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Barrier-Enhanced Parallel Homotopic Trajectory Optimization for Safety-Critical Autonomous Driving

Lei Zheng, Rui Yang, Michael Yu Wang, Jun Ma

TL;DR

This work tackles real-time safety-critical autonomous driving by integrating decision-making and planning through Barrier-Enhanced Parallel Homotopic Trajectory Optimization (BPHTO). It combines spatiotemporal control barrier functions with a bi-convex reformulation and over-relaxed ADMM to generate multiple maneuver-aligned trajectories in parallel, while adaptively tightening safety with a barrier coefficient that increases along the horizon. Using Bézier trajectory parameterization and reachability-informed warm starts, BPHTO yields feasible, smooth, and consistent trajectories with proactive HV interaction and robust safety recovery. Experimental results on IDM and NGSIM datasets show BPHTO outperforms baselines in terms of safety, task accuracy, and driving consistency, while maintaining real-time performance suitable for receding-horizon planning.

Abstract

Enforcing safety while preventing overly conservative behaviors is essential for autonomous vehicles to achieve high task performance. In this paper, we propose a barrier-enhanced parallel homotopic trajectory optimization (BPHTO) approach with the over-relaxed alternating direction method of multipliers (ADMM) for real-time integrated decision-making and planning. To facilitate safety interactions between the ego vehicle (EV) and surrounding vehicles, a spatiotemporal safety module exhibiting bi-convexity is developed on the basis of barrier function. Varying barrier coefficients are adopted for different time steps in a planning horizon to account for the motion uncertainties of surrounding HVs and mitigate conservative behaviors. Additionally, we exploit the discrete characteristics of driving maneuvers to initialize nominal behavior-oriented free-end homotopic trajectories based on reachability analysis, and each trajectory is locally constrained to a specific driving maneuver while sharing the same task objectives. By leveraging the bi-convexity of the safety module and the kinematics of the EV, we formulate the BPHTO as a bi-convex optimization problem. Then constraint transcription and the over-relaxed ADMM are employed to streamline the optimization process, such that multiple trajectories are generated in real time with feasibility guarantees. Through a series of experiments, the proposed development demonstrates improved task accuracy, stability, and consistency in various traffic scenarios using synthetic and real-world traffic datasets.

Barrier-Enhanced Parallel Homotopic Trajectory Optimization for Safety-Critical Autonomous Driving

TL;DR

This work tackles real-time safety-critical autonomous driving by integrating decision-making and planning through Barrier-Enhanced Parallel Homotopic Trajectory Optimization (BPHTO). It combines spatiotemporal control barrier functions with a bi-convex reformulation and over-relaxed ADMM to generate multiple maneuver-aligned trajectories in parallel, while adaptively tightening safety with a barrier coefficient that increases along the horizon. Using Bézier trajectory parameterization and reachability-informed warm starts, BPHTO yields feasible, smooth, and consistent trajectories with proactive HV interaction and robust safety recovery. Experimental results on IDM and NGSIM datasets show BPHTO outperforms baselines in terms of safety, task accuracy, and driving consistency, while maintaining real-time performance suitable for receding-horizon planning.

Abstract

Enforcing safety while preventing overly conservative behaviors is essential for autonomous vehicles to achieve high task performance. In this paper, we propose a barrier-enhanced parallel homotopic trajectory optimization (BPHTO) approach with the over-relaxed alternating direction method of multipliers (ADMM) for real-time integrated decision-making and planning. To facilitate safety interactions between the ego vehicle (EV) and surrounding vehicles, a spatiotemporal safety module exhibiting bi-convexity is developed on the basis of barrier function. Varying barrier coefficients are adopted for different time steps in a planning horizon to account for the motion uncertainties of surrounding HVs and mitigate conservative behaviors. Additionally, we exploit the discrete characteristics of driving maneuvers to initialize nominal behavior-oriented free-end homotopic trajectories based on reachability analysis, and each trajectory is locally constrained to a specific driving maneuver while sharing the same task objectives. By leveraging the bi-convexity of the safety module and the kinematics of the EV, we formulate the BPHTO as a bi-convex optimization problem. Then constraint transcription and the over-relaxed ADMM are employed to streamline the optimization process, such that multiple trajectories are generated in real time with feasibility guarantees. Through a series of experiments, the proposed development demonstrates improved task accuracy, stability, and consistency in various traffic scenarios using synthetic and real-world traffic datasets.
Paper Structure (34 sections, 2 theorems, 78 equations, 10 figures, 3 tables, 2 algorithms)

This paper contains 34 sections, 2 theorems, 78 equations, 10 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Spatiotemporal Control Barrier
  4. Trajectory Parameterization
  5. Spatiotemporal Control Barrier
  6. Safety Representations
  7. Invariably Safety Constraints
  8. Robustness of Safety
  9. Task-Oriented and Maneuver-oriented Motion
  10. Dynamic Feasibility
  11. Task-Oriented Motion
  12. Sampling Points Generation
  13. Barrier-Enhanced Parallel Homotopic Trajectory Optimization (BPHTO) with Over-relaxed ADMM
  14. Problem Reformulation
  15. Over-Relaxed ADMM
  16. ...and 19 more sections

Key Result

Theorem 1

Let $h$ be a discrete-time BF for the EV under Assumptions 1-2. Then, the EV, starting from an initial state $\mathbf{x}_{k-1} \in Int(\mathcal{S})$, can proactively avoid collisions with surrounding HVs with guaranteed safety if the constraint eq:spatiotemporal_barrier_cons is satisfied.

Figures (10)

  • Figure 1: Illustration of the motion of an EV (in red color) in a dynamic cluttered scenario with one lane under road construction ahead. The orange and blue vehicles represent perceived and unperceived HVs, respectively. The EV and the $i$-th HV are represented as ellipse-shaped convex compact set $\mathbb{X}$ and $\mathbb{O}_i$, respectively. The solid red line with an arrow represents the intended trajectory of the EV, while other solid lines denote alternative free-end homotopic candidate trajectories of the EV. Each trajectory shares the same initial state and corresponds to a specific driving maneuver denoted as $\xi$, with values of $\{0, 1, -1\}$, representing lane-keeping, left-lane-change, and right-lane-change behaviors, respectively.
  • Figure 2: Comparison of position, heading angle, and longitudinal jerk profiles when executing a cruise task using IDM dataset for surrounding HVs’ motion. The evolution of the heading angle profiles reveals how the EV adjusts its orientation to navigate through dense and dynamic traffic.
  • Figure 3: Evolution of the trajectory in the road construction scenario when executing a cruise task using IDM dataset for surrounding HVs' motion. The motion trajectory of the EV over a simulation duration of $20\,\text{s}$ is colored according to its longitudinal speed profile (yellow-red color).
  • Figure 4: A lane-merging conflict scenario. The red EV in lane 3 is in the process of executing a lane change to lane 2, while an imperceptible blue HV2 in lane 1 is initiating a lane change to lane 2 from a different direction, potentially leading to a hazardous situation.
  • Figure 5: Evolution of trajectory when executing a lane merge task under cruise scenario using NGSIM dataset for surrounding HVs' motion. It displays the trajectory of the EV, colored according to its longitudinal speed profile (yellow-red color).
  • ...and 5 more figures

Theorems & Definitions (12)

  • Definition 1
  • Remark 1
  • Remark 2
  • Definition 2
  • Definition 3
  • Theorem 1
  • Remark 3
  • Theorem 2
  • Remark 4
  • Remark 5
  • ...and 2 more