Appendix A
The differentiation of \( {}^{0}\overline{\text{A}}_{{\text{i}}} \) shown in Eq. (2.8) with respect to pose variables are given by:
$$ \frac{{\partial \left( {{}^{0}\overline{\text{A}}_{{\text{i}}} } \right)}}{{\partial {\text{t}}_{{\text{ix}}} }} = \left[ {\begin{array}{*{20}c} {{{\partial {\text{I}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{ix}}} } {\partial {\text{t}}_{{\text{ix}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{ix}}} }}} & {{{\partial {\text{J}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{ix}}} } {\partial {\text{t}}_{{\text{ix}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{ix}}} }}} & {{{\partial {\text{K}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{ix}}} } {\partial {\text{t}}_{{\text{ix}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{ix}}} }}} & {{{\partial {\text{t}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{ix}}} } {\partial {\text{t}}_{{\text{ix}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{ix}}} }}} \\ {{{\partial {\text{I}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{iy}}} } {\partial {\text{t}}_{{\text{ix}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{ix}}} }}} & {{{\partial {\text{J}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{iy}}} } {\partial {\text{t}}_{{\text{ix}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{ix}}} }}} & {{{\partial {\text{K}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{iy}}} } {\partial {\text{t}}_{{\text{ix}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{ix}}} }}} & {{{\partial {\text{t}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{iy}}} } {\partial {\text{t}}_{{\text{ix}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{ix}}} }}} \\ {{{\partial {\text{I}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{iz}}} } {\partial {\text{t}}_{{\text{ix}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{ix}}} }}} & {{{\partial {\text{J}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{iz}}} } {\partial {\text{t}}_{{\text{ix}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{ix}}} }}} & {{{\partial {\text{K}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{iz}}} } {\partial {\text{t}}_{{\text{ix}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{ix}}} }}} & {{{\partial {\text{t}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{iz}}} } {\partial {\text{t}}_{{\text{ix}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{ix}}} }}} \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$
(A.1)
$$ \frac{{\partial \left( {{}^{0}\overline{\text{A}}_{{\text{i}}} } \right)}}{{\partial {\text{t}}_{{\text{iy}}} }} = \left[ {\begin{array}{*{20}c} {{{\partial {\text{I}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{ix}}} } {\partial {\text{t}}_{{\text{iy}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iy}}} }}} & {{{\partial {\text{J}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{ix}}} } {\partial {\text{t}}_{{\text{iy}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iy}}} }}} & {{{\partial {\text{K}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{ix}}} } {\partial {\text{t}}_{{\text{iy}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iy}}} }}} & {{{\partial {\text{t}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{ix}}} } {\partial {\text{t}}_{{\text{iy}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iy}}} }}} \\ {{{\partial {\text{I}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{iy}}} } {\partial {\text{t}}_{{\text{iy}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iy}}} }}} & {{{\partial {\text{J}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{iy}}} } {\partial {\text{t}}_{{\text{iy}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iy}}} }}} & {{{\partial {\text{K}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{iy}}} } {\partial {\text{t}}_{{\text{iy}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iy}}} }}} & {{{\partial {\text{t}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{iy}}} } {\partial {\text{t}}_{{\text{iy}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iy}}} }}} \\ {{{\partial {\text{I}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{iz}}} } {\partial {\text{t}}_{{\text{iy}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iy}}} }}} & {{{\partial {\text{J}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{iz}}} } {\partial {\text{t}}_{{\text{iy}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iy}}} }}} & {{{\partial {\text{K}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{iz}}} } {\partial {\text{t}}_{{\text{iy}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iy}}} }}} & {{{\partial {\text{t}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{iz}}} } {\partial {\text{t}}_{{\text{iy}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iy}}} }}} \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$
(A.2)
$$ \frac{{\partial \left( {{}^{0}\overline{\text{A}}_{{\text{i}}} } \right)}}{{\partial {\text{t}}_{{\text{iz}}} }} = \left[ {\begin{array}{*{20}c} {{{\partial {\text{I}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{ix}}} } {\partial {\text{t}}_{{\text{iz}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iz}}} }}} & {{{\partial {\text{J}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{ix}}} } {\partial {\text{t}}_{{\text{iz}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iz}}} }}} & {{{\partial {\text{K}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{ix}}} } {\partial {\text{t}}_{{\text{iz}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iz}}} }}} & {{{\partial {\text{t}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{ix}}} } {\partial {\text{t}}_{{\text{iz}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iz}}} }}} \\ {{{\partial {\text{I}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{iy}}} } {\partial {\text{t}}_{{\text{iz}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iz}}} }}} & {{{\partial {\text{J}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{iy}}} } {\partial {\text{t}}_{{\text{iz}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iz}}} }}} & {{{\partial {\text{K}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{iy}}} } {\partial {\text{t}}_{{\text{iz}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iz}}} }}} & {{{\partial {\text{t}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{iy}}} } {\partial {\text{t}}_{{\text{iz}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iz}}} }}} \\ {{{\partial {\text{I}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{iz}}} } {\partial {\text{t}}_{{\text{iz}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iz}}} }}} & {{{\partial {\text{J}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{iz}}} } {\partial {\text{t}}_{{\text{iz}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iz}}} }}} & {{{\partial {\text{K}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{iz}}} } {\partial {\text{t}}_{{\text{iz}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iz}}} }}} & {{{\partial {\text{t}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{iz}}} } {\partial {\text{t}}_{{\text{iz}}} }}} \right. \kern-0pt} {\partial {\text{t}}_{{\text{iz}}} }}} \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$
(A.3)
$$ \begin{aligned} \frac{{\partial \left( {{}^{0}\overline{\text{A}}_{{\text{i}}} } \right)}}{{\partial \omega_{{\text{ix}}} }} = & \left[ {\begin{array}{*{20}c} {{{\partial {\text{I}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{ix}}} } {\partial \omega_{{\text{ix}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{ix}}} }}} & {{{\partial {\text{J}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{ix}}} } {\partial \omega_{{\text{ix}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{ix}}} }}} & {{{\partial {\text{K}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{ix}}} } {\partial \omega_{{\text{ix}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{ix}}} }}} & {{{\partial {\text{t}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{ix}}} } {\partial \omega_{{\text{ix}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{ix}}} }}} \\ {{{\partial {\text{I}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{iy}}} } {\partial \omega_{{\text{ix}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{ix}}} }}} & {{{\partial {\text{J}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{iy}}} } {\partial \omega_{{\text{ix}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{ix}}} }}} & {{{\partial {\text{K}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{iy}}} } {\partial \omega_{{\text{ix}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{ix}}} }}} & {{{\partial {\text{t}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{iy}}} } {\partial \omega_{{\text{ix}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{ix}}} }}} \\ {{{\partial {\text{I}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{iz}}} } {\partial \omega_{{\text{ix}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{ix}}} }}} & {{{\partial {\text{J}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{iz}}} } {\partial \omega_{{\text{ix}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{ix}}} }}} & {{{\partial {\text{K}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{iz}}} } {\partial \omega_{{\text{ix}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{ix}}} }}} & {{{\partial {\text{t}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{iz}}} } {\partial \omega_{{\text{ix}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{ix}}} }}} \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ = & \left[ {\begin{array}{*{20}c} 0 & {{\text{C}}\omega_{{\text{iz}}} {\text{S}}\omega_{{\text{iy}}} {\text{C}}\omega_{{\text{ix}}} + {\text{S}}\omega_{{\text{iz}}} {\text{S}}\omega_{{\text{ix}}} } & { - {\text{C}}\omega_{{\text{iz}}} {\text{S}}\omega_{{\text{iy}}} {\text{S}}\omega_{{\text{ix}}} + {\text{S}}\omega_{{\text{iz}}} {\text{C}}\omega_{{\text{ix}}} } & 0 \\ 0 & {{\text{S}}\omega_{{\text{iz}}} {\text{S}}\omega_{{\text{iy}}} {\text{C}}\omega_{{\text{ix}}} - {\text{C}}\omega_{{\text{iz}}} {\text{S}}\omega_{{\text{ix}}} } & { - {\text{S}}\omega_{{\text{iz}}} {\text{S}}\omega_{{\text{iy}}} {\text{S}}\omega_{{\text{ix}}} - {\text{C}}\omega_{{\text{iz}}} {\text{C}}\omega_{{\text{ix}}} } & 0 \\ 0 & {{\text{C}}\omega_{{\text{iy}}} {\text{C}}\omega_{{\text{ix}}} } & { - {\text{C}}\omega_{{\text{iy}}} {\text{S}}\omega_{{\text{ix}}} } & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], \\ \end{aligned} $$
(A.4)
$$ \begin{aligned} \frac{{\partial \left( {{}^{0}\overline{\text{A}}_{{\text{i}}} } \right)}}{{\partial \omega_{{\text{iy}}} }} = & \left[ {\begin{array}{*{20}c} {{{\partial {\text{I}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{ix}}} } {\partial \omega_{{\text{iy}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iy}}} }}} & {{{\partial {\text{J}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{ix}}} } {\partial \omega_{{\text{iy}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iy}}} }}} & {{{\partial {\text{K}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{ix}}} } {\partial \omega_{{\text{iy}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iy}}} }}} & {{{\partial {\text{t}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{ix}}} } {\partial \omega_{{\text{iy}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iy}}} }}} \\ {{{\partial {\text{I}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{iy}}} } {\partial \omega_{{\text{iy}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iy}}} }}} & {{{\partial {\text{J}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{iy}}} } {\partial \omega_{{\text{iy}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iy}}} }}} & {{{\partial {\text{K}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{iy}}} } {\partial \omega_{{\text{iy}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iy}}} }}} & {{{\partial {\text{t}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{iy}}} } {\partial \omega_{{\text{iy}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iy}}} }}} \\ {{{\partial {\text{I}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{iz}}} } {\partial \omega_{{\text{iy}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iy}}} }}} & {{{\partial {\text{J}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{iz}}} } {\partial \omega_{{\text{iy}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iy}}} }}} & {{{\partial {\text{K}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{iz}}} } {\partial \omega_{{\text{iy}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iy}}} }}} & {{{\partial {\text{t}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{iz}}} } {\partial \omega_{{\text{iy}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iy}}} }}} \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ = &\left[ {\begin{array}{*{20}c} { - {\text{C}}\omega_{{\text{iz}}} {\text{S}}\omega_{{\text{iy}}} } & {{\text{C}}\omega_{{\text{iz}}} {\text{C}}\omega_{{\text{iy}}} {\text{S}}\omega_{{\text{ix}}} } & {{\text{C}}\omega_{{\text{iz}}} {\text{C}}\omega_{{\text{iy}}} {\text{C}}\omega_{{\text{ix}}} } & 0 \\ { - {\text{S}}\omega_{{\text{iz}}} {\text{S}}\omega_{{\text{iy}}} } & {{\text{S}}\omega_{{\text{iz}}} {\text{C}}\omega_{{\text{iy}}} {\text{S}}\omega_{{\text{ix}}} } & {{\text{S}}\omega_{{\text{iz}}} {\text{C}}\omega_{{\text{iy}}} {\text{C}}\omega_{{\text{ix}}} } & 0 \\ { - {\text{C}}\omega_{{\text{iy}}} } & { - {\text{S}}\omega_{{\text{iy}}} {\text{S}}\omega_{{\text{ix}}} } & { - {\text{S}}\omega_{{\text{iy}}} {\text{C}}\omega_{{\text{ix}}} } & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], \\ \end{aligned} $$
(A.5)
$$ \begin{aligned} \frac{{\partial \left( {{}^{0}\overline{\text{A}}_{{\text{i}}} } \right)}}{{\partial \omega_{{\text{iz}}} }} = & \left[ {\begin{array}{*{20}c} {{{\partial {\text{I}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{ix}}} } {\partial \omega_{{\text{iz}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iz}}} }}} & {{{\partial {\text{J}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{ix}}} } {\partial \omega_{{\text{iz}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iz}}} }}} & {{{\partial {\text{K}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{ix}}} } {\partial \omega_{{\text{iz}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iz}}} }}} & {{{\partial {\text{t}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{ix}}} } {\partial \omega_{{\text{iz}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iz}}} }}} \\ {{{\partial {\text{I}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{iy}}} } {\partial \omega_{{\text{iz}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iz}}} }}} & {{{\partial {\text{J}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{iy}}} } {\partial \omega_{{\text{iz}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iz}}} }}} & {{{\partial {\text{K}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{iy}}} } {\partial \omega_{{\text{iz}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iz}}} }}} & {{{\partial {\text{t}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{iy}}} } {\partial \omega_{{\text{iz}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iz}}} }}} \\ {{{\partial {\text{I}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{iz}}} } {\partial \omega_{{\text{iz}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iz}}} }}} & {{{\partial {\text{J}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{iz}}} } {\partial \omega_{{\text{iz}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iz}}} }}} & {{{\partial {\text{K}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{iz}}} } {\partial \omega_{{\text{iz}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iz}}} }}} & {{{\partial {\text{t}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{iz}}} } {\partial \omega_{{\text{iz}}} }}} \right. \kern-0pt} {\partial \omega_{{\text{iz}}} }}} \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ =& \left[ {\begin{array}{*{20}c} { - {\text{S}}\omega_{{\text{iz}}} {\text{C}}\omega_{{\text{iy}}} } & { - {\text{S}}\omega_{{\text{iz}}} {\text{S}}\omega_{{\text{iy}}} {\text{S}}\omega_{{\text{ix}}} - {\text{C}}\omega_{{\text{iz}}} {\text{C}}\omega_{{\text{ix}}} } & { - {\text{S}}\omega_{{\text{iz}}} {\text{S}}\omega_{{\text{iy}}} {\text{C}}\omega_{{\text{ix}}} + {\text{C}}\omega_{{\text{iz}}} {\text{S}}\omega_{{\text{ix}}} } & 0 \\ {{\text{C}}\omega_{{\text{iz}}} {\text{C}}\omega_{{\text{iy}}} } & {{\text{C}}\omega_{{\text{iz}}} {\text{S}}\omega_{{\text{iy}}} {\text{S}}\omega_{{\text{ix}}} - {\text{S}}\omega_{{\text{iz}}} {\text{C}}\omega_{{\text{ix}}} } & {{\text{C}}\omega_{{\text{iz}}} {\text{S}}\omega_{{\text{iy}}} {\text{C}}\omega_{{\text{ix}}} + {\text{S}}\omega_{{\text{iz}}} {\text{S}}\omega_{{\text{ix}}} } & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]. \\ \end{aligned} $$
(A.6)
Appendix B
There are various means of specifying \( {}^{0}\overline{\text{A}}_{{\text{ej}}} \) [e.g., Eqs. (3.29a)–(3.29d)] and \( {}^{ej}\overline{\text{A}}_{{\text{i}}} \) [e.g., Eqs. (3.30a)–(3.30m)] based on different sequences of the rotation and translation operators in Sect. 1.5. However, if \( {}^{0}\overline{\text{A}}_{{\text{ej}}} \) is to be computed using computer code, a decision must be taken as to which form to apply. In the present study, the following formulation is applied:
$$ \begin{aligned} {}^{h}\overline{\text{A}}_{{\text{g}}} = & tran\left( {{\text{t}}_{{\text{x}}} ,{\text{t}}_{{\text{y}}} ,{\text{t}}_{{\text{z}}} } \right)rot\left( {\overline{z} ,\omega_{{\text{z}}} } \right)rot\left( {\overline{y} ,\omega_{{\text{y}}} } \right)rot\left( {\overline{\text{x}} ,\omega_{{\text{x}}} } \right) \\ =& \left[ {\begin{array}{*{20}c} {I_{{\text{x}}} } & {J_{{\text{x}}} } & {K_{{\text{x}}} } & {{\text{t}}_{{\text{x}}} } \\ {I_{{\text{y}}} } & {J_{{\text{y}}} } & {K_{{\text{y}}} } & {{\text{t}}_{{\text{y}}} } \\ {I_{{\text{z}}} } & {J_{{\text{z}}} } & {K_{{\text{z}}} } & {{\text{t}}_{{\text{z}}} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\text{C}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} } & {{\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} - {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{z}}} } & {{\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} - {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{z}}} } & {{\text{t}}_{{\text{x}}} } \\ {{\text{C}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} } & {{\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{z}}} + {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} } & { - {\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{z}}} + {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} } & {{\text{t}}_{{\text{y}}} } \\ { - {\text{S}}\omega_{{\text{y}}} } & {{\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} } & {{\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} } & {{\text{t}}_{{\text{z}}} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right], \\ \end{aligned} $$
(B.1)
where \( {}^{h}\overline{\text{A}}_{{\text{g}}} \) can be \( {}^{0}\overline{\text{A}}_{{\text{ej}}} \) or \( {}^{ej}\overline{\text{A}}_{{\text{i}}} \).
In Eqs. (3.29a)–(3.29d) and Eqs. (3.30a)–(3.30m), the arguments in matrices \( {}^{0}\overline{\text{A}}_{{\text{ej}}} \) and \( {}^{ej}\overline{\text{A}}_{{\text{i}}} \) can be written as \( \omega_{{\text{x}}} = {\text{a}}_{0} + \sum\limits_{{\text{v}}} = 1^{41} {{\text{a}}_{{\text{v}}} {\text{x}}_{{\text{v}}} } \), \( \omega_{{\text{y}}} = {\text{b}}_{0} + \sum\limits_{{\text{v}}} = 1^{41} {{\text{b}}_{{\text{v}}} {\text{x}}_{{\text{v}}} } \)and \( \omega_{{\text{z}}} = {\text{c}}_{0} + \sum\limits_{{\text{v}}} = 1^{41} {c_{{\text{v}}} {\text{x}}_{{\text{v}}} } \), respectively, where \( {\text{x}}_{{\text{v}}} \)is the vth component of \( \overline{\text{X}}_{{\text{sys}}} \), and \( {\text{a}}_{{\text{v}}} \), \( {\text{b}}_{{\text{v}}} \) and \( {\text{c}}_{{\text{v}}} \) ({\text{v}} = 1 to {\text{v}} = 41) are known constants. Therefore, \( {{\partial \left( {{}^{h}\overline{\text{A}}_{{\text{g}}} } \right)} \mathord{\left/ {\vphantom {{\partial \left( {{}^{{{\text{h}}}}\overline{\text{A}}_{{\text{g}}} } \right)} {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }} \) (where \( {}^{{\text{h}}}\overline{\text{A}}_{{\text{g}}} \) can be \( {}^{0}\overline{\text{A}}_{{\text{ej}}} \) or \( {}^{ej}\overline{\text{A}}_{{\text{i}}} \)) can be determined directly by differentiating Eq. (B.1) with respect to \( {\text{x}}_{{\text{v}}} \) to give
$$ \frac{{\partial \left( {{}^{h}\overline{\text{A}}_{{\text{g}}} } \right)}}{{\partial {\text{x}}_{{\text{v}}} }} = \left[ {\begin{array}{*{20}c} {{{\partial {\text{I}}_{{\text{x}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{x}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }}} & {{{\partial {\text{J}}_{{\text{x}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{x}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }}} & {{{\partial {\text{K}}_{{\text{x}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{x}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }}} & {{{\partial {\text{t}}_{{\text{x}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{x}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }}} \\ {{{\partial {\text{I}}_{{\text{y}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{y}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }}} & {{{\partial {\text{J}}_{{\text{y}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{y}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }}} & {{{\partial {\text{K}}_{{\text{y}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{y}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }}} & {{{\partial {\text{t}}_{{\text{y}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{y}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }}} \\ {{{\partial {\text{I}}_{{\text{z}}} } \mathord{\left/ {\vphantom {{\partial {\text{I}}_{{\text{z}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }}} & {{{\partial {\text{J}}_{{\text{z}}} } \mathord{\left/ {\vphantom {{\partial {\text{J}}_{{\text{z}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }}} & {{{\partial {\text{K}}_{{\text{z}}} } \mathord{\left/ {\vphantom {{\partial {\text{K}}_{{\text{z}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }}} & {{{\partial {\text{t}}_{{\text{z}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{z}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }}} \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$
(B.2)
where
$$ \frac{{\partial {\text{I}}_{{\text{x}}} }}{{\partial {\text{x}}_{{\text{v}}} }} = - {\text{b}}_{{\text{v}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} - {\text{c}}_{{\text{v}}} {\text{C}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} , $$
(B.3)
$$ \frac{{\partial {\text{I}}_{{\text{y}}} }}{{\partial {\text{x}}_{{\text{v}}} }} = - {\text{b}}_{{\text{v}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} + {\text{c}}_{{\text{v}}} {\text{C}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} , $$
(B.4)
$$ \frac{{\partial {\text{I}}_{{\text{y}}} }}{{\partial {\text{x}}_{{\text{v}}} }} = - {\text{b}}_{{\text{v}}} {\text{S}}\omega_{{\text{y}}} , $$
(B.5)
$$ \frac{{\partial {\text{J}}_{{\text{x}}} }}{{\partial {\text{x}}_{{\text{v}}} }} = {\text{a}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} + {\text{b}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} - {\text{c}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} + {\text{a}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{z}}} - {\text{c}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{z}}} , $$
(B.6)
$$ \frac{{\partial {\text{J}}_{{\text{y}}} }}{{\partial {\text{x}}_{{\text{v}}} }} = {\text{a}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} + {\text{b}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} + {\text{c}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} - {\text{a}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{z}}} - {\text{c}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{z}}} , $$
(B.7)
$$ \frac{{\partial {\text{J}}_{{\text{y}}} }}{{\partial {\text{x}}_{{\text{v}}} }} = {\text{a}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} - {\text{b}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} , $$
(B.8)
$$ \frac{{\partial {\text{K}}_{{\text{x}}} }}{{\partial {\text{x}}_{{\text{v}}} }} = - {\text{a}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} + {\text{b}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} - {\text{c}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} - {\text{a}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{z}}} - {\text{c}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{z}}} , $$
(B.9)
$$ \frac{{\partial {\text{K}}_{{\text{y}}} }}{{\partial {\text{x}}_{{\text{v}}} }} = - {\text{a}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} + {\text{b}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} + {\text{c}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} - {\text{a}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{z}}} + {\text{c}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{z}}} , $$
(B.10)
$$ \frac{{\partial {\text{K}}_{{\text{z}}} }}{{\partial {\text{x}}_{{\text{v}}} }} = - {\text{a}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} - {\text{b}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} , $$
(B.11)
$$ \frac{{\partial {\text{t}}_{{\text{x}}} }}{{\partial {\text{x}}_{{\text{v}}} }} = \frac{{\partial {\text{t}}_{{\text{x}}} }}{{\partial {\text{x}}_{{\text{v}}} }}, $$
(B.12)
$$ \frac{{\partial {\text{t}}_{{\text{y}}} }}{{\partial {\text{x}}_{{\text{v}}} }} = \frac{{\partial {\text{t}}_{{\text{y}}} }}{{\partial {\text{x}}_{{\text{v}}} }}, $$
(B.13)
$$ \frac{{\partial {\text{t}}_{{\text{z}}} }}{{\partial {\text{x}}_{{\text{v}}} }} = \frac{{\partial {\text{t}}_{{\text{z}}} }}{{\partial {\text{x}}_{{\text{v}}} }}. $$
(B.14)
Note that Eqs. (B.12), (B.13) and (B.14) simply indicate that \( {{\partial {\text{t}}_{{\text{x}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{x}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }} \), \( {{\partial {\text{t}}_{{\text{y}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{y}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }} \)and \( {{\partial {\text{t}}_{{\text{z}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{z}}} } {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }} \) can be obtained by directly differentiating their corresponding expressions (e.g., \( {{\partial {\text{t}}_{{\text{x}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{x}}} } {\partial x_{24} }}} \right. \kern-0pt} {\partial x_{24} }} = 0.5 \) for \( {}^{0}\overline{\text{A}}_{e2} \) from Eq. (3.29b) and \( {{\partial {\text{t}}_{{\text{y}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{y}}} } {\partial x_{14} }}} \right. \kern-0pt} {\partial x_{14} }} = {1 \mathord{\left/ {\vphantom {1 {\kappa_{1}^{2} }}} \right. \kern-0pt} {\kappa_{1}^{2} }} \) for \( {}^{e1}\overline{\text{A}}_{2} \) from Eq. (3.30b) for system in Fig. 3.12).
Having obtained \( {}^{0}\overline{\text{A}}_{{\text{ej}}} \), \( {}^{ej}\overline{\text{A}}_{{\text{i}}} \), \( {{\partial \left( {{}^{0}\overline{\text{A}}_{{\text{ej}}} } \right)} \mathord{\left/ {\vphantom {{\partial \left( {{}^{0}\overline{\text{A}}_{{\text{ej}}} } \right)} {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }} \) and \( {{\partial \left( {{}^{ej}\overline{\text{A}}_{{\text{i}}} } \right)} \mathord{\left/ {\vphantom {{\partial \left( {{}^{ej}\overline{\text{A}}_{{\text{i}}} } \right)} {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }} \), one can compute \( {{\partial \left( {{}^{0}\overline{\text{A}}_{{\text{i}}} } \right)} \mathord{\left/ {\vphantom {{\partial \left( {{}^{0}\overline{\text{A}}_{{\text{i}}} } \right)} {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }} \) from Eq. (5.85) and determine \( {{\partial {\text{t}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{ix}}} } {\partial \overline{\text{X}}_{{\text{sys}}} }}} \right. \kern-0pt} {\partial \overline{\text{X}}_{{\text{sys}}} }} \), \( {{\partial {\text{t}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{iy}}} } {\partial \overline{\text{X}}_{{\text{sys}}} }}} \right. \kern-0pt} {\partial \overline{\text{X}}_{{\text{sys}}} }} \), \( {{\partial {\text{t}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial {\text{t}}_{{\text{iz}}} } {\partial \overline{\text{X}}_{{\text{sys}}} }}} \right. \kern-0pt} {\partial \overline{\text{X}}_{{\text{sys}}} }} \), \( {{\partial \omega_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial \omega_{{\text{ix}}} } {\partial \overline{\text{X}}_{{\text{sys}}} }}} \right. \kern-0pt} {\partial \overline{\text{X}}_{{\text{sys}}} }} \), \( {{\partial \omega_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial \omega_{{\text{iy}}} } {\partial \overline{\text{X}}_{{\text{sys}}} }}} \right. \kern-0pt} {\partial \overline{\text{X}}_{{\text{sys}}} }} \), and \( {{\partial \omega_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial \omega_{{\text{iz}}} } {\partial \overline{\text{X}}_{{\text{sys}}} }}} \right. \kern-0pt} {\partial \overline{\text{X}}_{{\text{sys}}} }} \) accordingly.
Appendix C
Differentiating \( {{\partial \left( {{}^{h}\overline{\text{A}}_{{\text{g}}} } \right)} \mathord{\left/ {\vphantom {{\partial \left( {{}^{h}\overline{\text{A}}_{{\text{g}}} } \right)} {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }} \) in Eq. (B.2) (where \( {}^{h}\overline{\text{A}}_{{\text{g}}} \) can be \( {}^{0}\overline{\text{A}}_{{\text{ej}}} \) or \( {}^{ej}\overline{\text{A}}_{{\text{i}}} \)) with respect to \( {\text{x}}_{{\text{w}}} \) in \( \overline{\text{X}}_{{\text{sys}}} \) yields the following:
$$ \frac{{\partial^{2} \left( {{}^{h}\overline{\text{A}}_{{\text{g}}} } \right)}}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = \left[ {\begin{array}{*{20}c} {{{\partial^{2} {\text{I}}_{{\text{x}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{I}}_{{\text{x}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} & {{{\partial^{2} {\text{J}}_{{\text{x}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{J}}_{{\text{x}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} & {{{\partial^{2} {\text{K}}_{{\text{x}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{K}}_{{\text{x}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} & {{{\partial^{2} {\text{t}}_{{\text{x}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{t}}_{{\text{x}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \\ {{{\partial^{2} {\text{I}}_{{\text{y}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{I}}_{{\text{y}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} & {{{\partial^{2} {\text{J}}_{{\text{y}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{J}}_{{\text{y}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} & {{{\partial^{2} {\text{K}}_{{\text{y}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{K}}_{{\text{y}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} & {{{\partial^{2} {\text{t}}_{{\text{y}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{t}}_{{\text{y}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \\ {{{\partial^{2} {\text{I}}_{{\text{z}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{I}}_{{\text{z}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} & {{{\partial^{2} {\text{J}}_{{\text{z}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{J}}_{{\text{z}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} & {{{\partial^{2} {\text{K}}_{{\text{z}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{K}}_{{\text{z}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} & {{{\partial^{2} {\text{t}}_{{\text{z}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{t}}_{{\text{z}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$
(C.1)
where
$$ \frac{{\partial^{2} {\text{I}}_{{\text{x}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = - {\text{b}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} + 2{\text{b}}_{{\text{v}}} {\text{c}}_{{\text{v}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} - {\text{c}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} , $$
(C.2)
$$ \frac{{\partial^{2} {\text{I}}_{{\text{y}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = - {\text{b}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} - 2{\text{b}}_{{\text{v}}} {\text{c}}_{{\text{v}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} - {\text{c}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} , $$
(C.3)
$$ \frac{{\partial^{2} {\text{I}}_{{\text{y}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = - {\text{b}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{y}}} , $$
(C.4)
$$ \begin{aligned} \frac{{\partial^{2} {\text{J}}_{{\text{x}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = & - {\text{a}}_{{\text{v}}}^{2} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} + 2{\text{a}}_{{\text{v}}} {\text{b}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} - 2{\text{a}}_{{\text{v}}} {\text{c}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} - {\text{b}}_{{\text{v}}}^{2} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} \\ & - 2{\text{b}}_{{\text{v}}} {\text{c}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} - {\text{c}}_{{\text{v}}}^{2} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} + {\text{a}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{z}}} + 2{\text{a}}_{{\text{v}}} {\text{c}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{z}}} + {\text{c}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{z}}} , \\ \end{aligned} $$
(C.5)
$$ \begin{aligned} \frac{{\partial^{2} {\text{J}}_{{\text{y}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = & - {\text{a}}_{{\text{v}}}^{2} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} + 2{\text{a}}_{{\text{v}}} {\text{b}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} + 2{\text{a}}_{{\text{v}}} {\text{c}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} - {\text{b}}_{{\text{v}}}^{2} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} \\ & + 2{\text{b}}_{{\text{v}}} {\text{c}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} - {\text{c}}_{{\text{v}}}^{2} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} - {\text{a}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{z}}} + 2{\text{a}}_{{\text{v}}} {\text{c}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{z}}} - {\text{c}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{z}}} , \\ \end{aligned} $$
(C.6)
$$ \frac{{\partial^{2} {\text{J}}_{{\text{y}}} }}{{\partial
{\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = -
{\text{a}}_{{\text{v}}}^{2} {\text{S}}\omega_{{\text{x}}}
{\text{C}}\omega_{{\text{y}}} - 2{\text{a}}_{{\text{v}}}
{\text{b}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}}
{\text{S}}\omega_{{\text{y}}} - {\text{b}}_{{\text{v}}}^{2}
{\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} , $$
(C.7)
$$ \begin{aligned} \frac{{\partial^{2} {\text{K}}_{{\text{x}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = & - {\text{a}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} - 2{\text{a}}_{{\text{v}}} {\text{b}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} + 2{\text{a}}_{{\text{v}}} {\text{c}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} - {\text{b}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} \\ & - 2{\text{b}}_{{\text{v}}} {\text{c}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} - {\text{c}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} + {\text{a}}_{{\text{v}}}^{2} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{z}}} - 2{\text{a}}_{{\text{v}}} {\text{c}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{z}}} + {\text{c}}_{{\text{v}}}^{2} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{z}}} , \\ \end{aligned} $$
(C.8)
$$ \begin{aligned} \frac{{\partial^{2} {\text{K}}_{{\text{y}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = & - {\text{a}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} - 2{\text{a}}_{{\text{v}}} {\text{b}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} - 2{\text{a}}_{{\text{v}}} {\text{c}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} - {\text{b}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} \\ & + 2{\text{b}}_{{\text{v}}} {\text{c}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} {\text{C}}\omega_{{\text{z}}} - {\text{c}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} {\text{S}}\omega_{{\text{z}}} + {\text{a}}_{{\text{v}}}^{2} {\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{z}}} + 2{\text{a}}_{{\text{v}}} {\text{c}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{z}}} + {\text{c}}_{{\text{v}}}^{2} {\text{S}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{z}}} , \\ \end{aligned} $$
(C.9)
$$ \frac{{\partial^{2} {\text{K}}_{{\text{z}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = - {\text{a}}_{{\text{v}}} {\text{a}}_{{\text{v}}} {\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} + 2{\text{a}}_{{\text{v}}} {\text{b}}_{{\text{v}}} {\text{S}}\omega_{{\text{x}}} {\text{S}}\omega_{{\text{y}}} - {\text{b}}_{{\text{v}}}^{2} {\text{C}}\omega_{{\text{x}}} {\text{C}}\omega_{{\text{y}}} , $$
(C.10)
$$ \frac{{\partial^{2} {\text{t}}_{{\text{x}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = \frac{{\partial^{2} {\text{t}}_{{\text{x}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}, $$
(C.11)
$$ \frac{{\partial^{2} {\text{t}}_{{\text{y}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = \frac{{\partial^{2} {\text{t}}_{{\text{y}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}, $$
(C.12)
$$ \frac{{\partial^{2} {\text{t}}_{{\text{z}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = \frac{{\partial^{2} {\text{t}}_{{\text{z}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}. $$
(C.13)
Note that Eqs. (C.11), (C.12) and (C.13) simply indicate that \( {{\partial^{2} {\text{t}}_{{\text{x}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{t}}_{{\text{x}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} \), \( {{\partial^{2} {\text{t}}_{{\text{y}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{t}}_{{\text{y}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} \)and \( {{\partial^{2} {\text{t}}_{{\text{z}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{t}}_{{\text{z}}} } {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} \) can be obtained by directly differentiating their corresponding expressions (e.g., \( {{\partial^{2} {\text{t}}_{{\text{y}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{t}}_{{\text{y}}} } {\partial x_{14}^{2} }}} \right. \kern-0pt} {\partial x_{14}^{2} }} = {{ - 2} \mathord{\left/ {\vphantom {{ - 2} {\kappa_{1}^{3} }}} \right. \kern-0pt} {\kappa_{1}^{3} }} \) for \( {}^{{\rm{e}}1}\overline{\text{A}}_{2} \) from Eq. (3.30b) for system in Fig. 3.12). Having obtained \( {}^{0}\overline{\text{A}}_{{\text{ej}}} \), \( {}^{ej}\overline{\text{A}}_{{\text{i}}} \), \( {{\partial \left( {{}^{0}\overline{\text{A}}_{{\text{ej}}} } \right)} \mathord{\left/ {\vphantom {{\partial \left( {{}^{0}\overline{\text{A}}_{{\text{ej}}} } \right)} {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }} \), \( {{\partial \left( {{}^{ej}\overline{\text{A}}_{{\text{i}}} } \right)} \mathord{\left/ {\vphantom {{\partial \left( {{}^{ej}\overline{\text{A}}_{{\text{i}}} } \right)} {\partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{v}}} }} \), \( {{\partial^{2} \left( {{}^{0}\overline{\text{A}}_{{\text{ej}}} } \right)} \mathord{\left/ {\vphantom {{\partial^{2} \left( {{}^{0}\overline{\text{A}}_{{\text{ej}}} } \right)} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} \)and \( {{\partial^{2} \left( {{}^{ej}\overline{\text{A}}_{{\text{i}}} } \right)} \mathord{\left/ {\vphantom {{\partial^{2} \left( {{}^{ej}\overline{\text{A}}_{{\text{i}}} } \right)} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} \), one can compute \( {{\partial^{2} \left( {{}^{0}\overline{\text{A}}_{{\text{i}}} } \right)} \mathord{\left/ {\vphantom {{\partial^{2} \left( {{}^{0}\overline{\text{A}}_{{\text{i}}} } \right)} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right. \kern-0pt} {\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} \) from Eq. (5.108) and then determine \( {{\partial^{2} {\text{t}}_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{t}}_{{\text{ix}}} } {\partial \overline{\text{X}}_{{\text{sys}}}^{2} }}} \right. \kern-0pt} {\partial \overline{\text{X}}_{{\text{sys}}}^{2} }} \), \( {{\partial^{2} {\text{t}}_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{t}}_{{\text{iy}}} } {\partial \overline{\text{X}}_{{\text{sys}}}^{2} }}} \right. \kern-0pt} {\partial \overline{\text{X}}_{{\text{sys}}}^{2} }} \), \( {{\partial^{2} {\text{t}}_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial^{2} {\text{t}}_{{\text{iz}}} } {\partial \overline{\text{X}}_{{\text{sys}}}^{2} }}} \right. \kern-0pt} {\partial \overline{\text{X}}_{{\text{sys}}}^{2} }} \), \( {{\partial^{2} \omega_{{\text{ix}}} } \mathord{\left/ {\vphantom {{\partial^{2} \omega_{{\text{ix}}} } {\partial \overline{\text{X}}_{{\text{sys}}}^{2} }}} \right. \kern-0pt} {\partial \overline{\text{X}}_{{\text{sys}}}^{2} }} \), \( {{\partial^{2} \omega_{{\text{iy}}} } \mathord{\left/ {\vphantom {{\partial^{2} \omega_{{\text{iy}}} } {\partial \overline{\text{X}}_{{\text{sys}}}^{2} }}} \right. \kern-0pt} {\partial \overline{\text{X}}_{{\text{sys}}}^{2} }} \), and \( {{\partial^{2} \omega_{{\text{iz}}} } \mathord{\left/ {\vphantom {{\partial^{2} \omega_{{\text{iz}}} } {\partial \overline{\text{X}}_{{\text{sys}}}^{2} }}} \right. \kern-0pt} {\partial \overline{\text{X}}_{{\text{sys}}}^{2} }} \) accordingly.
Appendix D
$$ \frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{v}}} }} = \frac{{{\text{G}}_{{\text{i}}} }}{{{\text{F}}_{{\text{i}}} }}, $$
(D.1)
where
$$ {\text{F}}_{{\text{i}}} = {\text{I}}_{{\text{ix}}}^{2} + I_{{\text{iy}}}^{2} , $$
(D.1a)
$$ {\text{G}}_{{\text{i}}} = {\text{I}}_{{\text{ix}}} \frac{{\partial {\text{I}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{v}}} }} - {\text{I}}_{{\text{iy}}} \frac{{\partial {\text{I}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{v}}} }}. $$
(D.1b)
$$ \frac{{\partial \omega_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{v}}} }} =
\frac{{\text{G}}_{{\text{i}}} + {\text{H}}_{{\text{i}}} }{{{\text{F}}_{{\text{i}}} }}, $$
(D.2)
in which
$$ {\text{F}}_{{\text{i}}} = {\text{I}}_{{\text{iz}}}^{2} + \left( {I_{{\text{ix}}} {\text{C}}\omega_{{\text{iz}}} + I_{{\text{iy}}} {\text{S}}\omega_{{\text{iz}}} } \right)^{2} , $$
(D.2a)
$$ {\text{G}}_{{\text{i}}} = {\text{I}}_{{\text{iz}}} \left[ {( - I_{{\text{ix}}} {\text{S}}\omega_{{\text{iz}}} + I_{{\text{iy}}} {\text{C}}\omega_{{\text{iz}}} )\frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{v}}} }} + (\frac{{\partial {\text{I}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{v}}} }}{\text{C}}\omega_{{\text{iz}}} + \frac{{\partial {\text{I}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{v}}} }}{\text{S}}\omega_{{\text{iz}}} )} \right], $$
(D.2b)
$$ {\text{H}}_{{\text{i}}} = - \left( {I_{{\text{ix}}} {\text{C}}\omega_{{\text{iz}}} + I_{{\text{iy}}} {\text{S}}\omega_{{\text{iz}}} } \right)\frac{{\partial {\text{I}}_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{v}}} }}. $$
(D.2c)
$$ \frac{{\partial \omega_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{v}}} }} = \frac{{{\text{G}}_{{\text{i}}} {\text{H}}_{{\text{i}}} - U_{{\text{i}}} Q_{{\text{i}}} }}{{{\text{F}}_{{\text{i}}} }}, $$
(D.3)
where
$$ {\text{F}}_{{\text{i}}} = \left( {K_{{\text{ix}}} {\text{S}}\omega_{{\text{iz}}} - K_{{\text{iy}}} {\text{C}}\omega_{{\text{iz}}} } \right)^{2} + \left( { - J_{{\text{ix}}} {\text{S}}\omega_{{\text{iz}}} + J_{{\text{iy}}} {\text{C}}\omega_{{\text{iz}}} } \right)^{2} , $$
(D.3a)
$$ {\text{G}}_{{\text{i}}} = - \text{J}_{{\text{ix}}} {\text{S}}\omega_{{\text{iz}}} + \text{J}_{{\text{iy}}} {\text{C}}\omega_{{\text{iz}}} , $$
(D.3b)
$$ {\text{H}}_{{\text{i}}} = \left( {K_{{\text{ix}}} {\text{C}}\omega_{{\text{iz}}} + K_{{\text{iy}}} {\text{S}}\omega_{{\text{iz}}} } \right)\frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{v}}} }} + \left( {{\text{S}}\omega_{{\text{iz}}} \frac{{\partial {\text{K}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{v}}} }} - {\text{C}}\omega_{{\text{iz}}} \frac{{\partial {\text{K}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{v}}} }}} \right), $$
(D.3c)
$$ U_{{\text{i}}} = K_{{\text{ix}}} {\text{S}}\omega_{{\text{iz}}} - K_{{\text{iy}}} {\text{C}}\omega_{{\text{iz}}} , $$
(D.3d)
$$ Q_{{\text{i}}} = \left( { - J_{{\text{ix}}} {\text{C}}\omega_{{\text{iz}}} - J_{{\text{iy}}} {\text{S}}\omega_{{\text{iz}}} } \right)\frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{v}}} }} + \left( { - {\text{S}}\omega_{{\text{iz}}} \frac{{\partial {\text{J}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{v}}} }} + {\text{C}}\omega_{{\text{iz}}} \frac{{\partial {\text{J}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{v}}} }}} \right). $$
(D.3e)
$$ \frac{{\partial^{2} \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = \frac{{{\text{F}}_{{\text{i}}} \frac{{\partial {\text{G}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }} - {\text{G}}_{{\text{i}}} \frac{{\partial {\text{F}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }}}}{{{\text{F}}_{{\text{i}}}^{2} }}, $$
(D.4)
where \( {\text{F}}_{{\text{i}}} \),\( {\text{G}}_{{\text{i}}} \) are respectively given in Eqs. (D.1a), (D.1b) in this appendix, and
$$ \frac{{\partial {\text{F}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }} = 2\left( {\text{I}_{{\text{ix}}} \frac{{\partial {\text{I}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{w}}} }} + \text{I}_{{\text{iy}}} \frac{{\partial {\text{I}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{w}}} }}} \right), $$
(D.4a)
$$\frac{{\partial {\text{G}}_{{\text{i}}} }}{{\partial
{\text{x}}_{{\text{w}}} }} = {\text{I}}_{{\text{ix}}}
\frac{{\partial^{2} {\text{I}}_{{\text{iy}}} }}{{\partial
{\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} -
\text{I}_{{\text{iy}}} \frac{{\partial^{2} {\text{I}}_{{\text{ix}}}
}}{{\partial {\text{x}}_{{\text{w}}} \partial
{\text{x}}_{{\text{v}}} }} + \frac{{\partial
{\text{I}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{w}}}
}}\frac{{\partial {\text{I}}_{{\text{iy}}} }}{{\partial
{\text{x}}_{{\text{v}}} }} - \frac{{\partial
{\text{I}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{w}}}
}}\frac{{\partial {\text{I}}_{{\text{ix}}} }}{{\partial
{\text{x}}_{{\text{v}}} }}, $$
(D.4b)
$$ \frac{{\partial^{2} \omega_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = \frac{{{\text{F}}_{{\text{i}}} \left( {\frac{{\partial {\text{G}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }} + \frac{{\partial {\text{H}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }}} \right) - \left( {{\text{G}}_{{\text{i}}} + {\text{H}}_{{\text{i}}} } \right)\frac{{\partial {\text{F}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }}}}{{{\text{F}}_{{\text{i}}}^{2} }}, $$
(D.5)
where \( {\text{F}}_{{\text{i}}} \), \( {\text{G}}_{{\text{i}}} \), \( {\text{H}}_{{\text{i}}} \) are respectively given in Eqs. (D.2a), (D.2b), (D.2c) in this appendix, and
$$\frac{{\partial {\text{F}}_{{\text{i}}} }}{{\partial
{\text{x}}_{{\text{w}}} }} = 2I_{{\text{iz}}} \frac{{\partial
{\text{I}}_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{w}}} }} +
2\left( {\text{I}_{{\text{ix}}} {\text{C}}\omega_{{\text{iz}}} +
\text{I}_{{\text{iy}}} {\text{S}}\omega_{{\text{iz}}} }
\right)\left[ {\left( { - \text{I}_{{\text{ix}}}
{\text{S}}\omega_{{\text{iz}}} + \text{I}_{{\text{iy}}}
{\text{C}}\omega_{{\text{iz}}} )\frac{{\partial \omega_{{\text{iz}}}
}}{{\partial {\text{x}}_{{\text{w}}} }} +
({\text{C}}\omega_{{\text{iz}}} \frac{{\partial
{\text{I}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{w}}} }} +
{\text{S}}\omega_{{\text{iz}}} \frac{{\partial
{\text{I}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{w}}} }}}
\right)} \right], $$
(D.5a)
$$ \begin{aligned} \frac{{\partial {\text{G}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }}
= & \frac{{\partial {\text{I}}_{{\text{iz}}} }} {{\partial
{\text{x}}_{{\text{w}}} }} \left[ {\left( { - \text{I}_{{\text{ix}}}
{\text{S}}\omega_{{\text{iz}}} + \text{I}_{{\text{iy}}}
{\text{C}}\omega_{{\text{iz}}} } \right) \frac{{\partial
\omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{v}}} }} +
\left( {{\text{C}}\omega_{{\text{iz}}} \frac{{\partial
{\text{I}}_{{\text{ix}}} }} {{\partial {\text{x}}_{{\text{v}}} }} +
{\text{S}}\omega_{{\text{iz}}} \frac{{\partial
{\text{I}}_{{\text{iy}}} }}
{{\partial {\text{x}}_{{\text{v}}} }}} \right)} \right] \\
& + \text{I}_{{\text{iz}}} \left[ {\left( { - {\text{I}}_{{\text{ix}}}
{\text{C}}\omega_{{\text{iz}}} - \text{I}_{{\text{iy}}}
{\text{S}}\omega_{{\text{iz}}} } \right) \frac{{\partial
\omega_{{\text{iz}}} }} {{\partial {\text{x}}_{{\text{w}}} }}
\frac{{\partial \omega_{{\text{iz}}} }} {{\partial
{\text{x}}_{{\text{v}}} }} + \left( { -
{\text{S}}\omega_{{\text{iz}}}
\frac{{\partial {\text{I}}_{{\text{ix}}} }}
{{\partial {\text{x}}_{{\text{w}}} }}
+ {\text{C}}\omega_{{\text{iz}}}
\frac{{\partial {\text{I}}_{{\text{iy}}} }}
{{\partial {\text{x}}_{{\text{w}}} }}} \right)
\frac{{\partial \omega_{{\text{iz}}} }}
{{\partial {\text{x}}_{{\text{v}}} }}
+ \left( { - \text{I}_{{\text{ix}}} {\text{S}}\omega_{{\text{iz}}}
+ {\text{I}}_{{\text{iy}}} {\text{C}}\omega_{{\text{iz}}} } \right)
\frac{{\partial^{2} \omega_{{\text{iz}}} }}
{{\partial {\text{x}}_{{\text{w}}}
\partial {\text{x}}_{{\text{v}}} }} }\right. \\
& \left. + \left( { - {\text{S}}\omega_{{\text{iz}}}
\frac{{\partial {\text{I}}_{{\text{ix}}} }}
{{\partial {\text{x}}_{{\text{v}}} }}
+ {\text{C}}\omega_{{\text{iz}}}
\frac{{\partial {\text{I}}_{{\text{iy}}} }}
{{\partial {\text{x}}_{{\text{v}}} }}} \right)
\frac{{\partial \omega_{{\text{iz}}} }}
{{\partial {\text{x}}_{{\text{w}}} }}
+ \left( {{\text{C}}\omega_{{\text{iz}}}
\frac{{\partial^{2} {\text{I}}_{{\text{ix}}} }}
{{\partial {\text{x}}_{{\text{w}}}
\partial {\text{x}}_{{\text{v}}} }}
+ {\text{S}}\omega_{{\text{iz}}}
\frac{{\partial^{2} {\text{I}}_{{\text{iy}}} }}
{{\partial {\text{x}}_{{\text{w}}}
\partial {\text{x}}_{{\text{v}}} }}} \right) \right], \\ \end{aligned} $$
(D.5b)
$$ \frac{{\partial {\text{H}}_{{\text{i}}} }}{{\partial
{\text{x}}_{{\text{w}}} }} = - \left( {\text{I}_{{\text{ix}}}
{\text{C}}\omega_{{\text{iz}}} + \text{I}_{{\text{iy}}}
{\text{S}}\omega_{{\text{iz}}} } \right)\frac{{\partial^{2}
{\text{I}}_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{w}}}
\partial {\text{x}}_{{\text{v}}} }} - \left( { - \text{I}_{{\text{ix}}}
{\text{S}}\omega_{{\text{iz}}} + \text{I}_{{\text{iy}}}
{\text{C}}\omega_{{\text{iz}}} } \right)\frac{{\partial
\omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{w}}}
}}\frac{{\partial {\text{I}}_{{\text{iz}}} }}{{\partial
{\text{x}}_{{\text{v}}} }} - \left( {{\text{C}}\omega_{{\text{iz}}}
\frac{{\partial {\text{I}}_{{\text{ix}}} }}{{\partial
{\text{x}}_{{\text{w}}} }} + {\text{S}}\omega_{{\text{iz}}}
\frac{{\partial {\text{I}}_{{\text{iy}}} }}{{\partial
{\text{x}}_{{\text{w}}} }}} \right)\frac{{\partial
{\text{I}}_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{v}}} }}.
$$
(D.5c)
$$ \frac{{\partial^{2} \omega_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} = \frac{{{\text{F}}_{{\text{i}}} \left[ {{\text{H}}_{{\text{i}}} \frac{{\partial {\text{G}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }} + {\text{G}}_{{\text{i}}} \frac{{\partial {\text{H}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }} - \left( {Q_{{\text{i}}} \frac{{\partial U_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }} + \text{U}_{{\text{i}}} \frac{{\partial Q_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }}} \right)} \right] - \left( {{\text{G}}_{{\text{i}}} {\text{H}}_{{\text{i}}} - U_{{\text{i}}} Q_{{\text{i}}} } \right)\frac{{\partial {\text{F}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }}}}{{{\text{F}}_{{\text{i}}}^{2} }}, $$
(D.6)
where \( {\text{F}}_{{\text{i}}} \), \( {\text{G}}_{{\text{i}}} \), \( {\text{H}}_{{\text{i}}} \), \( \text{U}_{{\text{i}}} \), \( Q_{{\text{i}}} \) are respectively given in Eqs. (D.3a), (D.3b), (D.3c), (D.3d), (D.3e) in this appendix, and
$$ \begin{aligned} \frac{{\partial {\text{F}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }} = & 2\left( {K_{{\text{ix}}} {\text{S}}\omega_{{\text{iz}}} - K_{{\text{iy}}} {\text{C}}\omega_{{\text{iz}}} } \right)\left[ {\left( {K_{{\text{ix}}} {\text{C}}\omega_{{\text{iz}}} + K_{{\text{iy}}} {\text{S}}\omega_{{\text{iz}}} } \right)\frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{w}}} }} + \left( {{\text{S}}\omega_{{\text{iz}}} \frac{{\partial {\text{K}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{w}}} }} - {\text{C}}\omega_{{\text{iz}}} \frac{{\partial {\text{K}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{w}}} }}} \right)} \right] \\ & + 2\left( { - J_{{\text{ix}}} {\text{S}}\omega_{{\text{iz}}} + J_{{\text{iy}}} {\text{C}}\omega_{{\text{iz}}} } \right)\left[ {\left( { - J_{{\text{ix}}} {\text{C}}\omega_{{\text{iz}}} - J_{{\text{iy}}} {\text{S}}\omega_{{\text{iz}}} } \right)\frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{w}}} }} + \left( { - {\text{S}}\omega_{{\text{iz}}} \frac{{\partial {\text{J}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{w}}} }} + {\text{C}}\omega_{{\text{iz}}} \frac{{\partial {\text{J}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{w}}} }}} \right)} \right], \\ \end{aligned} $$
(D.6a)
$$ \frac{{\partial {\text{G}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }} = \left( { - J_{{\text{ix}}} {\text{C}}\omega_{{\text{iz}}} - J_{{\text{iy}}} {\text{S}}\omega_{{\text{iz}}} } \right)\frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{w}}} }} - {\text{S}}\omega_{{\text{iz}}} \frac{{\partial {\text{J}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{w}}} }} + {\text{C}}\omega_{{\text{iz}}} \frac{{\partial {\text{J}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{w}}} }}, $$
(D.6b)
$$ \begin{aligned} \frac{{\partial {\text{H}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }} = & \left( {{\text{C}}\omega_{{\text{iz}}} \frac{{\partial {\text{K}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{w}}} }} + {\text{S}}\omega_{{\text{iz}}} \frac{{\partial {\text{K}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{w}}} }}} \right)\frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{v}}} }} + \left( { - K_{{\text{ix}}} {\text{S}}\omega_{{\text{iz}}} + K_{{\text{iy}}} {\text{C}}\omega_{{\text{iz}}} } \right)\frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{w}}} }}\frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{v}}} }} + \left( {K_{{\text{ix}}} {\text{C}}\omega_{{\text{iz}}} + K_{{\text{iy}}} {\text{S}}\omega_{{\text{iz}}} } \right)\frac{{\partial^{2} \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} \\ & + \left( {{\text{S}}\omega_{{\text{iz}}} \frac{{\partial^{2} {\text{K}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} - {\text{C}}\omega_{{\text{iz}}} \frac{{\partial^{2} {\text{K}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right) + \left( {{\text{C}}\omega_{{\text{iz}}} \frac{{\partial {\text{K}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{v}}} }} + {\text{S}}\omega_{{\text{iz}}} \frac{{\partial {\text{K}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{v}}} }}} \right)\frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{w}}} }}, \\ \end{aligned} $$
(D.6c)
$$ \frac{{\partial U_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }} = \left( {K_{{\text{ix}}} {\text{C}}\omega_{{\text{iz}}} + K_{{\text{iy}}} {\text{S}}\omega_{{\text{iz}}} } \right)\frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{w}}} }} + {\text{S}}\omega_{{\text{iz}}} \frac{{\partial {\text{K}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{w}}} }} - {\text{C}}\omega_{{\text{iz}}} \frac{{\partial {\text{K}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{w}}} }}, $$
(D.6d)
$$ \begin{aligned} \frac{{\partial Q_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{w}}} }} = & \left( { - J_{{\text{ix}}} {\text{C}}\omega_{{\text{iz}}} - J_{{\text{iy}}} {\text{S}}\omega_{{\text{iz}}} } \right)\frac{{\partial^{2} \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} + \left( {J_{{\text{ix}}} {\text{S}}\omega_{{\text{iz}}} - J_{{\text{iy}}} {\text{C}}\omega_{{\text{iz}}} } \right)\frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{w}}} }}\frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{v}}} }} \\ & - \left( {{\text{C}}\omega_{{\text{iz}}} \frac{{\partial {\text{J}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{v}}} }} + {\text{S}}\omega_{{\text{iz}}} \frac{{\partial {\text{J}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{v}}} }}} \right)\frac{{\partial \omega_{{\text{iz}}} }}{{\partial {\text{x}}_{{\text{w}}} }} + \left( { - {\text{S}}\omega_{{\text{iz}}} \frac{{\partial^{2} {\text{J}}_{{\text{ix}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }} + {\text{C}}\omega_{{\text{iz}}} \frac{{\partial^{2} {\text{J}}_{{\text{iy}}} }}{{\partial {\text{x}}_{{\text{w}}} \partial {\text{x}}_{{\text{v}}} }}} \right). \\ \end{aligned} $$
(D.6e)
