Characterisation of Frenet-Serret and Bishop motions with applications to needle steering

Frenet-Serret and Bishop rigid-body motions have many potential applications in robotics, graphics and computer-aided design. In order to study these motions, new characterisations in terms of their velocity twists are derived. This is extended to general motions based on any moving frame to a space curve. Furthermore, it is shown that any such general moving frame motion is the product of a Frenet-Serret motion with a rotation about the tangent vector. These ideas are applied to a simple model of needle steering. A simple kinematic model of the path of the needle is derived. It is then shown that this leads to Frenet-Serret motions of the needle tip but with constant curvature. Finally, some remarks about curves with constant curvature are made. © Cambridge University Press 2013.