Analytical Derivatives for Efficient Mechanical Simulations of Hybrid Soft Rigid Robots

Abstract

Algorithms that use derivatives of governing equations have accelerated rigid robot simulations and improved their accuracy, enabling the modeling of complex, real-world capabilities. However, extending these methods to soft and hybrid soft-rigid robots is significantly more challenging due to the complexities in modeling continuous deformations inherent in soft bodies. A considerable number of soft robots and the deformable links of hybrid robots can be effectively modeled as slender rods. The Geometric Variable Strain (GVS) model, which employs the screw theory and the strain parameterization of the Cosserat rod, extends the rod theory to model hybrid soft-rigid robots within the same mathematical framework. Using the Recursive Newton-Euler Algorithm, we developed the analytical derivatives of the governing equations of the GVS model. These derivatives facilitate the implicit integration of dynamics and provide the analytical Jacobian of the statics residue, ensuring fast and accurate computations. We applied these derivatives to the mechanical simulations of six common robotic systems: a soft cable-driven manipulator, a hybrid serial robot, a fin-ray finger, a hybrid parallel robot, a contact scenario, and an underwater hybrid mobile robot. Simulation results demonstrate substantial improvements in computational efficiency, with speed-ups of up to three orders of magnitude. We validate the model by comparing simulations done with and without analytical derivatives. Beyond static and dynamic simulations, the techniques discussed in this paper hold the potential to revolutionize the analysis, control, and optimization of hybrid robotic systems for real-world applications.

Figures (17)

• Figure 1: Graphical summary of the GVS model: (a) Schematic of the implemented recursive scheme. (b) Block diagram showing the summary of the GVS $FD$ algorithm.
• Figure 2: Schematic of the RNEA for a single soft body showing the forward pass for kinematics and the backward pass for joint wrench computation. The joint motion subspace ($\bm{S}_\alpha$) projects the resultant wrench acting on the joint into the generalized coordinate space.
• Figure 3: (a) Schematics of cable-driven soft manipulator with cable routing. $L = 50\;\text{cm}$, base radius $r_b = 3 \;\text{cm}$, and tip radius $r_t = 1.5\;\text{cm}$. RGB arrows indicate $x$, $y$, and $z$ axes of the global frame. (b) Actuation input used for the simulation.
• Figure 4: Dynamic response of the CDM: (a) Snapshots of the manipulator at different times and the tip trajectory with and without using analytical derivatives (AD) of $FD$. (b) Mismatch between the tip positions ($\bm{r}_{tip}$). (c) State mismatch. Mismatch between numerical and analytical derivatives of $FD$ (d) with respect to $\bm{q}$ and (e) with respect to $\dot{\bm{q}}$
• Figure 5: Static simulation results: (a) Ten arbitrary equilibrium shapes. (b) The mismatch between tip positions. (c) The mismatch static equilibrium solutions.