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Wasserstein Distributionally Robust Chance Constrained Trajectory Optimization for Mobile Robots within Uncertain Safe Corridor

Shaohang Xu, Haolin Ruan, Wentao Zhang, Yian Wang, Lijun Zhu, Chin Pang Ho

TL;DR

Theoretically, the proposed DRSCCs are equivalent to a convex quadratic program, which is computationally efficient to deploy onto real robots and the simulation results show that the method enhances navigation safety by significantly reducing the infeasible motions compared to the baseline.

Abstract

Safe corridor-based Trajectory Optimization (TO) presents an appealing approach for collision-free path planning of autonomous robots, offering global optimality through its convex formulation. The safe corridor is constructed based on the perceived map, however, the non-ideal perception induces uncertainty, which is rarely considered in trajectory generation. In this paper, we propose Distributionally Robust Safe Corridor Constraints (DRSCCs) to consider the uncertainty of the safe corridor. Then, we integrate DRSCCs into the trajectory optimization framework using Bernstein basis polynomials. Theoretically, we rigorously prove that the trajectory optimization problem incorporating DRSCCs is equivalent to a computationally efficient, convex quadratic program. Compared to the nominal TO, our method enhances navigation safety by significantly reducing the infeasible motions in presence of uncertainty. Moreover, the proposed approach is validated through two robotic applications, a micro Unmanned Aerial Vehicle (UAV) and a quadruped robot Unitree A1.

Wasserstein Distributionally Robust Chance Constrained Trajectory Optimization for Mobile Robots within Uncertain Safe Corridor

TL;DR

Theoretically, the proposed DRSCCs are equivalent to a convex quadratic program, which is computationally efficient to deploy onto real robots and the simulation results show that the method enhances navigation safety by significantly reducing the infeasible motions compared to the baseline.

Abstract

Safe corridor-based Trajectory Optimization (TO) presents an appealing approach for collision-free path planning of autonomous robots, offering global optimality through its convex formulation. The safe corridor is constructed based on the perceived map, however, the non-ideal perception induces uncertainty, which is rarely considered in trajectory generation. In this paper, we propose Distributionally Robust Safe Corridor Constraints (DRSCCs) to consider the uncertainty of the safe corridor. Then, we integrate DRSCCs into the trajectory optimization framework using Bernstein basis polynomials. Theoretically, we rigorously prove that the trajectory optimization problem incorporating DRSCCs is equivalent to a computationally efficient, convex quadratic program. Compared to the nominal TO, our method enhances navigation safety by significantly reducing the infeasible motions in presence of uncertainty. Moreover, the proposed approach is validated through two robotic applications, a micro Unmanned Aerial Vehicle (UAV) and a quadruped robot Unitree A1.
Paper Structure (15 sections, 2 theorems, 26 equations, 4 figures, 1 table)

This paper contains 15 sections, 2 theorems, 26 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Prelimenary
  5. Safe Corridor Constraint
  6. Deterministic TO using Bernstein Basis Polynomial
  7. Main Method
  8. Uncertain Safe Corridor with Distributional Ambiguity
  9. DRSCC-TO
  10. Results and Discussion
  11. Numerical Simulation
  12. Robotic Applications
  13. UAV
  14. Quadruped Robot
  15. Conclusion

Key Result

Proposition 1

Suppose the trajectory $B_i(t)$ is a Bezier curve bezier_si. Then, the constraint eq_safe_corridor_constraint holds if where $\mathbf{e}_\mu$ is the standard basis in $\mathbb{R}^m$ where the entry associated with the $\mu$-th dimension is one.

Figures (4)

  • Figure 1: An example of TO with (red) and without (blue) considering uncertainty. When the safe corridor is exactly known (left), both methods are collision-free, and the blue trajectory is shorter and could minimize the control input. However, when the obstacle information is not perfect and/or there is an unknown obstacle region (right), the blue trajectory will be infeasible, leading to unacceptable collisions.
  • Figure 2: The robots used in our experiments: the quadruped robot Unitree A1 (left) and a micro UAV (right).
  • Figure 3: Experimental Results on our UAV. (a) Comparison of the optimized trajectories. (b) Nominal TO with the exact safe corridor. (c) DRSCC-TO with the exact safe corridor. Both methods are collision-free with the exact safety corridor, although the optimized trajectory of the nominal TO is closer to the boundary.
  • Figure 4: Experimental Results on Unitree A1. (a) The safe corridor is disturbed by moving the black obstacle. (b) Nominal TO with the disturbed safe corridor. (c) DRSCC-TO with the disturbed safe corridor.

Theorems & Definitions (6)

  • Proposition 1
  • Remark 1
  • Definition 1
  • Definition 2
  • Theorem 1
  • proof