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

Abstract

High motion performance and avoiding structural vibrations rely on smooth trajectory planning. The smoothness requirement depends on the robot's complexity in terms of the number of degrees of freedom and system order due to the flexibility of the joints and links. For industrial robots, the smoothness requirements have been determined and subsequently taken into account. Although unknown for soft continuum robots, the exact smoothness requirement can potentially be much larger than previously considered ones such as $\mathbf{\mathcal{C}^4}$ for collaborative robots and $\mathbf{\mathcal{C}^5}$ for quadcopter drones. To address this gap, we introduce a $\mathbf{\mathcal{C}^\infty}$-smooth trajectory generator that respects defined kinematic limits and generates infinitely differentiable trajectories. The generated path addresses and generalizes to all smoothness requirements, where a transition between sequential states is needed. Therefore, all trajectories have continuous velocity, acceleration, jerk, snap, crackle, pop, etc. Of particular interest is the case of null boundary conditions, i.e., all derivatives at the start and end of the trajectory are equal to zero. We verified and evaluated our proposed approach in simulations and on hardware.