Piecewise constant curvature is a popular kinematics framework for continuum robots. Computing the model parameters from the desired end pose, known as the inverse kinematics problem, is fundamental in manipulation, tracking and planning tasks. In this paper, we propose an efficient multi-solution solver to address the inverse kinematics problem of 3-section constant-curvature robots by bridging both the theoretical reduction and numerical correction. We derive analytical conditions to simplify the original problem into a one-dimensional problem. Further, the equivalence of the two problems is formalised. In addition, we introduce an approximation with bounded error so that the one dimension becomes traversable while the remaining parameters analytically solvable. With the theoretical results, the global search and numerical correction are employed to implement the solver. The experiments validate the better efficiency and higher success rate of our solver than the numerical methods when one solution is required, and demonstrate the ability of obtaining multiple solutions with optimal path planning in a space with obstacles.

翻译：分段恒定曲率是连续体机器人的一种常用运动学框架。根据期望末端位姿计算模型参数（即逆运动学问题）是操控、跟踪与规划任务中的基础问题。本文提出一种高效的多解求解器，通过理论降维与数值校正相结合的方法，解决三节恒曲率机器人的逆运动学问题。我们推导了分析条件，将原问题简化为单变量问题，并形式化证明了两个问题的等价性。此外，我们引入一种具有有界误差的近似方法，使得单变量问题可遍历求解，而其余参数可解析获得。基于这些理论结果，采用全局搜索与数值校正实现该求解器。实验验证了：当仅需单个解时，本求解器相比数值方法具有更高的效率与成功率；同时展示了其在存在障碍物的空间中获取多个解并进行最优路径规划的能力。