Work Note Kinematics: The Serret-Frenet frame and others. Roger Skjetne July 11, 2001 1 Introduction This work note will be an investigation of different kinematic frames that can possibly be used for guidance of a surface vessel. It is meant to be a preliminary work on a strategic method for nonlinear path following and trajectory tracking control of ships. 2 Kinematics 2.1 The Serret-Frenet frame As illustrated in figure (1) we see that the Serret-Frenet frame is defined by two axes in the plane, the tangent and the normal to the curve at a point s being the origin of the frame. x e y e T (1) N x t (2) y t x earth y e a r t h 2D hypothetical path Figure 1: A hypothetical reference path illustrating two different moving reference frames: (1) Serret-Frenet and (2) Parallel translating frame. The following derivation is taken from different texts. See for instance [1] for a simple treatise on the subject. Suppose a curve is given by C(x d ,y d ) with x d = x d (θ) and y d = y d (θ) where θ is an independent parametrization variable. We will now look into different properties of the curve C(x d ,y d ) and the Serret-Frenet frame. Let r d (θ,t)= £ x d (θ(t)) y d (θ(t)) ¤ T be the position vector of a point on the curve at time t. We assume that r d (θ) is a regular 1