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Martin Dieblich
31 lines
1.3 KiB
TeX
31 lines
1.3 KiB
TeX
The transformation from euler angles derivative to rates can be written as a matrix multiplication
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\begin{equation}
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\begin{pmatrix} p\\q\\r \end{pmatrix} =
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\begin{pmatrix}
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- \sin (\Roll) \dot{\Yaw} + \dot{\Roll} \\
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\sin (\Roll) \cos (\Pitch) \dot{\Yaw} + \cos (\Roll) \dot{\Pitch} \\
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\cos (\Roll) \cos (\Pitch) \dot{\Yaw} - \sin (\Roll) \dot{\Pitch}
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\end{pmatrix} \Leftrightarrow \begin{pmatrix} p\\q\\r \end{pmatrix} =
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\begin{pmatrix}
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1 & 0 & -\sin (\Roll) \\
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0 & \cos (\Roll) & \sin (\Roll) \cos (\Pitch) \\
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0 & -\sin (\Roll) & \cos (\Roll) \cos (\Pitch)
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\end{pmatrix} \multiplication \begin{pmatrix}
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\dot{\Roll} \\
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\dot{\Pitch} \\
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\dot{\Yaw}
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\end{pmatrix}.
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\end{equation}
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This can be solved easily to
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\begin{equation}
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\begin{pmatrix}\dot{\Roll} \\ \dot{\Pitch} \\ \dot{\Yaw} \end{pmatrix} =
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\begin{pmatrix}
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1 & \frac{ \sin^2 \Roll }{\cos \Pitch} & \frac{\sin \Roll \cos \Roll}{\cos \Pitch} \\
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0 & \cos \Roll & -\sin \Roll \\
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0 & \frac{\sin \Roll}{\cos \Pitch} & \frac{\cos \Roll}{\cos \Pitch}
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\end{pmatrix} \multiplication \begin{pmatrix} p\\q\\r \end{pmatrix}.
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\end{equation}
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Please note the singularity at the \emph{gimbal lock} ($\Pitch = \pm 90^{\circ }$)!
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\inHfile{INT32\_EULERS\_DOT\_OF\_RATES(ed, e, r)}{pprz\_algebra\_int}
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\inHfile{INT32\_EULERS\_DOT\_321\_OF\_RATES(ed, e, r)}{pprz\_algebra\_int}
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