2/12/20 Spacecraft Guidance Space System Design, MAE 342, Princeton University Robert Stengel • Oberth’s “Synergy Curve” • Explicit ascent guidance • Impulsive ΔV 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 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 1 2/12/20 Energy Gained from Propellant Specific energy = energy per unit weight Hermann Oberth V 2 E = h + 2g h :height; V :velocity Rate of change of specific energy per unit of expended propellant mass d dh V dV 1 ⎛ dh V dV ⎞ E = + = + dm dm g dm (dm dt) ⎝⎜ dt g dt ⎠⎟ 1 ⎛ dh 1 dv ⎞ 1 ⎛ 1 ⎞ = + vT = V sinγ + vT (T − mg) (dm dt) ⎝⎜ dt g dt ⎠⎟ (dm dt) ⎝⎜ g ⎠⎟ 1 ⎛ VT ⎞ = V sinγ + cosα −V sinγ (dm dt) ⎝⎜ mg ⎠⎟ 3 3 Oberth’s Synergy Curve γ : Flight Path Angle θ: Pitch Angle α: Angle of Attack dE/dm maximized when α = 0, or θ = γ, i.e., thrust along the velocity vector Approximate round-earth equations of motion dV T Drag = cosα − − gsinγ dt m m dγ T ⎛ V g ⎞ = sinα + ⎜ − ⎟ cosγ dt mV ⎝ r V ⎠ 4 4 2 2/12/20 Gravity-Turn Pitch Program With angle of attack, α = 0 dγ dθ ⎛ V g ⎞ = = − cosγ dt dt ⎝⎜ r V ⎠⎟ Locally optimal flight path Minimizes aerodynamic loads Feedback controller minimizes α or load factor 5 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 6 3 2/12/20 Feedback Control Law Errors due to disturbances and modeling errors corrected by feedback control Motor Gimbal Angle(t) ! δ G (t) = cθ [θdes (t) −θ(t)]− cqq(t) dθ θ = Desired pitch angle; q = = pitch rate des dt cθ ,cq :Feedback control law gains 7 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 8 4 2/12/20 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 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 10 5 2/12/20 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, iq ⎡ sini sinΩ ⎤ ⎢ d d ⎥ iq = ⎢ sinid cosΩd ⎥ ⎢ cosi ⎥ i :Desired inclination angle of final orbit ⎣ d ⎦ d Ωd :Desired longitude of descending node 11 11 Guidance Program Operation: Position and Velocity • Obtain thrust acceleration estimate, aT, from guidance system • Compute corresponding mass, mass flow rate, and throttle setting, δT r i = :Unit vector aligned with local vertical r r iz = ir × iq :Downrange direction Position ⎡ r ⎤ ⎡ r ⎤ ⎢ ⎥ ⎢ ⎥ −1 y = ⎢ rsin i • i ⎥ ⎢ ⎥ ⎢ ( r q ) ⎥ ⎢ z ⎥ ⎢ open ⎥ ⎣ ⎦ ⎣ ⎦ Velocity ⎡ ⎤ ⎡ r! ⎤ v IMU • ir ⎢ ⎥ ⎢ y ⎥ v i v :Velocity estimate in IMU frame ⎢ ! ⎥ = ⎢ IMU • q ⎥ IMU ⎢ z! ⎥ ⎢ ⎥ ⎣ ⎦ v IMU • iz 12 ⎣⎢ ⎦⎥ 12 6 2/12/20 Guidance Program: Velocity and Time to Go Effective gravitational acceleration 2 µ r × v g = − + eff r2 r 3 Time to go (to motor burnout) t t t gonew = goold − Δ Δt :Guidance command interval Velocity to be gained ⎡ r! − r! − g t / 2 ⎤ ⎢ ( d ) eff go ⎥ v go = ⎢ −y! ⎥ ⎢ ⎥ ⎢ z!d − z! ⎥ ⎣ ⎦ Time to go prediction (prior to acceleration limiting) m −v /c t 1 e go eff c :Effective exhaust velocity go = ( − ) eff m! 13 13 Guidance Program Commands Guidance law: required radial and cross-range accelerations a = a ⎡A + B t − t ⎤ − g Tr T ⎣ ( o )⎦ eff a = Net available acceleration a = a ⎡C + D t − t ⎤ T Ty T ⎣ ( o )⎦ Guidance coefficients, A, B, C, and D are functions of rd ,r,r!, tgo ( ) plus c , m m! , Acceleration limit eff (y, y!,tgo ) Required thrust direction, iT (i.e., vehicle orientation in (ir, iq, iz) frame ⎡ a ⎤ Tr ⎢ ⎥ a a T aT = ⎢ Ty ⎥; iT = ⎢ ⎥ aT what's left over ⎣⎢ ⎦⎥ Throttle command is a function of aT (i.e., acceleration magnitude) and acceleration limit 14 14 7 2/12/20 Impulsive ΔV Orbital Maneuver • If rocket burn time is short compared to orbital period (e.g., seconds compared to hours), impulsive ΔV 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 15 Orbit Change due to Impulsive ΔV • Maximum energy change accrues when ΔV 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 16 8 2/12/20 Assumptions for Impulsive Maneuver Instantaneous change in velocity vector v2 = v1 + Δvrocket Negligible change in radius vector r2 = r1 Therefore, new orbit intersects old orbit Velocities different at the intersection 17 17 Geometry of Impulsive Maneuver Change in velocity magnitude, |v|, vertical flight path angle, γ, and horizontal flight path angle, ξ ⎡ ⎤ ⎡ ⎤ ⎡ x! ⎤ vx V cosγ cosξ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v y v 1 = ⎢ ! ⎥ = ⎢ y ⎥ = ⎢ V cosγ sinξ ⎥ ⎢ z! ⎥ ⎢ v ⎥ ⎢ −V sinγ ⎥ ⎣ ⎦1 ⎢ z ⎥ ⎣ ⎦1 ⎣ ⎦1 ⎡ ⎤ ⎡ ⎤ ⎡ x! ⎤ vx V cosγ cosξ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v y v 2 = ⎢ ! ⎥ = ⎢ y ⎥ = ⎢ V cosγ sinξ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ z! vz −V sinγ ⎣ ⎦2 ⎢ ⎥ ⎣ ⎦2 ⎣ ⎦2 ⎡ v v ⎤ ⎡ ⎤ x2 − x1 Δvx ⎢ ( ) ⎥ ⎢ ⎥ ⎢ ⎥ Δv = Δvy = v − v ⎢ ⎥ ⎢ ( y2 y1 ) ⎥ ⎢ v ⎥ ⎢ ⎥ ⎢ Δ z ⎥ v − v ⎣ ⎦ ⎢ z2 z1 ⎥ ⎣ ( ) ⎦ 18 18 9 2/12/20 Required Δv for Impulsive Maneuver ⎡ v v ⎤ ⎡ ⎤ x2 − x1 Δvx ⎢ ( ) ⎥ ⎢ ⎥ ⎢ ⎥ Δv = Δvy = v − v ⎢ ⎥ ⎢ ( y2 y1 ) ⎥ ⎢ v ⎥ ⎢ ⎥ ⎢ Δ z ⎥ v − v ⎣ ⎦ ⎢ z2 z1 ⎥ ⎣ ( ) ⎦ ⎡ 1/2 ⎤ Δv2 + Δv2 + Δv2 ⎢ ( x y z )rocket ⎥ ⎡ ⎤ ⎢ ⎥ ΔVrocket ⎢ ⎛ v ⎞ ⎥ ⎢ ⎥ −1 Δ y ⎢ sin ⎜ 1/2 ⎟ ⎥ Δvrocket = ⎢ ξrocket ⎥ = ⎜ 2 2 ⎟ ⎢ ⎝ (Δvx + Δvy ) ⎠ ⎥ ⎢ γ ⎥ rocket ⎣⎢ rocket ⎦⎥ ⎢ ⎥ ⎢ −1 ⎛ Δvz ⎞ ⎥ sin ⎜ ⎟ ⎢ ⎝ ΔV ⎠ ⎥ ⎣⎢ rocket ⎦⎥ 19 19 Single Impulse Orbit Adjustment Coplanar (i.e., in-plane) maneuvers 1 2 2 2 2 E = v − µ r = e −1 µ h • Change energy 2 ( ) • Change angular 2 2 2 2 momentum µ (e −1) µ (e −1) h = = 2 • Change eccentricity E v 2 − µ r 2 2 e = 1+ 2Eh µ vnew ! vold + Δvrocket = 2(E new + µ r) • Required velocity increment 2 ⎡ e2 1 2 h2 r⎤ = ⎣( new − )µ new + µ ⎦ Δvrocket = vnew − vold 20 20 10 2/12/20 Single Impulse Orbit Adjustment Coplanar (i.e., in-plane) maneuvers • Change semi-major axis – magnitude 2 hnew µ – orientation (i.e., argument anew = 2 of perigee); in-plane 1− enew isosceles triangle rperigee = a(1− e) rapogee = a(1+ e) µ ⎛ 1+ e⎞ vperigee = ⎜ ⎟ a ⎝ 1− e⎠ • Change apogee or perigee 1 e – radius µ ⎛ − ⎞ vapogee = ⎜ ⎟ – velocity a ⎝ 1+ e⎠ 21 21 In-Plane Orbit Circularization Initial orbit is elliptical, with apogee radius equal to desired circular orbit radius Initial Orbit a = (rcir(target) + rinsertion ) 2 e = (rcir(target) − rinsertion ) 2a µ ⎛ 1− e⎞ vapogee = ⎜ ⎟ a ⎝ 1+ e⎠ Velocity in circular orbit is a function of the radius “Vis viva” equation: ⎛ 2 1 ⎞ ⎛ 2 1 ⎞ µ vcir = µ⎜ − ⎟ = µ⎜ − ⎟ = ⎝ rcir acir ⎠ ⎝ rcir rcir ⎠ rcir Rocket must provide the difference Δv = v − v rocket cir apogee 22 22 11 2/12/20 Single Impulse Orbit Adjustment Out-of-plane maneuvers • Change orbital inclination • Change longitude of the ascending node • v1, Δv, and v2 form isosceles triangle perpendicular to the orbital plane to leave in-plane parameters unchanged 23 23 Change in Inclination and Longitude of Ascending Node Longitude of Inclination Ascending Node Sellers, 2005 24 24 12 2/12/20 Two Impulse Maneuvers Transfer to Non- Phasing Orbit Intersecting Orbit st Rendezvous with trailing 1 Δv produces target orbit spacecraft in same orbit intersection nd At perigee, increase speed to 2 Δv matches target orbit increase orbital period Minimize (|Δv1| + |Δv2|) to At future perigee, decrease minimize propellant use speed to resume original orbit 25 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 v = ( cir1 cir2 ) cir1 r cir1 e = r − r 2a ( cir2 cir1 ) µ ⎛ 1+ e⎞ v = µ ptransfer ⎜ ⎟ v a ⎝ 1− e⎠ cir2 = rcir 2 µ ⎛ 1− e⎞ v = atransfer ⎜ ⎟ a ⎝ 1+ e⎠ 26 26 13 2/12/20 Outward Transfer Orbit Velocity Requirements Δv at 1st Burn Δv at 2nd Burn v v v v v v Δ 1 = ptransfer − cir1 Δ 2 = cir1 − atransfer ⎛ 2r ⎞ ⎛ 2r ⎞ cir2 cir1 = vcir ⎜ −1⎟ = vcir ⎜1− ⎟ 1 r + r 2 r + r ⎝ cir1 cir2 ⎠ ⎝ cir1 cir2 ⎠ r v