Coverage of a planar point set with multiple constrained robots

An important problem that arises in many applications is: Given k robots with known processing footprint to process a set of N points in the plane, find trajectories for each robot satisfying the geometric, kinematic, and dynamic constraints such that the time required to cover the points (processing time plus travel time) is minimized. This problem is a hybrid discrete-continuous optimization problem and is hard to solve optimally even for k = 1. One approach is to treat this as a two stage problem where the first stage is to find the best possible path satisfying the geometric constraints and then convert it into a trajectory satisfying the differential constraints. In this paper, we consider an industrial microelectronics manufacturing system of k(= 2) robots, with square footprints, that are constrained to translate along a line while satisfying proximity constraints. The points lie on a planar base plate that can translate along the plane normal to the direction of motion of the robots. We solve the geometric problem of path generation for the robots using a two step approach that yields a suboptimal solution: 1) Minimize the number of k-tuples subject to geometric constraints. 2) Solve a Traveling Salesman Problem (TSP) in the k-tuple space with an appropriately defined metric to minimize the total travel cost. We show that for k = 2, step 1 can be converted to a maximum cardinality matching problem on a graph and solved optimally in polynomial time. The matching algorithm takes O(N³) time in general and is too slow for large datasets. Therefore, we also provide a greedy algorithm for step 1 that takes O(N log N) time. We provide computational results comparing the two approaches and show that the greedy algorithm is very close to the optimal solution for large datasets. We also provide local search based heuristics to improve the TSP tour in the pair space and give preliminary implementation results showing an improvement of 1% to 2% in the resultant tour.