parallel high-fidelity trajectory optimization with application to … · 2018. 6. 20. ·...

Parallel High-fidelity Trajectory Optimization with Application to CubeSatDeployment in an Earth-moon Halo Orbit

Hongru Chen1) Mickaël Gastineau1) Daniel Hestroffer1) and Vishnu Viswanathan1)

1) Observatoire de Paris, IMCCE, 75014 Paris, France

(Received 24th April 2018)

The application of CubeSats to interplanetary missions is currently being exploited. However, trajectories ofinterplanetary CubeSats are greatly constrained by the very limited propulsion capacity. Electric propulsion systemsthat enjoy a high Isp as well as a high Δv budget is promising to CubeSats. However, the low thrust of electricsystems may not be applicable for some trajectories needing for quick response. In addition to propulsion, theephemeris and gravity of celestial bodies can exert positive or negative influence on CubeSat trajectories, whichresults in significant Δv saving or loss to the CubeSat. CubeSat trajectories should be designed as well as verified in ahigh-fidelity model accommodating the ephemeris and gravity of celestial bodies, and specifications of the CubeSatand the propulsion system. A parallel high-fidelity trajectory optimization tool, PHITO, is presented in this paper.How PHITO is applied to trajectory analysis and propulsion system sizing is discussed through a case study of thephasing trajectory of an Earth-moon halo orbit. The result of Δv reveals the effect of the full-ephemeris n-bodydynamics.

Key words: Trajectory Optimization, Interplanetary CubeSats, Parallel Computation

1. Introduction

The application of CubeSats to interplanetary spacemissions is currently exploited. Compared to Earth-orbiting CubeSats, interplanetary CubeSats generallyneed to perform orbit change and deployment. Forinstance, the EM-1 mission has selected 13 CubeSatsintended for a wide range of mission objectivesincluding landing on the moon, lunar observation,solar sail demonstration, and asteroid flyby(1. The 13CubeSats will be released at different mid-points, andhead to different destinations. By deploying CubeSats,collaborative operations, such as inter-satelliteranging, which is useful for autonomous navigation,and interferometric measurement, can be performed.For instance, the AIM mission is planed to rendezvouswith the binary asteroid 65803 Didymos and thendeploy a lander and two or more CubeSats to performinter-satellite network(2. There will be many novelexploration manners enabled by deployed CubeSats.However, upon release, CubeSats can only rely on thesmall propulsion system on board to reach desireddestinations. The propulsion capacity of a CubeSat is greatlyrestrained by its limited mass and volume budget, thusthe trajectory design of CubeSat is subject to strictconstraint compared to general interplanetaryspacecraft. Electric propulsion systems that generallycome with high specific impulse, Isp, are promising forinterplanetary CubeSats. A high Isp suggests that theCubeSat can enjoy a high Δv budget for a certain massbudget. One drawback of the electric propulsionsystem is the generally low thrust, which may not be

applicable for the trajectories requiring quickmaneuvers. In addition to propulsion, the gravity of celestialbodies exerts positive or negative influence on thetrajectory, depending on the mission date. Aspropellant cost is crucial to a CubeSat, it is necessaryto design and analyze CubeSat trajectories in the high-fidelity model accommodating the ephemeris andgravity of celestial bodies, and the specification of thepropulsion system. This paper presents a Parallel HIgh-fidelityTrajectory Optimization tool (PHITO). Same as otherdeveloped trajectory optimization tools, such asMALTO(3, GALLOP(4, and jTOP(5, the present tooladopts techniques of direct optimization and multipleshooting. The tool is programmed to be compatiblewith parallel computing environment. To be specific,the propagation of trajectory states and the partialderivatives of each trajectory segment is implementedin parallel, which greatly saves the running time. Thetool is applied to the low-thrust phasing trajectory of aCubeSat to an Earth-moon L2 halo orbit. Preliminarydesign in a simplified model shows this trajectoryimposes relatively high requirement of the thrust. Thetrajectories as well as required Δv are solved fordifferent mission starting dates. The feasibility of acertain type of propulsion system and the influence ofthe ephemeris on Δv are thus revealed.

Proceedings of iCubeSat 2018, the 7th Interplanetary CubeSat Workshop, Paris, France1

2. Structure of PHITO

2.1. High-fidelity Propagation

To have high-fidelity trajectories, the propagatortakes into account thrusting force, solar radiationpressure, the ephemeris and gravity field of thecelestial bodies, such as the Sun, the Earth, the moon,Jupiter, etc. The thrusting force is modeled as acontinuous force vector, FT, acting on the trajectorywhen the thruster is turned on. As the optimization isbased on direct optimization and multiple shootingmethods (see next section), where trajectory states andcontrols are discretized, FT is made constantduring the given period of propagation. Theequations of motion written in the form ofordinary differential equation, , are,

(1)

where n is the number of celestial bodies taken intoaccount, mi denotes the gravitational parameter of thei-th celestial body, rji denotes the relative positionfrom the j-th celestial body to the i-th celestial body,riSat denotes that from the i-th celestial body to theCubeSat, vjsat the relative velocity of CubeSat to the j-th celestial body, FSRP the solar radiation pressure, andm the mass of the CubeSat. In Eq. (1), the j-th celestialbody is chose to be the center of the coordinate. Thechoice of the center depends on the trajectoryscenario. For instance, if the trajectory is around themoon, then the moon is suitable for the central body.If the trajectory is mostly heliocentric, the Sun or solarsystem barycenter can be the center. In case of solarsystem barycenter, the second term of the second rowof the left-hand side is zero. FSRP is expressed as,

(2)

where Cr and A are the reflectivity coefficient and theexposed area of the CubeSat, respectively, AU is theastronomical unit, and P is the solar power at 1 AU.These parameters are considered constant by thecurrent propagator. Positions of celestial bodies areobtained from an ephemeris tool. The propellant losswith time due to thrusting is computed by Eq. (3),

(3)

where is the mass loss rate, Isp is the specificimpulse, and g0 is the standard gravity on the Earth.The input of the propagator consists of initialtrajectory states xi, including position and velocity,initial mass mi of the CubeSat, and FT. The propagatorintegrates Eq. (1) and Eq. (3). The output includes

final states [xf, mf]. In addition, to facilitateoptimization, the propagator also integrates the partialderivatives of [xf, mf] with respect to [xi, mi, FT],which consists of another 7 x 9 = 63 elements. Thetime derivatives of the partial derivatives aresummarized in Appendix A.

2.2. Optimization

2.2.1. Multiple shooting

The optimization is programmed based on directoptimization method and multiple shooting method.Both methods discretize trajectory states and controls.The multiple shooting method is illustrated in Fig. 1.The whole trajectory is divided into N segments. Thestate of the starting node of each segment, xk

-, is setfree for optimization. For trajectory continuity, theoptimizer tunes the initial state and control, in orderfor the final state to match the initial state of the nextsegment. Although the number of free variables andconstraints is increased, the sensitivity to initialguesses, however, is reduced, and thus the robustnessof the optimization is improved. The interested readeris referred to Ref. 6) and 7).

Figure 1. Concept of multiple shooting

2.2.2. Objectives, variables, and constraints

The objective of the trajectory optimization is tominimize the propellant loss, Δm. Free variablesgenerally include the states of the starting node, xk

-,and the control, FT,k, of each segment. Trajectorystates meet equality constraints at the initial and finalboundaries and the match points. FT is constrained bythe maximum thrust of the thruster, namely, ||FT|| ≤FTmax. The optimizer tunes free variables such that Δmis minimized and all constraints are met. The usedoptimizer is the Sparse Nolinear OPtimization Tool,SNOPT(8, (9.

2.2.3. Gradients

The optimizer needs gradients of constraints andobjective with respect to states and controls toimprove the feasibility and optimality of the solutionat each iteration. As mentioned in the previoussection, partial derivatives with respect to initial statesand controls are propagated along with trajectorystates. Using chained rule, gradients of constraints andobjective with respect to initial states and controls canbe easily derived. The obtained gradients are fed tothe optimizer. In this way, the accuracy, speed, and


( )fx x

robustness of the optimization process are greatlyimproved.

2.3. Parallel implementation and ephemeris tool

Since the propagation and evaluation of gradients ofeach segment is uncoupled, running time can begreatly saved by propagating of trajectory segmentsand the partial derivatives in parallel. However, forcalling ephemeris during propagation, the mostwidely-used ephemeris tool SPICE is not compatiblewith parallel computing environment. To implementthe parallel algorithm in a parallel environment, theephemeris tool CALCEPH(10 that is developed byIMCCE and designed to be compatible with parallelcomputing environment is used. The current PHITOcalls the propagator using the MATLAB parallelroutine PARFOR. The parallel implementation of theoptimization is thus verified.

3. Case Study: Halo Orbit Phasing Trajectory

3.1. Problem statement

PHITO is applied to the optimization of a halo orbitphasing trajectory to demonstrate its use for trajectoryanalysis. A previous work proposes a mission conceptof providing the positioning service for landers orrovers on the far side of the moon(11. The missionrequires that at least four CubeSats are deployedevenly along an Earth-moon L2 halo orbit. For thismission, the payloads of each CubeSat are thecommunication system that is capable of ranging, anda highly-stable clock. Components of a CubeSat arelisted in Table 1. The mass of the CubeSat without apropulsion system is 4.2 kg with a 3U structure, or 4.7kg with a 6U structure. The layout of the CubeSat isshown in Fig. 2. As it is costly to send all CubeSats bymultiple launches, it is desired that all CubeSats arereleased as a whole at the stable manifold converginginto the halo orbit or at the halo orbit, and then theCubeSats get separated using their propulsion systemson board. Two-impulsive low-Δv phasing trajectoriesto change the halo phase angles of CubeSats havebeen designed in the simplified Earth-moon circularrestricted 3-body problem (CR3BP)(12. The trajectoriesto separate four CubeSats evenly along a halo orbit ofa z-amplitude of 15,000 km is shown in Fig.3. Δv andtime of flight (ToF) for each CubeSat are listed inTable 2. The trajectory of Sat-3 requires relativelyquick response as it has to gain the highest Δv of 101m/s in a short ToF of 50 days, which can pose asevere constraint of the thrust magnitude. The low-thrust trajectory for Sat-3 is discussed in the rest ofthe section.

Table 1. Components of the CubeSat (without apropulsion system)

Fig.2.

Layout of the CubeSat

Fig.3. Phasing trajectories to separate the CubeSatsalong the Earth-moon halo orbit.

Table. 2. Deployment cost


Table 3. Suitable propulsion models

3.2. Solution existence and thrust magnitude

To gain a quick view of the solution existence interms of the thrust magnitude, many runs ofoptimization are performed in the simplified Earth-moon CR3BP with different magnitudes of thrustingacceleration, accthr. The result of Δv vs accthr is shownin Fig. 4. It can be seen that when accthr is high, Δv ismaintained at a low level, Δv starts to increase as accthr

is decreased to 5x10-4 m/s2, and there is no solution foraccthr < 7.4x10-5 m/s2, which basically has excludedmost models of micro electric propulsion systems. Asurvey of propulsion systems is carried out to findsuitable systems that meets both Δv and accthr

requirements, taking into account the mass of theCubeSat, the thrust, Isp, propellant, and dry mass of thepropulsion system. The suitable models are listed inTable 3. Among the candidates, the cold-gas systemoccupies larger mass and volume than the electrospraysystem, but can generate less Δv. The chemicalsystems are best in terms of all indices, but they areconsidered less stable than cold-gas and electricsystems. Assuming the mission has to use theelectrospray system, a further effort should be made toverify the low-thrust trajectory in the high-fidelitymodel.

Fig. 4. Δv vs accthr

3.3. Results in the high-fidelity model

Trajectories starting on the Julian dates sampledevery 10 days during the period from 2019-2020 areoptimized by PHITO. Because in the high-fidelitymodel, closed halo orbits do not exist, the boundaryconditions are slightly loosen. That is, the boundary xpositions of the halo orbit in the rotating frame areallowed to vary within a range of 3000 km, and the

boundary velocity vy are allowed to vary within arange of 10 m/s. The result of Δv is shown in Fig.5.The lunar phase angle is also displayed for reference.The result shows that the lowest Δv generally happenat the lunar true anomaly between 205 and 237 deg.The largest difference among the values is 21 m/s. Bychoosing a favorable mission date, such an amount ofΔv can be saved. However, this condition does notapply to most CubeSats as secondary payloads.Overall, Δv does not vary much as the standarddeviation is only 6 m/s. The mean value is 96 m/s. Ifthe trajectory is considered in the simplified CR3BPwithout considering propellant loss, the correspondingΔv obtained from Fig.4 is 104 m/s. That means thefull-ephemeris n-body dynamics generally bringspositive effect in this case. An example trajectorystarting on 01-01-2019 is shown in Fig. 6. Thenominal two-impulsive trajectory in the CR3BP isalso shown for comparison. The high-fidelitytrajectory deviates from the nominal evidently,especially on the halo-orbit leg.

4. Conclusion

This paper presents the high-fidelity trajectoryoptimization tool, PHITO, which can be implementedin a parallel environment with running time saved.The tool takes into account forces in the reality andthe specifications of the satellite and the propulsionsystem. How PHITO is applied to trajectory analysisand propulsion system sizing is discussed through acase study of the phasing trajectory an Earth-moonhalo orbit. The variation Δv for different mission datesreveals the influence of the planetary ephemeris,however, that influence is generally positive, as the Δvis generally lower than that obtained in the simplifiedmodel.

5. Acknowledgment

The first author wants to thank Dr. StefanoCampagnola (JPL) and Yazhe Meng (Technology andEngineering Center for Space Utilization, CAS) forthe useful discussions.


Fig. 5. Δv for different mission dates

Fig. 6. Low-thrust trajectory (red) in the high-fidelitymodel and nominal two-impulsive trajectory (black)

in the CR3BP.

References

1) Space Launch Systems (SLS) Scoundary Payload User’s Guide, NASA Technical Report, 2015

2) Kuppers, M., Carnelli, I., Galvez, A., Mellab, K., Michel, P., and the AIM team.: The Asteroid Impact Mission (AIM), Proceedings of European Planetary Science Congress 2015, 10, 2015

3) Sims, J., Finlayson, P., Rinderle, E., Vavrina, M., and Kowalkowski, T.: Implementation of a Low-Thrust Trajectory Optimization Algorithm for Preliminary Design, Proceedings of AIAA/AAS Astrodynamics Specialist Conference and Exhibit, 2006

4) Yam, C. H. and Kawakatsu, Y.: GALLOP: A Low-Thrust Trajectory Optimization Tool for Preliminary and High Fidelity Mission Design, Proceedings of 30th International Symposium on Space Technology and Science, Kobe, Japan,2015

5) Campagnola, S., Ozaki, N., Sugimoto, Y., Yam, C. H., Chen, H., Kawabata, Y., Ogura, S., Sarli, B., Kawakatsu, Y., Funase, R., and Nakasuka, S.: Low-Thrust Trajectory Design and Operations of PROCYON, The First Deep-space Micro-spacecraft, Proceedings of 25th International Symposium on Space Flight Dynamics, Munich, Germany, 2015

6) Betts, J. T. and Huffman, W. P.: Application of Sparse Nonlinear Programming to Trajectory Optimization, Journal of Guidance, 15 (1), pp. 198–206, 1990

7) Betts, J. T.: Survey of Numerical Methods for Trajectory Optimization, Journal of Guidance, Control, and Dynamics, 21(2), pp. 193–207, 1998. ISSN 0731-5090. doi:10.2514/2.4231.

8) Gill, P. E., Murray, W., Saunders, M. A., and Wong, E.: User’s Guide for SNOPT 7.7: Software for Large-Scale Nonlinear Programming, Center for Computational Mathematics Report CCoM 18-1, Department of Mathematics, University of California, San Diego, La Jolla,

CA, 2018.9) Gill, P. E., Murray, W., and Saunders, M. A.: SNOPT: An

SQP algorithm for large-scale constrained optimization, SIAM Rev., 47, pp. 99–131, 2005.

10) Gastineau, M., Laskar, J., Fienga, A., and Manche, H.: CALCEPH - C language -Release 3.1.0, IMCCE/Observatoire de Paris, 2018.

11) Chen, H., Liu, L., Meng, Y., Long, L., Xu, Z., and Liu, J.: Preliminary Mission Design and Analysis of a Lunar Far-side Positioning CubeSat Mission, Proceedings of 26th International Symposium on Space Flight Dynamics, Matsuyama, Japan, 2017.

12) Chen, H. and Jian, M.: Phasing Trajectories to Deploy a Constellation in a Halo Orbit, Journal of Guidance Control,and Dynamics, 10(40), pp. 2662–2667, 2017. doi: 10.2514/1.G002518.

Appendix A

The relationship between the current trajectory stateand initial state and control is expressed as,

(4)

The time derivative of Eq. (4) is expressed as,

(5)

To find , differentiating Eq.(1) and (3) yields,

(6)

Comparing Eq.(5) and (6) results in the expression,

(7)

The first term is expressed as,

(8)


G(rjsat) is the gradient of conservative forces, in otherwords, the hessian matrix of corresponding potential,and thus is symmetric. Elements of G(rjsat) arecomputed from,

(9)where xiSat , yiSat , and ziSat are the x, y, and zcomponents of the relative position of the CubeSat tothe i-th celestial body. The other non-zero terms inEq.(7) are computed from,

(10)

and

(11)

Finally, the propagator can integrate Eq. (7) alongwith Eq.(1) and (3) to obtain both trajectory states andthe partial derivatives with respect to initial states.


parallel high-fidelity trajectory optimization with application to … · 2018. 6. 20. ·...

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