Question

Problem 1. (15 points) Recall, the Eulerian angles that we defined in class as shown below. The axes (î, þ, â) are fixed in body frame B and the axes (Î, Ê, Â) are fixed in the inertial reference frame F. The orientation of B with respect to mathF is represented through the angles (0, 0, 4) 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. î 23 în 23 Ã z ñ3 Ø Frame F, Frame B (body-fixed) TV în 2 S. Frame F1 Frame F Y Frame F xoi (space-fixed) 1st Eulerian Angle în22 Frame F2 3rd Eulerian Angle Spin 0 în 22 2nd Eulerian Angle n21 Nutation ñ kov ñ Ky w ni Precession 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., ſ or î12, în23 or ġ, V. ñil or N21, 0 F F1 In this regard, F2 B 1. Write down the individual rotation matrices for each of the Eulerian angles, i.e. To, Te, Ty. 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 (în21, N22, ñ 23)

Answer 1