  The small size, high dexterity, and intrinsic compliance of continuum robots
(CRs) make them well suited for constrained environments. Solving the inverse
kinematics (IK), that is finding robot joint configurations that satisfy
desired position or pose queries, is a fundamental challenge in motion
planning, control, and calibration for any robot structure. For CRs, the need
to avoid obstacles in tightly confined workspaces greatly complicates the
search for feasible IK solutions. Without an accurate initialization or
multiple re-starts, existing algorithms often fail to find a solution. We
present CIDGIKc (Convex Iteration for Distance-Geometric Inverse Kinematics for
Continuum Robots), an algorithm that solves these nonconvex feasibility
problems with a sequence of semidefinite programs whose objectives are designed
to encourage low-rank minimizers. CIDGIKc is enabled by a novel
distance-geometric parameterization of constant curvature segment geometry for
CRs with extensible segments. The resulting IK formulation involves only
quadratic expressions and can efficiently incorporate a large number of
collision avoidance constraints. Our experimental results demonstrate >98%
solve success rates within complex, highly cluttered environments which
existing algorithms cannot account for.
