Online Minimum-Jerk Trajectory Generation

Robotic trajectory generation is reformulated as a controller design problem. For minimum-jerk trajectories, an optimal controller using the Hamilton-Jacobi-Bellman equation is derived. The controller instantaneously updates the trajectory in a closed-loop system as a result of the changes in the reference signal. The resulting trajectories coincide with piece-wise fifth-order polynomial trajectories for piece-wise constant target states. Since having hard constraints on the final time poses certain robustness issues, a smooth transition between the finite-horizon and an infinite-horizon problem is developed. This enables to switch softly to a tracking mode when a moving target is reached