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 conditions, and constant state and control constraints. The algorithm is recursive, as it constructs a high order trajectory using lower order trajectories. Multi-axis trajectories are computed by synchronizing independent single-axis trajectories to reach their respective targets at the same time.The algorithm's efficiency and ability to handle general initial and final conditions make it suitable for reactive real-time applications. Its ability to generate high order trajectories makes it suitable for applications requiring high trajectory smoothness. The algorithm is demonstrated in several examples for single- and dual-axis trajectories of orders 2-6.