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Differentiate trigonometric equation

  1. Jun 4, 2016 #1
    1. The problem statement, all variables and given/known data
    a) Differentiate the following equation with respect to:
    1) θ
    2) Φ
    3) ψ

    (Ua - Ub)' * C * r
    where:

    C is a 3 x 3 rotation matrix:
    [ cos θ cos ψ, -cos Φ sin ψ + sin Φ sin θ cos ψ, sin Φ sin ψ + cos Φ sin θ cos ψ]
    [ cos θ sin ψ, cos Φ cos ψ + sin Φ sin θ sin ψ, -sin Φ cos ψ + cos Φ sin θ sin ψ]
    [ -sin θ, sin Φ cos θ, cos Φ cos θ ]

    Ua is a 3x1 column vector:
    [Ua_x]
    [Ua_y]
    [Ua_z]
    Ub is a 3x1 column vector:
    [Ub_x]
    [Ub_y]
    [Ub_z]
    r is a 3 x 1 column vector:
    [r_x]
    [r_y]
    [r_z]
    ' means transpose

    Show your working out.

    2. Relevant equations
    derivative of:
    sin x is cos x
    cos x is -sin x

    3. The attempt at a solution
    Let:
    θ = theta
    Φ = phi
    ψ = psi

    Expanding:
    r_z*((Ua_x - Ub_x)*(sin(phi)*sin(psi) + cos(phi)*cos(psi)*sin(theta)) - (Ua_y - Ub_y)*(cos(psi)*sin(phi) - cos(phi)*sin(psi)*sin(theta)) + cos(phi)*cos(theta)*(Ua_z - Ub_z)) + r_y*((Ua_y - Ub_y)*(cos(phi)*cos(psi) + sin(phi)*sin(psi)*sin(theta)) - (Ua_x - Ub_x)*(cos(phi)*sin(psi) - cos(psi)*sin(phi)*sin(theta)) + cos(theta)*sin(phi)*(Ua_z - Ub_z)) + r_x*(cos(psi)*cos(theta)*(Ua_x - Ub_x) - sin(theta)*(Ua_z - Ub_z) + cos(theta)*sin(psi)*(Ua_y - Ub_y))


    Answer in book:
    (Ua - Ub)' * (skew(C * r))
    Where skew is the skew symmetric matrix
    ie skew (x y z) =
    [0 -z y]
    [z 0 -x]
    [-y x 0]
     
    Last edited: Jun 4, 2016
  2. jcsd
  3. Jun 5, 2016 #2

    Mark44

    Staff: Mentor

    The above is not an equation -- no = in it.
    The problem isn't clear to me. From your problem statement, are you supposed to find three separate derivatives or are you supposed to find the derivative with respect to θ, and then differentiate that function with respect to Φ, and then, finally, differentiate that function with respect to ψ?

    As I understand this problem, expanding the original expression as you did makes things worse. ##U_a## and ##U_b## don't involve θ, Φ, or ψ, so ##(U_a - U_b)^T## can be treated as a constant, using the differentiation rule ##\frac d {dx}(k f(x)) = k \frac d {dx} f(x)##. Also, r doesn't appear to involve θ, Φ, or ψ, so differentiating Cr would be a lot simpler.
     
  4. Jun 5, 2016 #3
    Hi,

    Sorry for not being specific enough. You can write
    y = (Ua - Ub)' * C * r;
    then find
    1) ∂y/∂θ
    2) ∂y/∂Φ
    3) ∂y/∂ψ
    so find three separate derivatives.
    Yes I see that (Ua - Ub)' is constant.
     
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