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## Homework Statement

a) Differentiate the following equation with respect to:

1) θ

2) Φ

3) ψ

(U

_{a}- U

_{b})' * 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.

## Homework Equations

derivative of:

sin x is cos x

cos x is -sin x

## 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:

(U

_{a}- U

_{b})' * (skew(C * r))

Where skew is the skew symmetric matrix

ie skew (x y z) =

[0 -z y]

[z 0 -x]

[-y x 0]

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