# I Body rates from Euler angles...

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1. May 23, 2016

### refrigerator

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.

2. May 23, 2016

### refrigerator

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.

3. May 24, 2016

### FactChecker

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.

4. May 24, 2016

### vanhees71

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, $t$.