Recovering a Rotation Matrix From Three Direction Cosines

In this paper, we propose a new parameterization method to represent rotation matrices using the angles $ϕ→$ recovered from the three direction cosines that lie on the diagonal. The map from the possible configuration space of the new variable $ϕ→$ to the solid ball model in axis-angle coordinates is constructed. We also introduce a bi-invariant metric and two left-invariant metrics for measuring the distance in configuration space which could be the foundation for path planning in $ϕ→$ space. We further analyze the Jacobian matrix and singularities to better understand the manipulability.