Non-coplanar rendezvous in near-circular orbit with the use a low thrust engine

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Introduction The problem of meeting in a near-circular orbit using low-thrust engines is important in the practice of spacecraft (SC) flights. Typical examples are the problem of rendezvous and docking of spacecraft, the implementation of a group flight of several spacecraft, the formation of a given configuration of satellite systems, during removal of space debris, during servicing of spacecraft and other astro- nautical missions involving more than one space- craft. Due to the great complexity of solving prob- lems of spacecraft meeting with greater accuracy, over the past few years many authors have deve- loped algorithms for solving the problem of space- craft meeting [1-2]. Currently, three main approaches are widely used in solving complex problems of multi-impulse maneuvering of spacecraft. In the first case, the problems of maneuvering in the orbital plane and the problems of rotating the orbital plane are solved independently. This scheme was used, for example, when approaching the Shuttle spacecraft with an orbital station, to control the movement of geo- stationary satellites [3], satellites within satellite systems [4-6], and so on. The advantage of this scheme lies in its simplicity and reliability, and the disadvantage is the excessive cost of characteristic velocity for maneuvering. In the second case, numerical methods are used to find optimal solutions for the most complex multi-impulse problems, taking into account a wide range of restrictions [7-8]. The simplex method is most often used [9-10]. In the third method, at the initial stage, the solution to the Lambert problem is used to deter- mine the parameters of the two-impulse solutions to the meeting problem. Then the behavior of the hodograph of the basis vector corresponding to the found solution is analyzed, and, if necessary, additional velocity impulses are added to obtain the optimal solution. This approach was first used in the works of Lion and Handelsman [11], Jezewski and Rozendaal [12]. There are also methods that are at the inter- section of different approaches. For example, in [13-14] numerical and analytical methods for solving the multi-impulse meeting problem are proposed, combining the advantages of the first and second of the previously listed approaches. They make it possible to use the results obtained in the early papers of T. Edelbaum [15], J.P. Marec [2], when solving modern practical problems. Since the 1960s, the process of using electric rocket engines (ERM) on spacecraft began. Thanks to their high specific impulse, electric propulsion engines can significantly reduce fuel costs for orbital maneuvering. However, the low (compared to traditional liquid rocket engines) thrust of electric propulsion engines leads to the need to take into account their long-term operation. Problems of this type take a special place a special place among the problems of optimal maneuvering of a spacecraft. A significant number of articles have been devoted to them, and several very interesting monographs have been published [16; 17]. Particularly noteworthy are the papers of V.G. Petukhov [18-20]. Due to the complexity of the problems in which it is assumed that maneuvering is carried out using a propulsion sys- tem (PS), they have traditionally been solved numerically and by methods using the Pontryagin maximum principle or the continuation method. In recent years, Yu.P. Ulybyshev has successfully used the interior point method to solve problems with long maneuvers [21]. In the method considered in this paper, the noncoplanar meeting problem is solved both in the impulse formulation and taking into account the long-term operation of the low-thrust engine [22-24]. To analyze the relative motion of a spacecraft in the vicinity of circular orbits, it is necessary to use special mathematical models of motion. The most popular mathematical model of the relative motion of a spacecraft in the vicinity of circular orbits is the Hill-Clohessy-Wiltshire (HCW) model. Linearized differential equations for the relative motion of a spacecraft in the vicinity of a circular orbit for the problem of rendezvous and docking were obtained by Clohessy-Wiltshire in 1960 [25], but back in the 19th century similar equations were used by Hill in his theory of lunar motion [26]. In this mathematical model, to obtain the equations of relative motion, a rotating (orbital) coordinate system and linearization of the differential equations of relative motion are used, based on the assumption that the distance between the considered spacecraft is small compared to the average orbital radius. This work uses linearized equations obtained by P.E. Elyasberg [27]. They were obtained using a cylindrical coordinate system and are significantly more convenient for solving the problem of long-duration encounters, in which there are significant deviations along the orbit. Due to the increase in the number of maneu- vering spacecraft and the increase in the efficiency of solving problems, there is currently a tendency to transfer the process of calculating maneuvers on board the spacecraft. This leads to the need to simplify the process of calculating maneuver para- meters and increase the reliability of this process. The algorithm considered in this paper has precisely these properties. 1. Formulation of the meeting problem The problem of calculating the parameters of transfer maneuvers between close near-circular orbits is solved in an approximate impulse formulation, within the framework of unperturbed Keplerian motion. The conditions for transferring with the help of N velocity impulses in a fixed time from the initial orbit to a given point of the final non-coplanar orbit (meeting problem) in a linear approximation can be written in the form Ilyin and Kuzmak [22]: N (ΔVrisinϕi +2ΔVti cosϕi )=Δex , (1) i=1 N (-ΔVri cos ϕi +2ΔVtisinϕ i )=Δe y , (2) i=1 N 2ΔVti =Δa, (3) i=1 N (2ΔVri (1- cosϕi )+Δ -Vti ( 3ϕi +4sinϕi ))=Δt, (4) i=1 ∑ -Δ sin φ = Δ , (5) ∑ Δ cos φ = Δ , (6) where Δех = efcosωf - e0cosω0, Δеу= efsinωf - e0sinω0, Δa=(аf-a0)/r0, Δt=λ0(tf-t0), Δz= z0/r0, ΔVz=ΔVz0/V0, . Here «f», «0» - the indices corresponding to the final and initial orbits, ef, e0 - the eccentricities of the orbits; аf, a0 - semi-axes major of orbits; ωf, ω0 - angles between the direction to the pericenter of the corresponding orbit and the direction to a point specified on the final orbit (the Ox axis is directed to this point); tf - the required time of arrival at a given point, t0 - the time at which, when moving along the initial orbit, the projection of the radius vector onto the plane of the final orbit hits the ray passing through the given meeting point; z0 - the deviation of the spacecraft in the initial orbit from the plane of the final orbit at time t0; Vz0 - lateral relative velocity at this moment; V0, λ0 - orbital and angular velocities of movement along the reference circular orbit of radius r0 (r0 = аf); N - number of velocity impulses; φi - the angle of application of the i-th velocity impulse, measured from the direction to a given meeting point in the direction of the SC motion; - transversal, radial and lateral components of the i-th velocity impulse, respectively. It is necessary to take into account that the angles φi - negative, because it was assumed that at a given point φf = 0. The problem of searching for parameters of optimal maneuvers can be formulated as follows: it is necessary to determine ΔVri, ΔVti, ΔVzi, φi (i = 1, …, N), at which the total characteristic velocity of maneuvers ΔV is minimal. Δ = Δ = Δ + Δ + Δ , under restrictions (1)-(6). In this paper problem of the meeting is solved in several stages. At the first stage, the problem of impulses transfer between non-coplanar orbits is solved (Section 2). The velocity impulses for solving the transfer problem are then distributed among the turns allowed for maneuvering to ensure that equation (4) is satisfied (Section 3). In sections (4) and (5), a solution to the low-thrust transfer problem is sought. The maneuver parameters can be refined using an iterative procedure to take into account all dis- turbances (the influence of compression of the Earth, atmosphere, etc.). 2. Algorithm for solving the transfer problem When solving the problem of transfer between non-coplanar orbits, five equations of the system (1)-(6) are used. The angle φ (the angle of application of the first velocity impulse) is searched and for each successive value of the angle the values of the velocity impulses and the angle φ are found: ∆ ∆ ∆ = , (7) ∆ ∆ ∆ ∆ ∆ = , (8) tanφ =, (9) and then from equations (5)-(6) the values of the lateral components of the velocity impulses are determined: ∆ ∆ ∆ = - , (10) ∆ ∆ ∆ = - . (11) ( ) From the entire set of solutions found, the one that provides the minimum total characteristic velocity is selected. Further, the parameters of this solution are indicated by the index «m» ΔVt1m, ΔVz1m, φ1m, ΔVt2m, ΔVz2m, φ2m. 3. Algorithm for solving the meeting problem When solving the meeting problem, the values of the velocity impulses ΔVt1, ΔVt2, determined when solving the transfer problem, are distributed among N turns allowed for maneuvering: N ΔV1tm = ΔV1ti ; (12) i=1 N ΔV2tm = ΔV2ti . (13) i=1 Here N is the number of turns on which maneu- vering is allowed. The lateral components are distributed in pro- portion to the transversal | | ΔV = Δ , | | and | | ΔV = Δ . (14) | | The further goal is to select such a distribution of the magnitudes of the velocity impulses along the turns so that equation (4) is satisfied. To significantly simplify the solution of the problem, we assume that the magnitudes of the velocity impulses along the turns change linearly: ΔV1ti =ΔV1 1t + + Δ( V1tN -ΔV1 1t )(i -1) / (N -1), (15) ΔV2ti =ΔV2 1t + + Δ( V2tN -ΔV2 1t )(i - 1) / (N - 1). (16) Here ΔV1t1, ΔV1tN и ΔV2t1, ΔV2tN are the magnitudes of the velocity impulses on the first and last permitted turns of maneuvering, which are a part of the first and second velocity impulses of solving the transfer problem. Substituting the values of velocity impulses calculated using formulas (15), (16) into (12) and (13) we obtain: Δ = ∑ Δ = 0.5 (Δ + Δ ); (17) Δ = ∑ Δ = 0.5 (Δ + Δ ). (18) Using (17) and (18), we obtain formulas for determining ΔV ,ΔV : 1tN 2tN Δ = - Δ ; . (19) Δ = - Δ . . (20) Substituting the found values ΔV1tN ,ΔV2tN into formulas (15) and (16), we obtain: ∆ = = 2∆ ( - 1) (- 1) + ∆ 1 - ( ) , (21) ∆ = = 2∆ ( - 1) (- 1) + ∆ 1 - ( ) . (22) Thus, we found the values of all velocity impulses, expressed only in terms of ΔV1t1 and ΔV2t1. Substituting them into equation (3), we obtain a linear equation with two unknowns ΔV1 1t , ΔV2 1t . The coefficients for velocity impulses are known, since their angles of application are known: ϕ1i =ϕ1m + π2 (N Ni - ) , (23) ϕ 2i = ϕ 2 m + π2 (N Ni - ) . (24) By sorting through the value of the variable ΔV1 1t , within the specified limits, for each value from equation (3) we find the value of the variable ΔV2 1t . Then, using (23) and (24), we find the values of all velocity impulses. Adding the modules of all velocity impulses, we find the total characteristic velocity of the next solution. The solution whose total characteristic velocity is minimal is accepted as a solution to the meeting problem. If the total characteristic velocity of the found solution coincides with the total characteristic velocity of the solution to the transfer problem, then a solution with the minimum possible total characteristic velocity was found. If the duration of the largest velocity impulse does not exceed 20°, then the solution is close to an impulse one and we consider that the problem has already been solved. Taking into account all disturbances (non-centrality of the gravitational field, the influence of the atmosphere, etc.), the operation of a real propulsion system can be carried out using the iterative procedure described in Section 5. If the duration of the maneuvers is significant, then we proceed to solving the problem taking into account low thrust PS. 4. Solving the problem with «low thrust» It is assumed that the orientation of the propulsion system during the execution of the maneuver is fixed in the orbital coordinate system. For each turn, we find what changes in eccentricity and semi-major axis produce the found velocity impulses determined at this turn Δ =2Δ cos φ + 2 cos φ , (25) Δ =2Δ sin φ + 2Δ sin φ , (26) Δ =2Δ + 2Δ . (27) Similarly for changing the lateral parameters on a turn Δ =Δ cos φ + Δ cos φ , (28) Δ = Δ sin φ + Δ sin φ . (29) Then we determine the required duration of low-thrust maneuvers that will produce the same change in these elements [20]: Δφ = 2 arcsin , (30) Δφ = 2 arcsin . (31) Thus, turn by turn we find the duration of all maneuvers. The low thrust problem has been correctly solved. If the arcsine argument is greater than 1, then there is no solution (with the existing thrust and mass of the spacecraft, it is impossible to solve the meeting problem for a given number of turns). The found solution with “low thrust” gives the same change in the eccentricity vector and orienta- tion of the orbital plane as the original impulse solution, because the midpoints of long maneuvers coincide with the moments of application of velo- city impulses. However, the difficulty is that the change in the semi-major axis becomes larger than necessary, since it changes with orbital orientation more effectively than eccentricity. Therefore, as a result of the maneuvers, an error remains in the formation of the required value of the semi-major axis, and to eliminate this error, you can use the iterative procedure described in [20]. Let us assume that the initial deviation of the semi-major axis was ∆ = - (for example, ∆ > 0), and the deviations ∆ , ∆ , ∆ , ∆ , ∆ϑ (the angle between the line of intersection of the orbital planes and the line of apses relative orbits) were used in determining the parameters of the maneuvers. As a result of performing the calculation maneuvers, the semi-major axis will be formed ( > ). In the next iteration, deviations ∆ = ∆ + - , ∆ , ∆ , ∆ , ∆ϑ will be used, at the next iteration ∆ = ∆ + - , etc., until the semi-major axis is formed with the given accuracy. Since at each turn the same change in the semimajor axis will be made as in the impulse solution, the meeting problem will be solved with the same accuracy. 5. Iterative procedure In the formulated meeting problem, linearized equations of motion are used, the non-centrality of the gravitational field, the influence of the atmosphere, etc. are not taken into account. This leads to the fact that the actual accuracy of fulfilling the terminal conditions in system (1)-(6) will be insufficient. To solve a problem with a given accuracy, you can use an iterative scheme [7-8], which consists of the following stages: 1. In the beginning of the next iteration, an “approximate” problem is solved: under the pre- viously accepted simplifying assumptions, the para- meters of maneuvers that ensure the formation of a “target” orbit are determined (at the first iteration, the “target” orbit coincides with the final orbit). 2. Then, taking into account the calculated maneuvers, using models of all necessary distur- bances, an “accurate” prediction of the spacecraft motion is carried out and the parameters of the formed orbit are found. 3. The deviations of the parameters of the for- med orbit from the corresponding parameters of the final orbit are calculated. 4. If the deviations exceed the permissible ones, then the parameters of the “target” orbit are changed by the value of the calculated deviations, and the next iteration is carried out. 5. The procedure ends when the terminal conditions are met with the specified accuracy. 6. For “accurate” forecasting, as a rule, nume- rical and/or high-precision numerical-analytical integration are used. It is possible to use different forecast methods at different iterations, but the accuracy of the forecast should increase with the number of the current iteration. 7. During numerical integration, the influence of the non-centrality of the gravitational field, atmosphere, light pressure, etc. is taken into account, the operation of the spacecraft engines is carefully modeled, therefore, despite the fact that the maneuver parameters are found at each iteration using the simplest motion model, but as a result of an iterative procedure, they ensure access to the final orbit with the required accuracy. 6. Example of solving the noncoplanar meeting problem Let us consider the motion of a spacecraft (SC) relative to point O, moving in an undisturbed near circular orbit with a radius of 6871 km. Let us take the gravitational parameter of the Earth equal to 3.9860044·1014 m3/s2. Let us consider the flight problem using N velocity impulses in a fixed time from the initial orbit to a given point in the final orbit from a point in phase space r0 = (10, 100, -5) km, v0 = (1, -10, 3) m /c to the origin, that is, to the point rf = (0, 0, 0) km, with a velocity vf = (0, 0, 0) m/s. For the problem, we will take the initial mass of the spacecraft equal to 1000 kg, the specific impulse of the spacecraft propulsion system is 220 seconds (2157.463 m/s), and the thrust (T) will be varied in the range from 1 to 100 N. The flight is carried out in N = 15 turns. Solution of the twoimpulse transfer problem Table 1 shows the results of calculations of the parameters of the optimal two-impulse transfer between non-coplanar orbits, that is, the values of the transversal and lateral components of the velocity impulses, the angles of application of the first and second impulses are given as well. The angle of application of the first velocity impulse was varied from 0 to 360° with a step of 0.75°. It can be seen that the minimum value of the characteristic velocity that a spacecraft (SC) must have for the transfer maneuver is 10.308 m/s. Multiimpulse solution to the meeting problem To obtain an impulse solution to the meeting problem, the velocity impulses of the two-impulse solution are distributed between 15 turns so that condition (4) is satisfied. For this purpose, the algorithm described in Section 3. The value of the first velocity impulse is varied within the range from -3.452 m/s to 0.5 m/s with a step of 0.023 m/s. Table 2 shows parameters of the distributed impulse solution Table 3 shows the deviations of orbital elements for each turn corresponding to the influence of distributed velocity impulses. This impulse solution can be transformed to take into account the real thrust of the engine. The process of obtaining a solution for 1N thrust is shown below. At the first stage, the durations of the maneu- vers are calculated, which for a real low thrust (1N) provide the changes in the orbital elements shown in Table 3 at each orbit (except for the semi-major axis). These durations are shown in Table 4. Then the change in the semi-major axis produced for a given duration of maneuvers is calculated and a new target value of the semi-major axis is formed for the next iteration. These data are shown in Table 5. The next iteration is performed and the para- meters of the new impulse solution, the duration of the maneuver and changes made of the semi-major axis under the influence of low thrust and errors in the correction of the semi-major axis are shown in Tables 6, 7 and 8. Table 1 Results of the calculation the parameters of the optimal noncoplanar impulse transfer problem ° ° ∆ , / ∆ , / ∆ , / ∆ , / ∆ , / ∆ , / , / , / , / 155 55.851 3.452 2.367 0.637 6.372 5.819 7.01 3.51 6.798 10.308 Table 2 Distribution of the twoimpulse solution by turns N ∆ , / ∆ , / ∆ , / ∆ , / ∆ , / ∆ , / , / , / , / 1 0.022 0.314 0.004 0.844 0.336 0.848 0.023 0.9 0.923 2 0.052 0.291 0.01 0.784 0.343 0.794 0.053 0.837 0.89 3 0.082 0.269 0.015 0.724 0.351 0.739 0.083 0.773 0.856 4 0.111 0.247 0.021 0.664 0.358 0.685 0.113 0.709 0.822 5 0.141 0.224 0.026 0.604 0.366 0.63 0.143 0.645 0.788 6 0.171 0.202 0.031 0.545 0.373 0.576 0.174 0.581 0.755 7 0.2 0.18 0.037 0.485 0.38 0.522 0.204 0.517 0.721 8 0.23 0.158 0.043 0.425 0.388 0.467 0.234 0.453 0.687 9 0.26 0.136 0.048 0.365 0.395 0.413 0.264 0.389 0.653 10 0.29 0.113 0.053 0.305 0.403 0.358 0.294 0.325 0.619 11 0.319 0.091 0.059 0.245 0.41 0.304 0.325 0.261 0.586 12 0.349 0.069 0.064 0.185 0.418 0.249 0.349 0.198 0.547 13 0.379 0.047 0.07 0.125 0.425 0.195 0.385 0.134 0.519 14 0.408 0.024 0.075 0.066 0.433 0.141 0.415 0.07 0.485 15 0.438 0.002 0.081 0.006 0.44 0.087 0.446 0.006 0.452 ∑ 3.452 2.367 0.637 6.372 5.819 7.01 3.51 6.798 10.308 Table 3 Results of deviations of orbital elements by turns N ∆ × ∆ × ∆ × ° ∆ × ∆ × ∆ × ° 1 0.41 0.71 0.816 59.877 0.765 0.627 0.915 55.578 2 0.306 0.691 0.755 66.096 0.629 0.589 0.847 55.163 3 0.203 0.675 0.705 73.302 0.492 0.552 0.779 54.68 4 0.099 0.66 0.667 81.466 0.356 0.514 0.711 54.112 5 0.005 0.644 0.644 89.598 0.219 0.476 0.642 53.435 6 0.108 0.629 0.638 80.254 0.083 0.439 0.574 52.612 7 0.212 0.614 0.649 70.979 0.053 0.401 0.506 51.594 8 0.315 0.598 0.676 62.228 0.19 0.364 0.438 50.302 9 0.419 0.583 0.718 54.318 0.326 0.326 0.37 48.609 10 0.522 0.568 0.771 47.388 0.463 0.288 0.302 46.299 11 0.626 0.552 0.835 41.432 0.599 0.251 0.234 42.976 12 0.729 0.537 0.906 36.362 0.736 0.213 0.166 37.832 13 0.833 0.521 0.983 32.057 0.872 0.176 0.097 29.028 14 0.936 0.506 1.064 28.395 1.009 0.138 0.029 11.991 15 1.04 0.491 1.15 25.267 1.145 0.1 0.039 21.159 Table 4 Duration of the maneuver for N = 15 N ∆ ° ∆ ° ∆ ° 1 1.427 59.874 61.301 2 3.347 55.244 58.591 3 5.268 50.71 55.978 4 7.19 46.259 53.449 5 9.115 41.881 50.996 6 11.041 37.567 48.608 7 12.971 33.306 46.277 8 14.905 29.093 43.998 9 16.843 24.92 41.763 10 18.786 20.78 39.566 11 20.734 16.667 37.401 12 22.689 12.575 35.264 13 24.65 8.5 33.15 14 26.618 4.436 31.054 15 28.595 0.377 28.972 N ∆ , / ∆ , / ∆ , / ∆ , / ∆ , / ∆ , / , / , / , / 1 0.04 0.317 0.045 0.843 0.357 0.888 0.060 0.901 0.961 2 0.066 0.294 0.029 0.791 0.36 0.82 0.072 0.844 0.916 3 0.092 0.271 0.014 0.73 0.363 0.744 0.093 0.779 0.872 … … … … … … … … … … 13 0.375 0.046 0.079 0.124 0.421 0.203 0.383 0.132 0.515 14 0.404 0.024 0.086 0.065 0.428 0.151 0.413 0.069 0.482 15 0.435 0.003 0.089 0.009 0.438 0.098 0.444 0.009 0.453 ∑ 3.501 2.378 0.501 5.402 5.879 7.073 3.585 6.82 10.405 Table 5 Changes made of the semimajor axis under the influence of low thrust and errors in the correction of the semimajor axis N ∆ × ∆ × × ∆ × 1 0.765 0.804 0.039 0.727 2 0.629 0.659 0.0304 0.598 3 0.492 0.516 0.0235 0.469 4 0.356 0.374 0.018 0.338 5 0.219 0.232 0.013 0.2065 6 0.083 0.092 0.0089 0.074 7 0.053 0.048 0.0056 0.059 8 0.19 0.187 0.00278 0.193 9 0.326 0.326 0.00036 0.327 10 0.463 0.464 0.0018 0.461 11 0.599 0.603 0.00375 0.596 12 0.736 0.741 0.00565 0.73 13 0.872 0.88 0.0076 0.865 14 1.009 1.018 0.0097 0.999 15 1.145 1.157 0.01203 1.133 Table 6 Parameters of the new impulse solution for N = 15 It can be seen that the accuracy of the semimajor axis formation has increased. It took four iterations to solve the problem. The information about the fourth iteration is given below (in Table 9). Fourth iteration. In the next iteration, an im- pulse solution is first sought for the deviations of the orbital elements at each turn. Then, the duration of the maneuvers is deter- mined and shown in Table 10. The change made in the semi-major axis is determined and shown in Table 11. The good accuracy of the semi-major axis formation was obtained, so the iterative procedure is completed. The duration of maneuvers is converted into impulse values. These results are shown in Table 12. Maneuvers are calculated in a similar way for various thrust values from a given range. The results are shown in the summary Table 13. Table 7 Duration of the maneuver for N = 15 N ∆ ° ∆ ° ∆ ° 1 3.82 60.588 64.408 2 4.57 55.777 60.347 3 5.944 51.103 57.047 … … … … 13 24.551 8.425 32.976 14 26.499 4.379 30.878 15 28.498 0.636 29.134 Table 8 Changes made of the semimajor axis under the influence of low thrust and errors in the correction of the semimajor axis N ∆ × ∆ × × ∆ × 1 0.765 0.717 0.00483 0.775 2 0.629 0.617 0.0116 0.61 3 0.492 0.494 0.0017 0.467 4 0.356 0.3614 0.0055 0.333 5 0.219 0.225 0.00569 0.201 6 0.083 0.0876 0.00461 0.0694 7 0.053 0.0503 0.00314 0.0622 8 0.19 0.188 0.00164 0.194 9 0.326 0.326 0.00022 0.327 10 0.463 0.454 0.0011 0.46 11 0.599 0.602 0.00233 0.593 12 0.736 0.723 0.0124 0.7425 13 0.872 0.869 0.00466 0.867 14 1.009 1.014 0.00566 0.993 15 1.145 1.15 0.00437 1.129 Table 9 Parameters of the next impulse solution for N = 15 N ∆ , / ∆ , / ∆ , / ∆ , / ∆ , / ∆ , / , / , / , / 1 0.031 0.315 0.019 0.848 0.346 0.867 0.036 0.905 0.941 2 0.061 0.293 0.015 0.788 0.354 0.803 0.063 0.841 0.904 3 0.093 0.271 0.016 0.73 0.364 0.746 0.094 0.779 0.873 … … … … … … … … … … 13 0.373 0.046 0.084 0.124 0.419 0.208 0.382 0.132 0.514 14 0.401 0.024 0.096 0.065 0.425 0.161 0.412 0.069 0.481 15 0.434 0.005 0.093 0.012 0.439 0.105 0.444 0.013 0.457 ∑ 3.496 2.381 0.517 6.382 5.877 7.039 3.562 6.834 10.396 Table 10 Duration of the maneuver for N = 15 N ∆ ° ∆ ° ∆ ° 1 2.273 60.18 62.453 2 3.991 55.573 59.564 3 5.999 51.128 57.127 … … … … 13 24.499 8.401 32.9 14 26.406 4.437 30.843 15 28.463 0.832 29.295 Table 11 Changes made of the semimajor axis under the influence of low thrust and errors in the correction of the semimajor axis N ∆ ∆ ∆ 1 0.765 0.774 0.00872 0.738 2 0.629 0.638 0.009 0.6 3 0.492 0.492 0.000198 0.468 4 0.356 0.356 0.0000162 0.333 5 0.219 0.219 0.0000054 0.2 6 0.083 0.0831 0.000121 0.0672 7 0.053 0.0532 0.000263 0.0646 8 0.19 0.19 0.0002505 0.385 9 0.326 0.326 0.0000469 0.327 10 0.463 0.463 0.000289 0.458 11 0.599 0.6 0.000481 0.589 12 0.736 0.734 0.00196 0.745 13 0.872 0.875 0.0029 0.857 14 1.009 1.01 0.001031 0.987 15 1.145 1.145 0.000124 1.127 Table 12 Parameters of the solution with low thrust for N = 15 N ∆ , / ∆ , / ∆ , / ∆ , / ∆ , / ∆ , / , / , / , / 1 0.035 0.33 0.006 0.888 0.365 0.894 0.036 0.947 0.983 2 0.062 0.305 0.011 0.82 0.367 0.831 0.063 0.875 0.938 3 0.093 0.28 0.017 0.755 0.373 0.772 0.095 0.805 0.900 4 0.12 0.256 0.022 0.688 0.376 0.71 0.122 0.734 0.856 5 0.148 0.231 0.027 0.623 0.379 0.650 0.150 0.664 0.815 6 0.176 0.207 0.032 0.558 0.383 0.59 0.179 0.595 0.774 7 0.204 0.184 0.038 0.494 0.388 0.532 0.208 0.527 0.735 8 0.232 0.16 0.043 0.431 0.392 0.474 0.236 0.460 0.696 9 0.261 0.137 0.048 0.368 0.398 0.416 0.265 0.393 0.658 10 0.29 0.114 0.054 0.306 0.404 0.36 0.295 0.327 0.622 11 0.319 0.091 0.059 0.245 0.41 0.304 0.324 0.261 0.586 12 0.349 0.07 0.064 0.187 0.419 0.251 0.355 0.200 0.554 13 0.379 0.046 0.07 0.124 0.425 0.194 0.385 0.132 0.518 14 0.409 0.024 0.075 0.065 0.433 0.14 0.416 0.069 0.485 15 0.441 0.005 0.081 0.012 0.446 0.093 0.448 0.013 0.461 ∑ 3.518 2.44 0.647 6.564 5.958 7.211 3.577 7.003 10.580 Table 13 Parameters of the solution with respect to maximal thrust magnitude T, N , m/s М, kg 1 10.580 4.892 2 10.377 4.798 5 10.32 4.772 10 10.318 4.771 100 10.308 4.766 Conclusion The paper describes an algorithm for calcula- ting the parameters of the multi-turn, multi-impulse meeting. The main advantage of the proposed algorithm is its simplicity and reliability, which allows it to be used not only in ground control centers, but also on board a spacecraft. In the same time, this algorithm makes it possible to obtain an optimal solution to the problem in the case when the initial phase belongs to the optimal phase range and the total characteristic velocity of solving the meeting problem coincides with the total character- ristic velocity of the optimal solution to the transfer problem. The algorithm makes it possible to obtain a solution even in the case when maneuvers are performed by low-thrust engines. Each stage of the algorithm is transparent for understanding and control. The examples given in the article confirm the performance of this algorithm and the high quality of the resulting solution.

About the authors

Andrey A. Baranov

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences

Email: andrey_baranov@list.ru
ORCID iD: 0000-0003-1823-9354
SPIN-code: 6606-3690

Candidate of Physical and Mathematical Sciences, Leading Researcher

Moscow, Russia

Adilson P. Olivio

RUDN University

Author for correspondence.
Email: pedrokekule@mail.ru
ORCID iD: 0000-0001-5632-3747

Postgraduate, Department of Mechanics and Control Processes, Academy of Engineering

Moscow, Russia

References