SMALL bodies of the solar system, such as Phobos, cannot

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1 INSTITUTO SUPERIOR TÉCNICO, LISBON, PORTUGAL, JUNE Controlled Approach Strategies on Small Celestial Bodies Using Approimate Analtical Solutions of the Elliptical Three-Bod Problem: Application to the Mars- Sstem Rui M. S. Martins Instituto Superior Técnico, Lisbon, Portugal Abstract In this thesis, quasi-satellite orbits in the elliptic restricted three-bod problem are studied from a preliminar mission design point of view. A description of such orbits and their long-term evolution is given, and the stabilit conditions investigated are used for the case of the Mars- sstem. A first-order solution on the osculating elements is obtained with the use of perturbation theories and validation is performed using the numerical model as comparison term. A trajector design algorithm is developed using the obtained analtical solutions with the purpose of studing different approach strategies to the Martian moon. An energ consumption analsis is performed to understand which strateg is more efficient in terms of fuel consumption. Due to the approimations made, the obtained analtical solutions have low validit when in close proimit to, as a result a control procedure is introduced onto the trajector algorithm, which performs timel corrections to the actual trajector. Two landing scenarios are considered providing some fleibilit in terms of mission design. Kewords Quasi-Satellite Orbits, Elliptic Restricted Three-Bod Problem, Variation of Parameters, Differential Evolution, Mars- Sstem. I. INTRODUCTION SMALL bodies of the solar sstem, such as, cannot be circumnavigated in Keplerian-tpe orbits because, in the contet of the three-bod problem 3BP), the Lagrange points are ver close to the surface of the small bod and the region of influence of the small bod ends below its surface [1]. This makes the problem of orbiting small bodies more difficult to solve because it is necessar to account for the gravitational attraction of the larger bod of the sstem, which in our case is Mars. The special interest for results from the fact that the Martian moon is one of the most prominent destinations for future scientific missions, and discovering its unknown origin and understanding its basic scientific nature could provide some insight about the evolution of planets and small bodies M. Sc. Candidate, Department of Mechanical Engineering, Avenue Rovisco Pais, Instituto Superior Técnico; rui.s.martins@ist.utl.pt The present work was performed under the supervision of Paulo J. S. Gil, Assistant Professor, Department of Mechanical Engineering, Instituto Superior Técnico, UTL, Avenue Rovisco Pais. of our solar sstem []. Also, the parameters of the problem make it particularl difficult to solve [1]. The eccentricit of orbit and its gravitational parameter µ P h are relativel high with respect to other small bodies of the solar sstem. Therefore, b solving the problem for case and testing its validit will enable the use of these solutions for other interesting bodies of the solar sstem, which have smaller µ and orbital eccentricit. To land on it is crucial to first determine an adequate landing site, which requires orbiting the moon and performing observations from a distance. Although it is not possible to orbit the moon in a Keplerian wa, it is still possible to do it using a special kind of orbits called QSO s [1]. Such orbits eist beond the Lagrange points, as seen from the snodic reference frame usuall used in the three-bod problem. The reference frame s origin is located at the centre of and rotates with the angular velocit of around Mars. The trajector that the S/C performs around does not fit well the concept of orbit in the usual sense of a perturbed Keplerian trajector so, for convenience of description, the term epiccle will be used to designate each orbit of the S/C around in the snodic reference frame while the term QSO will be used to describe the entire dnamics of the epiccles. In this configuration, it is possible to describe a coplanar QSO relative to the orbital plane of as a retrograde motion in elliptical epiccles that drift forward and backwards in the direction of the orbital velocit of the moon. In Fig. 1 it is presented the difference between two S/C orbits around when we take into account gravit and when the gravit of is neglected. As it is possible to observe in Fig. 1, the motion of a S/C in the vicinit of is considerabl different when we account for its gravit. This shows that gravit must be taken into account; in general it tends to stabilize the motion of the S/C around it, but depending on the initial conditions it ma have the opposite effect [1]. Theoreticall when the motion is stable, the epiccles continue to drift back and forth indefinitel. In practice, the perturbation forces cause the S/C to eventuall drift awa, which means that stabilit can onl be assured for a sufficientl long period of time.

2 INSTITUTO SUPERIOR TÉCNICO, LISBON, PORTUGAL, JUNE With Gravit Without Gravit using the program Wolfram Mathematica R to obtain one independent solution and then use the method described in [6], [7] to obtain the second independent solution from the first. The resulting solutions for and are too complicated to draw an conclusions about the motion of a S/C in the vicinit of. So given the fact that the eccentricit of the orbit of is small e = ), it is possible to simplif the obtained solutions using a series epansion with respect to the eccentricit, centered around e = 0, omitting the terms of the second and higher orders, resulting in f) =1 + e cos f)c 1 sin f + C cos f) ec + C 3 ec 3 cos f 3eC 3 f sin f, 400 Fig. 1: Difference between two S/C orbits around when we take into a account gravit and when the gravit of is neglected. Initial conditions:,, z) = 0, 50, 0) [] and v, v, v z ) = 5, 0, 0) [m/s]. II. ANALYTICAL SOLUTIONS The equations of motion for the three dimensional elliptic restricted three bod problem in pulsating variables are [3]: ) d df = d df e cos f R 3, ) d df = d df e cos f R 3, 1) d z df = z e cos f z R 3 ), where e and f represent the eccentricit and the true anomal of the orbit of, while R 1 and R are the distance of the third bod to Mars and, respectivel. f) = + e cos f)c 1 cos f C sin f) 1 ec 1 + 3eC f 3C 3 f + 8eC 3 sin f 6eC 3 f cos f + C 4, zf) =Z 1 cos f + Z sin f, where C i and Z i are integration constants that can be derived from the initial conditions of the problem. Equation 3) represent the analtical solution for the force free Hill s equations of the Three-Dimensional Elliptic Restricted Three Bod Problem ER3BP). Figure illustrates an eample of the solutions resultant from the epressions in Eq. 3). f 3 1 3) A. Force Free Hill s Equations When the orbit becomes large enough the term R 3 0 allowing us to neglect, as a first approimation, the last term of each equation on the right-hand side of 1) leading to [4], [5]: d df = d df e cos f, f 1 d df = d df, ) d z df = z. The resultant equations represent a set of ordinar differential equations with time-dependent periodic coefficients. There is no eplicit method of finding the solution of this tpe of sstems. However, it is possible to find the solution in and 3 Fig. : Parametric plot of the approimate analtical solutions of the unperturbed equations for e = , C 1 = 0.8, C = 0.3, C 3 = 0.03 and C 4 = 0.8.

3 INSTITUTO SUPERIOR TÉCNICO, LISBON, PORTUGAL, JUNE B. Osculating parameters The phsicall meaningful quantities in 3) are, rather than the C i and Z i parameters, the semi-minor ais A of the epiccle, the coordinates δ and δ of the centre of the epiccle, the phase difference φ between the motion of the S/C around and the motion of around Mars, the parameter γ for the amplitude of the harmonic oscillations with respect to the plane z = 0, and the parameter ψ as the phase of the oscillations along the z direction. The last parameter β describes the inclination relative to the -ais of the line of intersection between the plane of the epiccle and the plane. These parameters are graphicall represented in Fig. 3 and can be defined as: γ ψ 4 A C z δ O δ A =C 1 + C ) 1 φ = arctan C ) 1 C δ = ec + C 3 ec 3 cos f 3eC 3 f sin f Fig. 3: Geometric representation of the osculating parameters. O is the origin of the Cartesian reference frame and C is the epiccles center. Figure adapted from [1]. δ = 1 ec 1 + 3eC f 3C 3 f + 8eC 3 sin f 6eC 3 f cos f + C 4 4) γ =Z 1 + Z ) 1 ψ = arctan Z ) Z 1 β =ψ φ. The parameters in 4) are called osculating parameters because the describe an osculating orbit, which is the orbit that a S/C would assume if at some instant in time the S/C was not subject to perturbations. However, real orbits eperience perturbations which ma cause the osculating elements to evolve in time. As a result we consider the motion of the S/C as a continuous transition from one osculating orbit to another osculating orbit. Using 4) in Eqs.3) it is possible to rewrite the solutions of the force free Hill s equations as: f) = 1 + e cos f)a cosf + φ) + δ, f) = + e cos f)a sinf + φ) + δ, zf) = γ cosf + ψ). According to 4), we can assume that the terms A, φ, γ and ψ are constants and that onl δ and δ are time-dependent. Differentiating the epressions of δ and δ in 4) leads to: δ = ec 3 sin f 3eC 3 f cos f δ = 3C 3 + 3eA cos φ + ec 3 cos f + 6eC 3 f sin f. 5) 6) It is possible to observe from 6) that the derivatives of δ and δ are linearl dependent of the term ec 3 f. As a result, if the term C 3 0, the variation of both parameters will increase over time, which in turn will lead to orbital instabilit. To select stable solutions we must set C 3 = 0, ensuring that the position of the center of the epiccle does not present secular variation. Thus, the epressions for δ, δ and its respective derivatives can be rewritten as: δ = ea cos φ, δ = 1 ea sin φ + 3efA cos φ + C 4, δ = 0, δ = 3eA cos φ. C. Variation of Parameters The osculating elements were introduced in order to better describe the motion of a S/C orbiting. However, onl the case of sufficientl large orbits was considered. For such configuration it was assumed that the osculating parameters would be constant ecept for δ. Now, in order to account for the last term of each equation on the right-hand side of 1), it will be assumed that these osculating elements ma var in time. To obtain the approimate solutions of the ER3BP we use the method of variation of parameters to derive the differential equations that define the variation of the osculating elements [8]. However the result is too comple to conclude about the sstem behaviour. So, for the case of interest, i.e. the Mars- sstem, we simplif δ the problem b assuming that e 1, A 1, 1 and γ A 1. Doing a series epansion of the approimate solutions δ A 7)

4 INSTITUTO SUPERIOR TÉCNICO, LISBON, PORTUGAL, JUNE with respect to the mentioned quantities, centered around zero and considering onl the first order terms, and averaging the result over the period of π will result in the following average equations for the osculating parameters: Ā = 0, φ = K πā3, δ = 3πĀ3 K E) δ + 9πĀ 7E K) e sin φ, δ = 3 + 3πĀ3 K 4E) ) δ πā e cos φ, γ = 1 6πĀ3 5E K)γ sin β ), ψ = 1 6πĀ3 3E + 5E K) cos β)), β = 1 6πĀ3 3E 6K) + 5E K) cos β)), where the constants K and E are complete elliptic integrals of modulus k = 3. It is interesting to know that when the eccentricit is set to zero e = 0) this equations become identical to the ones of the zero eccentricit case [4], as epected. D. Solutions of the Averaged Equations After computing the average differential equations on the osculating parameters, it is possible to obtain a solution to describe the motion of a S/C orbiting. This is accomplished b integrating the sstem in 8). For the sake of simplicit the bar superscript was removed from the variables. The differential equations on A and φ are easil solvable and take the form A = constant φ = K πa 3 f f 0 ) + φ 0, where f 0 is the initial true anomal of the Orbit of and φ 0 the initial phase difference between the motion of the S/C about and the motion about Mars, as depicted in Fig. 4. Mars f 0 = π/ f 0 = 0 S/C ϕ 0 =π/4 S/C ϕ 0 =π/4 Fig. 4: Geometric representation of the parameters f 0 and φ 0 for two distinct cases. In the first one f 0 = 0 and φ 0 = π 4 and in the second f 0 = π and φ 0 = π 4. The remaining four equations can be separated into two subsstems, the first describing the drift oscillations the variables δ and δ ) while the second relates the precession of the plane 8) 9) of the epiccle the variables γ and ψ). Integration of the first subsstem gives the solution ] P δ =Q 1 cos P P 4 f + Q sin P P 4 f] P 4 P 1P 3 + P P 5 P 1 P P 4 e cos [P 1 f f 0 ) + φ 0 ] ] P4 δ =Q cos P P 4 f Q 1 sin P P 4 f] P + P 3P 4 + P 1 P 5 P 1 P P 4 e sin [P 1 f f 0 ) + φ 0 ], ) where the constants P 1, P, P 3, P 4, P 5, Q 1, Q ) are defined as: P 1 = K πa 3, P = K E), 3πA3 P 3 = 7E K), 9πA P 4 = 3 3πA 3 K 4E), P 5 = πa, ] Q 1 = δ 0 cos P P 4 f 0 P1 P 3 + P P 5 +e cos + Q = +e + P1 P P 4 P P 3 P 4 + P 1 P 5 P 4 P1 + P sin P 4 P4 δ 0 sin P P4 P 1 P 3 + P P 5 P P1 P P 4 P 3P 4 + P 1 P 5 P 1 P P 4 P ] δ 0 sin P P 4 f 0 P 4 ] P P 4 f 0 cos φ 0 P P 4 f 0 ] sin φ 0 ), ] ] P P 4 f 0 + δ 0 cos P P 4 f 0 cos ] sin P P 4 f 0 cos φ 0 P P 4 f 0 ] sin φ 0 ). 11) The second subsstem is independent on the variable β and its solution is P6 β = arctan + P 7 P6 P 7 tan P6 P 7 [f f 0) + β 0 ]) ], 1) where β 0 is a constant and P 6 and P 7 are: P 6 = 3E 6K 6πĀ3, P 7 = 5E K 6πĀ3. 13) It is clear from 1) that the intersection of the plane of the epiccle and the z = 0 plane rotates in the direction of the motion along the epiccle with an average velocit ω β = 3πA 3 K EK E A 3. 14)

5 INSTITUTO SUPERIOR TÉCNICO, LISBON, PORTUGAL, JUNE After the substitution of β the solution on γ is obtained γ 0 γ = P6 + P 7 cos β, 15) Period Π rad 4 Τ1 indicating that the amplitude of the z-oscillations var periodicall with frequenc ω β. Finall, using 9) - 15) and appling them in 5) we can write the approimate solutions of the ER3BP in osculating parameters: = 1 + e cos [f f 0 ])A cos [1 + P 1 )f f 0 ) + φ 0 ] ] P + Q 1 cos P P 4 f + Q sin P P 4 f] P 4 P 1P 3 + P P 5 P 1 P P 4 e cos [P 1 f f 0 ) + φ 0 ], = + e cos [f f 0 ])A sin [1 + P 1 )f f 0 ) + φ 0 ] ] P4 + Q cos P P 4 f Q 1 sin P P 4 f] 16) P z = + P 3P 4 + P 1 P 5 P 1 P P 4 e sin [P 1 f f 0 ) + φ 0 ], P γ 0 cos 1 + P 1 )f f 0 ) + φ 0 + arctan 6 +P7 P6 P 7 )]) tan P 6 P7 [f f 0) + β 0 ] P P 6 + P 7 cos arctan 6 +P7 P6 P 7 )]) tan P 6 P7 [f f 0) + β 0 ] E. Period Analsis With the obtained solutions 16) it is possible to identif 4 main periodic motions within the dnamics around. The revolution about the epiccle is one of the periodic motions and has a similar period to the period of the orbit of about Mars that is wh the orbits are denominated quasi-snchronous). Its value is τ 1 = π π = 1 + P K, 17) πa 3 the period of orbit is π in the dimensionless units used). For smaller amplitudes A) the difference to R, R, z R orbital period increases since the terms are no longer small, which eplains this approimation solutions low validit. The oscillations of the epiccle s center are composed b two distinct periodic motions. The first one has a period of τ = π A 3, 18) P P A3 3 1 Π A Fig. 5: Evolution of the revolution period τ 1 as a function of the epiccle s semi-minor ais A. The black dotted line represents the period of orbit. From Fig. 6 we notice that for increasing values of A both Period Π rad Τ Τ3 Π A Fig. 6: Evolution of the periods τ and τ 3 of the epiccle drift motion as a function of the epiccle s semi-minor ais A. The black dotted line represents the period of orbit. periods τ and τ 3 are much higher than the reference period of the orbit of around Mars, which suggests that the drifting movement of the epiccles along the direction tend to be slower for orbits with higher amplitudes. The remaining periodic motion is associated with the variations of the amplitude of the motion in the z direction which has frequenc ω β and period: τ 4 = 6π A 3 K EK E A3. 0) The period τ 4 behaves similarl to τ 3 meaning that for increasing values of the orbit s amplitude the period increases. and its amplitude is composed b terms that depend on the eccentricit of the orbit of and terms that depend on the initial position of the center of the epiccle δ 0 and δ 0 ), while the second periodic motion has a period of τ 3 = π φ = π K πa A 3. 19) III. APPROXIMATING PHOBOS Here we will describe the algorithm that was developed to approach : a sequential set of half epiccles. This algorithm will use the Differential Evolution method to minimize the algorithm cost function and compute the orbital parameters that provide the best cost value. After the nominal

6 INSTITUTO SUPERIOR TÉCNICO, LISBON, PORTUGAL, JUNE Period Π rad Π A Fig. 7: Evolution of the period τ 4 associated to the z direction as a function of the epiccle s semi-minor ais A. The black dotted line represents the period of orbit. trajector analtical model) has been computed we will use the calculated orbital parameters to obtain the orbit s initial velocities and use them as initial conditions for the integration of the numerical model. In the end a comparison between the approimate analtical model and the numerical model is made along with some discussion of the results. A. Approaching Strateg The first step of developing an algorithm is the definition of its goal and the strateg to obtain it. In the present work we set the objective of establishing an algorithm for a controlled approach to from an initial orbit but we have et to define the strateg to reach our goal. Here we present the strateg to be adopted, which consists on using consecutivel smaller epiccles to get closer to the Martian moon s surface. Departing from a point in the initial orbit observation orbit) the S/C will perform a burn V ) to reach a target point closer to in half epiccle. This process can then be repeated to approach the moon even further if desired. This strateg was chosen because it seemed fairl eas to eecute and because it allows for some trajector design freedom. This wa it is possible to include interesting points on the approach trajector to with the purpose of getting better video images of some point on the surface of the moon. The problem of finding the orbit arch that connects two different points in space at a given time is a form of two boundar value problem PBVP). Such a problem can be solved b means of optimization b computing the sstem parameters that present the lowest cost function value associated with the boundar requirements. The optimization method to be used in the approach algorithm is called Differential Evolution DE) method, which is a direct search method. B. Inputs and Outputs The inputs of the algorithm are the initial and final positions ), the initial true anomal rad) and the number of control steps to be performed during the algorithm. Although we can choose the initial and final positions we limited them to Τ4 coordinates on the z plane and z plane. This limitation was implied in part because in those planes it is much easier to compute an orbit transfer and using other points do not provide an advantage. The initial true anomal defines the position of relative to Mars at the start of the algorithm. Since the sstem is time dependent this means that for different initial true anomalies we will have different solutions because the position of in its orbit will be different and thus so it will be the influence of Mars on the trajector. The number of control steps basicall defines the frequenc at which the correction steps occur. The basic principle applied in this algorithm considers that each epiccle lasts half period to connect the initial position to the final position, nevertheless, as it was mentioned, the trajector of the real dnamics tend to diverge from the epiccle computed with the analtic model and so correction steps are needed. The number of control steps is used to divide the half period into equall spaced step times and for each step time a correction is performed. The higher the number of control steps the smaller the step time, increasing the precision but also the computational effort as well as the energ V ) consumed due to the higher number of corrections. A cost-benefit analsis can be made in order to decide which is the best number of control steps for each of the trajectories, if justified. The outputs are basicall the final positions and velocities of the non-corrected and corrected epiccles. These will be used as comparative parameters to assess the performance of the algorithm and the domain of validit of the developed model. The required V necessar to perform the corrections are also displaed in script while the algorithm is running and it is also an output that can be used to measure the cost-efficienc of the entire correction manoeuvre. IV. RESULTS We now present and discuss the results obtained with the deduced solutions of the three dimensional ER3BP along with the test of the designed trajector algorithm. A. Model Validation The comparison between model orbits and real orbits is important to assess the validit of the approimate analtical solutions obtained in section II-D. B comparing the differences between the two orbits for different parameters, we will be able to identif the regions for which the developed model can reproduce the real orbit behaviour described b the differential equations of the sstem 1). To perform this assessment we will use five eample orbits that are representative of the tpe of orbits that are likel to be used for an approimation trajector to, and we consider the moon as being a spherical bod with a radius of 13.5 [].In Table I we present the set of parameters dimensionless quantities) for the five orbits used to perform the comparisons. In Table I we can see that the orbit s amplitude varies from orbit to orbit, starting with a large amplitude Set A) and progressivel decreasing the amplitude for the following sets. This procedure will help us to understand how the amplitude

7 INSTITUTO SUPERIOR TÉCNICO, LISBON, PORTUGAL, JUNE Set A δ 0 δ 0 γ 0 φ 0 [rad] ψ 0 [rad] f 0 [rad] A B C D E F TABLE I: Initial conditions for the set of sample orbits used in the coordinate evolution stud. affects the orbits behaviour and identif the differences when changing its value Ε Set A Set B Set C Set D Set E Set F a) Difference between model and real orbit over π for each set of parameters. Error 15 5 Coordinate Coordinate f rad is also presented in Fig. 8b the maimum relative error for coordinate and registered for a period of π Here the error is directl related to the orbit s amplitude making it clear that for lower values of amplitude the difference between our model and the real dnamics start to increase significantl. This result can be a consequence of the approimations made when we applied the method of variation of parameters to the obtained solutions as a wa to account for gravitational attraction as a perturbation. Due to its complicated form onl first order terms were considered which might eplain the appearance of differences in the coordinates evolution when the orbit s amplitude becomes smaller. B. Approimation Trajector Design The net step will be the design of an approimation trajector to using onl, as initial and final positions, points laing on the z plane or z plane. This imposition will result in lower computational effort but it is also related to the fact that those are the two most interesting configurations to be studied, representing the cases when the orbit transfers occur at the etremities of the orbit. These two strategies were named as a reference to the points were the actual orbit transfers occur. The z plane transfers take as initial and final positions points in the z plane or simpl points that take 0 as coordinate while the z plane transfers take as initial and final positions points in the z plane or, in other words, points that take zero as coordinate. To perform the stud on how well the approimation strategies behave we will use the points in Table II and as initial true anomal we will use the value 0. Fig. 9 shows a Trajector Initial Pos. Interm. Pos. 1 Interm. Pos. Interm. Pos. 3 Final Pos. [] [] [] [] [] YZ1 0,150,0) 0,-15,0) 0,0,0) 0,-75,0) 0,50,0) XZ1 150,0,0) -15,0,0) 0,0,0) -75,0,0) 50,0,0) TABLE II: Initial, intermediate and final positions for the trajector design Amplitude b) Maimum relative error for the coordinates and Fig. 8: Difference between model and real orbit over π for each set of parameters and Maimum relative error for the coordinates and Figure 8a represents the evolution of the absolute difference ) registered between model and real orbits for the different sets of parameters described in Table I. We see that the differences presented grow from Set A through Set F, which follows the inverse trend of the orbit s amplitude that decreases from Set A through Set F. This suggests that in fact the amplitude of the orbit affects the precision of the developed approimation when compared to the real dnamics of the sstem. It is clear that for large amplitudes both orbits behave similarl, but when the value of the orbit s amplitude decreases the differences start to grow. To support this conclusion it graphical representation of the resultant trajectories, where the dotted black lines represent the nominal trajectories computed b the trajector algorithm, the gra lines represent the real spacecraft trajectories obtained b integration of 1) and the magenta lines represent the reference trajector to be followed b the spacecraft at each orbit transfer. A quick inspection of the figures shows that the nominal trajectories pass on the stipulated wa-points of Table II and that the real trajectories follow a similar behaviour b passing close to the required wa-points, indicating that the behaviour of the real dnamics is emulated fairl well b the nominal designed trajector. It is however worth mentioning that the closer we get to the moon, the bigger the differences between the nominal and real trajectories became, supporting the conclusions made in Section IV-A. Table III displas the specific energ necessar to perform the orbit transfers for the trajectories in Table II. The result is presented in specific energ J/kg) because changes in the kinetic and potential specific energies of the vehicle are independent of the mass of the vehicle. Analsing

8 INSTITUTO SUPERIOR TÉCNICO, LISBON, PORTUGAL, JUNE Real Orbit Reference Trajector 00 0 Real Orbit Reference Trajector Trajector Initial Pos. Interm. Pos. 1 Interm. Pos. Interm. Pos. 3 Final Pos. [] [] [] [] [] YZ3 0,150,) 0,-15,-) 0,0,) 0,-75,-) 0,50,) TABLE IV: Initial, intermediate and final positions for the 3D trajector design Trajector Specific Energ [J/kg] Note YZ % than YZ1 TABLE V: Specific energ spent on the trajectories YZ a) Trajector YZ1 b) Trajecor XZ1 Fig. 9: Graphical representation of Trajectories YZ1-XZ1. Table III we realise that the necessar specific energ is considerabl higher for orbit transfers occurring in the z plane, this result ma be eplained b the drift propert that the orbit presents in the -direction. If we consider that the orbit dnamics itself presents an oscillator motion on the - direction, than it is epected that performing an orbit transfer along such direction will be cheaper than performing it along a direction where there is no drift propert. That is mainl because a small difference in the spacecraft velocit when standing along the -direction will cause a drastic change in the trajector whereas the same is not true when the spacecraft is positioned along the -direction. The cost results evidence The results of the 3D trajector design are presented in Fig. suggesting that the introduction of the third coordinate does not change the planar orbit behaviour, which is epected since the oscillations in the z direction are much slower than Phobo s orbital period as proven in section II-E. This can be verified b simpl comparing Fig. a with the graphical representations of YZ1 in Fig. 9, which shows that the shape of the D projections of the trajectories are identical. Regarding the Real Orbit Reference Trajector Trajector Specific Energ [J/kg] YZ XZ TABLE III: Specific energ spent on the trajectories YZ1-XZ. 0 a) D b) 3D that the z plane transfers strateg comes with a lower price, which is alwas a factor to be considered in the field of mission design. Thus, if no other factors are taken into account we believe that the best strateg to use is the z plane transfer strateg, being the strateg used b default in the remaining of the present work. C. D vs 3D Until now we confined our analsis to planar cases D) but it is also important to get some insight on 3D cases and tr to understand weather the introduction of the third coordinate produces significant differences in terms of orbit behaviour and energ consumption. For that purpose we will define a new 3D trajectories and will be using trajector YZ1 a comparison term. The 3D trajector, YZ3, is defined in Table IV and is simpl the same as YZ1 but with a third coordinate different from 0. Fig. : Graphical representation of Trajector YZ3. specific energ consumption the results show that in fact there is an increase of energ consumption when comparing with the values registered in the planar trajector. This increment of 33% is not strange since in the 3D cases there is one more coordinate to control. D. Controlled Trajectories In order to compensate for the first order approimation of part of gravitational attraction we introduced a control strateg capable of computing the necessar corrections V ) to be applied to the real trajector so that it follows the reference trajector given b our model. We will appl the control strateg to the eample YZ1 and evaluate the resultant improvements as well as the cost associated with the corrections performed. As a result the given inputs for the algorithm are the set of positions in table II, the initial true

9 INSTITUTO SUPERIOR TÉCNICO, LISBON, PORTUGAL, JUNE anomal is zero and the number of control steps is 5. The initial true anomal value is zero to be in line with what was used in the computation of YZ1. The number of control steps to be performed throughout the control algorithm was chosen to be 5 so that a balance is kept between computational time and algorithm performance. 150 Real Orbit No Correction Real Orbit With Correction Trajector Orbit Transfer Energ [J/kg] Control Energ [J/kg] Total Energ [J/kg] YZ1 Correction) TABLE VI: Specific energ spent on the trajectories YZ1-YZ when control is applied. the S/C gets to the higher is the effort necessar to perform the trajector corrections, which is in line with the theor that the real orbit dnamics differs from the developed model dnamics when we get closer to a) D b) 3D Fig. 1: V evolution at each control step. Fig. 11: Graphical representation of Trajector YZ1 using the control strateg. The results obtained with the controlled algorithm Fig. 11) show that there is a clear improvement of the real orbit behaviour as epected, which now follows more accuratel the reference ). This is an important outcome for the purpose of trajector design because it allows us to compensate for the rough approimation of the gravitational potential made to obtain the analtical solution of the three dimensional ER3BP, and guarantee that the real orbit follows the predefined trajector computed with the developed solution. The analsis of the energ consumption resulting from the introduction of control steps returned some interesting results. Comparing the energ values required to perform the necessar orbit transfers Tables III and VI) we realise that the application of the control strateg resulted in lower values of energ consumption to perform the transfers. This outcome ma be a consequence of the improvement made on the trajector behaviour, suggesting that for the case where no control strateg is applied the orbit transfer V is higher in order to compensate for the divergences verified between the real orbit and the reference model orbit Fig. 9). On the other hand, for the controlled case we are appling timel corrections to guarantee that the real orbit is correctl tracking the reference model orbit, which also requires consumption of energ Table VI). The total energ consumption is, as epected, larger when controlled is used, however, it s higher value is still within the same order of magnitude as the value obtained when no controlled is applied, which indicates that the control strateg is affordable from the point of view of energ consumption. Figure 1 represents the evolution of the magnitude of the impulsive shots V s) for each control step showing that throughout the correction process the magnitude of the impulses tends to increase. This means that the closer E. Landing on We considered the possibilit of deorbiting from a QSO directl to the surface of because we felt that it would be an interesting case that is worth eamining. With the purpose of assessing the feasibilit of such approach we decided to test three different eamples. For all the eamples we selected 0, 50, 0) [] as the initial point and 13.5, 0, 0) [], 9.55, 9.55, 0) [] and 6.75, 6.75, 9.55) as landing points for eamples 1, and 3 respectivel. The results Fig. 13) are consistent and show that in fact it is possible to design a deorbit path capable of delivering the spacecraft directl from a QSO to the surface of the moon. It is visible that the corrected real orbit tracks the reference model orbit quite well in all the cases, which proves that the designed algorithm allows the selection of the desired landing position. The major concern about this strateg is related to the braking phase, which has not been studied in the present work. The S/C final velocit components are V, V, V z ) = , 11.87, ) [m/s] for the first eample, V, V, V z ) = ,.870, ) [m/s] for the second and V, V, V z ) = , , ) [m/s] for the last eample, but we believe the can be easil compensated using small braking burns in the last moments of the landing phase. In terms of energ costs all the eamples show a similar behaviour Fig. 14), starting with small V s at the beginning of the manoeuvre but with an increasing trend towards the end of the path. Further analsis should be made in order to evaluate which approach is the more suitable, however, with the results gatherer throughout our inspection and given the results obtained we believe it is viable the use of the developed algorithm as tool for trajector design in the vicinit of the Martian moon.

10 INSTITUTO SUPERIOR TÉCNICO, LISBON, PORTUGAL, JUNE Real Orbit With Correction 50 Real Orbit With Correction V's E.1 Approach V's E. Approach V J kg Control Steps a) Eample 1 b) Eample a) V evolution at each control step for E. 1 and V's E.3 Approach V J kg c) Eample 3 Fig. 13: Landing Approach. V. CONCLUSIONS The present work considered the development and implementation of an algorithm capable of computing an approimation trajector to approach the Martian moon. After having modelled the S/C dnamics near the moon, where onl the influence of Mars and Phobo s gravitational potential was taken into consideration, the associated analtical sstem of equations describing the problem was presented and an approimate analtical solution was obtained for the case of small gravitational parameter µ and eccentricit of the smaller primar orbit e. A strateg to approach from an observation QSO was designed using a Differential Evolution search method to solve the two point boundar value problem of obtaining suitable solutions given the desired initial and final positions of the desired trajector. A comparison between model and real orbits was used as validation method for the obtained analtical solution of the problem, which enabled the understanding of the limitations of the analtical approimation. A control strateg was designed as a method to etend the validit of the developed model to the vicinit of, region where the simple model has shown some flaws. Simulations of approimation trajectories were finished with success, which included an energ cost analsis proving the feasibilit of such solutions in terms of dnamical behaviour but also in terms of power needs. In future work it would be challenging the use of a secondorder approimation to the gravitational potential of, which could etend the validit of the attained solution to regions closer to the moon s surface providing a more accurate solution for mission design purposes. Also, the J of is large and should be included, as it will influence the orbit considerabl. Total Control Energ Spent J kg Control Steps b) V evolution at each control step for Eample Total Energ for E.1 Approch Total Energ for E. Approch Total Energ for E.3 Approch Control Steps c) Total energ evolution. Fig. 14: Energ cost for landing manoeuvre. REFERENCES [1] P. J. S. Gil and J. Schwartz, Simulations of Quasi-Satellite Orbits Around, Journal of Guidance, Control, and Dnamics, vol. 33, no. 3, 0. [] S. R. K. K. Mukhin, L. M. and A. Zakharov,, deimos mission, Concepts and Approaches for Mars Eploration, pp. 30 3, Jul 000. [3] V. G. Szebehel, Theor of Orbits, The Restricted Problem of Three Bodies. Academic Press, [4] A. Y. Kogan, Distant Satellite Orbits in the Restricted circular problem of Three Bodies, Cosmic Research, vol. 6, no. 6, pp , [5] D. Benest, Libration Effects for Retrograde Satellites in the Restricted Three-Bod Problem, Celestial Mechanics, vol. 13, no., pp , [6] W. Boce and R. DiPrima, Elementar Differential Equations and Boundar Value Problems. John Wile & Son, Inc., 005. [7] D. Zwillinger, Handbook of Differential Equations. Academic Press, [8] M. Efroimsk, Equations for the keplerian elements: Hidden smmetr, 00, preprint # 1844 of the IMA.