13.3 Basic Newton-Euler Mechanics

Mechanics is a vast and difficult subject. It is virtually impossible to provide a thorough introduction in a couple of sections. Here, the purpose instead is to overview some of the main concepts and to provide some models that may be used with the planning algorithms in Chapter 14. The presentation in this section and in Section 13.4 should hopefully stimulate some further studies in mechanics (see the suggested literature at the end of the chapter). On the other hand, if you are only interested in using the differential models, then you can safely skip their derivations. Just keep in mind that all differential models produced in this section end with the form ${\dot x}= f(x,u)$, which is ready to use in planning algorithms.

There are two important points to keep in mind while studying mechanics:

1. The models are based on maintaining consistency with experimental observations about how bodies behave in the physical world. These observations depend on the kind of experiment. In a particular application, many effects may be insignificant or might not even be detectable by an experiment. For example, it is difficult to detect relativistic effects using a radar gun that measures automobile speed. It is therefore important to specify any simplifying assumptions regarding the world and the kind of experiments that will be performed in it.
2. The approach is usually to express some laws that translate into constraints on the allowable velocities in the phase space. This means that implicit representations are usually obtained in mechanics, and they must be converted into parametric form. Furthermore, most treatments of mechanics do not explicitly mention action variables; these arise from the intention of controlling the physical world. From the perspective of mechanics, the actions can be assumed to be already determined. Thus, constraints appear as $g({\dot x},x) = 0$, instead of $g({\dot x},x,u) = 0$.

Several formulations of mechanics arrive at the same differential constraints, but from different mathematical reasoning. The remainder of this chapter overviews three schools of thought, each of which is more elegant and modern than the one before. The easiest to understand is Newton-Euler mechanics, which follows from Newton's famous laws of physics and is covered in this section. Lagrangian mechanics is covered in Section 13.4.1 and arrives at the differential constraints using very general principles of optimization on a space of functions (i.e., calculus of variations). Hamiltonian mechanics, covered in Section 13.4.4, defines a higher dimensional state space on which the differential constraints can once again be obtained by optimization.