Incremental Sampling-based Algorithm for Minimum-violation Motion Planning (1305.1102v2)

Abstract: This paper studies the problem of control strategy synthesis for dynamical systems with differential constraints to fulfill a given reachability goal while satisfying a set of safety rules. Particular attention is devoted to goals that become feasible only if a subset of the safety rules are violated. The proposed algorithm computes a control law, that minimizes the level of unsafety while the desired goal is guaranteed to be reached. This problem is motivated by an autonomous car navigating an urban environment while following rules of the road such as "always travel in right lane'' and "do not change lanes frequently''. Ideas behind sampling based motion-planning algorithms, such as Probabilistic Road Maps (PRMs) and Rapidly-exploring Random Trees (RRTs), are employed to incrementally construct a finite concretization of the dynamics as a durational Kripke structure. In conjunction with this, a weighted finite automaton that captures the safety rules is used in order to find an optimal trajectory that minimizes the violation of safety rules. We prove that the proposed algorithm guarantees asymptotic optimality, i.e., almost-sure convergence to optimal solutions. We present results of simulation experiments and an implementation on an autonomous urban mobility-on-demand system.

• The paper presents the MVRRT* algorithm, which incrementally constructs a durational Kripke structure to jointly minimize safety rule violations and trajectory costs.
• It leverages sampling-based methods like RRT* and PRM to adaptively synthesize control strategies for reliable autonomous urban navigation.
• Simulation experiments confirm asymptotic optimality and almost-sure convergence, underscoring the algorithm's practical potential in real-world autonomous vehicle applications.

Overview of Incremental Sampling-based Algorithm for Minimum-violation Motion Planning

This paper presents an algorithm that addresses the complex issue of motion planning for dynamical systems, specifically focusing on meeting reachability goals while minimizing the violation of predefined safety rules. The core challenge tackled by this research is synthesizing control strategies that ensure an autonomous vehicle, such as a car, can navigate urban environments while adhering as closely as possible to safety rules. The algorithm utilizes concepts from sampling-based motion-planning methods like Rapidly-exploring Random Trees (RRTs) and Probabilistic Roadmaps (PRMs), adapting these to create an optimal control strategy based on a weighted Kripke structure and automaton model.

Key Contributions

The paper's primary contribution is the Minimum-Violation RRT$^*$ (MVRRT$^*$) algorithm, which incrementally constructs a durational Kripke structure representing the dynamics and safety constraints of the system. The algorithm computes trajectories that optimize two main criteria:

1. Minimizing the violation of safety rules, quantified using a level of unsafety metric.
2. Minimizing a cost function for the trajectory, typically representing travel time or another weighted objective.

Methodology

The approach leverages a weighted automaton to evaluate safety rules, ensuring the solution not only finds a feasible path to a goal within the constraints but also quantifies and minimizes unsafety if some safety constraints need to be breached to achieve the goal. This method differs from traditional RRT$^*$ approaches by allowing for dynamic adaptation in environments where strict adherence to safety goals might not be feasible or optimal.

Numerical Results and Validation

One of the notable claims of this research is the proof of asymptotic optimality and almost-sure convergence to optimal solutions, which provides theoretical assurance for the algorithm's performance and effectiveness. The results from simulation experiments further support these claims, demonstrating the ability of MVRRT$^*$ to execute in real systems, such as an autonomous urban mobility-on-demand vehicle, with reduced levels of unsafety compared to traditional approaches.

Theoretical and Practical Implications

This research has significant implications for both theory and practice in the field of autonomous vehicles and motion planning:

• Theoretical Implications: The formal model linking dynamical system properties with motion-planning algorithms expands the capabilities of sampling-based methods to adhere to complex safety requirements without sacrificing optimality.
• Practical Applications: In practical terms, this model is adaptable and can be implemented in real-world scenarios, as evidenced by its potential application in autonomous vehicle navigation where constraints like lane keeping and collision avoidance are paramount.

Future Developments

Looking forward, the integration of MVRRT$^*$ with robust sensing and detection systems opens up avenues for more sophisticated applications, such as collaborative or multi-agent systems where coordinated motion planning is essential. Further studies could also address scenarios with dynamically changing safety rules or environments, enhancing the algorithm's robustness and applicability in non-static environments.

In conclusion, this work provides substantial advancements in the field of motion planning for autonomous systems, meeting tight performance and safety requirements in complex urban environments. The methodology outlined is not only theoretically sound but also practically viable, making a significant impact on the future of autonomous vehicle technology and planning algorithms.