Continuous path planning with multiple constraints

We examine the problem of planning a path through a low dimensional continuous state space subject to upper bounds on several additive cost metrics. For the single cost case, previously published research has proposed constructing the paths by gradient descent on a local minima free value function. This value function is the solution of the Eikonal partial differential equation, and efficient algorithms have been designed to compute it. In this paper we propose an auxiliary partial differential equation with which we can evaluate multiple additive cost metrics for paths which are generated by value functions; solving this auxiliary equation adds little more work to the value function computation. We then propose an algorithm which generates paths whose costs lie on the Pareto optimal surface for each possible destination location, and we can choose from these paths those which satisfy the constraints. The procedure is practical when the sum of the state space dimension and number of cost metrics is roughly six or below.