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This paper introduces a genetic algorithm (GA) planner that is able to rapidly determine optimal or near-optimal solutions for mobile robot path planning problems in environments containing moving obstacles. The method restricts the search space to the vertices of the obstacles, obviating the need to search the entire environment as in earlier GA-based approaches. The new approach is able to produce an off-line plan through an environment containing dynamic obstacles, and can also re-calculate the plan on-line to deal with any motion changes encountered. A particularly novel aspect of the work is the incorporation of the selection of robot speed into the GA genes. The results from a number of realistic environments demonstrate that planning changes in robot speed significantly improves the efficiency of movement through the static and moving obstacles.