We present a partial-differential-equation-based optimal path-planning framework for curvature constrained motion, with application to vehicles in 2-and 3-spatialdimensions. This formulation relies on optimal control theory, dynamic programming, and a Hamilton-Jacobi-Bellman equation. Many authors have developed similar models and work employed grid-based numerical methods to solve the partial differential equation required to generate optimal trajectories. However, these methods can be inefficient and do not scale well to high dimensions. We describe how efficient and scalable algorithms for solutions of high dimensional Hamilton-Jacobi equations can be developed to solve similar problems very efficiently, even in high dimensions, while maintaining the Hamilton-Jacobi formulation. We demonstrate our method with several examples.
We consider the problem of time-optimal path planning for simple nonholonomic vehicles. In previous similar work, the vehicle has been simplified to a point mass and the obstacles have been stationary. Our formulation accounts for a rectangular vehicle, and involves the dynamic programming principle and a time-dependent Hamilton-Jacobi-Bellman (HJB) formulation which allows for moving obstacles. To our knowledge, this is the first HJB formulation of the problem which allows for moving obstacles. We design an upwind finite difference scheme to approximate the equation and demonstrate the efficacy of our model with a few synthetic examples.
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