I Body rates from Euler angles....

AI Thread Summary
The discussion centers on the relationship between body rates [p, q, r] and Euler angle rates [φ_dot, θ_dot, ψ_dot]. The author struggles to derive this relationship, initially attempting to express body axes in terms of intermediate axes, but is corrected that x, y, z refer to positions, not rotation rates. The key insight is that the focus should be on expressing p, q, and r in terms of the Euler angles and their rates. A mathematical approach involving the angular momentum vector and the SO(3) matrix is suggested for clarity. The author seeks guidance on resolving their confusion regarding the derivation process.
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Referring to slides 3-4 (page 2) of this link: https://www.princeton.edu/~stengel/MAE331Lecture9.pdf

The author states the relationship between body rates [p q r] and Euler angle rates [φ_dot θ_dot ψ_dot]. I want to verify this but have been failing miserably...

My reasoning:
1) p, q, and r are angular rates about the body axes xB, yB, and zB.
2) φ, θ, and ψ are Euler angles about the intermediate axes x2, y1, and zI, respectively.

3) OK, so I just need to express xB, yB, and zB in terms of x2, y1, and zI, right?

4) p is trivial because xB already coincides with x2. Therefore, p = φdot.
5) When I do the same for yB, I get
yB = cosΦy1 + sinΦcosθzI + sinΦsinθx1.

This is awfully similar to the solution given, which is

q = cosΦ * θdot + sinΦcosθ * ψdot

What am I doing wrong? I've tried to find other sources online but they all just gloss over the derivation or present the result only.

Thank you in advance... I am completely at a loss and have been thinking unproductively about this for hours.[/SUB]
 
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I just saw the stickied thread about posting homework-type questions here... I hope this doesn't qualify as a homework-type question. Although it does involve debugging my thought process, it is also a conceptual question about why my reasoning is wrong. If this counts as a homework-type question, my apologies.
 
refrigerator said:
Referring to slides 3-4 (page 2) of this link: https://www.princeton.edu/~stengel/MAE331Lecture9.pdf

The author states the relationship between body rates [p q r] and Euler angle rates [φ_dot θ_dot ψ_dot]. I want to verify this but have been failing miserably...

My reasoning:
1) p, q, and r are angular rates about the body axes xB, yB, and zB.
2) φ, θ, and ψ are Euler angles about the intermediate axes x2, y1, and zI, respectively.

3) OK, so I just need to express xB, yB, and zB in terms of x2, y1, and zI, right?
No. x,y,z refer to positions in the axis systems, not rotation rates. If any x,y,z is in your answer, that is wrong. You want to express p, q, r in terms of φ, θ, ψ, φdot, θdot, and ψdot.
 
You can derive the angular momentum vector from
$$\omega_j=1/2 \epsilon_{jkl} (D^{-1} \dot{D})_{kl}=1/2 \epsilon_{jkl} (D^{\mathrm{T}}\dot{D})_{kl},$$
where ##D=D(\varphi,\vartheta,\psi)## is the SO(3) matrix in terms of Euler angles to be read as function of time, ##t##.
 

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