  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.
