Continuous-Time Gaussian Belief Trees for Motion Planning

2026-07-02

AI Summary

This work addresses motion planning for continuous-time stochastic systems under both process and observation uncertainties by proposing a sampling-based planning framework that enables continuous-time probabilistic safety verification over entire trajectories. The approach constructs an offline hybrid belief propagation model that integrates continuous-time ordinary differential equation (ODE) dynamics with discrete Kalman updates, and introduces a belief barrier function as a safety checker capable of detecting potential constraint violations between sampling instants—marking the first method to achieve such intra-interval safety guarantees. Integrated with RRT/SST planners, the framework demonstrates superior performance over conventional discrete-time methods across multiple benchmark scenarios, including narrow passages, achieving higher success rates, enhanced robustness, and stronger formal safety assurances.

Abstract

We address sampling-based motion planning for continuous-time stochastic systems under process and measurement uncertainty, with probabilistic guarantees on safety and performance. The robot dynamics are modeled as a continuous-time linear stochastic differential equation, while sensor measurements arrive at discrete time instants. We derive an offline hybrid belief propagation model in which the belief evolves according to continuous-time ODEs between measurements and undergoes discrete Kalman filter update jumps at measurement times. To ensure safety, we introduce a belief-barrier-function-based safety checker for segment-level probabilistic verification. This enables the planner to certify safety over entire continuous trajectory segments and detect inter-sample chance-constraint violations that are missed by conventional node-based checks. Together, these components provide a principled framework for sampling-based belief planning that accounts for both continuous-time uncertainty propagation and continuous-time safety requirements. We integrate the method with RRT and SST planners and evaluate it across multiple benchmark environments. The results show that the proposed method achieves high success rates and robust enforcement of chance constraints, including in narrow-passage scenarios where discrete-time counterparts fail due to missed inter-sample unsafe behavior.

Problem

• continuous-time motion planning
• stochastic systems
• chance constraints
• safety verification
• belief propagation

Innovation

• continuous-time belief propagation
• belief-barrier function
• chance-constrained planning
• hybrid Kalman update
• sampling-based belief space planning