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The problem of optimal path planning through narrow spaces in an unstructured environment is considered. The optimal path planning problem for a Dubins agent is formulated as a chance-constrained optimal control problem (CCOCP), wherein the uncertainty in obstacle boundaries is modelled using standard probability distributions. The chance constraints are transformed to deterministic equivalents using the inverse cumulative distribution function and subsequently incorporated into a deterministic optimal control problem. Due to multiple convex sub-regions introduced by the obstacles, the initial guess provided to optimal control solver is crucial for computation time and optimality of the solution. A constrained Delaunay triangulation mesh based approach is developed that ensures the initial guess to lie in the optimal sub-convex region. Finally, off-the-shelf software is used to transcribe the optimal control problem to a nonlinear program (NLP) using Gaussian quadrature orthogonal collocation and solved to obtain an optimal path that conforms to system dynamics. By varying the upper bound on probability of obstacle collision, a family of solutions is generated, parameterized by the risk associated with each solution. This approach enables discovery of special “keyhole trajectories” that can provide significant cost savings in a tightly-spaced obstacle field. Merits of this approach are illustrated by comparing it with the traditional bounded uncertainty approach. |
