Modeling & Analysis of the International Space Station

Transcription

1 Modeling & Analysis of the International Space Station 1

2 Physical Syste Solar Alpha Rotary Joints

3 Physical Syste Rotor Stator Gear Train Solar Array Inboard Body Outboard Body +x Solar Array 3

4 Physical Syste SARJ allows controlled rotation about the x-axis and constrains the other 5 DOF. Mass oent of inertia of the inboard body is significantly greater than the outboard body, so assue that the inboard body is echanically grounded. Interediate gear train inertia is ll copared to the rotor inertia. Torsional flexibility of the gear train is iportant as is the torsional stiffness of the solar arrays. All syste paraeters are approxiately constant over tie. Bearing friction and gear-train backlash can be neglected. Environent noise can be neglected except for plue loads. 4

5 Motivation Boss ys design a syste to keep the solar arrays pointed towards the sun as the station orbits the earth! Given: physical diensions of the station otor drive train How do you do that? What do you do first? 5

6 What Are the Steps You Take? Understand the Physical Syste Predict the Dynaic Behavior Develop a Physical Model Develop a Matheatical Model 6

7 Physical Model R V in + - i + e b J Stator Rotor q - Stator echanically grounded N:1 Gear Ratio K 1 K B 1 J J B q q q T d +x 7

8 Paraeters in Physical Model Sybol Meaning Units V in input voltage to arature volts i arature current aps R otor resistance ohs otor inductance henrys e b back ef voltage volts K t otor torque constant ft-lb/ap K e back ef constant volts-sec/ra N gear ratio unitless J otor inertia slug-ft θ otor position rad θ otor position reflected through gear train rad J outboard truss inertia slug-ft θ outboard body position rad J solar array inertia slug-ft θ solar array position rad K 1 gear train and truss torsional stiffness ft-lb/rad B 1 gear train and truss torsional daping ft-lb-s/rad K solar array torsional stiffness ft-lb/rad B solar array torsional daping ft-lb-s/rad T d plue disturbance torque ft-lb T otor torque ft-lb T otor torque reflected through gear train ft-lb 8

9 Siplifying Assuptions (a) one degree-of-freedo otion for each body (b) inboard inertia >> outboard inertia (c) gear train inertia << otor rotor inertia (d) luped eleents (e) neglect gear backlash (f) neglect bearing friction (g) constant paraeters (h) neglect all noise except plue load 9

10 Matheatical Siplifications (a), (b), (c): reduces the nuber and coplexity of the differential equations (d): leads to ordinary differential equations (e), (f): akes equations linear (g): leads to constant coefficients in the differential equations (h): avoids statistical treatent 10

11 Gearing up for the Math Model What are the constitutive physical relations needed? What are the equilibriu relations? What are the copatibility relations? What types of equations do we expect fro this syste? linear vs. nonlinear ODE vs. PDE What are the inputs to the syste? 11

12 Constitutive Physical Relations: T = J θ T = B T = Kθ V = ir V = di dt Equilibriu Relations: Kirchhoff s Current aw Newton s Second law Copatibility Relations: Kirchhoff s Voltage aw 1

13 Free Body Diagras b B θ θ g J θ a K θ θ f J +x b g a f T d J θ + B + K θ θ T = d 0 +θ 13

14 b B θ θ 1 g J θ b B θ θ g a f K 1 θ θ J +x a K θ θ f +θ b g a f b g a f J θ + B + K θ θ + B + K θ θ =

15 Gear Train Relations: N 1 θ θ N = N 1 T N T = 1 N N 1 N T θ N T θ 15

16 J θ K 1 a θ θ f 1 N T = K i t J +x B 1 b g 1 +θ N J b g a f B 1 N K 1 θ + θ θ + θ θ N K i 1 1 = t 0 16

17 Circuit Diagra R i + + V e in b eb = Ke θ - - V Ri di in = + + Ke θ dt 17

18 Matheatical Model J θ + B + K θ θ + B + K θ θ = J b g a f b g a f b g a f b g a f J θ + B + K θ θ T = 0 d 1 1 B 1 N K 1 θ + θ θ + θ θ N K i 1 1 = 0 t Ri di + + K = V dt e in 0 18

19 Since θ = Nθ = N = N b g a f b g a f b g a f b g a f J θ + B + K θ θ T = 0 d J θ + B + K θ θ + B + K θ θ = 1 1 J N θ + B + K θ θ NK i = t 0 Ri di + + NK = V dt e in 19

20 Matab / Siulink Block Diagra R/ 1/(N^*J) 1/s 1/s theta prie Gain1 Su1 Gain4 Integrator1 Integrator To Workspace Input: Vin 1/ Gain Su 1/s Integrator N*Kt Gain3 K1 Gain5 Su4 N*Ke/ Gain B1 Gain6 Su5 1/J 1/s 1/s theta_ Su Gain7 Integrator3 Integrator4 To Workspace1 K Gain8 Su6 B Gain9 Su7 1/J 1/s 1/s theta_ Input:Td Su3 Gain10 Integrator5 Integrator6 To Workspace 0

21 One More Approxiation In practice, a servo aplifier operating in torque ode is used to run the DC otor. This copentes for the back-ef effect by using current feedback. It generates a current proportional to applied voltage. The otor odel can be siplified to T = K t i where i = K ap V in and the otor current i now is an input rather than a state variable. Now, we need to solve the linear ODE s so we can predict the behavior of the actual syste...how do we do this? 1

22 State-Space Equations NM θ θ θ O QP = NM K1 K1 B1 B1 0 0 N J N J N J N J K1 K1 K K B1 B1 B B J J J J J J K K B B 0 0 J J J J Note: current i is an input since i = K ap V in NM y y y 1 3 O QP = NM O QP NM θ θ O M N QP O θ θ θ θ + NM QP x = Ax + Bu y = Cx + Du O + P Q NM NK t 0 N J J O NM QP i T d O QP O NM QP i T d O QP

23 Paraeter Nuerical Values Sybol Value Units i input aps K t 8.6 ft-lb/ap N 83 unitless J 9.4E-3 slug-ft θ output radians J 400 slug-ft θ output radians J 7.0E5 slug-ft θ output radians K 1 9.8E6 ft-lb/radian B 1 49 ft-lb-s/radian K.76E5 ft-lb/radian B 4.40E3 ft-lb-s/radian T d input ft-lb 3

24 Matheatical Analysis Poles: 0, ± 15.8i (ω n = rad/s, ζ = 0.11) ± i (ω n = rad/s, ζ = 0.04) Zeros: θ i θ i θ i ± i, -6.1 ± 154.4i (ω n = rad/s, rad/s) ± 0.679i (ω n = rad/s) None 4

25 Frequency Response Plots: Input i 0 Gain db -00 θ θ θ Frequency (rad/sec) 360 φ θ θ θ 5

26 Gain db θ i Frequency (rad/sec) 0 φ

27 Gain db θ i Frequency (rad/sec) 0 φ

28 Gain db θ i Frequency (rad/sec) 180 φ

29 Syste Behavior at w = 0.6 rad/s K J B θ J θ + B + K θ = 0 NM θ O NM QP = 0 K 1 B θ P N M O Natural Frequency: 0.6 rad/s J J θ Q P O P Q 9

30 Syste Behavior at w = 15.4 rad/s K J N +J J θ = θ B ( J N + J ) θ + B K B θ + θ = θ + Kθ J θ + B + K θ = B + K θ M N θ θ O P Q P = NM K B K B J J J J K B K B J N + J J N + J J N + J J N + J O QP NM θ θ θ O QP θ Natural Frequencies: 0, rad/s 30

31 Syste Behavior at w = rad/s K 1 K J N J θ B 1 θ B N J θ + B1 + K1θ = B1 + K1θ J θ + B + B + K + K θ = B + K θ NM θ θ O P Q P = a f a f O K1 B K B N J N J N J N J a f a f K1 B1 K + K1 B + B 1 NM J J J J QP NM θ θ O QP Natural Frequencies: 15.38, rad/s 31

32 Tie Response: i = cos(0.6t) θ Aplitude θ θ Tie (secs) 3

33 Tie Response: i = cos(15.39t) θ 0.0 Aplitude θ θ Tie (secs) 33

34 What did we do? Conclusion Understood physical syste Developed physical odel Developed atheatical odel Predicted dynaic behavior Matheatical odel based on justifiable approxiations! Siplify - reeber to verify your odel! Do the results ake sense? Garbage in = Garbage out Next step: Use the odel to design a control syste which will tisfy the boss! 34