Correction to "Smooth Parameterization of Rigid-Body Inertia"

Abstract

Equation (9) in [1] was calculated from $\mathbf {J}=\mathbf {U}^\top \mathbf {U}$ and is thus inconsistent with (7) in [1], which defines the decomposition as $\mathbf {J}=\mathbf {U}\mathbf {U}^\top$. The correct parameter mapping $\boldsymbol{\pi }=\mathbf {f}(\boldsymbol{\theta })$ consistent with (7) is \begin{equation*} \boldsymbol{\pi }= e^{2\alpha } \begin{bmatrix}1 \\ t_{1} \\ t_{2} \\ t_{3} \\ s_{23}^{2}+t_{2}^{2}+t_{3}^{2}+e^{2d_{2}}+e^{2d_{3}}\\ s_{12}^{2}+s_{13}^{2}+t_{1}^{2}+t_{3}^{2}+e^{2d_{1}}+e^{2d_{3}}\\ s_{12}^{2}+s_{13}^{2}+s_{23}^{2}+t_{1}^{2}+t_{2}^{2}+e^{2d_{1}}+e^{2d_{2}}\\ -s_{13}s_{23}-t_{1}t_{2}-s_{12}e^{d_{2}}\\ -t_{2}t_{3}-s_{23}e^{d_{3}}\\ -t_{1}t_{3}-s_{13}e^{d_{3}} \end{bmatrix} \tag{1} \end{equation*}