### Enhanced Stiffness Model of the 3 DJM on SO(3)  
According to the definition, the stiffness of the 3 DJM is evaluated by the variation of its load against its displacement. However, the loads and displacements for different poses of the 3 DJM are in different cotangent spaces and tangent spaces of SO(3). Considering the property of the motion of the 3 DJM, an affine connection called _Levi-Civita_ connection is introduced into \(SO(3)\) to connect different tangent spaces on SO(3) [32]. Given a curve \(\mathbf{R}(t)\) and a vector field \(\mathbf{V}(t)\) on \(SO(3)\), the affine connection specifies how a vector \(\mathbf{V}(t_{0})\) in the tangent space at point \(\mathbf{R}(t_{0})\) can be mapped to another vector \(\mathbf{V}^{t_{0}}(t_{1})\) of the tangent space at some other point \(\mathbf{R}(t_{1})\). The vector \(\mathbf{V}^{t_{0}}(t_{1})\) is called the paralleltransport of \(\mathbf{V}(t_{0})\) along the curve \(\mathbf{R}(t)\)[33]. Then, a differentiation operation can be defined on SO(3) as below  
\[\
abla_{\mathbf{\hat{R}}}\mathbf{V}\big{|}_{\mathbf{R}(t_{0})}=\lim_{t_{1}\to t_{0}} \frac{\mathbf{V}^{t_{0}}(t_{1})-\mathbf{V}(t_{0})}{t_{1}-t_{0}}, \tag{7}\]  
where \(\
abla_{\mathbf{\hat{R}}}\mathbf{V}\big{|}_{\mathbf{R}(t_{0})}\) is called the _covariant derivative_ of \(\mathbf{V}(t)\) along the curve \(\mathbf{R}(t)\). For two vectors, \(\mathbf{V}_{1}\) and \(\mathbf{V}_{2}\), in the tangent spaces of SO(3), \(\
abla_{\mathbf{V}_{2}}\mathbf{V}_{1}\) is employed to represent the covariant derivative of \(\mathbf{V}_{1}\) in the direction \(\mathbf{V}_{2}\). For a real-valued function \(f\) on \(SO(3)\), the covariant derivative \(\
abla_{\mathbf{V}}f\) is usually written as \(\mathbf{V}\circ f\). According to the property of covariant derivative, the covariant derivative of the load \(\mathbf{F}(t)\) to the instantaneous displacement \(\delta\mathbf{R}(t)\) can be written as  
\[\
abla_{\delta\mathbf{R}(t)}\mathbf{F}=\
abla_{\mathbf{\hat{R}}(t)\delta t}\mathbf{F}=\delta t \
abla_{\sum_{j=1}^{3}\omega^{j}\mathbf{L}_{j}}\mathbf{F}=\delta t\sum_{j=1}^{3}\omega ^{j}\
abla_{\mathbf{L}_{j}}\mathbf{F}=\delta t\mathbf{K}\mathbf{\omega}=\mathbf{K}\delta\mathbf{\xi}, \tag{8}\]  
and \(\mathbf{K}=(\
abla_{\mathbf{L}_{1}}\mathbf{F},\
abla_{\mathbf{L}_{2}}\mathbf{F},\
abla_{\mathbf{L}_{ 3}}\mathbf{F})\in\mathbb{R}^{3\times 3}\) is employed as the stiffness of the 3 DJM. The components of the stiffness matrix \(\mathbf{K}\), i.e., \(K_{jk}\) (\(j,k=1,2,3\)), yield  
\[K_{jk}=\langle\
abla_{\mathbf{L}_{k}}\mathbf{F},\mathbf{L}_{j}\rangle. \tag{9}\]  
According to the property of the Levi-Civita connection [30], we have  
\[K_{jk}=\mathbf{L}_{k}\circ\langle\mathbf{F},\mathbf{L}_{j}\rangle-\langle\mathbf{F},\
abla_{ \mathbf{L}_{k}}\mathbf{L}_{j}\rangle=\mathbf{L}_{k}\circ\tau_{j}-\frac{1}{2}\sum_{r=1}^{3 }\tau_{r}\gamma_{kj}^{r}, \tag{10}\]  
where the coefficients \(\gamma_{kj}^{r}\) (\(j,k,r=1,2,3\)) \(\in\mathbb{R}\) are zero except  
\[\gamma_{12}^{3}=\gamma_{31}^{2}=\gamma_{23}^{1}=2,\quad\gamma_{21}^{3}=\gamma _{13}^{2}=\gamma_{32}^{1}=-2. \tag{11}\]  
Furthermore, since the 3 DJM is a conservative mechanical system, its stiffness matrix \(\mathbf{K}\) can be proved to be symmetric at every pose, even if there is an external load applied on it [30].