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

Abstract

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.

Rational rotation-minimizing frames—Recent advances and open problems

“…In the Darboux vector (5), the term τ t specifies a rotation rate for the normal-plane vectors p, b about the tangent t. This term is superfluous to an adapted frame comprising the tangent and two vectors that span the normal plane at each point, and by eliminating it one can find normal-plane vectors u, v that do not rotate about t. This fact was observed in the paper There is more than one way to frame a curve [3] by Bishop, who couched the problem in terms of parallel transport of the vectors u, v along the curve -from any initial orientation, the variation of u, v should be only such as to suffice to keep them in the normal plane, i.e., their derivatives must always be parallel to t. Since the construction of such "relatively parallel" normal-plane vector fields is an initial-value problem, any curve admits a one-parameter family of rotation-minimizing adapted frames, also called [45,49,54] Bishop frames. The Frenet and rotation-minimizing frames are compared in Figure 1.…”

Persistent rigid-body motions and study’s “Ribaucour” problem

“…In [27] it was shown that Bishop motions are characterised by the fact that their body-frame velocity twist must always lie in a IIB three system of screw with modulus p = 0. The fact that this is a II system indicates that (almost) all the twists in the system have the same pitch, the B here means that the system contains a single exceptional twist with infinite pitch.…”

“…The most general rigid-body motion associated with a curve is a general frame motion which was called an aeroplane motion in [27]. Such a motion can be written as the product …”

A Class of Explicitly Solvable Vehicle Motion Problems

“…Theorem III.2. (After Brockett 1999) Consider a system as defined above; that is a left-invariant system defined by (11) with an objective function as given in (15) and subject to the restriction that the controls u(t) lie in a Lie triple system. The stationary solution to such a system is given by,…”

“…These are very similar to the better known Frenet-Serret motions and are sometimes referred to as frame-rotation minimising motions or natural motions in the computer aided design literature. In [11] it was shown that Bishop motions can be characterised as left-invariant systems on SE(3) where the control vector remains in a IIB(p = 0) screw system. Above it was shown that this screw system is a Lie triple system; the third row in table III.…”