TABLE OF CONTENTS First we must decide how generally to define the state spaces of the robotic systems we will consider. For example, we could treat a very general case, allowing the state space of the system to be any smooth manifold. This would allow us to study, e.g., the motion of a spherical pendulum. The configuration space of this system is the sphere S 2. Or we could limit our treatment to systems evolving on Lie groups, particularly matrix Lie groups. This would allow us to model the orientation of a satellite as a point in SO(3).
In this chapter, we restrict our attention even further to systems evolving on vector spaces 
. This allows us to get to the main results as quickly as possible. Also, any n-dimensional manifold is locally "similar" (diffeomorphic) to 
, so, equipped with a proper set of local coordinates, any n-dimensional manifold can be treated locally as 
. By making this simplification, we require the use of a local coordinate system in our computations, and we may lose information about the global structure of the space. As examples, the true configuration space of a 2R robot arm is the torus T 2 = S 1 S 1, which is doughnut-shaped while 
is not; and a global representation of the orientation of a satellite is SO(3), which is different from a local representation using three Euler angles ( 
). See figure 12.1 for another example.
