A system, medium, and method including obtaining a plurality of positions for multiple components defined by a plan； obtaining a set of constraints that express limitations for the multiple components at the plurality of positions, the constraints being applicable to a plan where the multiple components synchronously change their positions with time to traverse a prescribed sequence of the plurality of positions, at least one of the multiple components being further constrained to change its position over time by staying within a predefined tolerance to a predefined smooth function of position over time between different positions； determining a trajectory of position and a minimum duration in which the multiple components completely synchronously traverse the prescribed sequence of positions while satisfying the constraints for the multiple components； and generating a record of the determined trajectory of position and the minimum duration for the plurality of components.
