I'm not a mathematician, so please forgive my obvious naivety.


I'm interested in generating a representation in vector space of a two dimensional armature, or set of rigid linked elements, with each element articulated by rotation about a point defined on the previous (acyclic) element, about an axis orthogonal to the plane of the armature. What I require is a basis vector and a set of transform matrices that cover the configuration space of the armature. 


My understanding is that Lie algebras manipulate the identity homeomorphisms of Lie groups, in that, for example, the Lie algebra that generates the Lie group SO3 would define the local patches that contain the identity of a particular 3D rotation.


Also that an armature as described, with multiple articulations can be described in terms of a Lie algebra and that the algebra can be used to generate a representation and thereby a basis vector and transform matrices that cover the configuration space of the armature.


All my requirements are for mechanical systems, so constrained to special orthogonal groups. 


If anyone has suggestions regarding methodology or sources to assist in this pursuit, I would be very grateful.



modeling mechanical systems with Lie algebras
Probably your best bet is to read through Chapter 2 of [A Mathematical Introduction to Robotic Manipulation](http://www.cds.caltech.edu/~murray/mlswiki/index.php/Main_Page) by Murray et al. (free download at link). You seem to referring to "exponential coordinates" on SO(3) (see for example equation (2.16) on P.30). Although if your armature and rotations are confined to a plane, then you really want to consider direct products of the 2d Euclidean group $\mathrm{SE}(2) = \mathrm{SO}(2)\ltimes \mathbb{R}^2$ to cover your configuration space (one copy for each link). The 3D version $\mathrm{SE}(3)$ is covered in Appendix A.


Another place to look is Chapter 6 of [Harmonic Analysis for Engineers and Applied Scientists](http://store.doverpublications.com/0486795640.html) by Chirikjian and Kyatkin.


