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Moving Obstacle Collision Avoidance via Chance-Constrained MPC with CBF

Ming Li, Zhiyong Sun, Zirui Liao, Siep Weiland

TL;DR

This work tackles moving obstacle collision avoidance (MOCA) under stochastic perception by extending MPC-CBF with chance constraints, yielding probabilistic safety guarantees. It derives a tractable CC-MPC-CBF formulation through Gaussian moment matching, converting the probabilistic safety constraint into a deterministic surrogate that remains exact for certain horizons, and introduces an iterative convex optimization approach. To address feasibility challenges, the authors propose a sequential implementation: first solve a standard MPC for nominal performance, then apply a predictive safety filter (solved via an iterative convex algorithm) to enforce CBF constraints, improving feasibility at the potential cost of some optimality. The approach is validated on a double-integrator MOCA example, showing robustness to obstacle measurement noise, superior MOCA success rates, and faster computation with the sequential method, indicating practical applicability for real-world dynamic environments.

Abstract

Model predictive control (MPC) with control barrier functions (CBF) is a promising solution to address the moving obstacle collision avoidance (MOCA) problem. Unlike MPC with distance constraints (MPC-DC), this approach facilitates early obstacle avoidance without the need to increase prediction horizons. However, the existing MPC-CBF method is deterministic and fails to account for perception uncertainties. This paper proposes a generalized MPC-CBF approach for stochastic scenarios, which maintains the advantages of the deterministic method for addressing the MOCA problem. Specifically, the chance-constrained MPC-CBF (CC-MPC-CBF) technique is introduced to ensure that a user-defined collision avoidance probability is met by utilizing probabilistic CBFs. However, due to the potential empty intersection between the reachable set and the safe region confined by CBF constraints, the CC-MPC-CBF problem can pose challenges in achieving feasibility. To address this issue, we propose a sequential implementation approach that involves solving a standard MPC optimization problem followed by a predictive safety filter optimization, which leads to improved feasibility. Furthermore, we introduce an iterative convex optimization scheme to further expedite the resolution of the predictive safety filter, which results in an efficient approach to tackling the non-convex CC-MPC-CBF problem. We apply our proposed algorithm to a double integrator system for MOCA, and we showcase its resilience to obstacle measurement uncertainties and favorable feasibility properties.

Moving Obstacle Collision Avoidance via Chance-Constrained MPC with CBF

TL;DR

This work tackles moving obstacle collision avoidance (MOCA) under stochastic perception by extending MPC-CBF with chance constraints, yielding probabilistic safety guarantees. It derives a tractable CC-MPC-CBF formulation through Gaussian moment matching, converting the probabilistic safety constraint into a deterministic surrogate that remains exact for certain horizons, and introduces an iterative convex optimization approach. To address feasibility challenges, the authors propose a sequential implementation: first solve a standard MPC for nominal performance, then apply a predictive safety filter (solved via an iterative convex algorithm) to enforce CBF constraints, improving feasibility at the potential cost of some optimality. The approach is validated on a double-integrator MOCA example, showing robustness to obstacle measurement noise, superior MOCA success rates, and faster computation with the sequential method, indicating practical applicability for real-world dynamic environments.

Abstract

Model predictive control (MPC) with control barrier functions (CBF) is a promising solution to address the moving obstacle collision avoidance (MOCA) problem. Unlike MPC with distance constraints (MPC-DC), this approach facilitates early obstacle avoidance without the need to increase prediction horizons. However, the existing MPC-CBF method is deterministic and fails to account for perception uncertainties. This paper proposes a generalized MPC-CBF approach for stochastic scenarios, which maintains the advantages of the deterministic method for addressing the MOCA problem. Specifically, the chance-constrained MPC-CBF (CC-MPC-CBF) technique is introduced to ensure that a user-defined collision avoidance probability is met by utilizing probabilistic CBFs. However, due to the potential empty intersection between the reachable set and the safe region confined by CBF constraints, the CC-MPC-CBF problem can pose challenges in achieving feasibility. To address this issue, we propose a sequential implementation approach that involves solving a standard MPC optimization problem followed by a predictive safety filter optimization, which leads to improved feasibility. Furthermore, we introduce an iterative convex optimization scheme to further expedite the resolution of the predictive safety filter, which results in an efficient approach to tackling the non-convex CC-MPC-CBF problem. We apply our proposed algorithm to a double integrator system for MOCA, and we showcase its resilience to obstacle measurement uncertainties and favorable feasibility properties.
Paper Structure (26 sections, 3 theorems, 30 equations, 2 figures, 7 tables, 1 algorithm)

This paper contains 26 sections, 3 theorems, 30 equations, 2 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. System Models
  4. Discrete-Time Control Barrier Functions
  5. MPC with CBF
  6. Problem Statement
  7. Chance Constrained MPC-CBF
  8. CBF and Chance Constraints
  9. Linear Chance Constraints
  10. Chance-Constrained MPC-CBF
  11. CC-MPC-CBF with a Sequential Implementation
  12. Iterative Convex Optimization
  13. Application Examples and Simulation Results
  14. Dynamical Model of Robot
  15. Obstacle Model
  16. ...and 11 more sections

Key Result

Lemma 1

(Mean_Covariance_Results) Let $\mathbf{z}$ be a Gaussian variable, and the mean value and covariance are $\bm{\mu}$ and $\bm{\Sigma}$, respectively. $\mathbf{A}$ is a symmetric matrix. Then the expectation and variance of the quadratic form $\mathbf{z}^{\top}\mathbf{A}\mathbf{z}$ are given as follow where $\mathbb{E}(\cdot)$ and $\mathbf{Var}(\cdot)$ denote the operations for computing the expecta

Figures (2)

  • Figure 1: A snapshot of the tracking behaviors of the deterministic MPC-CBF and CC-MPC-CBF at time $k=90$ (top-down view): The red circle denotes the reference trajectory; the magenta solid point and black box maker represent the reference point and tracking point, respectively. Note that the black box is followed by a tail, which relates to the history tracking data.
  • Figure 2: A comparison of the tracking performance of CC-MPC-DC and the proposed CC-MPC-CBF (with different $N$ and $\gamma$) (top-down view): The red circle denotes the reference trajectory; The curves with different line styles and colors indicate the tracking trajectories with different parameters.

Theorems & Definitions (14)

  • Definition 1
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • Remark 2
  • Remark 3
  • Proposition 1
  • proof
  • Remark 4
  • ...and 4 more