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I'm doing my homework on flight mechanics and the first lesson is about non-inertial reference frames. I've learnt there are three basic reference frames to be account in the study of flight mechanics. Body (b), wind (w) and local horizon (h).

  • Body-local horizon orientation. With three angles: $\psi$ (yaw), $\theta$ (pitch), $\phi$ (roll).
  • Wind-local horizon orientation. With three angles: $\chi$ (wind yaw), $\gamma$ (wind pitch), $\mu$ (wind roll).
  • Body-wind orientation. With two angles: $\alpha$ (angle of attack), $\beta$ (slip angle).

I have determined the matrix of conversion to convert any vector from one to another frame of reference by using Euler angles, which is:

$$L_{fi}=$$ $$\begin{pmatrix} \cos \delta_2 \cdot \cos \delta_3 & \cos \delta_2\cdot \sin\delta_3 & -\sin\delta_2 \\ \sin\delta_1\cdot \sin\delta_2\cdot \cos\delta_3 - \cos\delta_1\cdot \sin\delta_3 & \sin\delta_1\cdot \sin\delta_2\cdot \sin\delta_3 + \cos\delta_1\cdot \cos\delta_3 & \sin\delta_1\cdot \cos\delta_2\\ \cos\delta_1\cdot \sin\delta_2\cdot \cos\delta_3 + \sin\delta_1\cdot \sin\delta_3 & \cos\delta_1\cdot \sin\delta_2\cdot \sin\delta_3 - \sin\delta_1\cdot \cos\delta_3 & \cos\delta_1\cdot \cos\delta_2 \end{pmatrix}$$

Where the subindex ($fi$) indicates the final reference frame (f) and the initial reference frame (i). And the angles $\delta_1,\delta_2,\delta_3$ are to be replaced for the corresponding angles depending on our conversion.

In such a way, that if we have a vector $\textbf{A}_w$ and we want to get $\textbf{A}_h$ we'd do: $$\textbf{A}_h=L_{hw} \cdot \textbf{A}_w$$

But I have found a problem, and here it's when my question comes, that asks you to write the following:

$$\gamma = f(\theta,\phi,\alpha,\beta)$$

That is, writing the angle $\gamma$ as function of the angles $\theta, \phi, \alpha, \beta$.

How can I do this? I have been thinking of how to do it, but how can I combine two reference frames to express the angle of another reference frame? Should I use the matrix of transformation or should I use another method?

Note: I'm not asking to get my homework done, just obtain a hint on how to do it since I've been stuck for a good deal of time.

user3780731
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1 Answers1

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One approach that uses the matrices you've already derived is to set $L_{bh}=L_{bw}L_{wh}$ and then solve for $\gamma$.

Rick
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  • Thank you, I have performed the product $L_{bw}L_{wh}$ and I have equated with $L_{bh}$. In order to solve for $\gamma$, can I choose whatever row/column? I mean, will the results be equivalent? – user3780731 Sep 27 '15 at 12:02
  • They should be. If not, then the rotation matrices are incorrect. – Rick Sep 27 '15 at 13:21