Kinematics of Nonzero Constant Mean Curvature Surface Models using a Kenmotsu-type Representation Formula

A simple functional expression giving multiple geometries is important for the kinematics of a smooth-surfaced robot. Such a representation gives the functions of tangent vectors and Jacobian matrices and reduces the computational time of forward and inverse kinematics. In this paper, we construct the kinematics of robotic surfaces using an equation for rotational surfaces called the Kenmotsu-type representation formula. The formulas in this paper switch between various nonzero constant mean curvature surfaces, namely, spheres, cylinders, unduloids, and nodoids. Forward kinematics and numerical simulations confirm the short computation time of these formulas; moreover, the local coordinate system on the robot is a Cartesian coordinate system. We also present an iterative algorithm for inverse kinematics based on the Newton-Raphson method.