Appendix 1: Numerical–symbolic expression of the inverse kinematic equation
sα=sin α, cα=cos α, sβ=sin β, cβ=cos β, k=sin θ, m=cos θ
For the convenience of studying the inverse kinematics of the 3-PRS parallel mechanism, a numerical–symbolic expression of the sliders’ travelling lengths h
1, h
2 and h
3 with regards to the structural parameters is given in Eq. 11:
$${\left[ {\begin{array}{*{20}c}
{{h_{1} }} \\
{{h_{2} }} \\
{{h_{3} }} \\
\end{array} } \right]} = {\left[ {\begin{array}{*{20}c}
{{{\text{Exp}}1}} \\
{{{\text{Exp}}2}} \\
{{{\text{Exp}}3}} \\
\end{array} } \right]}$$
(11)
where:
$${\text{Exp}}1 = - {\text{k}}r{\text{c}}\beta + R{\text{k}} - {\text{m}}r{\text{s}}\beta + {\text{m}}z + {\left( \begin{aligned}
& {\text{k}}^{2} r^{2} {\text{c}}\beta ^{2} - 2{\text{k}}^{2} r{\text{c}}\beta R + 2{\text{k}}r^{2} {\text{c}}\beta {\text{ms}}\beta \\
& - 2{\text{k}}r{\text{c}}\beta {\text{m}}z + R^{2} {\text{k}}^{2} - 2R{\text{km}}r{\text{s}}\beta + 2R{\text{km}}z \\
& + {\text{m}}^{2} r^{2} {\text{s}}\beta ^{2} - 2{\text{m}}^{2} r{\text{s}}\beta z + {\text{m}}^{2} z^{2} - r^{2} - R^{2} \\
& + 2zr{\text{s}}\beta + 2Rr{\text{c}}\beta + L^{2}_{1} - z^{2} \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} $$
$${\text{Exp2}} = \begin{array}{*{20}l}
{{{433} \mathord{\left/
{\vphantom {{433} {{\text{500}}}}} \right.
\kern-\nulldelimiterspace} {{\text{500}}}{\text{k}}^{2} r{\text{s}}\beta {\text{s}}\alpha + 1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2{\text{m}}r{\text{s}}\beta - 1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2{\text{k}}^{2} r{\text{c}}\beta + 1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2R{\text{k}}^{{\text{2}}} } \hfill} \\
{{ + {433} \mathord{\left/
{\vphantom {{433} {500}}} \right.
\kern-\nulldelimiterspace} {500}R{\text{km}} + {433} \mathord{\left/
{\vphantom {{433} {{\text{500}}}}} \right.
\kern-\nulldelimiterspace} {{\text{500}}}{\text{m}}r{\text{c}}\beta {\text{s}}\alpha - {433} \mathord{\left/
{\vphantom {{433} {{\text{500}}}}} \right.
\kern-\nulldelimiterspace} {{\text{500}}}{\text{km}}r{\text{c}}\alpha + {\text{mz}}} \hfill} \\
{{ + 1 \mathord{\left/
{\vphantom {1 {500}}} \right.
\kern-\nulldelimiterspace} {500}{\left( \begin{aligned}
& 62500{\text{m}}^{2} r^{2} {\text{s}}\beta ^{2} + 125000Rr{\text{c}}\beta - 249989r^{2} - 249989R^{2} \\
& + 374978R{\text{km}}^{2} r{\text{c}}\beta {\text{s}}\alpha - 374978R{\text{k}}^{2} {\text{m}}^{2} r{\text{c}}\alpha + 216500R{\text{k}}^{2} {\text{m}}r{\text{c}}\beta {\text{s}}\alpha \\
& - 216500R{\text{k}}^{3} {\text{m}}r{\text{c}}\alpha - 216500{\text{k}}^{4} r{\text{s}}\beta {\text{s}}\alpha R - 125000{\text{m}}r^{2} {\text{s}}\beta {\text{k}}^{2} {\text{c}}\beta \\
& + 62500{\text{k}}^{4} r^{2} {\text{c}}\beta ^{2} + 216500R^{2} {\text{k}}^{3} {\text{m}} + 187489R^{2} {\text{k}}^{2} {\text{m}}^{2} \\
& - 216500Rr{\text{s}}\beta {\text{s}}\alpha - 433000zr{\text{c}}\beta {\text{s}}\alpha + 250000L^{2}_{2} + 374978Rr{\text{c}}\alpha - 250000zr{\text{s}}\beta \\
& - 250000z^{2} + 374978{\text{k}}^{3} r{\text{s}}\beta {\text{s}}\alpha R{\text{m}} + 374978{\text{k}}^{2} r^{2} {\text{s}}\beta {\text{s}}\alpha ^{2} {\text{mc}}\beta \\
& - 374978{\text{k}}^{3} r^{2} {\text{s}}\beta {\text{s}}\alpha {\text{mc}}\alpha + 433000{\text{k}}^{2} r{\text{s}}\beta {\text{s}}\alpha {\text{m}}z + 125000{\text{m}}r{\text{s}}\beta R{\text{k}}^{2} \\
& + 216500{\text{m}}^{2} r{\text{s}}\beta R{\text{k}} + 216500{\text{m}}^{2} r^{2} {\text{s}}\beta {\text{c}}\beta {\text{s}}\alpha - 216500{\text{m}}^{2} r^{2} {\text{s}}\beta {\text{kc}}\alpha \\
& - 216500{\text{k}}^{3} r{\text{c}}\beta R{\text{m}} - 216500{\text{k}}^{2} r^{2} {\text{c}}\beta ^{2} {\text{ms}}\alpha + 216500{\text{k}}^{3} r^{2} {\text{c}}\beta {\text{mc}}\alpha \\
& + 216500{\text{k}}^{2} r^{2} {\text{s}}\beta ^{2} {\text{s}}\alpha {\text{m}} - 216500{\text{k}}^{4} r^{2} {\text{s}}\beta {\text{s}}\alpha {\text{c}}\beta + 187489{\text{k}}^{4} r^{2} {\text{s}}\beta {\text{s}}\alpha ^{2} \\
& + 250000{\text{m}}^{2} r{\text{s}}\beta z - 125000{\text{k}}^{4} r{\text{c}}\beta R + 250000R{\text{k}}^{2} {\text{m}}z + 433000R{\text{km}}^{2} z \\
& + 187489{\text{m}}^{2} r^{2} {\text{c}}\beta ^{2} {\text{s}}\alpha ^{2} + 187489{\text{k}}^{2} {\text{m}}^{2} r^{2} {\text{c}}\alpha ^{2} + 62500R^{2} {\text{k}}^{4} + 250000{\text{m}}^{2} z^{2} \\
& - 250000{\text{k}}^{2} r{\text{c}}\beta {\text{m}}z - 374978{\text{m}}^{2} r^{2} {\text{c}}\beta {\text{s}}\alpha {\text{kc}}\alpha + 433000{\text{m}}^{2} r{\text{c}}\beta {\text{s}}\alpha z \\
& - 433000{\text{km}}^{2} r{\text{c}}\alpha z \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} } \hfill} \\
\end{array} $$
$${\text{Exp3}} = \begin{array}{*{20}l}
{{{\text{m}}z + 1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2R{\text{k}}^{2} - {433} \mathord{\left/
{\vphantom {{433} {{\text{500}}}}} \right.
\kern-\nulldelimiterspace} {{\text{500}}}{\text{km}}r{\text{c}}\alpha + {433} \mathord{\left/
{\vphantom {{433} {{\text{500}}}}} \right.
\kern-\nulldelimiterspace} {{\text{500}}}R{\text{km}} + 1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2{\text{m}}r{\text{s}}\beta } \hfill} \\
{{ - 1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2{\text{k}}^{2} r{\text{c}}\beta - {433} \mathord{\left/
{\vphantom {{433} {500}}} \right.
\kern-\nulldelimiterspace} {500}{\text{k}}^{{\text{2}}} r{\text{s}}\beta {\text{s}}\alpha - {433} \mathord{\left/
{\vphantom {{433} {500}}} \right.
\kern-\nulldelimiterspace} {500}{\text{m}}r{\text{c}}\beta {\text{s}}\alpha } \hfill} \\
{{ + 1 \mathord{\left/
{\vphantom {1 {500}}} \right.
\kern-\nulldelimiterspace} {500}{\left( \begin{aligned}
& 374978Rr{\text{c}}\alpha + 125000Rr{\text{c}}\beta - 250000zr{\text{s}}\beta - 249989r^{2} - 249989R^{2} \\
& - 250000z^{2} + 62500{\text{m}}^{2} r^{2} {\text{s}}\beta ^{2} + 62500{\text{k}}^{4} r^{2} {\text{c}}\beta ^{2} - 433000{\text{m}}^{2} z{\text{k}}r{\text{c}}\alpha \\
& + 250000L^{2}_{3} + 216500Rr{\text{s}}\beta {\text{s}}\alpha + 433000zr{\text{c}}\beta {\text{s}}\alpha + 216500R^{2} {\text{k}}^{3} {\text{m}} + 187489R^{2} {\text{k}}^{2} {\text{m}}^{2} \\
& + 250000{\text{m}}zR{\text{k}}^{2} + 433000{\text{m}}^{2} zR{\text{k}} + 250000{\text{m}}^{2} zr{\text{s}}\beta \\
& - 125000R{\text{k}}^{4} r{\text{c}}\beta + 187489{\text{k}}^{2} {\text{m}}^{2} r^{2} {\text{c}}\alpha ^{2} + 187489{\text{k}}^{4} r^{2} {\text{s}}\beta ^{2} {\text{s}}\alpha ^{2} + 187489{\text{m}}^{2} r^{2} {\text{c}}\beta ^{2} {\text{s}}\alpha ^{2} \\
& + 250000{\text{m}}^{{\text{2}}} z^{2} + 62500R^{2} {\text{k}}^{4} - 250000{\text{m}}z{\text{k}}^{2} r{\text{c}}\beta - 433000{\text{m}}z{\text{k}}^{2} r{\text{s}}\beta {\text{s}}\alpha \\
& - 433000{\text{m}}^{2} zr{\text{c}}\beta s\alpha - 216500R{\text{k}}^{3} {\text{m}}r{\text{c}}\alpha + 125000R{\text{k}}^{2} {\text{m}}r{\text{s}}\beta - 216500R{\text{k}}^{4} r{\text{s}}\beta {\text{s}}\alpha \\
& - 216500R{\text{k}}^{2} {\text{m}}r{\text{c}}\beta {\text{s}}\alpha - 374978{\text{k}}^{2} {\text{m}}^{2} r{\text{c}}\alpha R - 216500{\text{km}}^{2} r^{2} {\text{c}}\alpha {\text{s}}\beta \\
& + 216500{\text{k}}^{3} {\text{m}}r^{2} {\text{c}}\alpha {\text{c}}\beta + 374978{\text{k}}^{3} {\text{m}}r^{2} {\text{c}}\alpha {\text{s}}\beta {\text{s}}\alpha + 374978{\text{km}}^{2} r^{2} {\text{c}}\alpha {\text{c}}\beta {\text{s}}\alpha \\
& + 216500R{\text{km}}^{2} r{\text{s}}\beta - 216500R{\text{k}}^{3} {\text{m}}r{\text{c}}\beta - 374978R{\text{k}}^{3} {\text{m}}r{\text{s}}\beta {\text{s}}\alpha \\
& - 374978R{\text{km}}^{2} r{\text{c}}\beta {\text{s}}\alpha - 125000{\text{m}}r^{2} {\text{s}}\beta {\text{k}}^{2} {\text{c}}\beta - 216500{\text{m}}r^{2} {\text{s}}\beta ^{2} {\text{k}}^{2} {\text{s}}\alpha \\
& - 216500{\text{m}}^{2} r^{2} {\text{s}}\beta {\text{c}}\beta {\text{s}}\alpha + 216500{\text{k}}^{4} r^{2} {\text{c}}\beta {\text{s}}\beta {\text{s}}\alpha + 216500{\text{k}}^{2} r^{2} {\text{c}}\beta ^{2} {\text{ms}}\alpha \\
& + 374978{\text{k}}^{2} r^{2} {\text{s}}\beta {\text{s}}\alpha ^{2} {\text{mc}}\beta \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} } \hfill} \\
\end{array} $$
Appendix 2: Numerical–symbolic expression for the determinant of the Jacobian matrix in singularity analysis (θ=0°, R=500, r/R=0.33 and L/R=2)
$${\text{Det}}{\left( {{\left[ {\mathbf{J}} \right]}} \right)} = \begin{array}{*{20}l}
{\begin{aligned}
& {\left( \begin{aligned}
& {\left( { - 165{\text{c}}\beta + {1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2} \mathord{\left/
{\vphantom {{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2} {{\left( {27225{\text{s}}\beta ^{2} + 722775 + 165000{\text{c}}\beta } \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }}} \right.
\kern-\nulldelimiterspace} {{\left( {27225{\text{s}}\beta ^{2} + 722775 + 165000{\text{c}}\beta } \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }{\left( {54450{\text{s}}\beta {\text{c}}\beta - 165000{\text{s}}\beta } \right)}} \right)} \\
& \times {\left( \begin{aligned}
& - {14289} \mathord{\left/
{\vphantom {{14289} {100{\text{c}}\beta {\text{c}}\alpha }}} \right.
\kern-\nulldelimiterspace} {100{\text{c}}\beta {\text{c}}\alpha } + {1 \mathord{\left/
{\vphantom {1 {1000}}} \right.
\kern-\nulldelimiterspace} {1000}} \mathord{\left/
{\vphantom {{1 \mathord{\left/
{\vphantom {1 {1000}}} \right.
\kern-\nulldelimiterspace} {1000}} {{\left( \begin{aligned}
& 10312500000{\text{c}}\beta + 180696799475 \\
& + 1701562500{\text{s}}\beta ^{2} - 5894212500{\text{c}}\beta {\text{s}}\alpha {\text{s}}\beta \\
& + 5104388025{\text{c}}\beta ^{2} {\text{s}}\alpha ^{2} + 17861250000{\text{s}}\beta {\text{s}}\alpha \\
& + 30935685000{\text{c}}\alpha \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }}} \right.
\kern-\nulldelimiterspace} {{\left( \begin{aligned}
& 10312500000{\text{c}}\beta + 180696799475 \\
& + 1701562500{\text{s}}\beta ^{2} - 5894212500{\text{c}}\beta {\text{s}}\alpha {\text{s}}\beta \\
& + 5104388025{\text{c}}\beta ^{2} {\text{s}}\alpha ^{2} + 17861250000{\text{s}}\beta {\text{s}}\alpha \\
& + 30935685000{\text{c}}\alpha \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} } \\
& \times {\left( \begin{aligned}
& - 5894212500{\text{c}}\beta {\text{c}}\alpha {\text{s}}\beta + 10208776050{\text{c}}\beta ^{2} {\text{s}}\alpha {\text{c}}\alpha + 17861250000{\text{s}}\beta {\text{s}}\alpha \\
& - 30935685000{\text{c}}\alpha \\
\end{aligned} \right)} \\
\end{aligned} \right)} \\
\end{aligned} \right)} \\
& + {\left( \begin{aligned}
& {\left( \begin{aligned}
& {14289} \mathord{\left/
{\vphantom {{14289} {100{\text{c}}\beta {\text{c}}\alpha }}} \right.
\kern-\nulldelimiterspace} {100{\text{c}}\beta {\text{c}}\alpha } + {1 \mathord{\left/
{\vphantom {1 {1000}}} \right.
\kern-\nulldelimiterspace} {1000}} \mathord{\left/
{\vphantom {{1 \mathord{\left/
{\vphantom {1 {1000}}} \right.
\kern-\nulldelimiterspace} {1000}} {{\left( \begin{aligned}
& 10312500000{\text{c}}\beta + 180696799475 \\
& + 1701562500{\text{s}}\beta ^{2} + 5894212500{\text{c}}\beta {\text{s}}\alpha {\text{s}}\beta \\
& + 5104388025{\text{c}}\beta ^{2} {\text{s}}\alpha ^{2} - 17861250000{\text{s}}\beta {\text{s}}\alpha \\
& + 30935685000{\text{c}}\alpha \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }}} \right.
\kern-\nulldelimiterspace} {{\left( \begin{aligned}
& 10312500000{\text{c}}\beta + 180696799475 \\
& + 1701562500{\text{s}}\beta ^{2} + 5894212500{\text{c}}\beta {\text{s}}\alpha {\text{s}}\beta \\
& + 5104388025{\text{c}}\beta ^{2} {\text{s}}\alpha ^{2} - 17861250000{\text{s}}\beta {\text{s}}\alpha \\
& + 30935685000{\text{c}}\alpha \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} } \\
& \times {\left( \begin{aligned}
& - 5894212500{\text{c}}\beta {\text{c}}\alpha {\text{s}}\beta + 10208776050{\text{c}}\beta ^{2} {\text{s}}\alpha {\text{c}}\alpha + 17861250000{\text{s}}\beta {\text{c}}\alpha \\
& - 30935685000{\text{s}}\alpha \\
\end{aligned} \right)} \\
\end{aligned} \right)} \\
& \times {\left( \begin{aligned}
& {165} \mathord{\left/
{\vphantom {{165} {2{\text{c}}\beta }}} \right.
\kern-\nulldelimiterspace} {2{\text{c}}\beta } + {14289} \mathord{\left/
{\vphantom {{14289} {100{\text{s}}\beta {\text{s}}\alpha }}} \right.
\kern-\nulldelimiterspace} {100{\text{s}}\beta {\text{s}}\alpha } + {1 \mathord{\left/
{\vphantom {1 {1000}}} \right.
\kern-\nulldelimiterspace} {1000}} \mathord{\left/
{\vphantom {{1 \mathord{\left/
{\vphantom {1 {1000}}} \right.
\kern-\nulldelimiterspace} {1000}} {{\left( \begin{aligned}
& 10312500000{\text{c}}\beta + 180696799475 \\
& + 1701562500{\text{s}}\beta ^{2} - 5894212500{\text{c}}\beta {\text{s}}\alpha {\text{s}}\beta \\
& + 5104388025{\text{c}}\beta ^{2} {\text{s}}\alpha ^{2} + 17861250000{\text{s}}\beta {\text{s}}\alpha \\
& + 30935685000{\text{c}}\alpha \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }}} \right.
\kern-\nulldelimiterspace} {{\left( \begin{aligned}
& 10312500000{\text{c}}\beta + 180696799475 \\
& + 1701562500{\text{s}}\beta ^{2} - 5894212500{\text{c}}\beta {\text{s}}\alpha {\text{s}}\beta \\
& + 5104388025{\text{c}}\beta ^{2} {\text{s}}\alpha ^{2} + 17861250000{\text{s}}\beta {\text{s}}\alpha \\
& + 30935685000{\text{c}}\alpha \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} } \\
& \times {\left( \begin{aligned}
& - 10312500000{\text{s}}\beta + 3403125000{\text{s}}\beta {\text{c}}\beta + 5894212500{\text{s}}\beta ^{2} {\text{s}}\alpha - 5894212500{\text{c}}\beta ^{2} {\text{s}}\alpha \\
& - 10208776050{\text{c}}\beta {\text{s}}\alpha ^{2} {\text{s}}\beta + 17861250000{\text{c}}\beta {\text{s}}\alpha \\
\end{aligned} \right)} \\
\end{aligned} \right)} \\
\end{aligned} \right)} \\
& - {\left( \begin{aligned}
& {\left( \begin{aligned}
& { - 165} \mathord{\left/
{\vphantom {{ - 165} {{\text{c}}\beta }}} \right.
\kern-\nulldelimiterspace} {{\text{c}}\beta } + {1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2} \mathord{\left/
{\vphantom {{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2} {{\left( {27225{\text{s}}\beta ^{2} + 722775 + 165000{\text{c}}\beta } \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }}} \right.
\kern-\nulldelimiterspace} {{\left( {27225{\text{s}}\beta ^{2} + 722775 + 165000{\text{c}}\beta } \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} } \\
& \times {\left( {54450{\text{s}}\beta {\text{c}}\beta - 16500{\text{s}}\beta } \right)} \\
\end{aligned} \right)} \\
& \times {\left( {{14289} \mathord{\left/
{\vphantom {{14289} {100{\text{c}}\beta {\text{c}}\alpha }}} \right.
\kern-\nulldelimiterspace} {100{\text{c}}\beta {\text{c}}\alpha } + {1 \mathord{\left/
{\vphantom {1 {1000}}} \right.
\kern-\nulldelimiterspace} {1000}} \mathord{\left/
{\vphantom {{1 \mathord{\left/
{\vphantom {1 {1000}}} \right.
\kern-\nulldelimiterspace} {1000}} {{\left( \begin{aligned}
& 10312500000{\text{c}}\beta + 180696799475 \\
& + 1701562500{\text{s}}\beta ^{2} + 5894212500{\text{c}}\beta {\text{s}}\alpha {\text{s}}\beta \\
& + 5104388025{\text{c}}\beta ^{2} {\text{s}}\alpha ^{2} - 17861250000{\text{s}}\beta {\text{s}}\alpha \\
& + 30935685000{\text{c}}\alpha \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }}} \right.
\kern-\nulldelimiterspace} {{\left( \begin{aligned}
& 10312500000{\text{c}}\beta + 180696799475 \\
& + 1701562500{\text{s}}\beta ^{2} + 5894212500{\text{c}}\beta {\text{s}}\alpha {\text{s}}\beta \\
& + 5104388025{\text{c}}\beta ^{2} {\text{s}}\alpha ^{2} - 17861250000{\text{s}}\beta {\text{s}}\alpha \\
& + 30935685000{\text{c}}\alpha \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }} \right)} \\
& \times {\left( \begin{aligned}
& - 5894212500{\text{c}}\beta {\text{c}}\alpha {\text{s}}\beta + 10208776050{\text{c}}\beta ^{2} {\text{s}}\alpha {\text{c}}\alpha - 17861250000{\text{s}}\beta {\text{c}}\alpha \\
& - 30935685000{\text{s}}\alpha \\
\end{aligned} \right)} \\
\end{aligned} \right)} \\
& - {\left( \begin{aligned}
& {\left( {{ - 14289} \mathord{\left/
{\vphantom {{ - 14289} {100{\text{s}}\beta {\text{s}}\alpha }}} \right.
\kern-\nulldelimiterspace} {100{\text{s}}\beta {\text{s}}\alpha } + {165} \mathord{\left/
{\vphantom {{165} {2{\text{c}}\beta }}} \right.
\kern-\nulldelimiterspace} {2{\text{c}}\beta } + {1 \mathord{\left/
{\vphantom {1 {1000}}} \right.
\kern-\nulldelimiterspace} {1000}} \mathord{\left/
{\vphantom {{1 \mathord{\left/
{\vphantom {1 {1000}}} \right.
\kern-\nulldelimiterspace} {1000}} {{\left( \begin{aligned}
& 10312500000{\text{c}}\beta + 180696799475 \\
& + 1701562500{\text{s}}\beta ^{2} + 5894212500{\text{c}}\beta {\text{s}}\alpha {\text{s}}\beta \\
& + 5104388025{\text{c}}\beta ^{2} {\text{s}}\alpha ^{2} - 17861250000{\text{s}}\beta {\text{s}}\alpha \\
& + 30935685000{\text{c}}\alpha \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }}} \right.
\kern-\nulldelimiterspace} {{\left( \begin{aligned}
& 10312500000{\text{c}}\beta + 180696799475 \\
& + 1701562500{\text{s}}\beta ^{2} + 5894212500{\text{c}}\beta {\text{s}}\alpha {\text{s}}\beta \\
& + 5104388025{\text{c}}\beta ^{2} {\text{s}}\alpha ^{2} - 17861250000{\text{s}}\beta {\text{s}}\alpha \\
& + 30935685000{\text{c}}\alpha \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }} \right)} \\
& \times {\left( \begin{aligned}
& - 10312500000{\text{s}}\beta + 3403125000{\text{s}}\beta {\text{c}}\beta - 5894212500{\text{s}}\beta ^{2} {\text{s}}\alpha + 5894212500{\text{c}}\beta ^{2} {\text{s}}\alpha \\
& - 10208776050{\text{c}}\beta {\text{s}}\alpha ^{2} {\text{s}}\beta - 17861250000{\text{c}}\beta {\text{s}}\alpha \\
\end{aligned} \right)} \\
& \times {\left( {{ - 14289} \mathord{\left/
{\vphantom {{ - 14289} {100{\text{c}}\beta {\text{c}}\alpha }}} \right.
\kern-\nulldelimiterspace} {100{\text{c}}\beta {\text{c}}\alpha } + {1 \mathord{\left/
{\vphantom {1 {1000}}} \right.
\kern-\nulldelimiterspace} {1000}} \mathord{\left/
{\vphantom {{1 \mathord{\left/
{\vphantom {1 {1000}}} \right.
\kern-\nulldelimiterspace} {1000}} {{\left( \begin{aligned}
& 10312500000{\text{c}}\beta + 180696799475 \\
& + 1701562500{\text{s}}\beta ^{2} - 5894212500{\text{c}}\beta {\text{s}}\alpha {\text{s}}\beta \\
& + 5104388025{\text{c}}\beta ^{2} {\text{s}}\alpha ^{2} + 17861250000{\text{s}}\beta {\text{s}}\alpha \\
& + 30935685000{\text{c}}\alpha \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }}} \right.
\kern-\nulldelimiterspace} {{\left( \begin{aligned}
& 10312500000{\text{c}}\beta + 180696799475 \\
& + 1701562500{\text{s}}\beta ^{2} - 5894212500{\text{c}}\beta {\text{s}}\alpha {\text{s}}\beta \\
& + 5104388025{\text{c}}\beta ^{2} {\text{s}}\alpha ^{2} + 17861250000{\text{s}}\beta {\text{s}}\alpha \\
& + 30935685000{\text{c}}\alpha \\
\end{aligned} \right)}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }} \right)} \\
& \times {\left( \begin{aligned}
& - 5894212500{\text{c}}\beta {\text{c}}\alpha {\text{s}}\beta + 10208776050{\text{c}}\beta ^{2} {\text{s}}\alpha {\text{c}}\alpha + 17861250000{\text{s}}\beta {\text{c}}\alpha \\
& - 30935685000{\text{s}}\alpha \\
\end{aligned} \right)} \\
\end{aligned} \right)} \\
\end{aligned} \hfill} \\
\end{array} $$