Spacecraft Guidance navigation and control

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Published Date:23-07-2017
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Spacecraft Guidance Space System Design, MAE 342, Princeton University Robert Stengel • Oberth s  Synergy Curve • Explicit ascent guidance • Impulsive "V maneuvers • Hohmann transfer between circular orbits • Sphere of gravitational influence • Synodic periods and launch windows • Hyperbolic orbits and escape trajectories • Battin’s universal formulas • Lambert’s time-of-flight theorem (hyperbolic orbit) • Fly-by (swingby) trajectories for gravity assist Copyright 2016 by Robert Stengel. All rights reserved. For educational use only. 1 http://www.princeton.edu/stengel/MAE342.html Guidance, Navigation, and Control • Navigation: Where are we? • Guidance: How do we get to our destination? • Control: What do we tell our vehicle to do? 2Energy Gained from Propellant Specific energy = energy per unit weight 2 Hermann Oberth V E =h+ 2g h:height; V :velocity Rate of change of specific energy per unit of expended propellant mass dE dh V dV 1dh V dV = + = + & dm dm g dm (dm dt)" dt g dt% 1dh 1 dv 1 1 T T = + v = Vsin' + v T(mg ( ) && dm dt" dt g dt% dm dt" g% ( ) ( ) 1 VT = Vsin' + cos)(Vsin' & dm dt" mg% ( ) 3 Oberth s Synergy Curve : Flight Path Angle ": Pitch Angle : Angle of Attack dE/dm maximized when = 0, or " = , i.e., thrust along the velocity vector Approximate round-earth equations of motion dV T Drag = cos""gsin dt m m ' d T V g = sin +" cos & ) dt mV% r V( 4 Gravity-Turn Pitch Program With angle of attack, = 0 ' d d" V g = = cos & ) % ( dt dt r V Locally optimal flight path Minimizes aerodynamic loads Feedback controller minimizes or load factor 5 Gravity-Turn Flight Path • Gravity-turn flight path is function of 3 variables – Initial pitchover angle (from vertical launch) – Velocity at pitchover – Acceleration profile, T(t)/m(t) Gravity-turn program closely approximated by tangent steering laws (see Supplemental Material) 6Feedback Control Law Errors due to disturbances and modeling errors corrected by feedback control Motor Gimbal Angle(t) t = c" (t)"(t)c q(t) ( ) G" des q d" " = Desired pitch angle; q = = pitch rate des dt c ,c :Feedback control law gains " q 7 Thrust Vector Control During Launch Through Wind Profile • Attitude control – Attitude and rate feedback • Drift-minimum control – Attitude and accelerometer feedback – Increased loads • Load relief control – Rate and accelerometer feedback – Increased drift 8Effect of Launch Latitude on Orbital Parameters Typical launch inclinations Space Launch Centers from Wallops Island • Launch latitude establishes minimum orbital inclination (without  dogleg maneuver) • Time of launch establishes line of nodes • Argument of perigee established by – Launch trajectory 9 – On-orbit adjustment Guidance Law for Launch to Orbit (Brand, Brown, Higgins, and Pu, CSDL, 1972) • Initial conditions – End of pitch program, outside atmosphere • Final condition – Insertion in desired orbit • Initial inputs – Desired radius – Desired velocity magnitude – Desired flight path angle – Desired inclination angle – Desired longitude of the ascending/ descending node • Continuing outputs – Unit vector describing desired thrust direction – Throttle setting, % of maximum thrust 10Guidance Program Initialization • Thrust acceleration estimate • Mass/mass flow rate • Acceleration limit ( 3g) • Effective exhaust velocity • Various coefficients • Unit vector normal to desired orbital plane, i q "% sini sin d d ' i = sini cos ' q d d ' cosi i :Desired inclination angle of final orbit d d & :Desired longitude of descending node d 11 Guidance Program Operation: Position and Velocity • Obtain thrust acceleration estimate, a , from guidance system T • Compute corresponding mass, mass flow rate, and throttle setting, T r i = :Unit vector alignedwithlocalvertical r r i = i i :Downrangedirection z r q Position r r & & '1 & y = rsin i•i ( ) r q & & z& open & "% "% Velocity v•i r IMU r & & v :VelocityestimateinIMUframe y = v•i IMU IMU q& & & & z v•i "% IMU z 12 & "% Guidance Program: Velocity and Time to Go Effective gravitational acceleration 2 µ r"v g = + eff 2 3 r r Time to go (to motor burnout) t =t"t t :Guidance command interval go go new old Velocity to be gained "% rr g t /2 ( ) d eff go ' v =y' go ' z z d ' & Time to go prediction (prior to acceleration limiting) m v /c go eff t = 1e c :Effective exhaust velocity ( ) eff go m 13 Guidance Program Commands Guidance law: required radial and cross-range accelerations a =a"A+B ttg ( ) T T o eff % r a = Net available acceleration T a =a"C+D tt ( ) T T o y% Guidance coefficients, A, B, C, and D are functions of r ,r,r,t ( ) d go plusc ,m m, Acceleration limit eff y,y,t ( ) go Required thrust direction, i (i.e., vehicle orientation in (i , i , i ) frame T r q z a T r & a T a a =&; i = T T T y a & T what's left over & "% Throttle command is a function of a (i.e., T acceleration magnitude) and acceleration limit 14Impulsive "V Orbital Maneuver • If rocket burn time is short compared to orbital period (e.g., seconds compared to hours), impulsive "V approximation can be made – Change in position during burn is zero – Change in velocity is instantaneous Vector diagram of Velocity impulse at apogee velocity change 15 Orbit Change due to Impulsive "V • Maximum energy change accrues when "V is aligned with the instantaneous orbital velocity vector – Energy change - Semi-major axis change – Maneuver at perigee raises or lowers apogee – Maneuver at apogee raises or lowers perigee • Optimal transfer from one circular orbit to another involves two impulses Hohmann transfer • Other maneuvers – In-plane parameter change – Orbital plane change 16Assumptions for Impulsive Maneuver Instantaneous change in velocity vector v =v +v 2 1 rocket Negligible change in radius vector r = r 2 1 Therefore, new orbit intersects old orbit Velocities different at the intersection 17 Geometry of Impulsive Maneuver Change in velocity magnitude, v, vertical flight path angle, , and horizontal flight path angle, % v Vcos' cos( x x & & & v v = y = = Vcos' sin( & y& 1 & & & z& )Vsin' v "% z 1& "% "% 1 1 v Vcos' cos( x x & & & v y v = = = Vcos' sin( y& & 2 & & & & z v)Vsin' "% z 2& "% "% 2 2 "% v(v ( ) "% x x v 2 1 ' x ' ' v v = = v(v y' ( ) y y 2 1' ' ' v z ' v(v & ( ) z z' 2 1 & 18Required "v for Impulsive Maneuver "% v(v "% ( ) x x v 2 1 ' x ' ' v = v = v(v ' y ( ) y y 2 1' ' ' v z ' v(v & ( ) z z ' 2 1 & 1/2 ' 2 2 2 v +v +v ( ) x y z &) rocket &) ' V+. rocket &) v 1 y &) sin-0 &) 1/2 v =" = 2 2 & rocket) rocket -0 v +v & ( )) , x y/ &) rocket &) rocket %( v +. &1) z sin -0 &) ,/ V rocket %( 19 Single Impulse Orbit Adjustment Coplanar (i.e., in-plane) maneuvers 1 2 2 2 2 E = vµ r = e1 µ h ( ) • Change energy 2 • Change angular 2 2 2 2 µ e1 µ e1 ( ) ( ) momentum h= = 2 E v 2µ r • Change eccentricity 2 2 e= 1+2Eh µ v v +v = 2 E +µ r ( ) new old rocket new • Required velocity 2 2 2 increment% = 2 e"1 µ h +µ r ( ) new new & v = v"v rocket new old 20Single Impulse Orbit Adjustment Coplanar (i.e., in-plane) maneuvers • Change semi-major axis 2 – magnitude h µ new a = new 2 – orientation (i.e., argument 1e new of perigee); in-plane isosceles triangle r = a 1e ( ) perigee r = a(1+e) apogee µ 1+e "% v = ' perigee & a 1e • Change apogee or perigee µ 1e "% – radius v = apogee' & a 1+e – velocity 21 In-Plane Orbit Circularization Initial orbit is elliptical, with apogee radius equal to desired circular orbit radius Initial Orbit a= r +r 2 ( ) cir(target) insertion e= rr 2a ( ) cir(target) insertion µ 1e "% v = ' apogee & a 1+e Velocity in circular orbit is a function of the radius “Vis viva” equation: "%"% 2 1 2 1 µ v = µ = µ = cir '' r a& r r& r cir cir cir cir cir Rocket must provide v = v"v the difference rocket cir apogee 22Single Impulse Orbit Adjustment Out-of-plane maneuvers • Change orbital inclination • Change longitude of the ascending node • v , "v, and v form isosceles triangle 1 2 perpendicular to the orbital plane to leave in-plane parameters unchanged 23 Change in Inclination and Longitude of Ascending Node Longitude of Inclination Ascending Node Sellers, 2005 24Two Impulse Maneuvers Transfer to Non- Phasing Orbit Intersecting Orbit Rendezvous with trailing st 1 "v produces target orbit spacecraft in same orbit intersection At perigee, increase speed to nd 2 "v matches target orbit increase orbital period Minimize ("v + "v ) to 1 2 At future perigee, decrease minimize propellant use speed to resume original orbit 25 Hohmann Transfer between Coplanar Circular Orbits (Outward transfer example) Thrust in direction of motion at transfer perigee and apogee Transfer Orbit µ a= r +r 2 ( ) cir cir 1 2 v = cir 1 r cir 1 e= rr 2a ( ) cir cir 2 1 µ 1+e "% v = ' p µ transfer & a 1e v = cir 2 r cir 2 µ 1e "% v = ' a transfer & a 1+e 26Outward Transfer Orbit Velocity Requirements st nd "v at 1 Burn"v at 2 Burn v = v"vv = v"v 1 p cir 2 cir a transfer 1 1 transfer && 2r 2r cir cir 2 1 = v"1 = v 1" cir%( cir%( 1 2 r +r r +r cir cir' cir cir' 1 2 1 2 r cir 1 v = v cir cir 2 1 r cir 2 Hohmann Transfer is energy-optimal for 2-impulse transfer between circular orbits and r /r 11.94 2 1 ), & 2r r r cir cir cir 2 1 1 v = v 1" +"1 +. total cir %( 1 r +r r r ' cir cir cir cir +. 1 2 2 2 - 27 Rendezvous Requires Phasing of the Maneuver Transfer orbit time equals target’s time to reach rendezvous point Sellers, 2005 28Solar Orbits • Same equations used for Earth-referenced orbits – Dimensions of the orbit – Position and velocity of the spacecraft – Period of elliptical orbits – Different gravitational constant 11 3 2 µ =1.332710 km /s Sun 29 Escape from a Circular Orbit Minimum escape trajectory shape is a parabola 30In-plane Parameters of Earth Escape Trajectories Dimensions of the orbit 2 h p= = "The parameter"or semi-latus rectum µ h= Angular momentum about center of mass E p e= 1+ 2 = Eccentricity 1 µ E = Specific energy, 0 p a= = Semi-major axis, 0 2 1"e r = a 1"e = Perigee radius ( ) perigee 31 In-plane Parameters of Earth Escape Trajectories Position and velocity of the spacecraft p r= = Radiusof thespacecraft 1+ecos = Trueanomaly 1 1 & V = 2µ" =Velocityof thespacecraft %( ' r 2a 2µ V) perigee r perigee 32Escape from Circular Orbit Velocity in circular orbit " 2 1% µ V = µ = c ' r r r & c c c Velocity at perigee of parabolic orbit ' 2 1 2µ V = µ = perigee &) % r (a")( r c c Velocity increment required for escape 2µ µ V =V"V =="0.414V escape perigee c c parabola r r c c 33 Earth Escape Trajectory "v to increase speed to escape velocity 1 Velocity required for transfer at sphere of influence 34Transfer Orbits and Spheres of Influence • Sphere of Influence (Laplace): – Radius within which gravitational effects of planet are more significant than those of the Sun • Patched-conic section approximation – Sequence of 2-body orbits – Outside of planet’s sphere of influence, Sun is the center of attraction – Within planet s sphere of influence, planet is the center of attraction • Fly-by (swingby) trajectories dip into intermediate object s sphere of influence for gravity assist 35 Solar System Spheres of Influence 2 5 m" m% Planet Planet for 1, r r SI PlanetSun ' m m & Sun Sun Planet Sphere of Influence, km Mercury 112,000 Venus 616,000 Earth 929,000 Mars 578,000 Jupiter 48,200,000 Saturn 54,500,000 Uranus 51,800,000 Neptune 86,800,000 36 Pluto 27,000,000-45,000,000Interplanetary Mission Planning • Example: Direct Hohmann Transfer from Earth Orbit to Mars Orbit (No fly-bys) 1) Calculate required perigee velocity for transfer orbit - Sun as center of attraction: Elliptical orbit 2) Calculate "v required to reach Earth’s sphere of influence with velocity required for transfer – Earth as center of attraction: Hyperbolic orbit 3) Calculate "v required to enter circular orbit about Mars, given transfer apogee velocity – Mars as center of attraction: Hyperbolic orbit 37 Launch Opportunities for Fixed Transit Time: The Synodic Period • Synodic Period, S : n The time between conjunctions – P : Period of Planet A A – P : Period of Planet B B • Conjunction: Two planets, A and B, in a line or at some fixed angle P P A B S = n PP A B 38 Launch Opportunities for Fixed Transit Time: The Synodic Period Synodic Period with Planet respect to Earth, days Period Mercury 116 88 days Venus 584 225 days Earth - 365 days Mars 780 687 days Jupiter 399 11.9 yr Saturn 378 29.5 yr Uranus 370 84 yr Neptune 367 165 yr Pluto 367 248 yr 39 Hyperbolic Orbits Orbit Shape Eccentricity, e Energy, E Circle 0 0 Ellipse 0 e 1 0 Parabola 1 0 Hyperbola 1 0 2 v µ µ E = = ,"a 0 2 r 2a Velocity remains positive as radius approaches vv " r" 2 v µ µ " E = , and v =% or a=% "" 2 2 a v " 40

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