Empirical Time Complexity Analysis of Tractrix-Based Inverse Kinematics Algorithms

In soft robotics, hyper-redundant kinematic chains can be an effective way of modeling compliant objects and structures. Unfortunately, increasing the number of joints, the computational complexity of the calculations needed to solve the inverse kinematics problem for such systems increases sharply, especially when adopting traditional algorithms for IK solving, such as the Jacobian pseudo-inverse method. A computational procedure with a potentially lower complexity as a function of joint amount is one based on the tractrix curve. More recently, this same algorithm was formalized to function in three dimensions, but its time complexity was not adequately evaluated for cases where additional constraints are added to the kinematic solver, such as the chain being bound to a mechanical ground. Here we present the results of an empirical evaluation of the time requirements for convergence.