Body rates from Euler angles....

[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 x_B, y_B, and z_B.
2) φ, θ, and ψ are Euler angles about the intermediate axes x₂, y₁, and z_I, respectively.

3) OK, so I just need to express x_B, y_B, and z_B in terms of x₂, y₁, and z_I, right?

4) p is trivial because x_B already coincides with x₂. Therefore, p = φdot.
5) When I do the same for y_B, I get
y_B = cosΦy₁ + sinΦcosθz_I + 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]

 

[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.

 

[FactChecker]

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 x_B, y_B, and z_B.
2) φ, θ, and ψ are Euler angles about the intermediate axes x₂, y₁, and z_I, respectively.

3) OK, so I just need to express x_B, y_B, and z_B in terms of x₂, y₁, and z_I, 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]

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, $t$.