Optimal Path-Planning With Random Breakdowns

Time-Optimal Paths for Simple Cars with Moving Obstacles in the Hamilton-Jacobi Formulation

“…Reverting back to the general framework from the beginning of section 2, we recall that for our models, the value function u(x, t) denotes the minimum achievable distance to the desired endpoint given that the vehicle is in position x at time t. An alternative approach to path-planning, like that used in [5,6,10,11,12,13,14], is to define the value function τ (x, t) to be the optimal remaining travel time to the desired endpoint given that the vehicle is in position x at time t. In the latter case, by definition, τ (x, t) = +∞ whenever there is no admissible path for a car which is at x at time t which can steer the car to the desired ending point before hitting the time horizon T , and thus τ (x, t) is finite if and only if there is an admissible path to the ending point in the allotted time. When there are no obstacles (or stationary obstacles), once τ (x, t) becomes finite, it will remain constant.…”

“…In particular, because we will solve these PDEs by translating them into optimization problems, it will be most convenient if we can avoid a formulation which requires boundary conditions, which would translate into difficult constraints in the optimization. Because of this, we opt for a level-set-type formulation in the vein of [7,8,9], as opposed to the control theoretic approach of [5,6,10,11,12,13,14]. These approaches are compatible, but different in philosophy.…”

“…In this paper, we present an optimal path-planning method for simple vehicles which maintains the Hamilton-Jacobi PDE formulation, but is very efficient even for problems with high-dimensional state spaces. This requires a reformulation of the standard control theoretic minimal-time path-planning problem analyzed in [5,6,10,11,12,13,14]. Our method represents a significant step toward fully interpretable PDE-based motion-planning algorithms which are real-time applicable.…”

“…To the authors' knowledge the problem was first analyzed using dynamic programming and PDE by Takei, Tsai and others [5,6]. Besides their work, there is a strong precedent in the literature for control theoretic, PDE-based optimal path-planning in a number of applications [7,8,9,10,11,12,13,14]. Trajectory generation methods which are rooted in PDE have the advantage that they are easy to implement, entirely interpretable, and can provide theoretical guarantees regarding optimality, robustness, and other concerns.…”