Optimization-Based Collision Avoidance (1711.03449v3)

Abstract: This paper presents a novel method for reformulating non-differentiable collision avoidance constraints into smooth nonlinear constraints using strong duality of convex optimization. We focus on a controlled object whose goal is to avoid obstacles while moving in an n-dimensional space. The proposed reformulation does not introduce approximations, and applies to general obstacles and controlled objects that can be represented in an n-dimensional space as the finite union of convex sets. Furthermore, we connect our results with the notion of signed distance, which is widely used in traditional trajectory generation algorithms. Our method can be used in generic navigation and trajectory planning tasks, and the smoothness property allows the use of general-purpose gradient- and Hessian-based optimization algorithms. Finally, in case a collision cannot be avoided, our framework allows us to find "least-intrusive" trajectories, measured in terms of penetration. We demonstrate the efficacy of our framework on a quadcopter navigation and automated parking problem, and our numerical experiments suggest that the proposed methods enable real-time optimization-based trajectory planning problems in tight environments. Source code of our implementation is provided at https://github.com/XiaojingGeorgeZhang/OBCA.

• The paper introduces an exact, non-conservative reformulation that smooths non-differentiable collision constraints using convex optimization.
• It applies a versatile framework modeling obstacles as unions of convex sets, such as polytopes and ellipsoids, to support diverse trajectory planning tasks.
• Numerical validations on quadcopter navigation and autonomous vehicle parking highlight the method’s efficiency in tight and complex environments.

Optimization-Based Collision Avoidance

The paper introduces a novel framework for addressing collision avoidance in trajectory planning, specifically through an optimization-based approach. The foundational premise of the research is to transform non-differentiable collision avoidance constraints into a set of smooth nonlinear constraints. This transformation leverages the strong duality property inherent in convex optimization. The paper's contributions are significant in the context of maneuvering objects within an $n$-dimensional space, especially when those objects and the obstacles can be modeled as unions of convex sets.

Key Contributions

1. Smooth Reformulation: The primary achievement is the exact and non-conservative reformulation of non-differentiable constraints into a smooth form using convex optimization. This facilitates the use of gradient- and Hessian-based optimization algorithms, which are essential for real-time trajectory planning.
2. General Applicability: The framework accommodates a wide range of obstacles and objects by modeling them with convex sets such as polytopes and ellipsoids. This is a broad approach that applies to various navigation and path planning tasks beyond simplistic geometric assumptions.
3. Signed Distance Methodology: By integrating the notion of signed distance, the paper further refines traditional trajectory generation algorithms, allowing the computation of least-intrusive paths when collision-free trajectories are unachievable.
4. Numerical Validations: The framework's efficacy is illustrated through practical applications involving quadcopter navigation and autonomous vehicle parking. These examples highlight the method’s capability to handle path planning in tight environments efficiently.

Theoretical Implications

The theoretical implications are profound as the paper bridges a gap in optimizing complex navigation problems with non-differentiable constraints. By ensuring that the reformulation process does not introduce approximations, the paper maintains the integrity of the model constraints, augmenting the potential accuracy and reliability of trajectory planning solutions. The utilization of duality principles in convex optimization for this transformation could inspire further exploration in applying these mathematical concepts to other areas of robotic and autonomous planning.

Practical Implications

From a practical standpoint, this research provides a robust framework that enhances real-time trajectory optimization for autonomous systems navigating in confined spaces. The framework's adaptability to cope with both collision avoidance and penetration constraints makes it particularly suitable for environments where spatial precision is critical, such as automated parking scenarios.

Future Directions

Potential future developments could involve extending the models to incorporate dynamic and moving obstacles, further increasing the real-world applicability of the framework to interactive and unpredictable environments. Additionally, the integration of machine learning techniques to improve initial state guesses for optimization routines presents a promising area for enhancing solution times and robustness in complex scenographic configurations.

Conclusion

This paper effectively contributes to the field of autonomous systems by providing a detailed and mathematically rigorous approach to trajectory planning with collision avoidance. While highly technical, the proposed methodology is both practical and theoretically grounded, offering valuable insights for researchers and practitioners striving to optimize path planning in sophisticated, obstacle-dense environments.