In Chapter 4 we introduce a new algorithm for bi-objective optimization on graphs. Similar to the approach of Chapter 3, there is the component of finding multiple paths subject to some additional constraint(s). However, in this situation these constraints are not based on computational time (like in the Anytime algorithms), but rather based on physical constraints posed on the paths themselves.
The application considered in this chapter is planning the shortest path (minimize the primary cost, e.g. distance) subject to constraints on the accumulated secondary cost (e.g. exposure to an enemy observer). First, a graph is built over the physical space using the Probabilistic RoadMap planner [40]. We then augment the state space to keep track of the remaining ‘budget’ for the secondary cost. Effectively this takes a copy of the graph generated to represent the physical space and makes a copy for every “allowable” budget level. If the secondary costs are strictly positive, this leads to an algorithm where an upward-sweep may be performed at a computational cost of O(mn) where n is the number of nodes in the original graph and m is the number of budget levels that have been discretized. However, if the secondary costs are allowed to be zero then Dijkstra’s algorithm must be applied, leading to a slightly higher computational cost of O(mn log(mn)).
The main objective of this section is to illustrate that our new algorithm can very quickly approximate what is known as the “Pareto Front”. The Pareto Front in the objective space consists of pairs of primary and secondary costs (P, S ) that correspond to a given path such that no other path can decrease one of the objective costs without increasing the other. We are interested in studying the convergence of the Pareto Front as the number of nodes in the graph grows or under refinement of the number of budget levels. The results show that our algorithm is not only fast and accurate, but recovers an approximation to the entire Pareto Front. A more common/competing approach based on the “scalarization algorithm” (see [51] and §4.4.4) is unable to recover the non- convex portions of the Pareto Front.
We illustrate the algorithm with numerical examples on a two-dimensional state space in §4.4 and well as present a field test done on a real robotics experiment in §4.5.
CHAPTER 2
CAUSAL DOMAIN RESTRICTION FOR EIKONAL EQUATIONS
2.1
Introduction
The Eikonal equation |∇u(x)| f (x) = 1 ∀x ∈Ω u(x) = q(x) ∀x ∈ Q ⊆∂Ω, (2.1)
arises naturally in many applications including continuous optimal path planning, com- putational geometry, photolithography, optics, shape from shading, and image process- ing [68]. One natural interpretation for the solution of (3.1) comes from isotropic time- optimal control problems. For a vehicle traveling through ¯Ω, f describes the speed of travel and q gives the exit time-penalty charged on Q. In this framework, u(x) is the value function; i.e., the minimum time to exit ¯Ω through Q if we start from a point x ∈Ω. The characteristic curves of the PDE (3.1) define the optimal trajectories for the vehicle motion.
The value function is Lipschitz continuous, but generally is not smooth onΩ. (The gradient of u is undefined at all points for which an optimal trajectory is not unique.) Correspondingly, the PDE (3.1) typically does not have a smooth solution and admits infinitely many Lipschitz continuous weak solutions. Additional conditions introduced in [19] are used to restore the uniqueness: the viscosity solution is unique and coincides with the value function of the above control problem.
In the last 20 years, many fast numerical methods have been developed to solve
This chapter is based on the paper Causal Domain Restriction for Eikonal Equations by Z. Clawson, A. Chacon, & A. Vladimirsky, SIAM Journal on Scientific Computing, Volume 36, Issue 5, pp. A2478- A2505, 2014. This publication is listed as [18] in the bibliography.
(3.1) on the entire domain ¯Ω; e.g., see [17, 67, 81, 88]. Many of these fast methods were inspired by classical label-correcting and label-setting algorithms on graphs; e.g., Sethian’s Fast Marching Method [67] mirrors the logic of the classical Dijkstra’s algo- rithm [22], which finds the minimum time to a target-node from every other node in the graph.
Our focus here is on a somewhat different situation, with the solution needed for one specific starting position only. On graphs, an A* modification of Dijkstra’s method [32] is widely used for similar single source/ single target shortest path problems.
There have been several prior attempts to extend A* techniques to algorithms for continuous optimal trajectory problems, but all of them have significant drawbacks: these methods either produce additional errors that do not vanish under numerical grid refinement [55, 57, 58, 59], or provide much more limited computational savings [26, 84, 85]. We believe that these disadvantages stem from an overly faithful mirroring of the “standard" A* on graphs. Our own approach is based on an alternative version of the A* algorithm [8] that has clear advantages in continuous optimal control problems. Numerical testing confirms that our method is both efficient (in terms of the percentage of domain restriction) and convergent under grid refinement.
We begin by reviewing two flavors of A* techniques on graphs in §2.2. We then describe the standard Fast Marching Method and its various A*-type modifications in §2.3. The numerical tests in §2.4 are used to compare the efficiency and accuracy of competing domain restriction techniques. §2.5 contains convergence analysis of the alternative A* under grid refinement, exploiting the probabilistic interpretation of the discretized equations. We discuss the limitations of our approach and directions of future work in §2.6.