On the Disentanglement of Tube Inequalities in Concentric Tube Continuum Robots

TL;DR

The paper addresses the challenge of interdependent tube-length inequalities in concentric tube continuum robots by deriving a unique affine transformation $oldsymbol{M}_oldsymbol{eta}$ that maps to independent box constraints via $oldsymbol{eta} = oldsymbol{M}_oldsymbol{eta}oldsymbol{eta}_{oldsymbol{U}}$ with $oldsymbol{eta}_{oldsymbol{U}} ext{ in }[-1,1]$. It proves the uniqueness of the low-triangular mapping and extends the framework to a minimal-length variant, enabling branchless sampling and easier integration into control and workspace estimation. Across sampling, workspace estimation, and control, the method yields 100% valid samples, substantial speedups (up to about $176 imes$), and simplified, more interpretable formulations, with vectorized implementations achieving further gains. These results enhance real-time planning and learning for CTCRs and potentially other variable-length continuum robots, by converting complex inequality constraints into tractable box constraints and enabling inverse-transform sampling.

Abstract

Concentric tube continuum robots utilize nested tubes, which are subject to a set of inequalities. Current approaches to account for inequalities rely on branching methods such as if-else statements. It can introduce discontinuities, may result in a complicated decision tree, has a high wall-clock time, and cannot be vectorized. This affects the behavior and result of downstream methods in control, learning, workspace estimation, and path planning, among others. In this paper, we investigate a mapping to mitigate branching methods. We derive a lower triangular transformation matrix to disentangle the inequalities and provide proof for the unique existence. It transforms the interdependent inequalities into independent box constraints. Further investigations are made for sampling, control, and workspace estimation. Approaches utilizing the proposed mapping are at least 14 times faster (up to 176 times faster), generate always valid joint configurations, are more interpretable, and are easier to extend.

Figures (5)

• Figure 1: Characteristics of the joint space of a 4dof CTCR with two nested tubes. While tube rotations $\alpha_i$ form a torus, therefore, can be represented as square, the tube translations $\beta_i$ result in a parallelogram. This parallelogram is bounded by the inequalities. Considering this geometric insight, it follows that an affine transformation $M_\mathcal{B}$ can be used to map a square $\mathcal{U}^2$ to the parallelogram $\mathcal{B}$. The geometric insight extends to $N$ tubes, and it should be clear that the transformed space, i.e., $\mathcal{U}^N$ for $N$ tubes, is more desirable.
• Figure 2: Distributions for different sampling method. The columns refer to the respective sampled joint values $\beta_i$, whereas the rows refer to the used sampling method. The rejection sampling methods (a) to (d) are described in Sec. \ref{['sec:rejection_sampling_via_branching']}. The direct sampling using $\boldsymbol{M}_\mathcal{B}$ is described in Sec. \ref{['sec:sampling_via_M_B']}.
• Figure 3: Affecting of distribution on two-dimensional workspace of toy examples. Two variables are randomly drawn from uniform distributions and used afterward to generate the workspace for a square, a disk, and a single-segment constant curvature continuum robot. Each workspace is divided into two areas -- (blue) vanilla approach and (green) proposed approach. The blue sampled points are linearly transformed, whereas, for the green sampled points, with the exception of the squared workspace, the additional transformation \ref{['eq:beta_sampling_area']} is used for the translational variable. As can be seen from the realized distribution, the green sampled points yield a more homogeneous distribution.
• Figure 4: Convergence of workspace estimation in percent over the number of samples. The top and bottom plot show the workspace estimation for a CTCR with geometrical parameters stated in GrassmannBurgner-Kahrs_et_al_IROS_2022 and Burgner-KahrsWebster_et_al_IROS_2014, respectively. The solid lines are the median values being limited by the minimum and maximum estimates. Ten different permutations of the ordered dataset are used to compute the median, minimum, and maximum area. Note that a new sampled point between two existing points at the boundary can reduce the computed area by the triangular area spanned by those three points and, therefore, the dashed 100% mark can be exceeded. Further note that the workspace estimation using $\boldsymbol{M}_\mathcal{B}$ has a faster overall time of convergence.
• Figure 5: Block diagram of the closed-loop system in state-space. The system encased in blue represents is a simple PT1 model of the simplified motor dynamics controlled by a PI controller. This system must account for the two interdependent inequalities \ref{['eq:inequality_beta']} and \ref{['eq:inequality_L']}. By using $M_\mathcal{B}$, they become independent box constraints that are easier to handle.