ASEN 3200 Attitude Dynamics - Final ExamNAMEASEN 3200 Orbital Mechanics and Attitude Dynamics Spring 2005ASEN 3200 Attitude Dynamics - Final ExamApril 30, 2005NAME On my honor, as a University of Colorado at Boulder student, I have neither given nor received unauthorized assistance on this work. SIGNATURE Closed book, Open notesPlease show your work clearly. Collaboration of any kind on this exam, including copying from or giving answers to another student, constitutes cheating and is not permitted under any circumstances.There are 4 questions on the exam with several parts each. The first three are worth 30 pts each, the fourth is worth 10 pts.Axelrad 1ASEN 3200 Orbital Mechanics and Attitude Dynamics Spring 20051) A spacecraft is in a circular orbit around the Earth with a radius of 7200 km (E=398600 km3/s2). At time t0, the quaternion that describes the orientation of the RIC (Radial, in-track, cross-track) frame with respect to the inertial frame I, is given by 0.3746000.9271IRICq� �� �� �=� �� �� � a) Compute the transformation matrix from I to RIC: [ ]IRICR =b) Compute the position and velocity vectors of the satellite expressed in the I-frame at this time. c) If the satellite is controlled to be nadir-pointing at all times, what is the angular velocity vector of the body-fixed frame with respect to the I frame in RIC coordinates [ ]/B IRICwr? Axelrad 230 ptsASEN 3200 Orbital Mechanics and Attitude Dynamics Spring 2005PROBLEM 2. - A solid cylindrical object of mass m=1 kg, radius R=0.5 m, and thickness t=0.1 m is spinning about its axis of symmetry and rotating (coning or precessing) about a fixed axis. The cylinder is in space with no forces or moments acting on it. At the instant shown, the axis of symmetry of the cylinder is observed to be coning abouta fixed axis at a fixed angle of 35 deg with a period of 0.2 s. a) Sketch the principal axes of the cylinder on the figure, labeling the axis of symmetry 3ˆp. Is 3ˆp the major, minor, or intermediate axis?b) Compute the inertia matrix of the cylinder about its center of mass, in the P-frame [IG]P = c) Calculate the angular velocity vector expressed in the P-frame [ ]/P IPwr for the cylinder at the instant shown. d) Calculate the angular momentum vector expressed in the P-frame /P IPH� �� �r for the cylinder at the instant shown. Does its magnitude and/or direction change with time? If yes, how and why?Axelrad 330 ptsASEN 3200 Orbital Mechanics and Attitude Dynamics Spring 2005e) Compute the inertia matrix for the same cylinder about point O in the C-frame as shown in the figure below. Point O is a distance 0.4 m away from the center of the cylinder. The angle between the axis of symmetry of the cylinder and 3ˆc is 30 deg, and the line connecting point O to the center of the cylinder lies in the plane of2ˆc and 3ˆc as shown. [IO]C = Axelrad 4O3ˆc2ˆc1ˆcASEN 3200 Orbital Mechanics and Attitude Dynamics Spring 2005PROBLEM 3. The pitch motion of a satellite is described by the following block diagram where A=5kg-m2, C=2 Nm/(rad/s), K=4 Nm/rad. Assume that the initial conditions are all zero.a) Find the transfer function from the input moment (s) to the output angle (s). Compute the transfer function poles and show their locations in the s-plane.b) Compute the overshoot, 5% settling time, and period of oscillations for this system.Axelrad 530 ptsMq+_+_C s21AsKASEN 3200 Orbital Mechanics and Attitude Dynamics Spring 2005c) To avoid overshoot, we will add a control moment c dM K q=-&. What is the minimum value of Kd required to produce zero overshoot? Compute the new transfer function poles and show their locations in the s-plane. Axelrad 6ASEN 3200 Orbital Mechanics and Attitude Dynamics Spring 20054) Math Questions a) Circle the equations below that represent an undamped harmonic oscillator( ) ( )2 22 0, 4 3, 7 4sin 3 , 3.5 2, cos 4ds X s X s y y x x t y tdtqq+ = =- + - = + = =&& && & &b) For unit vectors ˆaand ˆb and be shown below, compute the following:If your answer is a vector, please express it in terms of the I-frame shown in the figure with 3ˆi out of the paper.The dot product ˆˆ a b =g The cross product ˆˆ a b� = c) Circle all the matrices below that could represent an attitude or transformation matrix:1 0 0 1 0 1 0 0 1 0 0 10 0 1 0 0 1 1 0 0 cos sin 00 1 0 0 1 0 0 1 0 sin cos 0ff- -� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �- - -� � � � � � � �Axelrad 710 pts60 deg15