Analysis and simulation for a parallel drill point grinder

Abstract

To meet the quality and flexibility requirements of a hole-making process and to overcome the limitations of traditional manual grinding and cam-type automatic grinding in new drill point geometry development, a three degrees of freedom (3-DOF) parallel-mechanism-based drill point grinder is proposed in this paper. With the consideration of different inclination angle configurations of the grinder guideway, the inverse kinematics and the workspace analysis of the parallel grinder are developed; the relationship between the workspace and the structural parameters of the grinder, such as installation pattern, radius of joint distribution circle, length of the links etc. is discussed in detail. A numerical–symbolic method and a net search method are employed in the singularity analysis. The analysis of the kinematics, workspace and singularity provides the basis for understanding the characteristics of the new parallel grinder and also helps in the structural design of the proposed grinder.

Appendices

sα=sin α, cα=cos α, sβ=sin β, cβ=cos β, k=sin θ, m=cos θ

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} $$