GeoPro-VO: Dynamic Obstacle Avoidance with Geometric Projector Based on Velocity Obstacle

TL;DR

A geometric projector for dynamic obstacle avoidance based on velocity obstacle (GeoPro-VO) by leveraging the projection feature of the velocity cone set represented by VO and the augmented Lagrangian spectral projected gradient descent (ALSPG) algorithm.

Abstract

Optimization-based approaches are widely employed to generate optimal robot motions while considering various constraints, such as robot dynamics, collision avoidance, and physical limitations. It is crucial to efficiently solve the optimization problems in practice, yet achieving rapid computations remains a great challenge for optimization-based approaches with nonlinear constraints. In this paper, we propose a geometric projector for dynamic obstacle avoidance based on velocity obstacle (GeoPro-VO) by leveraging the projection feature of the velocity cone set represented by VO. Furthermore, with the proposed GeoPro-VO and the augmented Lagrangian spectral projected gradient descent (ALSPG) algorithm, we transform an initial mixed integer nonlinear programming problem (MINLP) in the form of constrained model predictive control (MPC) into a sub-optimization problem and solve it efficiently. Numerical simulations are conducted to validate the fast computing speed of our approach and its capability for reliable dynamic obstacle avoidance.

Figures (6)

• Figure 1: Velocity obstacle $\text{VO}_\text{R}^{\text{o}_i}(\bm{v}_{\text{o}_i})$ of robot $\text{R}$ induced by the obstacle $\text{o}_i$. $\text{CC}_\text{R}^{\text{o}_i}$ denotes the collision cone between them. If the relative velocity $\bm{v}_\text{R} -\bm{v}_{\text{o}_i} \in \text{CC}_\text{R}^{\text{o}_i}$ or the robot velocity $\bm{v}_\text{R} \in \text{VO}_\text{R}^{\text{o}_i}(\bm{v}_{\text{o}_i})$, a collision will occur between $\text{R}$ and $\text{o}_i$.
• Figure 2: Constructing the hyperplanes for $\text{VO}_\text{R}^{\text{o}_i}(\bm{v}_{\text{o}_i})$. $\text{R}$ can avoid collisions with $\text{o}_i$ if $\bigcup_{m \in \{1, 2\}} (\bm{N}_{\text{R},m}^{\text{o}_i})^T \bm{v}_\text{R} \geq c_{\text{R}, m}^{\text{o}_i}$, which means $\bm{v}_\text{R} \notin \text{VO}_\text{R}^{\text{o}_i}(\bm{v}_{\text{o}_i})$.
• Figure 3: If $\bm{v}_\text{R} \in \text{VO}_\text{R}^{\text{o}_i}(\bm{v}_{\text{o}_i})$, project the unsafe velocity to the nearest hyperplane.
• Figure 4: Robot navigation process
• Figure 6: Velocity and acceleration changes of our method GeoPro-VO and VO.