Abstract: Method and positioner apparatus are provided for minimizing the time required to move a tool along a continuous path such as over material conveyed continuously thereunder. The continuous path is discretized into geometric moves, pairs of which are preferably joined to minimize positioner stop and go. Ranges of velocities are analyzed for each discrete move and are adjusted to overlap for fitting a continuous velocity curve therethrough. Optimizing of the velocity ranges for maximum tool acceleration and velocity results in a time-optimized velocity profile for the tool. Velocity profiles for each discrete move are further optimized to maximize acceleration and deceleration resulting in the most time spent at maximum velocity. Throughout, the method is cognizant of the effects of time and displacement being measured in integer increments. Rounding is appropriately performed and move parameters are adjusted appropriately.