Toward Dynamic Control of Tendon-driven Continuum Robots using Clarke Transform

TL;DR

This work develops a Clarke transform–based dynamic model for tendon-driven continuum robots with multiple segments and an arbitrary number of tendons per segment, placing the dynamics on a 2DoF manifold embedded in joint space to intrinsically satisfy tendon constraints. Derived via Euler–Lagrange with the CC assumption, the model maps arc-space dynamics to the manifold and then to tendon actuations, enabling constraint-informed linear controllers (PID/PD) that require only two control parameters per segment. To handle physically infeasible negative tendon forces, three strategies—clipping, redistribution, and shifting—are analyzed, with shifting preserving manifold torques and enabling stiffness adjustments through pretension. Validation includes simulations and experiments on a 1-segment, 5-tendon prototype, showing accurate trajectory tracking at real-time rates and demonstrating robustness under dynamic actuation, cogging effects, and model simplifications. The approach broadens the design space for TDCRs by enabling efficient control for overactuated segments without increasing controller complexity, with potential impact on safer, stiffer, and more controllable soft robots.

Abstract

In this paper, we propose a dynamic model and control framework for tendon-driven continuum robots (TDCRs) with multiple segments and an arbitrary number of tendons per segment. Our approach leverages the Clarke transform, the Euler-Lagrange formalism, and the piecewise constant curvature assumption to formulate a dynamic model on a two-dimensional manifold embedded in the joint space that inherently satisfies tendon constraints. We present linear and constraint-informed controllers that operate directly on this manifold, along with practical methods for preventing negative tendon forces without compromising control fidelity. This opens up new design possibilities for overactuated TDCRs with improved force distribution and stiffness without increasing controller complexity. We validate these approaches in simulation and on a physical prototype with one segment and five tendons, demonstrating accurate dynamic behavior and robust trajectory tracking under real-time conditions.

Figures (8)

• Figure 1: TDCR prototype with one segment and five tendons. Each tendon is driven by a backdrivable actuation unit, which is controlled by a micro-controller Grassmann2024. The robot is controlled by linear controllers as described in this work.
• Figure 2: Mappings between kinematic spaces of a TDCR. The robot-independent mapping is denoted by $\boldsymbol{f}_{\mathrm{ind}}$, while the robot-dependent mapping is given by $\boldsymbol{f}_{\mathrm{dep}}=\boldsymbol{f}_{\mathrm{M,I}} \circ \boldsymbol{f}_{\mathrm{M,II}}$. The mappings from joint space to its manifold and from manifold to arc space are represented by $\boldsymbol{f}_{\mathrm{M,I}}$ and $\boldsymbol{f}_{\mathrm{M,II}}$, respectively. All inverse mappings are indicated by $\boldsymbol{f}^{-1}$.
• Figure 3: Physical interpretation of the Clarke coordinates. The blue line lies within the bending plane. The length difference to the arc length is the virtual displacement. The virtual displacement GrassmannSenykBurgner-Kahrs_submitted_2024 is equal to $\sqrt{q_{\mathrm{Re}}^2 + q_{\mathrm{Im}}^2}=\theta_{} r_{\mathrm{d}}$. The orange arrows lie within the respective projected plane corresponding to the $xz$-plane and $yz$-plane of the base. The virtual displacement can be projected onto the respective plane, resulting in the respective projected virtual displacement $q_{\mathrm{Re}}$ and $q_{\mathrm{Im}}$. The corresponding arc parameters $\theta$ and $\phi$ are also shown.
• Figure 4: Block diagram of the proposed controllers operating on the manifold: The desired trajectory, $q_{\mathrm{Re,des}}$ and $q_{\mathrm{Im,des}}$, serves as input to the controller, which outputs the generalized forces on the manifold $\boldsymbol{\tau}_{\mathrm{C}}$. These are mapped into joint space using the blue blocks, which implement (\ref{['eq:IM_Torques_JointSpace2Manifold']}). The orange block represents a saturation step, which prevents negative tendon forces by applying one of the proposed methods—clipping, redistribution, or shifting—as detailed in Sec. \ref{['sec:control_strategies']}. The measured tendon displacements $\boldsymbol{q}_{}$ are transformed onto the manifold using the green block, which follows (\ref{['eq:FM_JointSpace2Manifold']}). The diagram depicts the PID controller, whereas the PD controller omits the integral component in the controller gains.
• Figure 5: Translational and rotational energies of one segment during a step trajectory where the segment is controlled to bend to $\theta=\frac{\pi}{4}$, then rotate to $\phi=\frac{\pi}{4}$ while maintaining the bending, and finally return to its initial straight position. Note that the energy is scaled logarithmically.