Abstract
An open research question is how to define a metric on SE(n) that is as invariant as possible with respect to (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances. We present two techniques for approximating elements of the special Euclidean group SE(n) with elements of the special orthogonal group S0(n+1). These techniques are based on the singular value and polar decompositions (denoted as SVD and PD respectively) of the homogeneous transform representation of the elements of SE(n). The projection of the elements of SE(n) onto S0(n+1) yields hyperdimensional rotations that approximate the rigid-body displacements. Any of the infinite bi-invariant metrics on SO(n-fl) may then be used to measure the distance between any two spatial displacements. The results are PD and SVD based projection techniques that yield two approximately bi-invariant metrics on SE(n). These metrics have applications in motion synthesis, robot calibration, motion interpolation, and hybrid robot control.
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Larochelle, P.M., Murray, A.P., Angeles, J. (2004). SVD and PD Based Projection Metrics on SE(n). In: Lenarčič, J., Galletti, C. (eds) On Advances in Robot Kinematics. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-2249-4_2
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DOI: https://doi.org/10.1007/978-1-4020-2249-4_2
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