Recursive Model-agnostic Inverse Dynamics of Serial Soft-Rigid Robots

TL;DR

The paper tackles the inverse dynamics problem for serial soft-rigid robots built from heterogeneous reduced-order models by formulating a model-agnostic, recursive dynamics framework. It leverages Kane's method to produce a recursive Euler–Lagrange-like equation set in which the inertial contributions propagate backward along the chain, achieving an $O(N)$ complexity and enabling simultaneous computation of the mass matrix $\boldsymbol{M}(\boldsymbol{q})$ via a Mass Inertial Inverse Dynamics (MIID) variant. The approach treats kinematics as an input function, allowing integration of diverse ROMs such as Locally Volume Preserving primitives, Cosserat strain models, and volume-preserving deformations, while maintaining exact equivalence to traditional EL dynamics. Numerical results on simulated soft-robot systems—including a trimmed helicoid, a hybrid rigid-soft arm, and a PCC-based planar test—demonstrate scalability and practical applicability for real-time control and design optimization in heterogeneous soft robotics.

Abstract

Robotics is shifting from rigid, articulated systems to more sophisticated and heterogeneous mechanical structures. Soft robots, for example, have continuously deformable elements capable of large deformations. The flourishing of control techniques developed for this class of systems is fueling the need of efficient procedures for evaluating their inverse dynamics (ID), which is challenging due to the complex and mixed nature of these systems. As of today, no single ID algorithm can describe the behavior of generic (combinations of) models of soft robots. We address this challenge for generic series-like interconnections of possibly soft structures that may require heterogeneous modeling techniques. Our proposed algorithm requires as input a purely geometric description (forward-kinematics-like) of the mapping from configuration space to deformation space. With this information only, the complete equations of motion can be given an exact recursive structure which is essentially independent from (or `agnostic' to) the underlying reduced-order kinematic modeling techniques. We achieve this goal by exploiting Kane's method to manipulate the equations of motion, showing then their recursive structure. The resulting ID algorithms have optimal computational complexity within the proposed setting, i.e., linear in the number of distinct modules. Further, a variation of the algorithm is introduced that can evaluate the generalized mass matrix without increasing computation costs. We showcase the applicability of this method to robot models involving a mixture of rigid and soft elements, described via possibly heterogeneous reduced order models (ROMs), such as Volumetric FEM, Cosserat strain-based, and volume-preserving deformation primitives. None of these systems can be handled using existing ID techniques.

Figures (12)

• Figure 1: Schematic representation of the considered class of soft robotic systems, conceptualized as a sequence of $N$ deformable bodies $\mathcal{B}_{i}$ and joints $\mathcal{J}_{i}$. The purple and pale brown volumes illustrate the robot bodies in their stress-free and deformed configuration, respectively. The individual kinematics of each body $\mathcal{B}_{i}$ is modeled as a generic function $\boldsymbol{f}_{\mathcal{B}_{i}}(\boldsymbol{x}_{i}, {\boldsymbol{q}}_{\mathcal{B}_{i}}) \in \mathbb{R}^{3}$, where $\boldsymbol{x}_{i} \in \mathbb{R}^{3}$ denotes the (relative) position of $\mathcal{B}_{i}$ points in the stress-free configuration and ${\boldsymbol{q}}_{\mathcal{B}_{i}} \in \mathbb{R}^{n_{\mathcal{B}_{i}}}$ is a configuration vector parametrizing the body motions.
• Figure 2: For each body $\mathcal{B}_{i}$, we introduce two reference frames $\{ S_{i} \}$ and $\{ S_{\mathcal{J}_{i}} \}$. The frame $\{ S_{i} \}$ is attached at the connection point between $\mathcal{B}_{i}$ and its successor joint $\mathcal{J}_{i+1}$, and accounts for body deformability. On the other hand, $\{ S_{\mathcal{J}_{i}} \}$ is attached at the distal end of $\mathcal{J}_{i}$, which coincides with the connection point between $\mathcal{J}_{i}$ and $\mathcal{B}_{i}$, and is used to model the relative motion between $\mathcal{B}_{i-1}$ and $\mathcal{B}_{i}$ due to the joint. The reference frames are oriented as the contact areas between the corresponding body and joint.
• Figure 3: Graphical representation of how the transformation matrix from $\{ S_{i} \}$ to $\{S_{\mathcal{J}_{i}}\}$ can be constructed using the reduced-order kinematics, under the hypothesis that the bodies are rigidly connected to the joints.
• Figure 4: Simulation 1. Trimmed helicoid continuum soft robot in (a) its stress free configuration and (b) a sample configuration. Note the radial deformation of the bodies in the deformed configuration.
• Figure 5: Simulation 1. Time evolution of the configuration variables (in $[rad]$ and $[m]$) with a zoomed view in panel (b).

Theorems & Definitions (13)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Lemma 1
• Proposition 1
• Theorem 1
• Remark 5
• Remark 6
• Corollary 1