Global Optimization of Impulsive Interplanetary Transfers

Transcription

1 Global Optimization of Impulsive Interplanetary Transfers R. Armellin, Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano Taylor Methods and Computer Assisted Proofs Barcelona, June, 3 7, 2008

2 Motivation Space activities are expensive: Ariane 5 launch cost: 200 M$ Allowed Spacecraft Mass: kg = Cost per kilogram: $/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

3 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 Multiple Gravity Assist (MGA) space pruning

4 Dynamical Model: 2-Body Problem The 2-Body Problem considers two point masses in mutual orbit about each other m 1 The relative motion of the two masses is governed by: m 2 r = k r 3 r E.g. m1 m2 Sun Spacecraft Analytical solutions exist for the 2-Body Problem: Conic Arcs r = r(θ) explicit t = t(θ) implicit (Kepler s equation)

5 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

6 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 Future benchmark problems Multiple Gravity Assist interplanetary transfers E.g.: Cassini-Huygens ] U A [ Y Saturn 01/12/2005 Cassini-Huygens trajectory (Alternative solution) Earth 25/10/1997 Jupiter 17/02/2001 Earth (GA) 0 18/08/1999 Venus (GA2) 24/06/1999 Venus (GA1) 19/05/ X [AU]

7 Optimization Problem The optimization variables are the time of departure t 0 and the time of flight ttof The positions of the starting and arrival planets are computed through the ephemerides evaluation: (re, ve) = eph(t0, Earth) and (rm, vm) = eph(t0 +ttof, Mars) The starting velocity v 1 and the final one v2 are computed by solving the Lambert s problem

8 Optimization Problem The parking velocity and desired final velocities are v c E = µ E /r c E v c M = µ M /r c M The pericenter velocities of the escape and arrival hyperbola v p 1 = 2µ E /r c E + v 1 2 v p 2 = 2µ M /r c M + v 2 2 Objective function V = V 1 + V 2 Constraint: V 1 < V 1,max

9 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 (a, e, i, Ω, ω, M) The nonlinear equation can be analytically evaluated M = E e sin E is solved for the eccentric anomaly E The relation delivers tan θ 2 = 1 + e 1 e tan E 2 θ (Kepler s Eq) 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

10 Lambert s Problem (1/2) Given: initial position r 1 final position r 2 time of flight t tof Find the initial velocity, v1, the spacecraft must have to reach r2 in ttof The solution of the BVP exploits the analytical solution of the 2-body problem Given r1, r2 and ttof there exists only one conic arc connecting the two points in the given time

11 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 β(x) = 2 arcsin, and a(x) = 2a(x) 2(1 x 2 ) The value of s and c depend on r 1 and r2, so the nonlinear equation depends both on t0 and ttof

12 DA Solution of Parametric Implicit eqs Search the solution of p [p l, p u ] f(x, p) = 0 for p belonging to Use classical methods (e.g., Newton) to compute x0 solution of f(x, p 0 ) = 0 Initialize [x] = x 0 + x and [p] = p 0 + p as DA variables and expand f = M( x, p) Build the following map and invert it: ( ) f p = ( ) ( ) ( ) [M] x x [I p ] p p Force f = 0 so obtaining the Taylor expansion of of the solution w.r.t. the parameter: = x = x( p) ( ) 1 ( ) [M] f [I p ] p

13 Example: Mars Ephemerides Epoch interval: 40 days (a) 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 (b)

14 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 of the objective function Taylor representation error w.r.t. point-wise evaluation Box width: 40 days

15 Earth-Mars Direct Transfer Search space: [1000, 6000] [100, 600] Maximum departure impulse: V1 < 5 km/s Platform: Pentium IV 3.06 GHz laptop Objective function overview 16

16 DA Based Global Optimizer (1/2) DA based global optimization algorithm: Subdivide the search space in subintervals Suitably initialize the value of V opt t tof X For each subinterval : Initialize t0 and ttof as DA variables and compute a Taylor expansion of the objective function V and the constraint V1 on X Bound the value of V 1 on IF X min V 1 > V 1,max Bound the value of V on IF min V > V opt X X discard discard X X t 0

17 DA Based Global Optimizer (2/2) Build and invert the map of the objective function gradient: ( t0 V ttof V Localize the zero-gradient point IF ) x / X = M ( t0 t tof ) discard ( t0 t tof ) x = (t 0, t tof ) X = M 1 ( t0 V ttof V ) Evaluate V = V ( x ) V < V opt V opt x X IF update, and store and 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 x

18 Earth-Mars Direct Transfer Search space: [1000, 6000] [100, 600] Maximum departure impulse: V1 < 5 km/s Platform: Pentium IV 3.06 GHz laptop Solution 1: 10-day boxes + 5th order Pruning + Global Opt: s V opt = km/s x* = [ , ]

19 Earth-Mars Direct Transfer Search space: [1000, 6000] [100, 600] Maximum departure impulse: V1 < 5 km/s Platform: Pentium IV 3.06 GHz laptop Solution 1: 10-day boxes + 5th order Pruning + Global Opt: s V opt = km/s x* = [ , ] Solution 2: 100-day boxes + 5th order Pruning + Global Opt: 0.55 s V opt = km/s x* = [ , ]

20 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: Computation time: s Enclosure of the minimum: [ , ] km/s Enclosure of the solution: t 0 [ , ] t tof [ , ]

21 Planet-to-Planet Transfer Number of steps: Computation time: s Enclosure of the minimum: [ , ] km/s Enclosure of the solution: t 0 [ , ] t tof [ , ] Most of the computational time is spent in splitting box containing discontinuities CutOff [km/s] Boxes containing discontinuities with size lower than a given threshold are rejected The solution is mathematically not rigorous Earth-Venus Iteration # x 10 4

22 MGA Space Pruning MGA transfers problems are characterized by High number of variables (+1 for each GA) Constraints on gravity assist radii and VGA Increased multimodality and more discontinuities Jupiter, T3, V3 Earth, T1, V1 Mars, T2, VGA Validated solution is almost impractical (we have obtained solution for 1 GA) The objective function is simply given by V = V1 + VGAi + Vn IDEA: use DA expansion and polynomial bounders just to prune the search space

23 DA Based Pruning of MGA For the first arc: i = 1 Subdivide T 1 and T 2 in subintervals For each subinterval X : X - Compute the expansion of V 1 on X - Bound the polynomial expansion of V 1 on X - IF min V 1 > V 1,max discard X and all its subsequent combinations For each gravity assist: i = 2,..., n 1 Subdivide T i+1 in subintervals For each subinterval remaining from the previous arc and for each combination with the subinterval on T i+1, X - Compute the expansion of V GAi and r pi on X - Bound the polynomial expansion of and on X V GAi r pi

24 DA Based Pruning of MGA - IF min V GAi > V GAi,max or discard X and all its subsequent combinations max r pi < r pi,min For the last arc: i = n For each subinterval remaining from the previous arc V n - Compute the expansion of on - Bound the polynomial expansion of on min V n > V n,max - IF discard X V n X X

25 Cassini-like transfer Problem Definition: Planet E Search Space [day] Int. size [day] V Cut-off [km/s] Dep. Epoch [MJD2000] V tof V tof E tof J tof S tof

26 Cassini-like transfer Tot. boxes Remaining 1085 (0.003%) Obj Fcn km/s This work has been done under ESA Ariadna contract Final report available at ariadna/completed.htm

27 Global Optimization of Impulsive Interplanetary Transfers R. Armellin, Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano Taylor Methods and Computer Assisted Proofs Barcelona, June, 3 7, 2008

28 Conclusions and Future Work DA and TM global optimizers are effective tools for the global optimization of planet-to-planet transfers Efficient management of regions with singularities is needed for TM global optimization with COSY-GO Validated management of nonlinear constraints will be required to apply COSY-GO to MGA transfers DA is a promising technique for search space pruning of high dimensional problems such Multiple Gravity Assist (MGA) interplanetary transfers

29 Verified Implicit Eq Solution - 1D Suppose to have the (n +1) differentiable function f over the domain D = [-1, 1] and its n-th order Taylor model P(x) + I so that f(x) P (x) + I for all x D Consider the enclosure R of P(x) + I over D and suppose P (x) >d >0 on D with P(0) = 0 Find the Taylor Model C(y) + J on R so that any solution of the problem f(x) = y lies in C(y) + J Algorithm: First compute C(y), the n-th order polynomial inversion of P(x), so that P (C(y)) = n y Using Taylor model computation, obtain where P (C(y)) y + J includes the terms of order exceeding n in P(C(y)), and thus scales with at least order n +1 J

30 Verified Implicit Eq Solution - 1D Use the consequences of small correction x to C(y) to find the rigorous reminder J for C(y) so that all the solutions of f (x) = y lie in C(y) +J. According to the mean value theorem: f(c(y) + x) y P (C(y) + x) y + I for suitable ξ [C(y), C(y) + x] Since P is bounded below by d, the set x P (ξ) + I + J will never contain the zero except for the interval J = I + J which is the desired interval d = P (C(y)) + x P (ξ) y + I y + J + x P (ξ) y + I = x P (ξ) + I + J

31 Verified Implicit Eq Solution - vd Let P(x) + I be a n-th order Taylor model of the (n +1) times differentiable function f over the domain f (x) P(x) + I for all x D Indicate with L(x) the linear part of P(x) D = [ 1, 1] v so that: Instead of the original problem and in analogy with the 1D case, consider the problem of finding a verified enclosure of the inverse of L 1 f where L is analytically inverted P(x) + J The Taylor model enclosure of L 1 f over D is: P + J = L 1 (P + I)

32 Verified Implicit Eq Solution - vd It is worth observing that: P 1 = x 1 + h.o.t. we can bound P 1 from below x 1 P 2 = x 2 + h.o.t. we can bound P 2 from below x 2 etc. Consequently we can proceed as in the 1D case on When the solution has been obtained for right-compose with L 1 Application to the solution of f (x, p) = 0: { y = f (x, p) p = p L 1 f L 1 f Once a validated inversion of the system is achieved, just set y = 0

33 Orbital parameters The orbital parameters are: (a, e, i, Ω, ω, θ) The position and the velocity (r, v) in cartesian coordinates are obtained from the orbital parameters by simple algebraic relations