Spacecraft Guidance!

Transcription

1 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 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? 2

2 Hermann Oberth Energy Gained from Propellant Specific energy = energy per unit weight E = h + V 2 2g h :height; V :velocity = Rate of change of specific energy per unit of expended propellant mass 1 dm dt ( ) de dm = dh dm + V dv g dm = 1 dm dt ( )! dh dt + 1 dv g vt dt = 1 dm dt = 1 dm dt ( )!! dh dt + V dv g dt ( ) V sin + 1 ( g vt T ( mg)! VT V sin + cos) ( V sin 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 dt = T Drag cos! m m gsin d dt = T mv sin! + V r g ) cos V ( 4

3 Gravity-Turn Pitch Program With angle of attack,! = 0 d! dt = d dt = V r g V ) cos! ( 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) 6

4 Feedback Control Law Errors due to disturbances and modeling errors corrected by feedback control Motor Gimbal Angle(t)!! G ( t) = c [ des (t) (t)] c q q(t) des = Desired pitch angle; q = d = pitch rate dt c,c q :Feedback control law gains 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 8

5 Effect of Launch Latitude on Orbital Parameters Typical launch inclinations from Wallops Island Space Launch Centers Launch latitude establishes minimum orbital inclination (without dogleg maneuver) Time of launch establishes line of nodes Argument of perigee established by! Launch trajectory! On-orbit adjustment 9 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 10

6 Guidance 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 i q = sini d sin! d sini d cos! d cosi d i d :Desired inclination angle of final orbit! d :Desired longitude of descending node 11 Guidance Program Operation: Position and Velocity Obtain thrust acceleration estimate, a T, from guidance system Compute corresponding mass, mass flow rate, and throttle setting, T i r = r :Unit vector aligned with local vertical r i z = i r! i q :Downrange direction Position!! r r y = rsin 1 ( i r i q ) z open Velocity!!r!y!z! v IMU i r = v IMU i q v IMU i z v IMU :Velocity estimate in IMU frame 12

7 Guidance Program: Velocity and Time to Go Effective gravitational acceleration g eff =! µ r + r v 2 2 r 3 Time to go (to motor burnout) t gonew = t goold! t!t :Guidance command interval Velocity to be gained v go = (!r d!!r )! g eff t go / 2 Time to go prediction (prior to acceleration limiting) t go = m!m 1! e!v go/c ( eff )!!y!z d!!z c eff :Effective exhaust velocity 13 Guidance Program Commands Guidance law: required radial and cross-range accelerations ( ) ( ) a Tr = a T A + B t! t o! g eff a Ty = a T C + D t! t o Guidance coefficients, A, B, C, and D are functions of ( r d,r,!r, t go ) plus c eff, m!m, Acceleration limit y,!y,t go ( ) a T = Net available acceleration Required thrust direction, i T (i.e., vehicle orientation in (i r, i q, i z ) frame! a Tr a T = a Ty whats left over ; i T = a T a T Throttle command is a function of a T (i.e., acceleration magnitude) and acceleration limit 14

8 Impulsive 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 Velocity impulse at apogee Vector diagram of 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 16

9 Assumptions for Impulsive Maneuver Instantaneous change in velocity vector v 2 = v 1 +!v rocket Negligible change in radius vector r 2 = r 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 1 =!x!y!z! = 1 v x v y v z! = 1 V cos cos( V cos sin( )V sin 1! v 2 =!x!y!z! = 2 v x v y v z! = 2 V cos cos( V cos sin( )V sin 2!v =!v x!v y!v z v x2 ( v ( x1 ) = ( v y2 ( v y1 ) ( v z2 ( v z1 ) 18

10 Required v for Impulsive Maneuver!v =!v x!v y!v z = ( v x2 ( v x1 ) ( v y2 ( v y1 ) ( v z2 ( v z1 )!v rocket =!V rocket rocket rocket ) ) = ) (!v 2 x +!v 2 2 1/2 ( y +!v z ) rocket + sin *1, -!v. y!v 2 2 ( x +!v y ) 1/2 / 0!v sin *1 + z., -!V / 0 rocket rocket ) ) ) ) ) ) ) ) ( 19 Single Impulse Orbit Adjustment Coplanar (i.e., in-plane) maneuvers Change energy Change angular momentum Change eccentricity E = 1 2 v2! µ r = ( e 2!1)µ 2 h 2 h = µ 2 ( e 2!1) E = e = 1+ 2Eh 2 µ 2 ( ) µ 2 e 2!1 v 2 2! µ r Required velocity increment 2( E new + µ r) v new! v old +!v rocket = 2 = 2 ( e new 1)µ 2 2 h new + µ r!v rocket = v new v old 20

11 Single Impulse Orbit Adjustment Coplanar (i.e., in-plane) maneuvers Change semi-major axis! magnitude! orientation (i.e., argument of perigee); in-plane isosceles triangle Change apogee or perigee! radius! velocity a new = h 2 new µ 2 1! e new ( ) ( ) r perigee = a 1! e r apogee = a 1+ e v perigee = v apogee = µ 1+ e a 1! e µ 1! e a 1+ e 21 In-Plane Orbit Circularization Initial orbit is elliptical, with apogee radius equal to desired circular orbit radius a cir Initial Orbit ( ) 2 ( ) 2a a = r cir(target) + r insertion e = r cir(target)! r insertion v apogee = r cir µ 1! e a 1+ e Velocity in circular orbit is a function of the radius Vis viva equation: v cir = µ 2! 1 r cir = µ 2! 1 r cir = µ r cir Rocket must provide the difference!v rocket = v cir v apogee 22

12 Single Impulse Orbit Adjustment Out-of-plane maneuvers Change orbital inclination Change longitude of the ascending node v 1, v, and v 2 form isosceles triangle perpendicular to the orbital plane to leave in-plane parameters unchanged 23 Change in Inclination and Longitude of Ascending Node Inclination Longitude of Ascending Node Sellers,

13 Two Impulse Maneuvers Transfer to Non- Intersecting Orbit Phasing Orbit 1 st v produces target orbit intersection 2 nd v matches target orbit Minimize ( v 1 + v 2 ) to minimize propellant use Rendezvous with trailing spacecraft in same orbit At perigee, increase speed to increase orbital period At future perigee, decrease 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 v cir1 = v cir2 = µ r cir1 µ r cir2 Transfer Orbit ( ) 2 ( ) 2a a = r cir1 + r cir2 e = r cir2! r cir1 v ptransfer = µ a v atransfer = µ a 1+ e 1! e 1! e 1+ e 26

14 Outward Transfer Orbit Velocity Requirements v at 1 st Burn v at 2 nd Burn!v 1 = v ptransfer v cir1!v 2 = v cir1 v atransfer 2r cir2 = v cir1 1( r cir1 + r cir2 = v cir2 1 2r cir1 r cir1 + r cir2 ( v cir2 = v cir1 r cir1 r cir2 Hohmann Transfer is energy-optimal for 2-impulse transfer between circular orbits and r 2 /r 1 < ) 2r cir2!v total = v cir1 1 r cir 1 r cir1 + r cir2 r ( cir2 + r, cir * + r cir Rendezvous Requires Phasing of the Maneuver Transfer orbit time equals target s time to reach rendezvous point Sellers,

15 Solar 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 µ Sun = !10 11 km 3 /s 2 29 Escape from a Circular Orbit Minimum escape trajectory shape is a parabola 30

16 In-plane Parameters of Earth Escape Trajectories Dimensions of the orbit p = h2 µ = The parameteror semi-latus rectum h = Angular momentum about center of mass e = 1+ 2 E p µ = Eccentricity! 1 E = Specific energy,! 0 p a = = Semi-major axis, < e r perigee = a 1 e ( ) = Perigee radius 31 In-plane Parameters of Earth Escape Trajectories r = Position and velocity of the spacecraft p 1+ ecos! = Radius of the spacecraft! = True anomaly V = 2µ 1 r 1 2a ( = Velocity of the spacecraft V perigee ) 2µ r perigee 32

17 Escape from Circular Orbit Velocity in circular orbit V c = µ 2! 1 r c r c = µ r c Velocity at perigee of parabolic orbit V perigee = µ 2! r c 1 (a )( ) = 2µ r c Velocity increment required for escape!v escape = V perigeeparabola V c == 2µ r c µ r c 0.414V c 33 Earth Escape Trajectory v 1 to increase speed to escape velocity Velocity required for transfer at sphere of influence 34

18 Transfer 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 for m Planet m << 1, r SI! r Planet Planet!Sun m Sun m Sun 2 5 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 Pluto 27,000,000-45,000,000 36

19 Interplanetary 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 A : Period of Planet A! P B : Period of Planet B Conjunction: Two planets, A and B, in a line or at some fixed angle S n = P AP B P A! P B 38

20 Launch Opportunities for Fixed Transit Time: The Synodic Period Planet Synodic Period with respect to Earth, days Period Mercury days Venus days Earth days Mars days Jupiter yr Saturn yr Uranus yr Neptune yr Pluto yr 39 Hyperbolic Orbits Orbit Shape Eccentricity, e Energy, E Circle 0 <0 Ellipse 0 < e <1 <0 Parabola 1 0 Hyperbola >1 >0 E = v2 2! µ r =! µ 2a, a < 0 Velocity remains positive as radius approaches v! v r! E = v 2 2, and v = µ a or a = µ 2 v 40

21 Hyperbolic Encounter with a Planet Trajectory is deflected by target planet s gravitational field! In-plane! Out-of-plane Velocity w.r.t. Sun is increased or decreased! : Miss Distance, km : Deflection Angle, deg or rad Kaplan 41 Hyperbolic Orbits Asymptotic Value of True Anomaly Polar Equation for a Conic Section ( ) p r = 1+ ecos! = a 1 e 2 1+ ecos! cos! = 1 a( 1 e 2 ) 1( e r (!! r! = cos 1 1 * ( ) e+, 42

22 Hyperbolic Orbits h = Constant = v! = µ p = µa 1 e 2 e = Angular Momentum ( ) = µ 2 ( e 2 1) Eccentricity v! h2 E = 1+ v 4! 2 µ 2 Perigee Radius µ 2 r p = a( 1! e) = µ 2 ( e!1) v Eccentricity e = 1+ r v 2 p! µ 43 Hyperbolic Mean and Eccentric Anomalies H : Hyperbolic Eccentric Anomaly! t M = esinh H! H Newton s method of successive approximation to find H from M, similar to solution for E (Lecture 2) ( ) = 2 tan 1 e +1 ( ) e 1 tanh H t 2 ( ) r = a 1! ecosh H ( see Ch. 7, Kaplan,

23 Battin s Universal Formulas for Conic Section Position and Velocity as Functions of Time v( t 2 ) = ( ) = 1! 2 ( ) C 2 + a r t 2 µ ) * r( t 1 )r t 2 ( ) r t 1, (.r t 1 - )! 3 a S! 2 a (!, +.r t 1 * - ) + * ( ) + t 2! 3 µ S 2 ) a, (.v t 1 - ( ) + 1(! 2 ( ) C! 2 + a * r t 2 ( ),.v t 1 - ( ) *,,! = +,, -, ( ) E( t 1 ) ( ) H ( t 1 ) ( ) ( ) a E t 2, Ellipse a H t 2, Hyperbola p tan t 2 ( 2 tan t 1 2 ), Parabola C(.) and S(.) see Ch. 7, Kaplan, 1976; also Battin, Lambert s Time-of-Flight Theorem (Hyperbolic Orbit) ( t 2! t 1 ) =!a3 µ ( sinh! ) + ( sinh! ) where r!! 2sinh r 2 + c r ;! 2sinh r 2 c 4a 4a see Ch. 7, Kaplan, 1976; also Battin,

24 Asteroid Encounter Hyperbolic trajectory Launch window V to establish trajectory V for rendezvous or impact 47 Swing-By/Fly-By Trajectories Hyperbolic encounters with planets and the moon provide gravity assist! Shape, energy, and duration of transfer orbit altered! Potentially large reduction in rocket V required to accomplish mission Why does gravity assist work? 48

25 Effect of Target Planet s Gravity on Probe s Sun-Relative Velocity Deflection Velocity Reduction 49 Effect of Target Planet s Gravity on Probe s Sun-Relative Velocity Deflection Velocity Addition 50

26 Planet Capture Trajectory Hyperbolic approach to planet s sphere of influence v to decrease speed to circular velocity 51 Effect of Target Planet s Gravity on Probe s Velocity Probe Departure Velocity (wrt Sun) Probe Approach Velocity (wrt Sun) Probe Approach Velocity (wrt Planet) Kaplan Probe A Departure Velocity (wrt Planet) Target Planet Velocity 52

27 Next Time:! Spacecraft Environment! 53 Supplemental Material! 54

28 Phases of Ascent Guidance Vertical liftoff Roll to launch azimuth Pitch program to atmospheric exit! Jet stream penetration! Booster cutoff and staging Explicit guidance to desired orbit! Booster separation! Acceleration limiting! Orbital insertion 55 Tangent Steering Laws Neglecting surface curvature tan!(t) = tan! o 1 t t BO Open-loop command, i.e., no feedback of vehicle state ( Accounting for effect of Earth surface curvature on burnout flight path angle * tan!(t) = tan! o 1 t tan t -, )/ + t BO t BO (. 56

29 Jet Stream Profiles Launch vehicle must able to fly through strong wind profiles Design profiles assume 95 th -99 th -percentile worst winds and wind shear 57 Longitudinal (2-D) Equations of Motion for Re-Entry Differential equations for velocity (x 1 = V), flight path angle (x 2 = ), altitude (x 3 = h), and range (x 4 ) Angle of attack (!) is optimization control variable x 1 =!D(x 1, x 3,)/m! gcos x 2 x 2 = [ g/ x 1! x 1 /( R + x 3 )]sin x 2! L(x 1, x 3,)/mx 1 x 3 = x 1 cos x 2 x 4 = x 1 sin x 2 / 1 + x 3 /R ( ) D = drag L = lift M = pitch moment R = Earth radius Equations of motion define the dynamic constraint x (t) = f [ x(t),!(t) ] 58

30 A Different Approach to Guidance: Optimizing a Cost Function Minimize a scalar function, J, of terminal and integral costs t f J =![ x(t f )]+ L[ x(t), u(t) ] dt t o L[.]: Lagrangian with respect to the control, u(t), in (t o,t f ), subject to a dynamic constraint x (t) = f[x(t),u(t)], x(t o ) given dim(x) = n x 1 dim(f) = n x 1 dim(u) = m x 1 59 Guidance Cost Function! x(t f ) L[ x(t),u(t) ] Terminal cost, e.g., in final position and velocity Integral cost, e.g., tradeoff between control usage and trajectory error Minimization of cost function determines the optimal state and control, x* and u*, over the flight path duration min u J = min u t f t f )! x(t f ) + L [ x(t),u(t) + ) ( ] dt, *) t o -) [ ] =! x*(t f ) + L x*(t),u*(t) dt. x*(t),u*(t) t o [ ] 60

31 Example of Re-Entry Flight Path Cost Function J = a[ V (t f )!V d ] 2 + b[ r(t f )! r d ] 2 + c (t) t f [ ] 2 dt t o Cost function includes! Terminal range and velocity! Penalty on control use! a, b, and c tradeoff importance of each factor Minimization of this cost function! Defines the optimal path, x*(t), from t o to t f! Defines the optimal control,!*(t), from t o to t f 61 Extension to Three Dimensions Add roll angle as a control; add crossrange as a state For the guidance law, replace range and crossrange from the starting point by distance to go and azimuth to go to the destination 62

32 Optimal Trajectories for! Space Shuttle Transition Altitude vs. Velocity Range vs. Cross-Range ( footprint ) 63 Angle of Attack and Roll Angle vs. Specific Energy Optimal Controls for Space Shuttle Transition 64

33 Optimal Guidance System Derived from Optimal Trajectories Angle of Attack and Roll Angle vs. Specific Energy Diagram of Energy-Guidance Law 65 Guidance Functions for! Space Shuttle Transition Angle of Attack Guidance Function Roll Angle Guidance Function 66

34 Lunar Module Navigation, Guidance, and Control Configuration 67 Lunar Module Transfer Ellipse to Powered Descent Initiation 68

35 Lunar Module Powered Descent 69 Lunar Module Descent Events 70

36 Lunar Module Descent Targeting Sequence Braking Phase (P63) Approach Phase (P64) Terminal Descent Phase (P66) 71 Characterize Braking Phase By Five Points 72

37 Lunar Module Descent Guidance Logic (Klumpp, Automatica, 1974) Reference (nominal) trajectory, r r (t), from target position back to starting point (Braking Phase example)! Three 4 th -degree polynomials in time! 5 points needed to specify each polynomial r r (t) = r t + v t t + a t t j t t s t t 4 24! x(t) r(t) = y(t) z(t) 73 Coefficients of the Polynomials r r (t) = r t + v t t + a t t 2 r = position vector v = velocity vector a = acceleration vector j = jerk vector (time derivative of acceleration) s = snap vector (time derivative of jerk) 2 + j t! x r = y z! a = a x a y a z t s t! j = t 4 24! x! v = y = z j x j y j z v x v y v z! s = s x s y s z 74

38 Corresponding Reference Velocity and Acceleration Vectors v r (t) = v t + a t t + j t t s t a r (t) = a t + j t t + s t t 2 a r (t) is the reference control vector! Descent engine thrust / mass = total acceleration! Vector components controlled by orienting yaw and pitch angles of the Lunar Module 2 t Guidance Logic Defines Desired Acceleration Vector If initial conditions, dynamic model, and thrust control were perfect, a r (t) would produce r r (t) a r (t) = a t + j t t + s t t 2 2! r r(t) = r t + v t t + a t t j t t s t t but they are not Therefore, feedback control is required to follow the reference trajectory 76

39 Guidance Law for the Lunar Module Descent Linear feedback guidance law a command (t) = a r (t) + K V [ v measured (t)! v r (t)] + K R r measured (t)! r r (t) K V :velocity error gain K R :position error gain [ ] Nominal acceleration profile is corrected for measured differences between actual and reference flight paths Considerable modifications made in actual LM implementation (see Klumpp s original paper on Blackboard) 77