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An alternative to the exponential map is the Cayley transform, $$\begin{equation} \boldsymbol{\Lambda}\left(\boldsymbol{\chi}\right)=\cay\left[\hat{\boldsymbol{\chi}}\right]=\mathbf{I}+\frac{2}{1+\left(\frac{1}{2}\chi\right)^{2}}\left(\frac{1}{2}\hat{\boldsymbol{\chi}}+\frac{1}{4}\hat{\boldsymbol{\chi}}^{2}\right)\label{eq:rbr-Cayley-transform} \end{equation}$$ which is a second order approximation to the exponential map. This formula is a correction to that appearing in [62] (which has $\frac{1}{2}\chi^{2}$ in the denominator). According to Puso [62], 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 $\exp\left[\hat{\boldsymbol{\theta}}\right]$ using (6.3.1.1-6), we find that $\boldsymbol{\chi}$ and $\boldsymbol{\theta}$ are related via $$\begin{equation} \boldsymbol{\chi}=2\tan\frac{\theta}{2}\,\mathbf{n}\,.\label{eq:exponential-Cayley-relation} \end{equation}$$