Displacement-Actuated Continuum Robots: A Joint Space Abstraction

TL;DR

The paper advances displacement-actuated continuum robots (DACR) as a unified joint-space abstraction aligned with Clarke coordinates, addressing the lack of a general, linear mapping from joint displacements to a low-dimensional manifold. It extends the DACR framework to include variable segment length (Type-I) and twisting (Type-II/III), and discusses multi-segment configurations, providing forward/inverse mappings that remain compact and linear under the Clarke transform. Key contributions include formalizing kinematic design parameters, detailing joint representations, and deriving mappings for symmetric and generalized joint layouts, with guidance on hardware-dependent implementations. The approach enables closed-form, computationally efficient kinematics across a broad class of continuum robots, fostering scalable design, control, and planning while clarifying implicit assumptions and future directions toward dynamics and richer geometries.

Abstract

The displacement-actuated continuum robot as an abstraction has been shown as a key abstraction to significantly simplify and improve approaches due to its relation to the Clarke transform. To highlight further potentials, we revisit and extend this abstraction that features an increasingly popular length extension and an underutilized twisting. For each extension, the corresponding mapping from the joint values to the local coordinates of the manifold embedded in the joint spaces is provided. Each mapping is characterized by its compactness and linearity.

Figures (4)

• Figure 1: Abstraction leads to mathematical models. The model of the physical world informs the abstraction and this representation.
• Figure 2: Displacement-actuated continuum robot. Assuming fully constrained actuation paths, the distance $d_i$ and angle $\psi_i$ of each joint location on the cross-section is constant along the arc length.
• Figure 3: Geometric smoothness $\mathcal{G}$. The smooth backbone informs both fully constrained actuation paths that are parallel curves. The minimal distance between them is always $d$. Left side: The backbone of length $l^{\left(j\right)}$ is a curve with continuous tangent vectors. Here, $l_\text{left} < l^{\left(j\right)}$ and $l_\text{right} > l^{\left(j\right)}$ differs to $l^{\left(j\right)}$ by a same absolute among, i.e., $|d\phi|$. Right side: The construction of the parallel curves creates degenerate cases with loops and cusps between straight lines, see Pham Pham_CAD_1992 for visualizations. Furthermore, the fully constrained actuation paths are composed of straight and curved pieces.
• Figure 4: Helical fully constrained actuation path. This actuation path lies in a plane that can be rolled out, revealing a linear function and a Pythagorean equation.