$\mathcal{C}^\infty$ Smooth Trajectory Planning with Application to Soft Robots

This paper presents a trajectory planning method that generates infinitely differentiable paths ($\mathcal{C}^\infty$) to ensure high motion performance and avoid structural vibrations in soft continuum robots. The approach generalizes existing smoothness requirements (such as $\mathcal{C}^4$ for collaborative robots and $\mathcal{C}^5$ for drones) and ensures continuous derivatives including velocity, acceleration, jerk, snap, crackle, pop, etc., even under null boundary conditions where all derivatives at start and end are zero. The method was verified and evaluated through simulations and hardware experiments.