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I Body rates from Euler angles...

  1. May 23, 2016 #1
    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]
  2. jcsd
  3. May 23, 2016 #2
    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.
  4. May 24, 2016 #3


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    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.
  5. May 24, 2016 #4


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    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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