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.

翻译：同心管连续体机器人利用嵌套式管道，这些管道需满足一系列不等式约束。当前处理不等式的方法依赖分支结构（如if-else语句），这可能导致不连续性、形成复杂的决策树、计算耗时较长且无法向量化，进而影响下游方法（如控制、学习、工作空间估计与路径规划等）的行为与结果。本文研究了一种映射方法以缓解分支方法带来的问题。我们推导出下三角变换矩阵来实现不等式的解耦，并证明了该矩阵的唯一存在性，该方法将相互依赖的约束转化为独立的有界约束。进一步针对采样、控制和工作空间估计进行了研究。采用所提映射的方法速度至少提升14倍（最高可达176倍），总能生成有效的关节构型，具有更高的可解释性，且更易于扩展。