# Body rates from Euler angles...

• I
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]

## Answers and Replies

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.

FactChecker
Science Advisor
Gold Member
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.

vanhees71
Science Advisor
Gold Member
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##.