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. Further 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.