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 
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
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
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