This work presents an approach for the modeling and numerical optimization of ball joints within a Marker-less Motion Capture (MoCap) framework. In skeleton based approaches, kinematic chains are commonly used to model 1 DoF revolute joints. A 3 DoF joint (e.g. a shoulder or hip) is consequently modeled by concatenating three consecutive 1 DoF revolute joints. Obviously such a representation is not optimal and singularities can occur. Therefore, we propose to model 3 DoF joints with spherical joints or ball joints using the representation of a twist and its exponential mapping (known from 1 DoF revolute joints). The exact modeling and numerical optimization of ball joints requires additionally the adjoint transform and the logarithm of the exponential mapping. Experiments with simulated and real data demonstrate that ball joints can better represent arbitrary rotations than the concatenation of 3 revolute joints. Moreover, we demonstrate that the 3 revolute joints representation is very similar to the Euler angles representation and has the same limitations in terms of singularities.
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