$$
\begin{aligned}
    (L_{ijr}^{T}L_{ijr}+\left[\begin{array}{cc}q_{11} & 0\\0 & 0 \end{array}\right])\mathbf{v}_{er,x}
    +\left[\begin{array}{cc}q_{12} & 0\\0 & 0 \end{array}\right]\mathbf{v}_{er,y}
    +\left[\begin{array}{cc}q_{13} & 0\\0 & 0 \end{array}\right]\mathbf{v}_{er,z}
    +\left[\begin{array}{c}v_{e,g}q_{14}+q_{15}\\0\end{array}\right]
    +L_{ijr}^{T}b_{x} = 0 \\
    \left[\begin{array}{cc}q_{12} & 0\\0 & 0\end{array}\right]\mathbf{v}_{er,x}
    +(L_{ijr}^{T}L_{ijr}+\left[\begin{array}{cc}q_{22} & 0\\0 & 0\end{array}\right])\mathbf{v}_{er,y}
    +\left[\begin{array}{cc}q_{23} & 0\\0 & 0\end{array}\right]\mathbf{v}_{er,z}
    +\left[\begin{array}{c}v_{e,g}q_{24}+q_{25}\\0\end{array}\right]
    +L_{ijr}^{T}b_{y} = 0 \\
    \left[\begin{array}{cc}q_{13} & 0\\0 & 0\end{array}\right]\mathbf{v}_{er,x}
    +\left[\begin{array}{cc}q_{23} & 0\\0 & 0\end{array}\right]\mathbf{v}_{er,y}
    +(L_{ijr}^{T}L_{ijr}+\left[\begin{array}{cc}q_{33} & 0\\0 & 0\end{array}\right])\mathbf{v}_{er,z}
    +\left[\begin{array}{c}v_{e,g}q_{34}+q_{35}\\0\end{array}\right]
    +L_{ijr}^{T}b_{z} = 0 \\
    \left[\begin{array}{cc}q_{14} & 0\end{array}\right]\mathbf{v}_{er,x}
    +\left[\begin{array}{cc}q_{24} & 0\end{array}\right]\mathbf{v}_{er,y}
    +\left[\begin{array}{cc}q_{34} & 0\end{array}\right]\mathbf{v}_{er,z}
    +q_{44}v_{e,g}
    +q_{45} = 0
\end{aligned}
$$
