Clarke Transform -- A Fundamental Tool for Continuum Robotics

TL;DR

This paper addresses the challenge of connecting high-dimensional displacement actuation in continuum and soft robots to task-space behavior. It introduces the Clarke transform, mapping an $n$-dof actuator space to a two-dimensional manifold via a generalized matrix $\boldsymbol{M}_\mathcal{P}$, and derives branchless, closed-form robot-dependent mappings for forward and inverse kinematics that are singularity-free. By grounding Clarke coordinates in arc-space parameters $(\kappa,\theta)$ and showing their relation to a virtual displacement, the authors unify existing state representations and enable efficient kinematics, sampling, and control on the 2dof manifold. The framework supports real-time applications, provides a physically interpretable and amplitude-invariant representation, and paves the way for unified modeling across tendon-driven, pneumatic, and soft continuum robots.

Abstract

This article introduces the Clarke transform and Clarke coordinates, which present a solution to the disengagement of an arbitrary number of coupled displacement actuation of continuum and soft robots. The Clarke transform utilizes the generalized Clarke transformation and its inverse to reduce any number of joint values to a two-dimensional space without sacrificing any significant information. This space is the manifold of the joint space and is described by two orthogonal Clarke coordinates. Application to kinematics, sampling, and control are presented. By deriving the solution to the previously unknown forward robot-dependent mapping for an arbitrary number of joints, the forward and inverse kinematics formulations are branchless, closed-form, and singular-free. Sampling is used as a proxy for gauging the performance implications for various methods and frameworks, leading to a branchless, closed-form, and vectorizable sampling method with a 100 percent success rate and the possibility to shape desired distributions. Due to the utilization of the manifold, the fairly simple constraint-informed, two-dimensional, and linear controller always provides feasible control outputs. On top of that, the relations to improved representations in continuum and soft robotics are established, where the Clarke coordinates are their generalizations. The Clarke transform offers valuable geometric insights and paves the way for developing approaches directly on the two-dimensional manifold within the high-dimensional joint space, ensuring compliance with the constraint. While being an easy-to-construct linear map, the proposed Clarke transform is mathematically consistent, physically meaningful, as well as interpretable and contributes to the unification of frameworks across continuum and soft robots.

Figures (11)

• Figure 1: Schematics of a displacement-actuated robot with sufficient smooth center-line with fixed length $l$. Each of the displacement-actuated joints is equally distributed, described by $d_i$ and $\psi_i$. Its value is described by a displacement $\rho_i$.
• Figure 2: Commutative diagram-like overview. To circumvent the explicit consideration of the displacement constraint, i.e., $\sum \rho_i = 0$, approaches and methods should be considered on the 2dof manifold embedded in the $n\,$dof joint space. For the transform, linear maps $\boldsymbol{M}_\mathcal{P}$ and $\boldsymbol{M}_\mathcal{P}^{-1}$ are used and any output denoted by $\left(\rho_\text{Re}^*, \rho_\text{Im}^*\right)$ can subsequently mapped back leading to $\rho^*$.
• Figure 3: Visual aid for property \ref{['eq:properties_hotvec']} and \ref{['eq:properties_circle_manifold']}. Regarding the property \ref{['eq:properties_hotvec']} illustrated in the sequence (a) to (c), a one-hot vector $\mathbbm{1}_{n \times 1}^{(k)}$ amplifies after mapping back from the manifold to the displacement. Its magnitude remains the same, whereas all other displacements are either equal or smaller than a previous one-hot vector. Regarding the property \ref{['eq:properties_circle_manifold']} illustrated in (b) and (c), the $\overline{\boldsymbol{\rho}}$ in \ref{['eq:properties_circle_manifold']} parameterized by $\alpha$ describes a circle. Tracing the tip of all possible displacement parameterized by $\psi \in \left[0, 2\pi\right)$ creates an ellipse. Its semi-major and semi-minor axes are $\sqrt{d^2 + \rho_\mathrm{Re}^2 + \rho_\mathrm{Im}^2}$ and $d$ length, respectively. The maximum displacement achievable coincides with the angle $\alpha$, hinting at its geometric interpretation.
• Figure 4: Physical interpretation of the Clarke coordinates. The magenta line lies within the bending plane. The length difference to the arc length is the virtual displacement. The virtual displacement is equal to $\sqrt{\rho_\text{Re}^2 + \rho_\text{Im}^2} = dl\kappa = d\phi$. The yellow arrows lie within the respective projected plane corresponding to the $xz$-plane and $yz$-plane of the base. Similar to the curvature components $\kappa_x$ and $\kappa_y$, the virtual displacement can be projected onto the respective plane, resulting in the respective projected virtual displacement.
• Figure 5: Spaces and their mappings. To map between the actuator, e.g., motor angles to joint values, e.g., tendon displacements, $f_\text{dyn}$ and $f_\text{dyn}^{-1}$ are used. Note that it is crucial to treat actuation space and joint space as different spaces as $f_\text{dyn}$ is not necessarily a linear map. The robot-dependent mapping is denoted by $f_\text{dep}$, whereas robot-independent mapping is denoted by $f_\text{ind}$. Given both mappings, the forward and inverse kinematics can be found using $f_\text{ind}\circ f_\text{dep}$ and $f_\text{dep}^{-1}\circ f_\text{ind}^{-1}$, respectively.

Theorems & Definitions (9)

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