Clarke Transform and Encoder-Decoder Architecture for Arbitrary Joints Locations in Displacement-Actuated Continuum Robots

TL;DR

The paper addresses the constraint of symmetric joint layouts in displacement-actuated continuum robots by generalizing the Clarke transform to unbalanced, arbitrary joint locations and by introducing an encoder–decoder architecture to map joint values across different designs. It derives a generalized transformation matrix $\boldsymbol{M}_\mathcal{P}$, relates Clarke coordinates to arc-parameters, and enables lossless latent representations for cross-design transfer. The authors demonstrate feasible joint-value generation, $\mathcal{C}^4$-smooth trajectory planning, and PD control in the Clarke frame, with simulation showing accurate tracking across diverse morphologies and asymmetric layouts. The approach promises flexible hardware design, improved manipulability, and explicit paths for error analysis and knowledge transfer between continuum and soft robotics domains.

Abstract

In this paper, we consider an arbitrary number of joints and their arbitrary joint locations along the center line of a displacement-actuated continuum robot. To achieve this, we revisit the derivation of the Clarke transform leading to a formulation capable of considering arbitrary joint locations. The proposed modified Clarke transform opens new opportunities in mechanical design and algorithmic approaches beyond the current limiting dependency on symmetric arranged joint locations. By presenting an encoder-decoder architecture based on the Clarke transform, joint values between different robot designs can be transformed enabling the use of an analogous robot design and direct knowledge transfer. To demonstrate its versatility, applications of control and trajectory generation in simulation are presented, which can be easily integrated into an existing framework designed, for instance, for three symmetric arranged joints.

Figures (11)

• Figure 1: Joint location and improved joint representation. The kinematics of a displacement-actuated continuum robot with fixed segment length is mainly influenced by the number of joints $n$ and their location in the cross-section in terms of polar coordinates. For the $i\textsuperscript{th}$ joint, the polar coordinates is described by the distance $d_i$ to the center-line and the angle $\psi_i$. Joint representations, e.g., AllenAlbert_et_al_RoboSoft_2020DellaSantinaBicchiRus_RAL_2020GrassmannSenykBurgner-Kahrs_arXiv_2024GrassmannBurgner-Kahrs_ICRA_EA_2024, have been proposed to consider various arrangement. Our approach generalizes and covers all cases.
• Figure 2: Kinematic design parameters of a displacement-actuated continuum robot. (Image credit: Grassmann et al.GrassmannSenykBurgner-Kahrs_arXiv_2024)
• Figure 3: Possible designs of arbitrary asymmetric joint location.
• Figure 4: Encoder-decoder architecture. Joint values of one robot type (robot A) with $n$-dimensional joint space can be transformed into joint values of a different robot type (robot B) with $m$-dimensional joint space. The latent space representation is encoded as Clarke coordinates. It is worth noticing that the compression is a lossless compression that allows joint values to be uniquely reconstructed from the Clarke coordinates.
• Figure 5: Workflow of the evaluation.