Case study: motion planning algorithm

Business problem: An industrial engineer programs a robotic manufacturing device to insert bolts in sequential receptacles on a widget, then attach and tighten nuts onto each bolt. The robot carriers a different tool for each operation, and can switch between them automatically on command. However, the process of switching between tools adds considerable time to the overall workflow, especially when the tool is switched back and forth over each bolt. This has been identified as a source of inefficiency, and the engineer wants to improve the speed of the process, reducing the overall time required to complete each unit.

In order to eliminate the latency introduced by switching between tools repeatedly, the engineer decides to program the robot to install all of the bolts first, and then switch tools before returning and installing all of the nuts. To further improve performance, he does not want the robot to reset to it's original starting position, but...