Calculating the Meeting Point of Scattered Robots on Weighted Terrain Surfaces

In this paper we discuss the problem of determining a meeting point of a set of scattered robots R = {r1, r2, . . . , rs} in a weighted terrain P which has n > s triangular faces. Our algorithmic approach is to produce a discretization of P by producing a graph G = {V , E} which lies on the surface of P . For a chosen vertex p ∈ V , we define ‖Π(ri, p)‖ as the minimum weight cost of traveling from ri to p . We show that minp′∈V G{max1≤i≤s{‖Π(ri, p )‖}} ≤ minp∗∈P{max1≤i≤s{‖Π(ri, p )‖}}+W |L| where L is the longest edge of P , W is the maximum cost weight of a face of P , and p is the optimal solution. This error of W |L| is an upper bound which can be stated more precisely as W |L|(1+2k) 2(m+1) , where m is an adjustable non-zero parameter and k is the maximum number of segments of Π(ri, p ) ∀ 1 ≤ i ≤ s. Our algorithm requires O(snm log(snm)+ snm) time to run, where m = n in the Euclidean metric and m = n in the weighted metric. However, we show through experimentation that only a constant value of m is required (e.g., m=8) in order to produce very accurate solutions (< 1% error). Hence, for typical terrain data, the expected running time of our algorithm is O(sn log(sn)). Also, as part of our experiments we show that by using geometrical subsets (i.e., 2D/3D convex hulls, 2D/3D bounding boxes and random selection) of the robots we can improve the running time for finding p, with minimal or no additional accuracy error when comparing p to p.