Using Clarke Transform to Create a Framework on the Manifold: From Sampling via Trajectory Generation to Control

TL;DR

The paper tackles the lack of a morphology-agnostic framework for displacement-actuated continuum robots and proposes a Clarke-coordinate formulation on a 2dof manifold with coordinates $(\rho_{\mathrm{Re}}, \rho_{\mathrm{Im}})$. A smooth trajectory generator on the manifold, a direct mapping between joint-space and manifold trajectories via the Clarke transform, and an encoder–decoder interface for compatibility with non-Clarke components are developed. Key contributions include a kinematic link between $\overline{\boldsymbol{\rho}}$ and arc parameters, a sampling method in Clarke space, and a simulation demonstrating a four-segment actuation. The framework enables modular, efficient, and transferable development across robotic platforms, without assuming constant curvature.

Abstract

We present a framework based on Clarke coordinates for spatial displacement-actuated continuum robots with an arbitrary number of joints. This framework consists of three modular components, i.e., a planner, trajectory generator, and controller defined on the manifold. All components are computationally efficient, compact, and branchless, and an encoder can be used to interface existing framework components that are not based on Clarke coordinates. We derive the relationship between the kinematic constraints in the joint space and on the manifold to generate smooth trajectories on the manifold. Furthermore, we establish the connection between the displacement constraint and parallel curves. To demonstrate its effectiveness, a demonstration in simulation for a displacement-actuated continuum robot with four segments is presented.

Figures (10)

• Figure 1: Surrogate displacement-actuated continuum robot.
• Figure 2: Arc length of the surrogate displacement-actuated continuum robot. For convenience, the absolute values of $\widehat{\rho}_1$ and $\widehat{\rho}_2$ are used. The bending angle $\phi$ is identical to the tip orientation. Note that the bending plane angle $\theta$ is other tip orientation. Both angles are well-known and reflected in the rotation matrix of the kinematics for the constant curvature model.
• Figure 3: Parallel curves. Dashed lines are parallel curves and the solid lines between them are the center-line. The displacement depends on the angle $\phi$ and the distance $d$. (left) 2dof case of Cavalieri's principle, where the tip and base orientations are identical, resulting in $\widehat{\rho}_i = 0$. (middle) parallel curves resulting in $\widehat{\rho}_i = \pm d\phi$. (right) full circle, where the tip and base poses are identical, resulting in $\widehat{\rho}_i = \pm 2\pi d$.
• Figure 4: Encoder-decoder architecture. Joint values of one robot type (robot A) can be transformed into joint values of a different robot type (robot B). The latent space representation is encoded as Clarke coordinates. Note that the compression is lossless allowing joint values to be uniquely reconstructed from the Clarke coordinates. (Image credit: Grassmann & Burgner-Kahrs GrassmannBurgner-Kahrs_arXiv_2024)
• Figure 5: Controller scheme utilizing Clarke transform. For the sake of compactness, we omit potential derivative terms such as velocity.