Collisionless Multi-Agent Path Planning in the Hamilton-Jacobi Formulation

Abstract

We present a method for collisionless multi-agent path planning using the Hamilton-Jacobi-Bellman equation. Because the method is rooted in optimal control theory and partial differential equations, it avoids the need for hierarchical planners and is black-box free. Our model can account for heterogeneous agents and realistic, high-dimensional dynamics. We develop a grid-free numerical method based on a variational formulation of the solution of the Hamilton-Jacobi-Bellman equation which can resolve optimal trajectories even in high-dimensional problems, and include some practical implementation notes. In particular, we resolve the solution using a primal-dual hybrid gradient optimization scheme. We demonstrate the method's efficacy on path planning problems involving simple cars and quadcopter drones.

Figures (3)

• Figure 3: Two cars navigate around obstacles to their final destinations. Notice they are careful to narrowly avoid colliding at time $2$.
• Figure 4: Six cars navigate to antipodal points around a circle. Gray circles are undiscovered obstacles, black circles are discovered obstacles. Dotted lines are the currently resolved paths. These are recomputed to avoid obstacles as needed.
• Figure 5: Four quadcopter drones navigate around obstacles and each other to their final destinations. Here the obstacles are the yellow cylinders. Notice that the drones on the left in the initial picture (green and pink lines) allow themselves to drop a bit at the beginning, so that they can then fly unobstructed between the obstacles. This is to avoid collisions which are harder to prevent in a dynamic model compared to a kinematic model, and hence need to be considered long in advance.