Efficient Motion Planning for Problems Lacking Optimal Substructure

Abstract

We consider the motion-planning problem of planning a collision-free path of a robot in the presence of risk zones. The robot is allowed to travel in these zones but is penalized in a super-linear fashion for consecutive accumulative time spent there. We suggest a natural cost function that balances path length and risk-exposure time. Specifically, we consider the discrete setting where we are given a graph, or a roadmap, and we wish to compute the minimal-cost path under this cost function. Interestingly, paths defined using our cost function do not have an optimal substructure. Namely, subpaths of an optimal path are not necessarily optimal. Thus, the Bellman condition is not satisfied and standard graph-search algorithms such as Dijkstra cannot be used. We present a path-finding algorithm, which can be seen as a natural generalization of Dijkstra’s algorithm. Our algorithm runs in O ((n B · n) log(n B · n) + n B · m) time, where n and m are the number of vertices and edges of the graph, respectively, and n B is the number of intersections between edges and the boundary of the risk zone. We present simulations on robotic platforms demonstrating both the natural paths produced by our cost function and the computational efficiency of our algorithm.

Asymptotically optimal inspection planning via efficient near-optimal search on sampled roadmaps

“…We can speed up this naïve algorithm using the notion of dominance , which is used in many shortest-path algorithms (see, e.g., Salzman et al (2017)). In our context, given two paths P , P ′ in our original roadmap G that start and end at the same vertices, we say that P dominates P ′ if ℓ ( P ) ≤ ℓ ( P ′) and S(P)⊇S(P′).…”

An Integrated Localization, Motion Planning and Obstacle Avoidance Algorithm in Belief Space

“…Uncertain obstacles are modeled as polytopes with Gaussian-distributed faces in [28]. Planning a collision-free path in the presence of "risk zones" is considered in [27] by penalizing the time spent in these risk zones. Risk contour maps which give the risk information (uncertainties in location, size and geometry of obstacles) in uncertain environments are used in [9] to obtain safe paths with bounded risks.…”