Orbit Transfer Optimization for Multiple Asteroid Flybys

Orbit Transfer Optimization for Multiple Asteroid Flybys Bruno Victorino Sarli 1 and Yasuhiro Kawakatsu 2 1 Department of Space and Astronautical Science, The Graduate University for Advance Studies, Sagamihara, Japan Tel: +81-50-336-23042; E-mail: sarli@ac.jaxa.jp 2 Department of Space Flight Systems, Japan Aerospace Exploration Agency JAXA, Sagamihara, Japan Tel: +81-50-336-27836; E-mail: kawakatsu.yasuhiro@jaxa.jp Abstract: This paper presents a method of trajectory optimization based on a well known theory, primer vector, which is modified to accommodate weights in the cost function; arising from the need of a more accurate analysis that takes into account the velocity increment used for the planet s departure and, particularly for flyby missions, the disregard of the last rendezvous impulse. A detailed derivation of the weighted cost function and its gradient is presented, followed by a discussion on the adjustment of the weights specifically for flyby missions. In order to test the optimization method, a realistic test case is selected and its results are compared against a trajectory using the solution of the Lambert problem and classical primer vector optimization. It is possible to clearly evaluate from the results the advantages of the proposed method, enabling to design a trajectory with a midcourse impulse which costs less than the trajectories calculated by the other methods and provides a more realistic analysis of the planet s departure. Keywords: trajectory optimization, flyby mission, asteroid, primer vector. 1. INTRODUCTION Over the years the missions to asteroids have enhanced our knowledge on many aspects of these bodies. Such a growing interest on them is due to many reasons which can go from purely scientific, such as, understanding the formation mechanisms and composition of our early solar system, to more mundane matters like planetary protection and the possibility of mining rare materials. Particularly, flyby missions present some interesting aspects which are not found in any other types of mission, such as: cost effective, cheaper with respect to the use of propellant, flexible and, due to that, more effective. The framework of the proposed problem consists in generating a trajectory composed by impulses that allows multiple asteroid flybys based on Keplerian orbits around the Sun. The initial estimation for the multiple flyby sequence optimization is the Lambert solutions for each transfer arc: planet-asteroid 1, asteroid 1- asteroid 2, and so on. The methodology used to generate the final optimal flyby sequence is then a succession of optimizations calculating the best location and time for a midcourse impulse at each segment of the sequence respecting its initial and final positions, as well as, the transfer time. The process that generates the optimal transfer arc or segment consists in adding a midcourse impulse to each arc using the primer vector theory PVT, an indirect method of trajectory optimization based on impulsive maneuvers. In this work, the classical cost used in the PVT is modified to better accommodate the reality of the transfer by applying weights in the cost function s elements and its gradient. Initially developed by Derek Frank Lawden in 1963 1 and later complemented with the works of Lion & Handelsman, 1968 2, Jezewiski & Rozendaal, 1968 3, and Jezewski & Faust, 1971 4, the PVT provides time and position for adding a midcourse thrust impulse that minimizes the cost. In the context of space missions, a low cost transfer requires, among others, a minimum velocity increment at the planet s LEO to generate hyperbolic excess velocity and minimum fuel usage for deep space maneuvers; both requirements can be expressed in terms of velocity increments, v. Therefore, it makes the optimization more realistic if the cost can be associated with these two points; for this, it is modified to accommodate weights that can be adjusted accordingly. This proposed method of optimization is here called weighted method. As an application example, the design of a flyby sequence to two main asteroids of the Phaethon Geminid Complex PGC, Phaethon and 2005UD, is performed. Among the many possible targets for a multiple flyby mission the study of the Geminid meteor shower can be of special interest since it may hold the answers for fundamental questions in the early solar system. Perhaps the most important asteroid related to the Geminid is 3200 Phaethon, a B-type asteroid which is supposed to be its parent. Due to similar orbit properties, the asteroids 2005UD is believed to be a fragment originated from Phaethon 6. Such is the importance of the PGC that Phaethon was a target candidate for NASA s Deep Impact and OSIRIS-Rex missions. As it follows, section 2 presents a background of the classical linearization method used on the transfer trajectory and a short historical background of the primer vector theory with its most important equations. Section 3, presents the originality of this work where a weighted cost function and its gradient are derived, showing the difference in the necessary conditions for optimality between the weighted and the classical methods. Section 4, presents possible values for the weights particularly for single or multiple asteroid flyby cases. Section 5 deals with a Phaethon-2005UD flyby test case comparing the results with a Lambert and a classical primer vector solutions. Finally, section 6 presents the conclusions derived from the previous chapters.

2. CLASSICAL THEORY 2.1 Linearization The linearization of the orbit is necessary to calculate the evolution of the primer vector and some variables of interest. A perturbed trajectory is evaluated in three points of interest: beginning, 0, a generic midcourse point, m, and the end, f. Figure 1 presents these points, as well as, the initial velocity perturbation, δv 0, final velocity perturbation, δv f, the midcourse perturbation on the position, δr m, and the perturbed velocities before, δv m, and after, δv + m, it. The state transition matrix for a Fig. 1 Trajectory representation generic elliptical orbit can be obtained from the work of Glandorf 7 which bases the linearization in an inversesquare gravitational field. Having the state transition matrix, Φ, the perturbations can be derived in the linear system caused by a position displacement δr m at the point m, however, maintaining the initial and final points the same, δr 0 = 0 and δr f = 0. δrm δv m δr0 δrf δrm = Φ m0 ; = Φ δv 0 δv fm f δv + m Where, Φ m0 is the state transition matrix form the beginning, point 0, until the generic point m, Φ m0 = Φt m, t 0. From Eq. 1 we obtain the velocity variation at the beginning and end. For simplicity the matrix M N Φ will be subdivided Φ =, S T δv 0 = Nm0 1 m δv m = T m0 Nm0 1 m δv f = S fm T fm N 1 fm M fmδr m 1 δv + m = N 1 fm M fmδr m 2 Finally, the difference between the velocities at the point m, v m, can be calculated as v m = v + m v m = v m + δv + m v m + δv m, which making used of Eqs. 2 result in v m = N 1 fm M fm + T m0 Nm0 1 δr m 3 The time derivative of the state transition matrix, as it will be seen further in the text, will be used to calculate the cost s gradient. Φ can be obtained, according to reference 8, as Φt, t 0 = DfxtΦt, t 0, where, Dfxt is the Jacobian matrix evaluated along the non-perturbed trajectory. 2.2 Primer Vector Theory The primer vector theory comprehends an indirect method of trajectory optimization, determining the necessary conditions and sufficient conditions for optimality. Particularly for impulsive trajectories, the primer vector provides information on if the trajectory s cost can be decreased by a midcourse impulse, as well as, the optimal direction, time and position of this impulse. In 1963 Lawden 1 gave birth to the theory and the term primer vector by defining the necessary conditions for an optimal impulsive trajectory, these conditions passed to be known as Lawden s necessary conditions for an optimal impulsive trajectory. Following the work of Lawden, Lion & Handelsman 2 developed in 1968 a criterion that improves a reference impulsive trajectory; in this way, reducing the cost. Jezewski & Rozendal 3 in 1968 developed an efficient method for computing a two-body optimal N-impulse trajectory using the primer vector. Finally in 1971 Jezewski & Faust 4 developed a theory which describes how a general differential cost function can be evaluated by using inequality constraints on the states and the control variables based on a penalty function approach, also known as cost well. As carefully derived in 5, for the case of impulsive maneuvers, the thrust arc can be approximated as an impulse represented by the Dirac delta, δ, having unbounded magnitude and zero duration. The important equations of the PVT used in this work are presented as follows: The thrust acceleration a T tut, where a T is the thrust magnitude and u is its direction component that can be evaluated as ut = λ vt λ v t where, u is being multiplied by the adjoint vector of the spacecraft s equation of motion λ T v. In order to minimize a generic Mayer cost function, J = J t f, u and λ T v are chosen parallel in opposite directions, generating the largest possible negative value. Due to the importance of the vector λ v Lawden named it primer vector; Ṁ Ṅ Φ = Ṡ T Ṁ t, t 0 = S t, t 0 Ṅ t, t 0 = T t, t 0 Ṡ t, t 0 = GrM t, t 0 T t, t 0 = GrN t, t 0 where, Gr is the gravity gradient matrix, gr λv t λv t 0 = Φ t, t λ v t 0 λ v t 0 r ; 4 5 6 and it can be shown that for every trajectory the following relation is true, λδv λδv = const 7 3. WEIGHTED COST FUNCTION The cost function used in this work takes into account the transfer terminal constraints; sum of weighted v s

along the trajectory J = K 1 v 0 + K 2 v m + K 3 v f 8 The velocity increments are weighted by the constants K because for each part of the trajectory the addition of velocity has a different physical delivery system, more details and the evaluations of the K constants are addressed on section 5. As performed in the works 2-4, the midcourse impulse, v m, can be written as a function of the control parameters. Starting from Eq. 4 we have that for the optimality the direction of the impulse needs to be parallel to the primer vector, which leaves the magnitude of the impulse, ξ, to be calculated freely and unbounded, v m = ξ λ vt m λ v t m = ξu t m 9 clearly depicting the two variables to be optimized: the time of the impulse, t m and its position, r m. On other propulsion systems, such as, power-limited and lowthrust engines do have the magnitude bounded with the position. Based on these two variables and the definition of the linear system, the cost function can be obtained by solving two Lambert problems from r 0, t 0 r m + δr m and r m + δr m r f, t f, which will provide v 0, v f and v m. Then, the magnitude of the impulse can be obtained by evaluating the solution generated by the combination of the t m and r m. The cost s gradient J = J t m J r m 10 can be calculated by using the following relations: the first order relation between the non-linear and linear systems, dr m = r t m + dt m r t m dr m = δr m + ṙ m dt 11 and the relation derived by Jezewski 5 for a generic vector w, w δw w = wt δw 12 w in the case that w is the velocity variation, v δ v = δv, the relation becomes v δv v = vt v δv = λt v δv 13 Based on the above relations, it is assumed a comparison between a non-perturbed, v m = 0, and a perturbed orbit, v m = δv + m δv m 0, as Therefore, the difference of the costs is dj = J pert J non pert = K 1 v 0 + δv 0 +K 2 δv + m δv m +K 3 v f δv f K 1 v 0 K 3 v f = K 1 v 0 + δv 0 v 0 + K 2 δv + m δv m + K 3 v f δv f v f 14 where, the v represent the values in the non-perturbed trajectory. Using Eq. 13 in the above relation it becomes dj = K 1 λ T v t 0 δv 0 + K 2 δv + m δv m K 3 λ T v t f δv f 15 Mean while, using relation 7 at the trajectory s beginning or the end until the point of the midcourse impulse λ T v t 0 δv 0 λ T v t 0 δr 0 = λ T v t m δv m λ T v t f δv f λ T v t f δr f = λ T v t m δv + m λ T v λ +T v t m δr m t m δr + m 16 and applying on Eq. 15, remembering that the initial and final positions remain the same, δr 0 = δr f = 0, dj = K 1 λ T v t m δv T m λ v t m δr m + K 2 δv + m δv m + K 3 λ T v t m δv + +T m + λ v t m δr + m 17 and knowing that, as λ v t m is a unit vector, at the time of the impulse δv + m δv m = λ T v t m δv + m δv m, Eq. 17 becomes dj = λ T v t m K 1 K 2 δv m + K 2 K 3 δv + m + K 3 λ+t v t m δr + m K 1 λ T v t m δr m 18 Applying Eq. 11 to the perturbed trajectory we obtain { δr m = dr m v mdt m δr + m = dr m v + mdt m 19 with this and Eqs. 2 into Eq. 18 we obtain dj = + where, Λ 1 + K 3 λ+t v t m K λ T 1 v t m Λ 2 + K λ T 1 v t m v m K λ+t 3 v t m v + m Λ 1 = λ T v t m K 1 K 2 T m0 N 1 m0 + K 3 K 2 N 1 fm M fm Λ 2 = λ T v t m K 2 K 1 T m0 N 1 m0 v m + K 2 K 3 N 1 fm M fmv + m Finally, the gradient can be easily calculated as J = Λ 2 + K 1 λ T v Λ 1 + K 3 λ+t v t m v m K 3 λ+t v dr m dt m 20 t m v + m t m K 1 λ T v t m 21 22 The above gradient may generate a discontinuity on the primer vector s derivative and Hamiltonian depending on the choice of the weights. According to Lawden 1 this discontinuities would violate the necessary conditions for

optimality characterizing a non-optimal trajectory. However, Lawden s necessary conditions LNC were derived using the classical, non-weighted K 1 = K 2 = K 3 = 1, cost function that has its gradient J Lawden = λ T v t m v +T m λ v t m v + m λ + v t m λ v t m Hm H m + = λ + v t m λ v t m which in the optimal condition gives J Lawden = Hm H m + 0 λ + v t m λ = v t m 0 { Hm = H m + λ + v t m = λ v t m 23 24 Therefore, LNC are still valid for Eq. 22 if K 1 = K 2 = K 3 = 1, but other wise they do not provide the continuity on the Hamiltonian and primer vector s derivative, because these conditions were derived with a different cost function that is not applicable anymore. Nevertheless, it is important to point out that the gradient is still being converged to zero and the Hamiltonian in the weighted gradient is still being minimized. 4. WEIGHTING CONSTANTS In the first part of the trajectory where the spacecraft is exiting the planet, v 0 can be understood as the departure s velocity, v 0 v planet, it can be considerably changed by small velocity increment at the perigee of the departing orbit. v f is the increment provided at the end of the transfer which, for a flyby mission, typically needs to be provided only if the flyby velocity is too high for the spacecraft s instruments perform measurements or to generate a connection with a new orbit. v m, therefore, is the most critical element since it can only be controlled by a direct engine burn. The constant K 1 is directly related with v 0, which, in the first arc of the transfer, is the planet s hyperbolic excess velocity calculated based on the perigee of the departing orbit. The fact that v 0 can be altered with a relative small velocity increment at the perigee makes it less critical for the cost. The constant K 3 relates to the final part of the transfer, which typically for single flyby missions is does not required any v allowing K 3 = 0 and simplifying the problem by making it sensitive only to the initial and midcourse impulses. Nevertheless, for multiple asteroid flybys, K 3 maybe important as one of the strategies for achieving the second flyby can be to place the spacecraft, after the first flyby, into a new transfer orbit that will eventually flyby the second asteroid. The constant K 2 related to the midcourse impulse can be seen as the most important element, for it can only be altered by a direct engine burn. v m is the main reason why the new trajectory represent an advantage with respect to the two-impulse one, therefore, it is necessary to make sure that a second impulse will, in fact, improve the transfer. Due to its degree of importance, K 2 can be assigned the value 1 and the other two will follow based on this value. As for K 1, form the classical celestial mechanics the relation between the excess velocity and a circular planetary parking orbit is v inj = v p v park = 2µ µ + v0 2 r 25 p r p where, v p is the velocity at the perigee of the escape orbit, v inj is the velocity increment at the orbit s perigee, r p is the perigee s radius and µ is the planet s gravitational parameter. Eq. 25 allows to make the relation between the injection and excess velocities taking into account that an increase of v 0 can be easily made at the perigee. The relation for K 1 can be, then, evaluated as K 1 = v inj v inj0 v 0 K 2 26 where, v inj0 can be calculated by Eq. 25 with v 0 = 0. This makes K 1 a function of the perigee s radius and the magnitude of the excess velocity, K 1 = K 1 r p, v 0. As the v 0 is always larger than the injection velocity, K 1 will always be smaller than one. Interesting to notice that by this evaluation, hypothetically, the cost function compares the actual velocity increment provided at the perigee of the escape orbit since the v 0 from K 1 and from the cost function will cancel each other J = v inj v inj0 v 0 K 2 v 0 + K 2 v m + K 3 v f = K 2 v inj v inj0 + K 2 v m + K 3 v f 27 For the case of multiple flybys, the last weight, K 3, can be understood as the relation between the final velocity of the previous transfer and the velocity of the new orbit. However, if it is considered that there is no velocity change at the end of the first transfer, K 3 can be zero if the K 1 of the new orbit is adjusted properly. In this way, K 1 of the new orbit will assume the role of adjusting the importance of the velocity increment delivery at the asteroid flyby point; its value will be related to a deep space maneuver conferring it the value of 1. A trajectory of multiple flybys can be then evaluated taking in account each transfer segment individually, and analyzing it by properly adjusting the weighting constants. Then, an N- impulse trajectory can be analyzed using the same idea and cost function, breaking the problem into segments and properly adjusting the weights. Finally, it is important to point out that the reduction of the total transfer v = v 0 + v m + v f is not assured by this procedure, it may present a large sum of vs if the relation between the Ks provide a lower cost.

5. TEST CASE This section is dedicated to present a test case that provides a better understanding of the advantages of the weighted method making use of a real case scenario, the exploration of the asteroids Phaethon and 2005UD. In this application example, the Earth and asteroids positions were obtained using NASA s Horizon system from January 2020 until December 2029. For simplicity, the transfer solutions are limited to segments with a maximum duration of two years and an Earth hyperbolic excess velocities, v 0, of no more than 3 km/s. Initially, a simple ballistic transfer from Earth to Phaethon is obtained by solving successive Lambert problems for every Earth departure time combined with every Phaethon arrival time. From those transfers a second arc is design connecting Phaethon with 2005UD. The combined sequence Earth-Phaethon-2005UD that requires in total the smallest v is then selected, figure 2. The selected two-flyby transfer departures with an excess velocity of around 2.85 km/s and performs a deep space maneuver of 0.13 km/s at Phaethon in order to make the second flyby at 2005UD. The PVT can be applied to the two-impulse Earth- Phaethon transfer transfer using the classical K 1 = K 2 = K 3 = 1 generating the solution depicted on figure 3, where the excess velocity is reduced to 0.588 km/s with a large midcourse impulse. This result is due to the fact that the cost function was also considering the final v in the minimization process, which in this particular case is not delivered. Moreover, the initial excess velocity is typically achieved by accelerating the spacecraft from a low Earth orbit LEO, which makes the reduction of the excess velocity less important. An improvement in the classical PVT optimization can be made by setting K 3 = 0 since the rendezvous with the asteroid is not required. The resulting transfer, figure 4, presents considerable decrease in the excess velocity with a reduction of near 1.04 km/s to reach Phaethon. Following, K 1 can also be considered as calculated by Eq. 26, where the perigee is assumed to be at 300 km from Earth s surface, figure 5, where the result shows the same solution as the two-impulse transfer. This means that by evaluating Earth s departure more realistically, the use of the midcourse impulse does not reduce the total v, which maybe surprising at first but it is expected since the resulting K 1 = 0.1282 makes cheaper to provide the v at the Earth s departure rather than at any other part of the trajectory. However, this result presents a very important point, if the variable K 1 is used as 1, like previously, one may wrongfully assume that the midcourse impulse would in fact present an advantage; therefore, analyzing the problem more realistically allows to obtain a better comprehension of the system. Nevertheless, the usefulness of this method is not diminished by this particular example K 1 can be altered based on the mission s need or a different mission may result in a trajectory that requires a considerable midcourse impulse. The spacecraft can extend its mission by performing Fig. 2 Earth-Phaethon-2005UD minimum v transfer sequence a second flyby at 2005UD, which would require another trajectory maneuver. For simplicity, it is assumed that the second transfer will originate directly from the first flyby arc, provided by an impulsive v starting at Phaethon flyby and going directly to 2005UD. In this second arc, Phaethon-2005UD, the first v is now a deep space maneuver which makes K 1 = K 2 = 1 and, again without the rendezvous at the end, K 3 = 0, figure 6. There is no necessity to readjust the weights of the first arc because making K 1 = 1 already takes into account the velocity change at the Phaethon flyby. The resulting values of the complete two flyby transfer can be seen on table 1 for the two-impulse, classical and weighted cases, where all the velocity increments are depicted. Table 1 Comparison between the analyzed cases Transfer Ballistic Classical method Weighted method v km/s Earth- Phaethon 2.8497 0.58786+4.287 = 4.87486 0.89853+0.91234 = 1.81087 Phaethon- 2005UD 0.12949 not evaluated 0.11936+0.007888 = 0.127248 Total 2.97918 more than 4.874 1.93812 6. CONCLUSIONS During this work an optimization method using the primer vector theory to analyze a weighted cost function was derived. A detailed derivation of the cost function and its gradient was made and a comparison of the necessary conditions for optimality was performed against the classical non-weighted one. Finally, a test case was used to provide a better understanding of its advantages. Comparing the weighted method against the classical one in a flyby trajectory design show clearly its usefulness by obtaining a more realistic and cheaper trajectory in terms of v, weighted K 3. Particularly for the single flyby mission used, the method presents the same solution as a two-impulse transfer, weighted K 1, this result emphasis an error that one could occur in making the

Fig. 3 Classical solution for the Earth-Phaethon arc Fig. 6 Weighted K 1, K 3 for the Phaethon-2005UD arc missions; for example on rendezvous missions or transfers with a high change in the inclination plane, it may present a significant improvement because it allows to take in account the actual velocity increment used for Earth s hyperbolic escape and provides a decrease in the v used for the rendezvous or circularization by adding a deep space maneuver. Particularly for flyby cases, another way to use this method for improving the solution is to devise a transfer to a different orbit that either passes through two asteroids or at least one that has a lower fuel consumption, such strategies will be addressed in future works. Fig. 4 Weighted K 3 for the Earth-Phaethon arc Fig. 5 Weighted K 1, K 3 for the Earth-Phaethon arc designing without a more realistic analysis of the excess velocity. Also, the weighted method was shown advantageous for multiple flyby missions using all Ks weighted, achieving a lower total v for the whole mission. Moreover, this method is useful for a variety of other types of REFERENCES 1 D. F. Lawden, Optimal Trajectories for Space Navigation, Butterworths, London, 1963. 2 P. M. Lion and M. Handelsman, Primer Vector on Fixed-Time Impulsive Trajectories, AIAA Journal, Vol. 6, No. 1, pp. 127 132, 1968. 3 D. J. Jezewski and H. L. Rozendaal, An Efficient Method for Calculating Optimal Free-Space N-impulsive Trajectories, AIAA Journal, Vol. 6, No. 11, pp. 2160 2165, 1968. 4 D. J. Jezewski and N. L. Faust Inequality Constraints in Primer-Optimal, N-Impulse Solutions, AIAA Journal, Vol. 9, No. 4, pp. 760 763, 1971. 5 D. J. Jezewski, Primer-Vector Theory and Applications, Technical Report NASA TR R-454, Lyndon B. Johnson Space Center, Texas, 77058, 1975. 6 K. Ohtsuka, T. Sekiguchi, D. Kinoshita, J. Watanabe, T. Ito, H. Arakida, T. Kasuga, Apollo asteroid 2005 UD: split nucleus of 3200 Phaethon? Astron. Astrophys. 450, L25 L28, 2006. 7 D. R. Glandorf, Lagrange Multipliers and the State Transition Matrix for Coasting Arcs, AIAA Journal, 7, Vol. 2, pp. 363 365, 1968. 8 O. Montenbruck and E. Gill, Satellite Orbits: Models, Methods and Applications, Springer, ISBN 978-3540672807, 2011.