MAE 142 Homework #5 Due Friday, March 13, 2009

Transcription

1 MAE 142 Homework #5 Due Friday, March 13, 2009 Please read through the entire homework set before beginning. Also, please label clearly your answers and summarize your findings as concisely as possible. The problems are designed to be solved. Therefore, please do not hand in pages and pages of computer printouts without an answer. Figure 1: Spacecraft Coordinate Frame The body frame coordinates of the spacecraft are shown above. The roll axis (x 1 ) is nominally in the direction of the orbit s tangential linear velocity. The yaw axis (x 3 ) or nadir is always pointing toward the Earth s surface. This implies that the spacecraft is spinning about the pitch axis (x 2 ) with the angular velocity of ω o to maintain nadir pointing. If the Earth s equator lies within the spacecraft s orbital plane, then the pitch axis would be pointing in the same direction as the Earth s south pole. If there are no disturbances and the spacecraft spins purely about pitch, the spacecraft body frame will be perfectly aligned with the inertial frame once per orbit. 1

2 1. Equations of Motion (20pts) (a) Recall the methodology of the second problem in the first homework assignment. Find the direction cosine matrix for an Euler rotation sequence. Find the relationship between the angular velocities in body axis coordinates and the Euler angular velocities (i.e. Ω = [ ] θ). (b) The equations of motion for a rigid spacecraft with n reaction wheels can be written directly from Euler s equations: J Ω + ḣ + Ω (JΩ + h) = τ dist where Ω IR 3 is the angular velocity in body frame coordinates, h IR 3 is the cumulative momentum of the reaction wheel cluster, and J = diag (J 1, J 2, J 3 ). As we are interested in controlling the body through the relative torque of the reaction wheels against the body, we can also write the equations in this form: J Ω + Ω JΩ = u + τ dist ḣ + Ω h = u Substitute the Euler angles and angular velocities for the body frame angular velocities (Ω) in the equations above. Linearize the system about θ 1 = 0, θ 2 = 0, θ 3 = 0, θ1 = 0, θ2 = ω o, θ3 = 0 in a similar fashion to the midterm/homework #3. You may have to make small angle approximations as well. You should get a system of equations in the following form: a 1 θ 1 + a 2 θ 1 + a 3 θ 3 = u 1 + τ dist1 a 4 θ 2 + a 5 θ 2 = u 2 + τ dist2 a 6 θ 3 + a 7 θ 3 + a 8 θ 1 = u 3 + τ dist3 ḣ 1 + a 9 h 3 = u 1 ḣ 2 = u 2 ḣ 3 + a 10 h 1 = u 3 We can now append some appendages to our linearized model through hybrid modal coordinates so that our final system is: 2

3 a 1 θ 1 + a 2 θ 1 + a 3 θ 3 + 2σ 1 q 1 = u 1 + τ dist1 a 4 θ 2 + a 5 θ 2 + 2σ 2 q 2 = u 2 + τ dist2 a 6 θ 3 + a 7 θ 3 + a 8 θ 1 + 2σ 3 q 3 = u 3 + τ dist3 ḣ 1 + a 9 h 3 = u 1 ḣ 2 = u 2 ḣ 3 + a 10 h 1 = u 3 q 1 + ω 2 1q 1 + 2σ 1 θ 1 = 0 q 2 + ω 2 2q 2 + 2σ 2 θ 2 = 0 q 3 + ω 2 3q 3 + 2σ 3 θ 3 = 0 This system is usually referred to as a linear flex model in industry. Notice that the structural modes have no damping. Aerospace structures are usually very stiff with little damping, especially structures on spacecraft as they are mostly carbon-fiber laminates. Therefore, damping is usually assumed to be 1% or neglected alltogether. For clarity, we will now drop the primed notation for the Euler angles in the rest of the homework. (c) Bonus points (+5): If the orbital angular velocity (ω o ) of the spacecraft is equal to rad/sec, what can you quantitatively say about the ephemeris of the spacecraft? 3

4 2. Classical Control (35pts) Note that the reaction wheel momentum is decoupled from the spacecraft body. This was done on purpose so that the controls design would be simplified as we can now ignore the momentum terms. (However, in reality momentum management schemes are needed so that the wheel momentum buildup is minimized and the scheme must also account for wheel momentum s effective stiffening of the inertia tensor.) Notice that in the equations of motion the pitch (θ 2 ) is decoupled from the roll (θ 1 ) and yaw (θ 3 ). For the aircraft aficionados, this might look like a familiar result. The pitch equation is similar to the short period longitudinal mode. The phugoid mode is missing as the air is rather thin at our spacecraft s altitude and lift is negligible. However, another new mode is now present in the longitudinal equation that is a consequence of the spinning body. Like the aircraft, the lateral roll/yaw equations are coupled a la the Dutch roll mode. (a) Using the Laplace transform, combine the pitch structural mode equation with the pitch equation and find the transfer function: G p (s) = Θ 2(s) U 2 (s) (b) Plug in the parameters that are in the appendix. Is G p (s) stable? Look at the Bode plot for G p (s) in the frequency range of to 50 rad/sec. What do each of the features represent? (c) Using classical loop shaping (lead/lag/notch/etc.), design a controller that meets the following criteria: i. Stability: 6dB GM and 45deg PM ii. Tracking Performance: Step response rise time 1.5sec, max overshoot 1%, and settling time 2.5sec iii. Disturbance Rejection: Maximum excursion of θ 2 1rad due to a unit impulse at the input (d) Write down your final control law, clearly mark the stability margins, and plot the step and impulse responses. 4

5 3. Modern Control (35pts) As a spacecraft orbits the Earth, the line of sight to the sun is blocked once per orbit (depending on season). These eclipses cause the spacecraft to undergo drastic and rapid thermal changes. A control problem that arises from this is called the thermal snap of the solar arrays. As each solar panel is a laminate of multiple materials with different coefficients of thermal expansion, the solar array warps as the different materials cool at different rates during eclipse. Once the spacecraft is out of eclipse, the array snaps back as it quickly warms. The control problem is to maintain accurate pointing during this phenomenon. (a) Arrange the coupled roll/yaw equations into state-space form where the two inputs are u 1, u 2 and the two outputs are θ 1, θ 3. Hint, recall from your Diff Eq/FEM/vibrations classes that: M and for ẋ = Ax + Bu: θ 1 θ 3 q 1 q 3 + C θ 1 θ 3 q 1 q 3 + K θ 1 θ 3 q 1 q 3 [ ] 0 I A = M 1 K M 1 C 0 [ 4 2 ] B = M 1 I = x = ( ) T θ 1 θ 3 q 1 q 3 θ1 θ3 q 1 q 3 ( ) u1 u = u 3 (b) We can model the thermal snap as a unit impulse disturbance to the spacecraft roll/yaw inputs. Design a LQR full state feedback controller for the roll/yaw equations that meets the following criteria: i. θ 1 3milliradians, θ 3 5milliradians in response to a unit impulse applied simultaneously to u 1 and u 2. ii. θ 1 0.1milliradians, θ 3 0.1milliradians within 10 seconds in response to a unit impulse applied simultaneously to u 1 and u 2. iii. All modes of the closed loop system (A + BK) should be less than 10 Hz (magnitudes of all eigenvalues less than 20πrad/sec) iv. Minimize the wiggles that result from the structural flexibility. u 1 u

6 You will have to dial in the Q x and Q u to meet the performance requirements. Some good initial guesses for Q x and Q u are: Q x = 1/x 2 1 max /x 2 2 max 0 0 Q u = /x 2 8 [ ] max 1/u 2 1max 0 0 1/u 2 2 max where x 1max is approximately the maximum excursion in state x 1 expected in the system response to a unit impulse input and similarly u 1max is the maximum control effort expected for u 1. (c) Write down your final control law and plot the impulse responses. 6

7 4. Systems Engineering (10pts) The analysis so far as been strictly in the continuous time domain. In reality, the actual plant dynamics are in the continuous time domain and the control law is implemented digitally based on feedback from digital sensors. For the problems above, the sensor necessary for feedback would be an IMU (inertial measuring unit), which is essentially a 3-axis angular rate gyroscope. Recall that the response of a gyroscope is similar to a first order low pass filter. To meet all of the requirements of the controllers above, which IMU from the choices below is best suited for the application and why? What are the advantages/disadvantages of each IMU? (a) IMU 1: 120 Hz bandwidth, 1 khz sampling rate, milliradians RMS noise, $1 million (b) IMU 2: 50 Hz bandwidth, 100 Hz sampling rate, 0.01 milliradians RMS noise, $100,000 (c) IMU 3: 50 Hz bandwidth, 50 Hz sampling rate, 0.02 milliradians RMS noise, $75,000 (d) IMU 4: 1 khz bandwidth, 10 khz sampling rate, 100 milliradians RMS noise, $1000 (Note: The gyro output is actually an angular rate. However, the RMS noise listed is the residual noise in the angle estimate after going through an attitude estimator/kalman filter) 7

8 Parameters J 1 = 110kg m 2 J 2 = 10kg m 2 J 3 = 100kg m 2 ω o = rad/sec ω 1 = 13rad/sec ω 2 = 10rad/sec ω 3 = 8rad/sec σ 1 = 0.1 kg m 2 σ 2 = 0.01 kg m 2 σ 3 = 0.1 kg m 2 If you can not obtain the expression in problem 1, use the following values for the a i parameters: a 1 = 110 a 2 = e 4 a 3 = a 4 = 10 a 5 = e 5 a 6 = 100 a 7 = e 4 a 8 = a 9 = a 10 =