If the direct cosine matrix for the transformation from xyz to $x^{′} y^{ ′} z^{′}$ is the same as it was in Example 4.5,

$[ Q] = \begin{bmatrix} 0.64050 & 0.75319 & -0.15038 \\ 0.76736 & – 0.63531 & 0.086824 \\ -0.030154 & -0.17101 & -0.98481 \end{bmatrix}$

find the angles α , β and γ of the yaw, pitch, and roll sequence

Question

If the direct cosine matrix for the transformation from xyz to [latex] x^{′} y^{ ′} z^{′} [/latex] is the same as it was in Example 4.5,

[latex] [ Q] = \begin{bmatrix} 0.64050 & 0.75319 & -0.15038 \ 0.76736 & - 0.63531 & 0.086824 \ -0.030154 & -0.17101 & -0.98481 \end{bmatrix} [/latex]

find the angles α , β and γ of the yaw, pitch, and roll sequence

Accepted Answer

Use Algorithm 4.4.

Step 1:

[latex]\alpha= an^{-1}\!\!\frac{Q_{12}}{Q_{11}}= an^{-1}\!\left(\frac{0.75319}{0.64050} ight)[/latex]

Since both the numerator and the denominator are positive, α must lie in the first quadrant. Thus ,

[latex] an^{-1}\left({\frac{0.75319}{0.64050}} ight)= an^{-1}1.1759=49.62°[/latex]

Step 2:

[latex]\beta=\sin^{-1}(-Q_{13})=\sin^{-1}[-(-0.15038)]=\sin^{-1}(0.15038)=8.649°[/latex]

Step 3:

[latex]\gamma= an^{-1}\!\!{\frac{Q_{23}}{Q_{33}}}= an^{-1}\left({\frac{0.086824}{-0.98481}} ight)[/latex]

The numerator is positive and the denominator is negative, so γ lies in the second quadrant,

[latex] an^{-1}\left({\frac{0.086824}{-0.98481}} ight)= an^{-1}\left(-0.088163 ight)=-5.0383^{\circ}\Rightarrow γ = 174.96°[/latex]

Question 4.6: If the direct cosine matrix for the transformation from xyz ......