The author constructs optimal paths as one parameter groups defined in terms of matrix exponentiation, proves matrix groups are manifolds, explores the properties of the

Lie bracket, and uses maximal tori to prove several theorems about familiar compact matrix groups.

A unicycle model or a two-wheel car model, shown in Figure 1, is basically a three-dimensional nonholonomic system having two inputs and three states with depth-one

Lie bracket. A two-wheel car kinematic model is defined as [12]

The usual

Lie bracket on vector fields induces the bracket on [GAMMA](A), and the anchor is given by [rho] = dt: A [right arrow] TM.

where the

Lie bracket on Gr(G) is given by the group commutator.

Recall that a

Lie bracket on a vector space V is a bilinear binary product [*, *]: V x V [right arrow] V such that for all x, y, z [member of] V,

The tangent space of G at identity is the Lie algebra g, where the

Lie bracket is defined.

To compute the

Lie Bracket amounts to compute the

Lie Bracket over [[universal].sub.x[member of]u] [L.sub.x]M U [subset or equal to] M.

Given two polynomials P, Q [member of] K<A>, their Lie product is the

Lie bracket [P, Q] = PQ - QP.

Our convention is that the

Lie bracket IX, Y] of vector fields on a manifold M is given by [X, Y]f = Y(Xf) -X(Yf) for f [element of] [C.sup.[infinity]](M).

The notion of semidirect sums [bar.g] = g [??] [g.sub.c] means that g and [g.sub.c] satisfy [mathematical expression not reproducible] denoting the

Lie bracket of [bar.g].

The tangent bundle (TM,t,M) with the usual

Lie bracket and [rho] equal to the identity map form a Lie algebroid.

An algebroid is an ALS ([theta],D, [[*,*].sub.[theta]]) which enjoys the property that [DX,DY ] = D[[X, Y ].sub.[theta]], ([for exists])X, Y [member of] [GAMMA]([theta]), where the first bracket is the

Lie bracket on X(M).