A matrix Lie group, which is also a differentiable manifold simultaneously, attracts more and more researchers' attention from both theoretic interest and its applications [1-5].

The most general matrix Lie group is the general linear group GL(n, R) consisting of the invertible n x n matrices with real entries.

Let S denote a matrix Lie group and s its Lie algebra.

The exponential of a matrix plays a crucial role in the theory of the Lie groups, which can be used to obtain the Lie algebra of a matrix Lie group, and it transfers information from the Lie algebra to the Lie group.

The matrix Lie group also has the structure of a Riemannian manifold.

Compact Matrix Lie Group. A Lie group is compact if its differential structure is compact.

Let G be a

matrix Lie group, g its Lie algebra and a : R x G - g.

Later, the system was studied as a left invariant control system on a matrix Lie group in [2].

The dynamics of the car with two trailers have been studied as a mechanical problem on a matrix Lie group. Following [2],

A lot of mechanical problems, like the cinematic model of an automobile with (n-3) trailers [7], the underwater vehicle dynamics [1], the spacecraft dynamics [8], the molecular motion in the context of coherent control of quantum dynamics [9], and the ball-plate problem [10] or the control tower problem from air traffic [11] or the Lagrange system [12] have been modeled as a set of differential equations with the configuration space on a matrix Lie group. Finding a Hamilton-Poisson formulation for these systems is an important first step.

Walsh, "Optimal path planning on matrix lie groups," in Proceedings of the 33rd IEEE Conference on Decision and Control, pp.

Duan, Information geometry of

matrix Lie groups and applications [Ph.D.