The dynamic equations of many continuum and soft robot designs can be succinctly formulated as a set of partial differential equations (PDEs) based on classical Cosserat rod theory, which includes bending, torsion, shear, and extension. In this work we present a numerical approach for forward dynamics simulation of Cosserat-based robot models in real time. The approach implicitly discretizes the time derivatives in the PDEs and then solves the resulting ordinary differential equation (ODE) boundary value problem (BVP) in arc length at each timestep. We show that this strategy can encompass a wide variety of robot models and numerical schemes in both time and space, with minimal symbolic manipulation required. Computational efficiency is gained owing to the stability of implicit methods at large timesteps, and implementation is relatively simple, which we demonstrate by providing a short MATLAB-coded example. We investigate and quantify the tradeoffs associated with several numerical subroutines, and we validate accuracy compared with dynamic rod data gathered with a high-speed camera system. To demonstrate the method’s application to continuum and soft robots, we derive several Cosserat-based dynamic models for robots using various actuation schemes (extensible rods, tendons, and fluidic chambers) and apply our approach to achieve real-time simulation in each case, with additional experimental validation on a tendon robot. Results show that these models capture several important phenomena, such as stability transitions and the effect of a compressible working fluid.
Real-time dynamics of soft and continuum robots based on Cosserat rod models