Before revision: Motion planning is a fundamental problem in robotics and machine perception.Sampling-based planners find accurate solutions by exhaustively exploring the space, but are inefficient and tend to produce jerky motions. Optimization and learning-based planners are more efficient and produce smooth trajectories. However, a significant hurdle that these approaches face is constructing a differentiable cost function that simultaneously minimizes path length and avoids collisions. These two objectives are conflicting by nature -- path length is continuous and well-behaved, but collisions are discrete non-differentiable events. Reconciling these terms has been a significant challenge in optimization-based motion planning. The main contribution of this paper is a novel cost function that guarantees collision-free shortest paths are found at its minimum.
After revision: We demonstrate that challenging shortest path problems can be solved via direct spline regression from a neural network, trained in an unsupervised manner (i.e. without requiring ground truth optimal paths for training). To achieve this, we derive a geometry-dependent optimal cost function whose minima guarantees collision-free shortest paths are found at its minimum.