Planning high order trajectories with general initial and final conditions and asymmetric bounds

This paper presents a trajectory planning algorithm for linear multi-axis systems.It generates smooth trajectories of any order subject to general initial and final con-ditions, and constant state and control constraints. The algorithm is recursive, as itconstructs a high order trajectory using lower order trajectories. Multi-axis trajecto-ries are computed by synchronizing independent single-axis trajectories to reach theirrespective targets at the same time.The algorithm’s efficiency and ability to handle general initial and final conditionsmake it suitable for reactive real time applications. Its ability to generate high ordertrajectories makes it suitable for applications requiring high trajectory smoothness. Thealgorithm is demonstrated in several examples for singleand two-axis trajectories oforders 2− 6.