Problem 1. (15 points)

Recall, the Eulerian angles that we defined in class as shown below. The axes (i.j. k) are fixed in

body frame B and the axes (Î, Ĵ, K) are fixed in the inertial reference frame F. The orientation

of B with respect to mathF is represented through the angles (o, 0, v) using a sequence about

intermediate z, intermediate y and intermediate z-axis again to obtain body-fixed frame B in the

final configuration, from the inertial reference frame F. This is often referred to as 3-2-3 sequence

or z-y-z sequence.

X

Z M13

Frame F₁

M₁2

Frame F

(space-fixed)

1st Eulerian Angle

Precession

jor 12,

F

In this regard,

F₁

Frame F₂

11 or 21, 0

M13

F₂

Frame F₁

M₂2

2nd Eulerian Angle

n21 Nutation

M₁2

1₂1

You are hired as an control and navigation engineer at a satellite manufacturing firm. The

satellite is equipped with thrusters which can rotate it about all the possible axes in the intermediate

body-fixed frame at any given instant instead of just the y and z-axes. Now, there is a requirement

to use a 2-1-3 sequence or the y-x-z sequence of successive rotations about the intermediate axes,

i.e.,

23 or

or 3, v

B

123

k

Frame B

(body-fixed)

Ĵ

ܕܕܐ

Frame F₂

3rd Eulerian Angle

î Spin

1. Write down the individual rotation matrices for each of the Eulerian angles, i.e. To. To, T.

2. Write the simplest angular velocity vector wB/F in terms of Euler angles according to the

2-1-3 rotation sequence. Remember, the simplest expression is always written using basis

vectors of intermediate reference frames.

3. Express the angular velocity vector wB/F in the second intermediate reference frame, F2, i.e.

in terms of (21, 22, 23)