Position and Displacement

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Chapter 1 Position and Displacement

Prior to studying the kinematics of rigid bodies, we must address the problem of defining their position in space. Parametrizing the position of a rigid body is equivalent to parametrizing the position of one of its points and to parametrizing its orientation (or attitude). The first task is accomplished by adopting one of several coordinate systems. Much of this chapter will be devoted to the second task. We shall show that the orientation of a rigid body is parametrized by at most three independent parameters which can be defined by employing one of three methods:

by using rotation operators whose matrix representations rely on direction cosines,
by using one of several Euler sequences defined in terms of three angles,
by using a single rotation defined in terms an equivalent angle/direction.

We will show how these methods relate to each other, and point out their relative benefits and drawbacks. We will also show that quaternions are useful tools to represent spatial rotations. Finally, we end this chapter with the study of finite displacements of rigid bodies.