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U-OBCA: Uncertainty-Aware Optimization-Based Collision Avoidance via Wasserstein Distributionally Robust Chance Constraints

Zehao Wang, Yuxuan Tang, Han Zhang, Jingchuan Wang, Weidong Chen

TL;DR

The results demonstrate that U-OBCA significantly mitigates the conservatism in trajectory planning and achieves higher navigation efficiency compared to existing baseline methods, particularly in narrow and cluttered environments.

Abstract

Uncertainties arising from localization error, trajectory prediction errors of the moving obstacles and environmental disturbances pose significant challenges to robot's safe navigation. Existing uncertainty-aware planners often approximate polygon-shaped robots and obstacles using simple geometric primitives such as circles or ellipses. Though computationally convenient, these approximations substantially shrink the feasible space, leading to overly conservative trajectories and even planning failure in narrow environments. In addition, many such methods rely on specific assumptions about noise distributions, which may not hold in practice and thus limit their performance guarantees. To address these limitations, we extend the Optimization-Based Collision Avoidance (OBCA) framework to an uncertainty-aware formulation, termed \emph{U-OBCA}. The proposed method explicitly accounts for the collision risk between polygon-shaped robots and obstacles by formulating OBCA-based chance constraints, and hence avoiding geometric simplifications and reducing unnecessary conservatism. These probabilistic constraints are further tightened into deterministic nonlinear constraints under mild distributional assumptions, which can be solved efficiently by standard numerical optimization solvers. The proposed approach is validated through theoretical analysis, numerical simulations and real-world experiments. The results demonstrate that U-OBCA significantly mitigates the conservatism in trajectory planning and achieves higher navigation efficiency compared to existing baseline methods, particularly in narrow and cluttered environments.

U-OBCA: Uncertainty-Aware Optimization-Based Collision Avoidance via Wasserstein Distributionally Robust Chance Constraints

TL;DR

The results demonstrate that U-OBCA significantly mitigates the conservatism in trajectory planning and achieves higher navigation efficiency compared to existing baseline methods, particularly in narrow and cluttered environments.

Abstract

Uncertainties arising from localization error, trajectory prediction errors of the moving obstacles and environmental disturbances pose significant challenges to robot's safe navigation. Existing uncertainty-aware planners often approximate polygon-shaped robots and obstacles using simple geometric primitives such as circles or ellipses. Though computationally convenient, these approximations substantially shrink the feasible space, leading to overly conservative trajectories and even planning failure in narrow environments. In addition, many such methods rely on specific assumptions about noise distributions, which may not hold in practice and thus limit their performance guarantees. To address these limitations, we extend the Optimization-Based Collision Avoidance (OBCA) framework to an uncertainty-aware formulation, termed \emph{U-OBCA}. The proposed method explicitly accounts for the collision risk between polygon-shaped robots and obstacles by formulating OBCA-based chance constraints, and hence avoiding geometric simplifications and reducing unnecessary conservatism. These probabilistic constraints are further tightened into deterministic nonlinear constraints under mild distributional assumptions, which can be solved efficiently by standard numerical optimization solvers. The proposed approach is validated through theoretical analysis, numerical simulations and real-world experiments. The results demonstrate that U-OBCA significantly mitigates the conservatism in trajectory planning and achieves higher navigation efficiency compared to existing baseline methods, particularly in narrow and cluttered environments.
Paper Structure (35 sections, 8 theorems, 70 equations, 10 figures, 7 tables)

This paper contains 35 sections, 8 theorems, 70 equations, 10 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. The Preliminary Results
  4. Optimization-Based Collision Avoidance Without Uncertainties
  5. Bounds on Disjunctive Chance Constraints
  6. Worst-Case Chance Constraints with the Wasserstein Distributional Ambiguity
  7. Wasserstein distance and Wasserstein ambiguity set
  8. Elliptical distributions
  9. Uncertainty-aware Optimization-Based Collision Avoidance
  10. Round-Shaped Obstacles
  11. Polygon-Shaped Obstacles
  12. Simulation Results
  13. Global Planning: Parallel Parking Trajectory Planning for a Four Wheel Steering Vehicle
  14. The Simulation Settings
  15. The Simulation Results
  16. ...and 20 more sections

Key Result

Theorem 1

(OBCA OBCA): On one hand, for the $j$-th round-shaped obstacle, the minimum distance constraint $\operatorname{dist}(\mathbb{V}_k,\mathbb{O}^j_k)\geq{\rm d}_{\min}$ is equivalent to where $\mu^j_k \in \mathop{\mathrm{\mathbb{R}}}\nolimits^{l_v}$ are dual variables. On the other hand, for the $j$-th polygon-shaped obstacle, the minimum distance constraints $\operatorname{dist}(\mathbb{V},\mathbb{O

Figures (10)

  • Figure 1: Motivation: collision risk assessment between polygonal shapes under pose uncertainty.
  • Figure 2: The distance between the vehicle and the obstacle.
  • Figure 3: The simulation results in the parallel parking scenario.
  • Figure 4: The simulation results in the narrow corridor scenario.
  • Figure 5: The smart wheelchair platform.
  • ...and 5 more figures

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Theorem 5
  • Corollary 6
  • proof
  • Theorem 7
  • proof
  • Lemma 8
  • ...and 6 more