Body rates from Euler angles....

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Discussion Overview

The discussion revolves around the relationship between body rates [p, q, r] and Euler angle rates [φ_dot, θ_dot, ψ_dot], as presented in a specific lecture document. Participants are exploring the derivation and understanding of these relationships, focusing on the conceptual and mathematical aspects involved.

Discussion Character

  • Conceptual clarification
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the relationship between body rates and Euler angles, attempting to derive expressions for p, q, and r based on their understanding of the axes involved.
  • Another participant corrects the first by stating that the variables x, y, z refer to positions in the axis systems rather than rotation rates, emphasizing the need to express p, q, r in terms of the angular rates φ_dot, θ_dot, and ψ_dot.
  • A later post introduces a formula for deriving the angular momentum vector, linking it to the SO(3) matrix in terms of Euler angles, but does not clarify its relevance to the initial confusion about body rates.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there is a clear disagreement regarding the interpretation of the variables involved in the derivation. The initial participant's reasoning is challenged, indicating uncertainty and differing perspectives on the correct approach.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the relationships between the axes and the rates, as well as the mathematical steps involved in the derivation. The discussion does not resolve these issues.

refrigerator
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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##.