Question here
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^AR^B_{X' Y' Z'} ( \alpha, \beta, \gamma) = \left[
\begin{array}{ccc}
c \beta c \gamma & - c \beta s \gamma & s \beta \\
s \alpha s \beta c \gamma + c \alpha s \gamma & - s \alpha s \beta s \gamma + c \alpha c \gamma & - s \alpha c \beta \\
- c \alpha s \beta c \gamma + s \alpha s \gamma & c \alpha s \beta s \gamma + s \alpha c \gamma & c \alpha c \beta
\end{array}
\right] |
Using the examples from class as a primer, and using the atan2 function, please derive the formulas necessary to extract the values of angles , , and from the above rotation matrix. For simplicity, you may assume that no angles equal either 0° or 90°. Please express your answers in terms of elements of the rotation matrix (Frame A and point lie on the plane :
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Note that unit vector | Mathinline |
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| body | --uriencoded--z_%7B\mathrm%7BA%7D%7D |
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is pointing out of the screen. The coordinates of point are (2.1, 0.5, 0) when expressed in frame A. Frame B lies in the same plane and has the orientation shown below:
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(a) Find the rotation matrix | Mathinline |
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| body | --uriencoded--%7B%7D%5e%7B\mathrm%7BA%7D%7D\!R_%7B\mathrm%7BB%7D%7D |
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that relates the orientation of frames A and B. Use any strategy to check your answer and explain why it is correct.
(b) For part (b) only, suppose the origins of frames A and B are coincident. Use your answer from part (a) and the expression | Mathinline |
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| body | --uriencoded--p_%7B\mathrm%7BA%7D%7D = %7B%7D%5e%7B\mathrm%7BA%7D%7D\!R_%7B\mathrm%7BB%7D%7D \ p_%7B\mathrm%7BB%7D%7D |
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to compute | Mathinline |
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| body | --uriencoded--p_%7B\mathrm%7BB%7D%7D |
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. Recall that rotation matrices are orthogonal, which means that | Mathinline |
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| body | --uriencoded--\left( %7B%7D%5e%7B\mathrm%7BA%7D%7D\!R_%7B\mathrm%7BB%7D%7D \right)%5e%7B-1%7D = \left( %7B%7D%5e%7B\mathrm%7BA%7D%7D\!R_%7B\mathrm%7BB%7D%7D \right)%5e%7B\operatorname%7BT%7D%7D |
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.
(c) Now suppose the origin of frame B is at | Mathinline |
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| body | --uriencoded--3.7 x_%7B\mathrm%7BA%7D%7D + 1.1 y_%7B\mathrm%7BA%7D%7D + 0 z_%7B\mathrm%7BA%7D%7D |
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relative to the origin of frame A. What is the transformation matrix | Mathinline |
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| body | --uriencoded--r_%7B11%7D |
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, | %7B%7D%5e%7B\mathrm%7BA%7D%7DT_%7B\mathrm%7BB%7D%7D |
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that relates frames A and B?
(d) Use your answer from part (c) and the expression below to compute | Mathinline |
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| body | --uriencoded--r_%7B23%7D |
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, etc.) and clearly state any assumptions that you make:
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\begin{Bmatrix}p_{\mathrm{A}}\\1\end{Bmatrix} = {}^{\mathrm{A}}T_{\mathrm{B}} \begin{Bmatrix}p_{\mathrm{B}}\\1\end{Bmatrix} |
(e) Sketch frame A, frame B, and point to confirm that your answer to part (d) is correct.
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| title | Solution (only visible by instructors; please contact us to request access) |
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| S7.1 Extracting Euler angles from a rotation matrixS7.1 Extracting Euler angles from a rotation matrix3 Introduction to transformation matrices |
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| S7.3 Introduction to transformation matrices |
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