This chapter was synthesized from numerous sources. Many important, related subjects were omitted. For some mechanics of bodies in contact and manipulation in general, see [681]. Three-dimensional vehicle models were avoided because they are complicated by ; see [433]. For computational issues associated with simulating dynamical systems, see [247,863].

For further reading on velocity constraints on the C-space, see [596,725] and Sections 15.3 to 15.5. For more problems involving rolling spheres, see [527] and references therein. The rolling-ball problem is sometimes referred to as the Chaplygin ball. A nonholonomic manipulator constructed from rolling-ball joints was developed and analyzed in [729]. The kinematics of curved bodies in contact was studied in [632,716]. For motion planning in this context, see [101,103,223,676]. Other interesting nonholonomic systems include the snakeboard [473,629], roller racer [556], rollerblader [214], Trikke [213], and examples in [112] (e.g., the Chaplygin sled).

Phase space representations are a basic part of differential equations, physics, and control theory; see [44,192].

Further reading in mechanics is somewhat complicated by two different levels of treatment. Classical mechanics texts do not base the subject on differential geometry, which results in cumbersome formulations and unusual terminology (e.g., generalized coordinates). Modern mechanics texts overcome this problem by cleanly formulating everything in terms of geodesics on Riemannian manifolds; however, this may be more difficult to absorb for readers without background in differential geometry. An excellent source for modern mechanics is [39]. One of the most famous texts for classical mechanics is [397]. For an on-line book that covers the calculus of variations, including constrained Lagrangians, see [790]. The constrained Lagrangian presentation is based on Chapter 3 of [789], Section 2.4 of [397], and parts of [405]. Integral constraints on the Lagrangian are covered in [790], in addition to algebraic and differential constraints. Lagrangian mechanics under inequality constraints is considered in [789]. The presentation of the Hamiltonian in Section 13.4.4 is based on Chapter 7 of [397] and Section 15 of [39]. For advanced, modern treatments of mechanics in the language of affine connections and Christoffel symbols, see [3,156,677]. Another source, which is also heavily illustrated, is [359]. For further reading on robot dynamics, see [30,204,725,856,907,994]. For dynamics of automobiles, see [389].

For further reading on differential game theory, primary sources are [59,423,477]; see also [34,57,783,985,991,992,993,997]. Lower bounds for the algorithmic complexity of pursuit-evasion differential games are presented in [821].

Steven M LaValle 2012-04-20