TABLE OF CONTENTS This section discusses second-order mechanical control systems in greater depth, leading to simplified controllability tests and ideas that lead to reduced-complexity motion planning. We introduce the minimal set of ideas from Riemannian geometry that allows us to do this. We do not attempt to be rigorous or thorough in our treatment of mechanical systems from a geometric viewpoint. The motivated reader is instead referred to the books by Abraham and Marsden [10], do Carmo [131], Marsden and Ratiu [308], Bloch [50], Bullo and Lewis [77], and Boothby [60] for further study of differential geometry in mechanics and control. The results of this section are used in subsection 12.5.7, but otherwise this section can be skipped without affecting the reading of the rest of the chapter.
In chapter 10 we derived equations of motion of the form
| (12.7) |  |
where f is a generalized force vector, T( q) defines the action of f on the coordinates, M( q) is the inertia matrix, 
are Coriolis and centrifugal terms, and g( q) are potential terms. Recall that 
, where ?( q) is the set of 
Christoffel symbols of the inertia matrix M( q), and the computation 
is described in chapter 10. [2] We restrict our discussion in this section to simple mechanical control systems of this form with g( q) = 0.
Since we are considering underactuated systems, we can...
