Spacecraft attitude Determination and control

Spacecraft Attitude Control, spacecraft attitude control using magnetic actuators, autonomous spacecraft attitude control using magnetic torquing only, pdf free download
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Dr.TomHunt,United States,Teacher
Published Date:23-07-2017
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Spacecraft Attitude Control Space System Design, MAE 342, Princeton University Robert Stengel • More on Rotation Matrices • Direction cosine matrix • Quaternions • Yo-yo De-Spin • Continuously Variable Torque Controllers • On/Off-Torque Controllers Copyright 2016 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/stengel/MAE342.html 1 Attitude Control System Fortescue 2UARS Attitude Control System 3 Spacecraft Attitude Control Inputs • On-Board Sensors – Inertial Measurements • Accelerometers • Angle sensors • Angular-rate sensors – Optical Sensors • Star sensors • Sun sensors • Horizon sensors • Off-Board Observations – Ground-Based Tracking • Radar • Navigation beacons (VOR/DME, LORAN, …) – Spaced-Based Tracking • GPS, GLONASS, … 4Potential Accuracies of External Attitude Measurements Fortescue 5 Spacecraft Attitude Control Outputs • Continuous Control Torques –Control Moment/Reaction Wheel Gyros –Magnetic Torquers –Solar Panels • Pulsed Control Torques –Reaction Control Thrusters (RCS) • One-Shot Devices –RCS Spin-up –Yo-Yo De-Spin 6Spacecraft Attitude Disturbances • External Torques – Solar radiation pressure – Gravity gradient – Magnetic fields – Aerodynamics – Can be put to good use if related to attitude control objectives • Vehicle-Based Torques – Mass movement – Elasticity – Out-gassing 7 More on Rotation Matrices and Quaternions 8Direction Cosine Matrix Cosines of angles between each I axis and each B axis Projections of vector components in one frame on the other "% cos cos cos 11 21 31 ' B H = cos cos cos ' I 12 22 32 ' cos cos cos 13 23 33 & B r =H r B I I 9 Euler’s Rotation Formula Angular orientation of one axis system, B, with respect to another, I Vector transformation B r =H r B I I T T " = a r a+ r a r a cos&+sin& r'a ( ) ( ) ( ) I I I I % T = cos&r + 1cos& a r asin& a'r ( ) ( ) ( ) I I I a 1 & a a & 2 & a 3 "% 10Euler’s Formula B T r =H r = cosr + 1"cos a r a"sin ar ( ) ( ) ( ) B I I I I I Identity T T a r a = aa r ( ) ( ) I I Rotation matrix B T H = cosI +(1"cos)aa"sina I 3 11 Quaternion Derived from Euler Rotation Angle and Orientation Quaternion vector 4 parameters based on Euler’s formula (+ a q 1 1 & & - & a a sin(' 2) q 2 &' 2- && q = = &- a q q & 3& 3), 4 &"% & q & 4 cos' 2 ( ) "%& "% Not singular at = ±90° 4-parameter representation of 3 parameters; hence, a constraint must be satisfied T 2 2 2 2 q q=q +q +q +q 1 2 3 4 2 2 =sin 2 +cos 2 =1 ( ) ( ) 12Rotation Matrix Expressed with Quaternion From Euler’s formula B 2 T T % H = q a a I +2a a2q a ( ) I 4"" 3"" 4" & Rotation matrix from quaternion B H = I 2 2 2 2 "% qqq +q 2 qq +q q 2 qqq q ( ) ( ) 1 2 3 4 1 2 3 4 1 3 2 4 ' 2 2 2 2 ' 2 qqq qq +qq +q 2 q q +qq ( ) ( ) 1 2 3 4 1 2 3 4 2 3 1 4 ' 2 2 2 2 2 qq +q q 2 q qqqqq +q +q ( ) ( )' 1 3 2 4 2 3 1 4 1 2 3 4 & 13 Quaternion Expressed from Elements of Rotation Matrix 1 q = 1+h +h +h 4 11 22 33 2 Assuming that q " 0 4 "% h(h "% ( ) q 23 32 1 ' ' 1 ' a = q = h(h ( ) 2' 31 13 4q ' 4 ' q h(h 3 ( ) ' 12 21 & & Pisacane, 2005 14Successive Rotations Expressed by Products of Quaternions and Rotation Matrices B q : Rotation from A to B A Rotation from Frame C A to Frame C through q : Rotation from B to C B Intermediate Frame B C q : Rotation from A to C A Matrix Multiplication Rule C C C C B B H q =H q H q ( ) ( ) ( ) A A B B A A Quaternion Multiplication Rule C C C B "% B C B "% q a + q a( a" a ( ) ( ) ( ) a 4A 4BA B A' B C C B ' q = = q q A B A ' T C B C B q ' 4 q q( a a ( ) ( )' & ( ) 4 4BA A B A & 15 Quaternion Vector Kinematics ODE is linear in both q and B "% " "% q(()(( a a 4 B B d 1 ' ' q= = T dt 2' q)(( a ' 4 B & & 0'('' q z y x q 1 1 & && (' 0'' & q q 1 z x y && 2 2 = & && q 2'(' 0' q 3 y x z 3 & && & q q && ('('(' 0 4 4 x y z "%"% "% B Pisacane, 2005 16Propagate Quaternion Vector 0' (t)(' (t)' (t) q t q t ( ) z y x ( ) 1 1 & && (' t 0' t' t& q t& ( ) ( ) ( ) q t& ( ) ( ) dq(t) 1 z x y 2 2 = =& && dt 2 q t' t(' t 0' t q t ( ) ( ) ( ) ( ) ( ) & 3& y x z 3& & && q t q t ( )(' (t)(' (t)(' (t) 0 ( ) 4 x y z 4 &&& "%"% "% B Digital integration to compute q(t ) k t k dq(") q t =q t + d" ( ) ( ) int k k1 dt t k1 Normalize q(t ) to enforce constraint k T q t =q t q t q t ( ) ( ) ( ) ( ) k int k int k int k 17 Quaternion Interface with Euler Angles • Quaternion and its kinematics unaffected by Euler angle convention B • Definition of H makes the connection I • Specify Euler angle convention (e.g., 1-2-3 or 3-1-3) ; for (1-2-3), B h h h 11 12 13 & B H = h h h I 21 22 23& & h h h 31 32 33 "% I cos'cos( cos'sin()sin' & =)cossin( +sinsin'cos( coscos( +sinsin'sin( sincos' & & sinsin( +cossin'cos()sincos( +cossin'sin( coscos' "% • Apply equations on earlier slide to find q(0) • Perform trigonometric inversions as indicated to generate "(t ), (t ),(t ) from q(t ) k k k k 18Yo-Yo De-Spin 19 Mars Odyssey Launch Phases Booster Separation Stage 2 Separation Stage 2 Ignition Heat Shield Separation Stage 3 Spinup Yo-Yo De-Spin 20Yo-Yo De-spin Kaplan • Satellite is initially spinning at rad/s z • Angular momentum and rotational energy of satellite plus expendable masses are conserved • Masses are released, moment of inertia increases, and angular velocity of satellite decreases • With proper cord length (independent of initial spin rate), satellite is de-spun to zero angular velocity 21 Yo-Yo De-spin Angular momentum R= spacecraft radius 2 2 % h = I +mR +" +" l= tether length ( ) z zz z z z & 2 mR + I zz c= 2 2 Rotational energy mR m= mass of 2 deployable objects 2 1 1 2 2 2 2 % I = satellite moment of inertia T = I + mR +" +" ( ) zz zz z z z & 2 2 = angle between split hinge and tangent point Simultaneous solution for final angular rate 2 2 & cR"l = =0 if l= R c final initial % 2 2( cR +l' Spaceloft 7 Sounding Rocket De-Spin https://www.youtube.com/watch?v=5ZqbjQ9ASc8 22Continuously Variable Torque Controllers 23 Overview of Control Single- or multi-axis interpretation 24Single-Axis “Classical” Control of Non-Spinning Spacecraft Pitching motion (about the y axis) is to be controlled I' I q t r t / I ( ) ( ) ( ) M t / I p(t) ( ) zz yy xx x xx & & & & q t = M (t) / I&' I' I p t r t / I ( )& ( ) ( ) ( ) y yy xx zz yy & && & r t ( ) M (t) / I I' I p t q t / I & z zz& ( ) ( ) ( ) yy xx zz "% "% & "% • For motion about the y axis q t = M t / I ( ) ( ) y yy only, this reduces to • Pitch angle equation(t)= q(t) 25 Single-Axis Angular Rate Control of Non-Spinning Spacecraft • Small angle and angular rate perturbations • Linear actuator, e.g., – Momentum wheel • Linear measurement, e.g., – Angular rate gyro Simplified Control Law (C = Control Gain) e(t)= q (t)q(t) c u(t)=Ce(t) 26Angular Rate Control t t t g Cg Cg A A A q(t)= u(t)dt = e(t)dt = q"q(t) dt c I I I yy yy yy 0 0 0 • I : moment of inertia yy • q(t): angular rate • q (t): desired angular rate c • g : actuator gain A • g u(t): control torque A 27 Step Response of Angular Rate Controller Stepinput : " 0, t 0 q (t)= c 1, t 0 % "% Cg A (+ t ' t I yy&.t (/+ - (+ q(t)= q 1e = q 1e = q 1e c c c ), - ), - ), • where "" = –Cg /I = eigenvalue or A yy root of the system (rad/s) – = I /Cg = time constant of yy A the response (s) 28Angle Control of the Spacecraft • Small angle and angular rate perturbations • Linear actuator, e.g., – Momentum wheel • Linear measurement, e.g., – Earth horizon sensor Angle Control Law (C = Control Gain) e(t)= (t)"(t) c u(t)=Ce(t) 29 Model of Dynamics and Angle Control nd • 2 -order ordinary differential equation 2 d(t) Cg A =(t)="(t) c 2 dt I yy • Output angle, (t), as a function of time t t t t t t g Cg Cg A A A (t)= u(t)dtdt = e(t)dtdt =(t) dtdt c """""" I I I yy yy yy 0 0 0 0 0 0 30nd Rewrite 2 -Order Model as Two st 1 -Order Equations (t)=q(t) Cg A q(t)="(t) c I yy "% "%"% "%(t) 0 (t) 0 1 '= + C (t)((t) '' ' c g / I q(t) q(t) 0 0 A yy ' &'' && & "%"% "%"% 0 1(t) 0 (t) '= + ''' c (Cg / I 0 Cg / I q(t) q(t) A yy A yy ' ''' &&& & 31 Simulation of Step Response with Angle Feedback Objective is to control angle to 1 rad, but solution oscillates about the target Cg /I = 1, 0.5, and 0.25 32 A yyWhat Went Wrong? • No damping • Solution: Add rate feedback • Control law with u(t) =c (t)"(t)"c q(t) 1 c 2 rate feedback Closed-loop dynamic equation "% "%"%"% 0 1(t) 0 (t) '= + ''' c (c g / I(c g / I c g / I q(t) q(t) 1 A yy 2 A yy 1 A yy ' ' '' &&& & 33 Step Response with Angle and Rate Feedback c g /I = 1 1 A yy c g /I = 0, 1.414, 2.828 2 A yy 34nd 2 -Order Dynamics Oscillation and damping are induced by linear feedback control "% "%"%"% 0 1(t) 0 (t) '= + ''' c (c g / I(c g / I c g / I q(t) q(t) 1 A yy 2 A yy 1 A yy ' ''' &&& & "% "%"%"% 0 1(t) 0 (t) '= + ''' 2 2 c ()(2) q(t)) q(t) ' n n n ''' &&& & Natural frequency and damping ratio = cg / I n 1 A yy " = c g / I /2 =c /2 c g I ( ) 2 A yy n 2 1 A yy 35 Effect of Damping on Eigenvalues, Damping Ratio, and Natural Frequency c g /I = 1 1 A yy c g /I = 0, 1.414, 2.828 2 A yy Eigenvalues Damping Ratio, Natural Frequency "" , "" = 1 2 = = (rad/s) n 0 + 1.0000i 0 1 0 - 1.0000i -0.7070 + 0.7072i 0.707 1 -0.7070 - 0.7072i -0.4143 Overdamped -2.4137 36Control System Design to Adjust Roots Choose control gains to satisfy desirable eigenvalue range 37 Control System Design to Adjust Transient Response Choose control gains to satisfy step response criteria 38Control System Design to Adjust Frequency Response Choose control gains to satisfy frequency response criteria 39 Laplace Transform of the State Vector Neglecting the initial condition Adj sIF ( ) x(s)= Gu(s) "(s) Applied to the closed-loop system && c g I c g I 1 A yy 1 A yy %(%( u(s) sc g / I sc g / I %(%( & 1 A yy 1 A yy "(s) '' %(=u(s)= 2 q(s)(s) s + c g I s +c g I ( ) ( ) ( ) %( 2 A yy 1 A yy ' 40

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