Earth-Mars Halo to Halo Low Thrust

Transcription

1 Earth-Mars Halo to Halo Low Thrust Manifold Transfers P. Pergola, C. Casaregola, K. Geurts, M. Andrenucci New Trends in Astrodynamics and Applications V 3 June / -2 July, 28 Milan, Italy

2 Outline o Introduction o Mathematical ti Fundamentals o Dynamic Systems o Periodic Orbits & Manifolds o Methodology Analysis o Optimization Approach o Global Approach o Local results o Methodology Application o Examples of Application o Conclusions o Inputs o Trajectory o Outputs

3 Introduction Dynamic structures created by a three body model can be used to reduce the fuel mass fraction. Electric Propulsion can further reduce propellant requirements, due to its intrinsic characteristics. Theapplicationofthiscombination can be applied to design Earth-Planet transfers characterized by long transfer times and low propellant requirements. The aim is to develop a methodology to design low thrust interplanetary transfers which link two periodic orbits around Earth and Mars libration points employing the ballistic structures enabled by the three body model.

4 Main Idea Powered phase which h modifies the s/c energy such that t upon thrust completion the s/c adheres the phase-space conditions for the insertion on the destination manifold. Ballistic escape phase on the Earth manifold from a periodic orbit around an Earth-Sun libration point. Ballistic capture phase where the three body dynamics of Sun-Mars guide the s/c onto a periodic orbit around the libration point of Mars.

5 The principal dynamics model considered is the Bi- Circular Model (BCM). It is an extension of the classic Circular restricted Three Body Problem (CR3BP) with the inclusion of another attractor moving in circular motion around the center of mass of the other two primaries. It is a non autonomous system. Dynamic Systems Mars is considered the gravitational perturbation in the Earth-Sun three body model.

6 The Earth escape and Mars capture phase are analyzed by the CR3BP: The other planet s perturbation is very small. It is possible to compute and use the dynamical structures enabled by this model. Dynamic Systems BCM CR3BP L L 2 The BCM is used for the 2 interplanetary powered body phase. More complete than other models (more realistic major propellant mass).

7 In the CR3BP periodic orbits around libration points exist. Halo orbits are considered. These orbits are family of solutions parameterized by the out-of-plane amplitude, for the three dimensional case. ] EL 2 Ax [km] ML Ax [km] 2.35 Periodic Orbits 2.4 x A x 2. 5 EL Az [km] 2 x 4 Earth EL 2 Ay [km] 6.82 x 5 A y ] EL Az [km] 2 x 4 EL 2 Period [dy ys] x x A x.47 5 ML Az [km] x 4 Mars ML Ay [km] A y ML x 4 Az [km] ML Per riod [dys] Period EL Az [km] x Period ML Az [km] x 4

8 Manifolds Associated with the periodic orbits there are two x 4 2 manifolds which identify the L envelopeofallthepossible 2 L M 2 trajectories which lead 2 towards (stable) or move 3 away from (unstable) the periodic solution. y [DU] x [DU] Considering dynamics and geometry of the problem: The L2 unstable Earth manifold is considered d for the escape phase. The L stable Mars manifold is considered for the capture phase

9 Optimization Approach Hybrid optimization approach: elements of the calculus of variations and direct minimization of the performance index. The thrust angles and thruster on/off control laws are computed by the Euler-Lagrange equations, the Pontryagin minimum principle and the Legendre-Clebsh condition. The dynamics of the adjoint variables are derived in the frame of the BCM and integrated together with the dynamics to compute the instantaneous value of the optimum laws [initial guess to assure the convergence].

10 Optimization Approach Also other elements, which do not admit an explicit optimal law, are included in the control vector. In the control vector are integrated also: Mars initial angle (Ω M ) Permanence time on the Earth manifold (t Dep ) Permanence time on the Mars manifold (t Arr ) Duration of the thrusting phase (t Pow ) Amplitudes of departure and arrival Halo orbits (A z ) Initial and final positions on both orbits It is a 4-dimensional vector (6+8) which identifies the search space for a Sequential Quadratic Programming.

11 A Global Approach The SQP method implemented often results in local solutions near the initial guess supplied. A global l approach based on a Genetic Algorithm scheme has been implemented in order to identify the region of the global minimum and comparing local solutions. The final constraints are imposed directly in the fitness function by two (position error / velocity error) weights. Also side constraints are included at each generation. Based on the results of the global approach some local results are derived considering the output as initial guesses and compared with local solutions.

12 Methodology Application In order to apply the described approach some initial parameters specifying the s/c have been imposed. Initial mass [kg] in Earth Halo orbit Engine Specific Impulse [sec] 3 Ion thruster Engine Power [W] decreasing ~/r 2 Thrust Efficiency [-].55 constant Based on these fixed values the control vector has been updated, from an initial guess, in order to minimize the propellant mass to link the two manifold and satisfy final non linear match constraints.

13 Methodology Application Several transfers are implemented following two kind of approaches: Simple Approach: the out of plane amplitudes of the two Halo orbits are considered constant and are removed from the optimization scheme (useful for particular mission with fixed initial and/or final conditions) Advanced Approach: the two amplitudes are reincluded in the control vector and the optimization scheme is applied to the complete

14 Simple Approach: Methodology Application Optimization A z Amplitudes Final Mass Transfer Time Approach [km] [kg] [yrs] Local 5e Global 5e Local from Global 5e The solutions obtained by the genetic algorithm never satisfy perfectly the condition required for the manifold transition. The results for the propellant mass and the total transfer time must be intended as indicative results identifying the region of the global minimum.

15 Advanced Approach: Methodology Application Optimization A z Amplitudes Final Mass Transfer Time Approach Guess [km] [kg] [yrs] Local 4e Local 6e Global Local from Global The initial population is chosen uniformly distributed over the search space defined by boundaries on the various elements of the control vector assuring the feasibility of the solution (in both approaches).

16 Methodology Application Solutions obtained using the local by global strategy are very close to the local solutions for both approaches. The final masses are comparable and also the total transfer time (mainly dependent from the arrival time) range is quite narrow. The transfers resulting lie in the region of the absolute minima for the propellant mass. The method converges also without performing a preliminary global search. The behavior of the thruster on/off variable results always in a thrust-coast-thrust strategy, usually considered as the optimal strategy, although it is not a priori imposed.

17 Examples of Application: Input Data A representative transfer between two generic Halo orbits is shown. It is obtained with a Local solution in an Advanced Approach: Two initial A z of 4[km] are chosen and a departure, arrival and powered time of [yrs] guessed, moreover an initial Mars angle of 2[deg]. It is one of the solutions with the minor total transfer time found.

18 Trajectory: Earth Frame Red Earth L2 unstable manifold Blue Mars L stable manifold Synodic Frame Sun Earth System [μ= 3.9e 6] 2.5 Synodic Frame Sun Earth System [μ= 3.9e 6] 2.5 Arrival Phase y [DU].5.5 Sun EL EL 2 Earth y [DU].5.5 Sun Coasting Phase EL EL 2 Earth x [DU].5 Powered Phase x [DU]

19 Trajectory : Earth Frame The resulting transfer completes approximately one revolution on the Halo orbit, then it follows the unstable manifold up to the powered phase. y [DU] Synodic Frame Sun Earth System [μ= 3.9e 6] EL EL 2 Earth x [DU] Synodic Frame Sun Earth System [μ= 3.9e 6] x 4.5 y [DU] Synodic Frame Sun Earth System [μ= 3.9e 6] EL EL 2 Earth x [DU] Synodic Frame Sun Earth System [μ= 3.9e 6] x 4 Earth (or Mars) is always visible during the permanence on the Halo orbit. z [DU].5 EL.5.5 x 3.5 EL EL 2 2 EL Earth Earth y [DU] x [DU] 5 x 3 y [DU] z [DU]. x [DU].2

20 Trajectory : Mars Frame Red Earth L2 unstable manifold Blue Mars L stable manifold Synodic Frame Sun Mars System [μ= e 7] Synodic Frame Sun Mars System [μ= e 7] Departure Phase.5.5 Coasting Phase y [DU DU] Sun ML Mars y [DU DU] Sun ML Mars x [DU] Powered Phase.5.5 x [DU]

21 Trajectory : Mars Frame The solution realizes the insertion conditions on the stable Mars manifold quite far from the Halo orbit. y [DU] Synodic Frame Sun Mars System [μ= e 7] Synodic Frame Sun Mars System [μ= e 7] ML ML Mars Mars y [DU] x [DU] x [DU] Synodic Frame Sun Mars System [μ= e 7] Synodic Frame Sun Mars System [μ= e 7] In some solutions the powered phase ends directly in the Halo orbit [in this way the total transfer time can be reduced]. z [DU].5.5 x 4.99 ML Mars y [DU] x 3 x [DU] z [DU] x ML Mars x [DU] y [DU] x 3

22 Trajectory : Inertial Frame y [DU] Mission Outputs: Inertial Frame Mars Sun Earth Az [km]earth Halo Az [km]mars Halo Δt [yrs] Departure.395 Δt [yrs] Powered Δt [yrs] Arrival.7283 Mars Initial Angle [deg] Δt [yrs] Total Final Mass [kg] x [DU]

23 Trajectory : main phases Coasting Phase On Off On Earth Halo Escape Mars Halo Capture

24 Transfer Analysis: Optimum laws Thruster off Thrust-Coast-Thrust

25 Transfer Analysis: Position & Velocity Earth Escape Escape Capture Mars Capture

26 Conclusions A methodology to design a manifolds-low thrust Halo to Halo interplanetary transfer has been defined and applied. Derivation and application of the adjoint equations for an indirect optimization approach in the BCM. A global search, followed by several local solutions results in minimum propellant representative transfers. The propellant mass fraction computed for the various transfers results always in the order of 2% with a transfer time of the order of 3-4.5[yrs]. Significant saving if compared with results obtained without thruster on/off and power decrease [M prop p 3 %]. Data suitable just for cargo-like missions.