This paper introduces the design and complexity analysis of a parallel genetic algorithm to generate a "best" path for a robot arm to follow, given a starting position and a goal in three-dimensional space. Path generation takes into account any obstacles near the arm. This algorithm uses multiple optimization criteria, independent cross-pollinating populations, and handles multiple hard constraints. Individuals in the population consist of multiple chromosomes. The complexity of the algorithm is the number of generations processed times O(N ²) where N is the total number of individuals used for path generation on all of the optimizations. This research is being sponsored by NASA grant NAG 9-1401.