control Approach
  3. Sum-of-Norm Approximation
  4. Numerical Examples
  5. Representative Result (Closed Loop)
  6. Linear Systems (Open Loop)
  7. Nonlinear Systems (Open Loop)
  8. Conclusion

Key Result

Theorem III.1

Assume that the optimal trajectories of Problem pro:problem4 are given as $u^{**}$ and $x^{**}$, respectively. Then, for any $i'$, when $x^{i'}(k)^{**}$ becomes $0$ at $k = k^{i'}_0, \,\, 1 \leq k^{i'}_0 \leq \Gamma$, $x^{i'}(k)^{**}$ remains $0$ at any $k$ such that $k^{i'}_0 \leq k \leq \Gamma$.

Figures (6)

  • Figure 1: Overview: The proposed framework minimizes the expected terminal (mission) time to reach a prescribed destination under uncertainties at the initial state. Solving it in a closed loop enables to consider measurement uncertainties.
  • Figure 2: The $L_2$ norm distance time histories to the prescribed destination using the proposed approach and the deterministic minimum-time optimal control. It also displays the convergence time steps of those two methods (blue and red dotted lines). This indicates the capability of our proposed approach to achieve faster convergence in contrast to the deterministic controller.
  • Figure 3: 4D free-flying spacecraft system.
  • Figure 4: Resulting distributions of terminal time-steps for each trajectory from the proposed approach and the deterministic minimum-time optimal control. Each figure also displays the mean terminal time-steps (black dotted line). The mean terminal time-steps for the proposed approach is less than that for the other.
  • Figure 5: Distributions of the terminal time-steps for each trajectory show that our proposed approach outperforms the deterministic minimum-time optimal control approach in terms of the mean terminal time-step (black dotted line).
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem III.1
  • proof