Rigorous Global Optimization of Impulsive Planet-to-Planet Transfers R. Armellin, P. Di Lizia, Politecnico di Milano K. Makino, M. Berz Michigan State University 5th International Workshop on Taylor Model Methods Toronto, May 20 – 23, 2008
Motivation ‣ Space activities are expensive: Ariane 5 launch cost: 200 M$ ÷ Allowed Spacecraft Mass: 10000 kg = Cost per kilogram: 20000 $/kg ‣ Propellant represents the main contribution to s/c mass: • Propellant is on average 40% of spacecraft mass we want to reduce the required propellant ‣ The goal of the trajectory design is to find the best solution in terms of propellant consumption while still achieving the mission goals Roberto Armellin 2
Outline ‣ Dynamical Model ‣ Patched-Conics Approximation ‣ Two-Impulse Transfers • Ephemerides Evaluation • Lambert´s Problem Solution ‣ Differential Algebra Based Global Optimization ‣ Rigorous Global Optimization with COSY-GO Roberto Armellin 3
Dynamical Model: 2-Body Problem ‣ The 2-Body Problem considers two point masses in mutual orbit about each other The relative motion of the two m 1 masses is governed by: r = − k ¨ m 2 � r 3 � r E.g. m 1 Sun m 2 Spacecraft Analytical solutions exist for the 2-Body Problem: Conic Arcs • explicit r = � r ( θ ) � • implicit (Kepler´s equation) t = t ( θ ) Roberto Armellin 4
Patched-Conics Approximation ‣ The whole interplanetary transfer is divided in several arcs ‣ Each arc is the solution of a 2-Body Problem considering the spacecraft and only one other planet at a time E.g.: 2-impulse Earth-Mars transfer 3 conic arcs Earth escape Heliocentric phase Mars capture Roberto Armellin 5
2-Impulse Planet-to-Planet Transfer ‣ 2-impulse Earth-Mars transfer has been selected as first benchmark problem • Applied for preliminary design of Earth-Mars (any planet to planet transfer) interplanetary transfers • Objective function characterized by several comparable local minima 10 ‣ Future benchmark problems 8 Saturn 01/12/2005 6 Jupiter 17/02/2001 • Multiple Gravity Assist 4 Earth interplanetary transfers 25/10/1997 2 E.g.: Cassini-Huygens Earth (GA) 0 18/08/1999 Venus (GA2) 24/06/1999 Venus (GA1) (11 conic arcs) 19/05/1998 -2 -8 -6 -4 -2 0 2 4 Roberto Armellin 6
Optimization Problem ‣ The optimization variables are the time of departure t 0 and the time of flight t tof ‣ The positions of the starting and arrival planets are computed through the ephemerides evaluation: ( r E, v E ) = eph ( t 0 , Earth ) and ( r M, v M ) = eph ( t 0 +t tof , Mars ) ‣ The starting velocity v 1 and the final one v 2 are computed by solving the Lambert´s problem Roberto Armellin 7
Optimization Problem ‣ The parking velocity and desired final velocities are � � v c v c µ M /r c µ E /r c M = E = E M ‣ The pericenter velocities of the escape and arrival hyperbola � � v p v p 2 2 µ E /r c 2 µ M /r c 2 E + v ∞ M + v ∞ 1 = 2 = 1 2 ‣ Objective function ∆ V = ∆ V 1 + ∆ V 2 ‣ Constraint: ∆ V 1 < ∆ V 1 ,max Roberto Armellin 8
Ephemerides Evaluation ‣ Polynomial interpolations of accurate planetary ephemerides (JPL-Horizon) are used for the preliminary phase of the space trajectory design ‣ Given an epoch and a celestial body, its orbital parameters can be analytically evaluated ( a, e, i, Ω , ω , M ) ‣ The nonlinear equation (Kepler’s Eq) M = E − e sin E is solved for the eccentric anomaly E � ‣ The relation delivers θ tan θ 1 − e tan E 1 + e 2 = 2 ‣ The position and the velocity ( r , v ) of the celestial body in inertial frame reference frame are computed We have to solve an implicit equation: Kepler´s equation Roberto Armellin 9
Lambert´s Problem (1/2) Given: ‣ initial position r 1 Find the initial velocity, v 1 , ‣ final position r 2 the spacecraft must have to ‣ time of flight t tof reach r 2 in t tof • The solution of the BVP exploits the analytical solution of the 2-body problem • Given r 1 , r 2 and t tof there exists only one conic arc connecting the two points in the given time Roberto Armellin 10
Lambert´s Problem (1/2) ‣ Several algorithms have been developed for the identification and characterization of the resulting conic arc ‣ We used an algorithm developed by Battin (1960) ‣ A nonlinear equation must be solved (Lagrange´s equation for the time of flight): f ( x ) = log( A ( x )) − log( t tof ) = 0 in which , A ( x ) = a ( x ) 3 / 2 (( α ( x ) − sin( α ( x ))) − ( β ( x ))) � s − c � s , and β ( x ) = 2 arcsin a ( x ) = 2(1 − x 2 ) 2 a ( x ) ‣ The value of s and c depend on r 1 and r 2 , so the nonlinear equation depends both on t 0 and t tof Roberto Armellin 11
DA Solution of Parametric Implicit eqs ‣ Search the solution of for p belonging to f ( x, p ) = 0 p ∈ [ p l , p u ] ‣ Use classical methods (e.g., Newton) to compute x 0 f ( x, p 0 ) = 0 solution of ‣ Initialize and as DA [ p ] = p 0 + ∆ p [ x ] = x 0 + ∆ x variables and expand ∆ f = M ( ∆ x, ∆ p ) ‣ Build the following map and invert it: − 1 � � ∆ f � � [ M ] � � ∆ x � � � � � � ∆ x [ M ] ∆ f = = [ I p ] [ I p ] ∆ p ∆ p ∆ p ∆ p ‣ Force ∆ f = 0 so obtaining the Taylor expansion of of the solution w.r.t. the parameter: ∆ x = ∆ x ( ∆ p ) Roberto Armellin 12
Example: Mars Ephemerides Epoch interval: 40 days (a) (b) Errors on position, (a), and velocity, (b), between the DA and the point-wise evaluation of Mars ephemerides Errors drastically decrease when the order of the Taylor series increases Roberto Armellin 13
Example: Objective Function ‣ The DA evaluation of the planetary ephemerides and the Lambert´s problem solution enables the Taylor expansion of the objective function Taylor representation Taylor representation error of the objective function w.r.t. point-wise evaluation Box width: 40 days Roberto Armellin 14
Earth-Mars Direct Transfer Search space: [1000 , 6000] × [100 , 600] Maximum departure impulse: ∆ V 1 < 5 km/s Platform: Pentium IV 3.06 GHz laptop Objective function overview Roberto Armellin 15 16
DA Based Global Optimizer (1/2) DA based global optimization algorithm: ‣ Subdivide the search space in subintervals � t tof X ‣ Suitably initialize the value of ∆ V opt t 0 � For each subinterval : X ‣ Initialize t 0 and t tof as DA variables and compute a Taylor expansion � of the objective function ∆ V and the constraint ∆ V 1 on X � ‣ Bound the value of on ∆ V 1 X � IF discard min ∆ V 1 > ∆ V 1 ,max X � ‣ Bound the value of on ∆ V X � IF discard min ∆ V > ∆ V opt X Roberto Armellin 16
DA Based Global Optimizer (2/2) ‣ Build and invert the map of the objective function gradient: � ∇ t 0 ∆ V � t 0 � t 0 � ∇ t 0 ∆ V � � � � = M − 1 = M ∇ t tof ∆ V ∇ t tof ∆ V t tof t tof x ∗ = ( t ∗ ‣ tof ) Localize the zero-gradient point � 0 , t ∗ x ∗ / ∈ � � X IF discard � X ∆ V ∗ = ∆ V ( � ‣ Evaluate x ∗ ) ∆ V ∗ < ∆ V opt � x ∗ IF update , and store and � ∆ V opt X ‣ x ∗ If necessary, a more accurate identification of the actual optimum � can be finally achieved using a higher order DA computation on the last � stored subinterval X Roberto Armellin 17
Earth-Mars Direct Transfer Search space: [1000 , 6000] × [100 , 600] Maximum departure impulse: ∆ V 1 < 5 km/s Platform: Pentium IV 3.06 GHz laptop Solution 1: • 10-day boxes + 5th order • Pruning + Global Opt: 59.98 s • = 5.6973 km/s ∆ V opt • x* = [3573.188, 324.047] Roberto Armellin 18
Earth-Mars Direct Transfer Search space: [1000 , 6000] × [100 , 600] Maximum departure impulse: ∆ V 1 < 5 km/s Platform: Pentium IV 3.06 GHz laptop Solution 1: • 10-day boxes + 5th order • Pruning + Global Opt: 59.98 s • = 5.6973 km/s ∆ V opt • x* = [3573.188, 324.047] Solution 2: • 100-day boxes + 5th order • Pruning + Global Opt: 0.55 s • = 5.6974 km/s ∆ V opt • x* = [3573.530, 323.371] Roberto Armellin 19
Verified GO of Earth-Mars Transfer ‣ Implicit equations can be solved in a verified way enabling the Taylor Model evaluation of the objective function ‣ COSY-GO is applied for the global optimization of an impulsive Earth-Mars transfer ‣ Number of steps: 216911 ‣ Computation time: 4954.39 s ‣ Enclosure of the minimum: [5.6974155, 5.6974159] km/s ‣ Enclosure of the solution: t 0 ∈ [3573 . 176 , 3573 . 212] t tof ∈ [324 . 034 , 324 . 088] Roberto Armellin 20
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