An open research question is how to define a useful metric on the special Euclidean group SE$(n)$ with respect to: (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances that is useful for the synthesis and analysis of mechanical systems. We discuss a technique for approximating elements of $SE(n)$ with elements of the special orthogonal group $SO(n+1)$. This technique is based on using the singular value decomposition (SVD) and the polar decompositions (PD) of the homogeneous transform representation of the elements of $SE(n)$. The embedding of the elements of $SE(n)$ into $SO(n+1)$ yields hyperdimensional rotations that approximate the rigid-body displacements. The bi-invariant metric on $SO(n+1)$ is then used to measure the distance between any two displacements. The result is a left invariant PD based metric on $SE(n)$.

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