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Theory Manual Version 3.6
 Subsubsection 6.3.1.1: Exponential Map Up Subsection 6.3.1: Rigid Body Rotation Subsubsection 6.3.1.3: Linearization Along Rotational Increment 

6.3.1.2 Cayley Transform

An alternative to the exponential map is the Cayley transform, which is a second order approximation to the exponential map. This formula is a correction to that appearing in [83] (which has in the denominator). According to Puso [83], the Cayley transform must be used to enforce conservation of momentum and energy in a midpoint rule discretization scheme, whenever the rigid body is connected to a deformable body, or whenever two rigid bodies are connected by a joint. Comparing the above expression to the exponential map using (6.3.1.1-6), we find that and are related via
 Subsubsection 6.3.1.1: Exponential Map Up Subsection 6.3.1: Rigid Body Rotation Subsubsection 6.3.1.3: Linearization Along Rotational Increment