Modified Algorithm for Calculating the Parameters of Maneuvers of Coplanar Meeting of Spacecraft in a Near-Circular Orbit Using Low-Thrust Engines

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Introduction The rendezvous of spacecraft (SC) in near-circular orbits is a highly intricate and technically demanding problem in astronautics. Its complexity arises from the interplay of nonlinear orbital dyna-mics, gravitational perturbations, and control con-straints, all of which must be carefully managed to achieve mission success. The precise execution of spacecraft rendezvous is fundamental to a wide range of space operations, including satellite ser-vicing, space station resupply, and autonomous docking maneuvers. The choice of methodology for spacecraft ren-dezvous is strongly influenced by mission-specific objectives, which can vary significantly depending on operational requirements. These objectives dictate the selection of optimal control strategies, trajectory planning techniques, and guidance algo-rithms, all of which must balance fuel efficiency, time constraints, and navigational accuracy. As advancements in space technology continue to push the boundaries of autonomous operations, the development of robust and efficient rendezvous strategies remains a critical area of research in astronautics. For instance, the trajectory optimi-zation strategies used for SC rendezvous in near-circular orbits differ fundamentally from those applied in atmospheric observation missions. These disparities arise from variations in space-craft modeling and the corresponding control system architecture. In the realm of commercial and operational spaceflight, SC rendezvous problems exhibit notable similarities. While fundamental rendezvous algo-rithms have been developed and successfully implemented, continuous refinement is necessary to enhance their precision and efficiency. Typically, rendezvous operations involve two spacecraft: an active vehicle executing maneuvering procedures and a passive target following a free-flight trajectory. This paper aims to propose a modified algorithm for optimizing spacecraft rendezvous in near-circular orbits, ensuring maximum efficiency and accuracy while adhering to operational constraints. The problem of spacecraft rendezvous in near-circular orbits using low-thrust propulsion is of critical importance in contemporary spaceflight. It plays a fundamental role in various applications, including coordinated spacecraft formations, satellite constellation deployment, active debris removal, and on-orbit servicing missions. Since the mid-20th century, electric propulsion systems have been extensively employed due to their high specific impulse, which significantly reduces pro-pellant consumption rate for orbital maneuvers. However, the inherently low thrust of these systems results in prolonged maneuver durations, which must be meticulously accounted for in mission planning and control strategies. Optimal low-thrust maneuvering has been ex-tensively investigated in previous studies [1-12]. Of particular relevance are the contributions in [3-7], which address trajectory optimization under stringent constraints. Due to the mathematical com-plexity of low-thrust trajectory planning, numerical approaches based on Pontryagin’s Maximum Principle and the continuation method have been traditionally employed. More recently, interior point methods [8] have gained prominence, demonstrat-ing efficacy in solving largescale maneuvering problems. Over the past two decades, spacecraft rendez-vous has remained an active field of research [3-5; 9; 10]. Initial studies predominantly focused on high-thrust rendezvous strategies in near-circular orbits [11; 12], successfully addressing short-duration rendezvous (within three orbital revolut-ions) and classical mid-term rendezvous scenarios in coplanar circular orbits. Given the evolving landscape of space operations, continued advance-ments in rendezvous algorithms are essential to support emerging mission architectures and operational requirements. Currently, the problem of multi-impulse space-craft maneuvers remains one of the central chal-lenges in astrodynamics, requiring the development of increasingly efficient and reliable computational methods. Due to the inherent complexity of this problem, contemporary approaches typically adopt a multi-stage resolution framework that combines analytical and numerical techniques. The primary difficulty in formulating and solving these problems arises from the need to model space trajectories under multiple dynamic and operational constraints. To address these challenges, various algorithms have been proposed to decompose the solution into structured steps, as demonstrated in studies [13-21]. Analytical methods, as presented in [13-17], are widely used to solve orbital maneuvering and orbital plane rotation problems independently. Although this approach may lead to an increase in the total characteristic velocity required for maneuvers, it is advantageous due to its simplicity and operational reliability. Additionally, numerical methods have been employed to determine optimal solutions in highly complex multi-impulse scenarios, taking into account specific constraints, as detailed in [18; 19]. The authors of [20; 21] developed efficient algo-rithms for maneuver parameter calculations, which are widely used due to their accuracy and applicability. An alternative based on solving Lambert’s problem was presented in [21]. In this approach, the parameters corresponding to a two-impulse trajectory are initially determined, followed by an analysis of the behavior of the hodograph of the base vector associated with the solution. If ne-cessary, additional velocity impulses are introduced to refine the trajectory and ensure an optimized solution. Finally, the studies [4; 22] propose hybrid numerical-analytical methods for solving multi-impulse rendezvous problems, aiming to effectively address contemporary practical challenges. These approaches integrate the principles established in previous studies [13-21], providing a more com-prehensive solution adapted to the demands of modern astrodynamics. This paper presents a modified algorithm with enhanced capabilities for addressing the rendezvous problem of two SC in near-circular orbits under low-thrust propulsion. The proposed modification of this algorithm aims to overcome the limitations of existing methods, which often fail to obtain viable solutions in low-thrust regimes, as demon-strated in study [9]. With the implemented improve-ments, it becomes possible to successfully execute maneuvers even under minimal thrust conditions, thereby expanding the algorithm’s applicability. In this study, the SC rendezvous problem is analyzed both from the perspective of impulsive maneuvers and considering the continuous opera-tion of low-thrust propulsion systems. In contrast to previous studies [3-6; 9], which employed diffe-rent strategies for solving the rendezvous problem in coplanar orbits, the proposed algorithm offers a more comprehensive approach, being applicable both in ground-based control centers and onboard satellites, thus enabling greater operational auto-nomy. Various specialized mathematical models are used to describe the relative motion of spacecraft in near-circular orbits. One of the most widely employed is the Hill-Clohessy-Wiltshire (HCW) model [23; 24], which assumes that the separation between SC is small compared to the orbital radius. However, in this study, we adopt an alter-native linearized formulation derived in [25], which provides greater accuracy and applicability for maneuver planning in low-thrust regimes. With the increasing number of SC and the growing demand for real-time problem-solving, there is a significant shift toward onboard com-putation of maneuver parameters. This necessitates the development of computationally efficient and highly reliable algorithms. The proposed method meets these requirements, ensuring computational robustness and enhancing the feasibility of auto-nomous execution of orbital maneuvers. 1. Mathematical Formulation of the Rendezvous Problem The maneuver planning for a spacecraft trans-ferring between two closely spaced near-circular orbits is analyzed within the framework of unper-turbed Keplerian motion. The problem is approached using an approximate impulsive model, where the trajectory is discretized into N velocity impulses applied over a predefined time horizon. By em-ploying a linearized approximation, the conditions governing the transition from an initial orbit to a target coplanar orbit can be expressed as follows [26; 27]: + (1) - (2) (3) (4) where Here, аf and a0 represent semi-major axes of the orbits. The initial and final time is given by tf and t0. The reference circular orbit, characterized by a radius r0 such that r0 = аf, imposes the constraints V0 and n0, which respectively represent the orbital velocity and angular velocity of the spacecraft’s motion. The maneuvering strategy consists of N discrete velocity impulses, each applied at an angle φi, -measured from the line connecting the spacecraft to the target point in the direction of motion. The i-th velocity correction is decomposed into its transverse and radial components, each playing a critical role in shaping the transfer trajectory. The maneuver optimization problem is defined as minimizing the total characteristic velocity DV associated with the executed maneuvers: under restrictions (1)-(6). 2. Algorithm for Solving the Rendezvous Problem The rendezvous problem is solved based on the resolution of the orbital transfer problem. To achieve this, an algorithm presented in papers [9; 27] is employed, where the authors assume that the correction of the eccentricity vector and the impulse application angles can be performed by applying velocity impulses at optimal points along the trajectory. The determination of these points is formalized by the following expressions: where αе is the angle defining the optimal direction for correcting deviations in the eccentricity vector. The optimization conditions result in three distinct categories of solutions, as presented in [4; 27]. The optimal impulse magnitudes can be obtained from the analysis of the first three equations of system (1)-(4), following the approach described below: (5) (6) Once the optimal impulse magnitudes are determined, they will be used as initial approxi-mations to solve the rendezvous problem. Sub-sequently, the velocity impulses ΔVt1 and ΔVt2 are distributed over the N available orbital revolutions designated for maneuver execution [9; 27]: (7) (8) The next goal is to determine the distribution of velocity impulses over the turns in a manner that satisfies equation (4). We will adopt a significant simplification, assuming that the variation of velocity impulses over the turns occurs linearly to make the analysis more tractable, that is, allowing the approximation to reduce the complexity of the meeting problem: (9) (10) Therefore, substituting the values of the velocity impulses determined using expressions (9) and (10) into equations (7) and (8), we will obtain the follow-ing equations: (11) (12) Consequently, substituting the obtained values into equations (9) and (10), we obtain: (13) (14) Thus, we found the values of all velocity impulses expressed only through DV1t1 and DV2t1. Substituting them into equation (3), we obtain a linear equation with two unknowns , . The coefficients of the velocity impulses are known, since their angles of application are known: (15) (16) By iterating the value of the variable , within the specified interval, we determine the corresponding value of the variable for each case based on equation (3). Next, the values of all velocity impulses are calculated based on equations (13) and (14). The sum of the magnitudes of these impulses defines the total characteristic velocity for each obtained solution. The solution corresponding to the lowest total characteristic velocity is considered the optimal rendezvous trajectory. If the total characteristic velocity of the selected solution matches that of the transfer problem, it can be inferred that the tra-jectory with the minimum achievable characteristic velocity has been determined. In the next step, the duration of each identified maneuver is estimated using the following formula: (17) where is the represents the centripetal acceleration of the reference circular orbit, is the acceleration generated by the propulsion system, m denotes the mass of the active SC, T is the thrust of its engine. If (the duration of the largest velocity impulse), the solution can be considered approxi-mately equivalent to an impulsive-based approach, and the problem is considered resolved. However, when the maneuver duration becomes significant, the solution shifts to one that involves low thrust. 3. Solving the Problem With “Low Thrust” The velocity impulses applied for each turn result, in a certain way, in the change in eccentricity and semi-major axis, so to take these changes into account we will use the following expressions: (18) (19) Therefore, we calculate the necessary duration of low-impulse maneuvers that will result in the same change in these elements [25]: (20) Thus, the duration of each maneuver is determined iteratively, turn by turn, ensuring the successful resolution of the low-thrust problem. If the arcsine argument exceeds unity, no feasible solution exists under the given thrust constraints and spacecraft mass for the specified number of orbital turns. The computed low-thrust solution exhibits a similar evolution of the semi-major axis and ec-centricity vector compared to the corresponding impulsive transfer. Equation (4) is satisfied with high accuracy, as the midpoints of the extended-duration maneuvers coincide with the instants at which velocity impulses are applied in the impulsive case. This alignment ensures a comparable modifi-cation of the orbit’s major axis and guarantees arrival at the designated rendezvous point within the required timeframe. However, the rendezvous problem has been solved using linearized equations of motion, which neglect perturbative effects such as the non-centrality of the gravitational field, atmospheric drag, and other external influences. Consequently, the accuracy in satisfying the terminal conditions defined in system (1)-(6) remains insufficient. To enhance precision, an iterative correction scheme may be necessary [18; 19]. Furthermore, the previously proposed algo-rithm proved inadequate for rendezvous maneuvers involving spacecraft equipped with low-thrust engines of very small thrust magnitude [9]. As the thrust level decreases, the duration of certain critical maneuvers extends beyond the required correction time for the eccentricity vector. To mitigate this issue, the duration of these maneuvers is con-strained to the upper bound at which the eccentri-city correction remains maximized (180° change), while increasing the number of intermediate maneuvers. If these adjustments do not sufficiently impact the arrival time at the rendezvous point, an additional velocity impulse can be introduced at a specific orbital position. To compensate for the residual trajectory deviation at the rendezvous point in the absence of impulsive corrections, we sub-tract the effect of pre-defined discrete impulses (typically 4, 5, or 6, depending on their influence on the final arrival time). This approach ensures a more precise alignment with the target conditions while maintaining the feasibility of the low-thrust transfer strategy. 4. Algorithm for Solving the Meeting Problem When Fixing Velocity Impulses The formulated rendezvous problem uses an algorithm that consists of the following stages: 1. For long-term maneuvers, we calculate how fixed (N impulses) maneuvers on the outer turns change the eccentricity and major semi-axis; 2. Then we solve the transfer problem for the remaining misses in eccentricity and major semi-axis; 3. We calculate the change in arrival time due to the influence of N impulses; 4. Then we distribute two new calculated velocity impulses between the remaining turns to correct the time miss remaining after the fixed impulses; 5. We determine the change in eccentricity and major semi-axis on each turn; 6. We take into account the duration of maneu-vers; 7. If the new internal impulses are also greater than the permissible value, then the procedure is repeated, new fixed velocity impulses appear; 8. We calculate the total costs. 5. Examples of Solving the Coplanar Rendezvous Problem When Recording Velocity Impulses Let us analyze the motion of a SC relative to a reference point O, which follows a near-circular orbit of radius 6871 km around the Earth, under the assumption of an unperturbed gravitational field. The Earth’s gravitational parameter is taken as 3.9860044∙1014 m3/s2. The objective is to resolve the problem of a flight, where the spacecraft performs N velocity impulses within a fixed time interval to transition from an initial orbit to a target location in phase space. The initial conditions are: r0 = (10, 100, -5) km and vf = (0, 0, 0) m/s with the goal of reaching the origin of the reference frame, i.e., rf = (0, 0, 0) km and vf = (0, 0, 0) m/s in the tenth turn N=10. The spacecraft has an initial mass of 1000 kg, and its propulsion system operates with a specific impulse of 220 s, corresponding to an effective exhaust velocity of 2157.463 m/s. The thrust magnitude varies within the range 0.19 to 0.362 N. Table 1 presents the parameters of the coplanar transition maneuvers, where the first velocity impulse is braking, and the second is accelerating. This occurs because the orbits intersect. When addressing the rendezvous problem, the velocity impulses were not only distributed across the turns, as shown in Figure 1, but also optimized with respect to a single parameter, ensuring com-pliance with the time constraint. Solution to the problem with low thrust: In certain instances, the previous solution algo-rithm is unavailable because the argument of the arcsine falls outside the range (-1; 1). Additionally, as thrust increases, the maneuver duration dec-reases, and the total velocity costs for the low-thrust solution with thrust entrainment align with those of the impulse-based solution. Table 2 shows the correction of eccentricity and semi-major axis using velocity impulses for each turn. After correcting the orbital elements, Table 3 shows the calculated results for the problem with low thrust (T = 0.362N) for N = 10. Figure 1. Distribution of the two-impulse optimal maneuvers by turns S o u r c e: made by A.A. Baranov, A.P. Olivio Table 1 Results of the calculation the parameters of coplanar transition maneuvers -2.785 1. 7 4.485 6.4 186.4 366.4 S o u r c e: made by A.A. Baranov, A.P. Olivio Table 2 Results of the correction of eccentricity and semi-major axis by turns N 1 -0.5492 0.5485 2 -0.6888 0.3633 3 -0.8285 0.1781 4 -0.9682 -0.00711 5 -1.108 -0.1923 6 -1.248 -0.3775 7 -1.387 -0.5627 8 -1.527 -0.7479 9 -1.667 -0.9331 10 -1.806 -1.118 S o u r c e: made by A.A. Baranov, A.P. Olivio Table 3 Results of calculation of the problem with low thrust “T = 0.362 N” for N = 10 N 1 -0.002 0.211 0.213 -0.342 36.991 37.333 2 -0.064 0.202 0.266 -11.166 35.44 46.606 3 -0.126 0.193 0.319 -22.044 33.944 55.988 4 -0.188 0.186 0.374 -33.029 32.554 65.583 5 -0.252 0.179 0.431 -44.173 31.324 75.497 6 -0.317 0.173 0.49 -55.54 30.317 85.857 7 -0.383 0.169 0.552 -67.2 29.602 96.802 8 -0.452 0.167 0.619 -79.237 29.265 108.502 9 -0.523 0.168 0.691 -91.758 29.412 121.17 10 -0.598 0.172 0.77 -104.902 30.181 135.083 -2.903 1.818 4.721 -509.392 319.029 828.421 S o u r c e: made by A.A. Baranov, A.P. Olivio It can be stated that with such a thrust the problem is solved optimally and the introduction of fixed impulses is not required. Further, in order to move to an algorithm with fixed maneuvers, it is necessary to solve the problem with such a low thrust that impulses appear greater than the permissible value. However, we are interested in solutions with a thrust of 0.24, 0.22, 0.21, 0.2 and 0.19 N, and thrusts less than 0.19 N are not interesting. Figure 2 presents the results of calculating the problem with low thrusts for T = 0.24, 0.22, 0.21, 0.2, 0.19 N. It can be seen that there is no solution, since the first maneuver is larger than permissible. We can observe the evolution of the velocity impulses and durations of the maneuvers at the different thrust levels. With the increase in thrust, the velocity impulses and durations of the maneuvers in the last turns will not exist (this is visualized in the Figures of levels 2 to 5). In these same figures we can observe the reduction of the values of N, in reality there was no reduction, but rather the non-existence of the values of the velocity impulses and duration of the maneuvers in the last turns. Next, we take one of the solutions presented in Figure 2 and transform it into a solution with fixed impulses. We fix the impulses on the last turn. The duration of the first impulse on the turn is -180 degrees, and the duration of the second impulse is 72 degrees. It is necessary to calculate and present in the table the influence of these impulses on the difference in the orbital elements (OE), compare the values of the orbital elements before and after the maneuvers are performed. After that, the standard algorithm for the first 9 turns presented in [9] can be applied. Table 4 presents the orbital elements (eccent-ricity, semi-major axis, and flight time) before and after applying these fixed impulses. It is important to note that, to calculate the parameters for co-planar transfer maneuvers, new orbital elements must be calculated from the initial orbital elements. Here and everywhere, deviations (Da and Dt) will be presented in dimensionless variables. Afterward, we need to multiply by r0 and λ0 to convert them back to original units. These updated values will be defined in Table 5, and so on. Table 4 Difference of orbital elements Orbital elements (OE) OE1 OE2 Difference of OE -1.806 -1.655 -0.151 -1.118 -0.9823 -0.1357 -12 -1.51 -0.1049 Note: OE1 = Orbital elements before impulse fixation and OE2 = Orbital elements after impulse fixation. S o u r c e: made by A.A. Baranov, A.P. Olivio Table 5 Results of calculation of parameters of coplanar transfer maneuvers -2.283 1.572 3.855 S o u r c e: made by A.A. Baranov, A.P. Olivio Subsequently, the changes in eccentricity and the semi-major axis for each turn are determined to satisfy the spacecraft's flight time condition, with the results presented in Table 6. Table 6 Results of the correction of eccentricity and semi-major axis by turns N 1 -0.499 0.4983 2 -0.6554 0.3219 3 -0.8119 0.1454 4 -0.9683 -0.0311 5 -1.125 -0.2074 6 -1.281 -0.3838 7 -1.438 -0.5603 8 -1.594 -0.7368 9 -1.75 -0.9132 10 -1.655 -0.9821 S o u r c e: made by A.A. Baranov, A.P. Olivio Figure 3 shows the parameters of the optimal solution to the meeting problem for N = 10. Figure 2. Results of calculation of the problem with low thrust for N = 10 S o u r c e: made by A.A. Baranov, A.P. Olivio Figure 2. Results of calculation of the problem with low thrust for N = 10 (continuation) S o u r c e: made by A.A. Baranov, A.P. Olivio Figure 3. Distribution of the two-impulse optimal maneuvers by turns for the rendezvous problem S o u r c e: made by A.A. Baranov, A.P. Olivio Table 7 Results of calculation of the problem with low thrust “T = 0.22 N” for N = 10 N 1 -0.004 0.194 0.198 -1.133 55.898 57.031 2 -0.067 0.19 0.257 -19.414 54.786 74.2 3 -0.132 0.187 0.319 -37.966 53.945 91.911 4 -0.198 0.186 0.384 -57.068 53.654 110.722 5 -0.267 0.188 0.455 -77.076 54.269 131.345 6 -0.341 0.195 0.536 -98.493 56.293 154.786 7 -0.423 0.21 0.633 -122.144 60.551 182.695 8 -0.519 0.238 0.757 -149.693 68.707 218.4 9 -0.645 0.297 0.942 -186.245 85.866 272.111 10 -0.623 0.249 0.872 -180 72 252 -3.219 2.134 5.353 -929.232 615.97 1545.202 S o u r c e: made by A.A. Baranov, A.P. Olivio As shown in Figure 2, no solution was found for the tenth turn, as the maneuver durations exceeded the permissible limit. However, after applying the new algorithm, the results presented in Table 7 show that a solution for the tenth turn is now available. This indicates that the algorithm was successful. Conclusion The paper proposes a modified algorithm for calculating the parameters of a multi-turn rendez-vous. The main advantage of the proposed algo-rithm is its simplicity and reliability, which allows it to be used not only in ground control centers, but also on board a spacecraft. At the same time, this modification of the algorithm for calculating the parameters of a multi-turn rendezvous allows us to obtain a solution to the problem at low thrust. The examples given in the article confirm the oper-ability of this modified algorithm and the high quality of the resulting solution. Furthermore, the algorithm’s ability to adapt to varying mission conditions, such as changes in thrust or trajectory, demonstrates its versatility and potential for broa-der applications in future space missions. This en-hancement could significantly contribute to improv-ing mission efficiency and accuracy, particularly for long-duration spaceflights requiring precise maneuvering.

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

4, Miusskaya square, Moscow, 125047, Russian Federation

Adilson P. Olivio

RUDN University

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

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

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

References

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