Differential Correction Algorithm Research Articles

Five-impulse low-energy earth-moon transfer using manifolds

The libration points in the Earth-Moon and Sun-Earth systems are widely regarded as cost-effective gateways into interstellar space. The associated manifold tubes have been utilized by numerous researchers in the development of transfers between distinct systems. In this paper, a five-impulse transfer method, based on the manifold, has been proposed for designing the low-energy transfer trajectory for the Earth-Moon system. The Sun-Earth-Moon-Spacecraft four-body problem is decomposed into two circular restricted three-body problems: Sun-Earth-Spacecraft and Earth-Moon-Spacecraft. Moreover, a time parameter is employed to describe the positional relationship between the Sun-Earth and Earth-Moon systems, thereby facilitating their conversion. A phase angle is utilized to characterize the position of the Poincare section, and multiple orbital elements, such as orbital altitude and inclination, are considered in the constraints. To solve the design of multi-constrained transfer trajectories, sequential differential correction algorithms are proposed. The results demonstrate that this method can be used to create compliant low-energy transfer trajectories for various combinations of time parameters and phase angles. Additionally, excellent velocity increments and time-of-flight metrics can be found within a single Sun-Earth-Moon conjunction cycle. In parallel, the method is validated by reconstructing transfer trajectories in a bicircular restricted four-body problem model (BCR4BP) of the Sun-Earth/Moon system.

A trajectory design and optimization framework for transfers from the Earth to the Earth-Moon triangular L4 point

• A synthetic computation procedure for transfer families using tangential velocity increments is presented. • A fast optimization method for direct transfer is constructed based on primer vector theory. • The necessary optimality conditions for a transfer utilizing powered lunar gravity assist under a minimum swing-by height constraint are deriver. • An optimization procedure for the optimal transfer utilizing powered lunar gravity assist is presented. The triangular libration points in the Earth-Moon system are considered to be potential candidates for future space missions because of their unique positions and dynamics. This paper presents a comprehensive transfer trajectory design and optimization framework for transfers from the Earth to L 4 in the Earth-Moon system, including direct transfer and transfer utilizing powered lunar gravity assist. In this framework, first, a synthetic procedure combining an initial guess, a differential correction algorithm and numerical continuation for generating tangential transfer families is illustrated. Subsequently, the optimization models for optimal transfer trajectories with the minimum velocity increment requirement are constructed. For direct transfer, the optimal direct transfer trajectory is obtained using primer vector theory. For the optimal transfer utilizing powered lunar gravity assist under a minimum swing-by height constraint, the necessary optimality conditions are found through optimal control theory, and an optimization procedure that allows for numerical convergence to the optimal solution is constructed. Numerical results indicate that the proposed transfer trajectory design and optimization framework has sufficient effectiveness to provide various transfer trajectories associated with different insertion points and fuel-optimal transfers.

Orbit Estimation of GEO-missions Applicable to NavIC Data

Indian Regional Navigation Satellite System (IRNSS) is India’s own navigation system. It is a seven satellite constellation with the operational name NavIC – Navigation with Indian Constellation. Satellite Orbit determination (OD) estimates the position and velocity of orbiting satellite. The two main estimation algorithms widely used in Global Navigation Satellite Systems (GNSS) are Extended Kalman Filter (EKF) and Least Squares. This study on Batch Least Squares (BLS) - Differential Correction (DC) algorithm demonstrates precise orbit estimation of any GEO missions using only range measurements with crude initial state parameters. Currently, the study is based on simulated inputs and the satellite orbits are successfully estimated with position error in sub-centimeters level. Further, the work will be extended to live data of IRNSS satellites.

A multi-arc approach for chaotic orbit determination problems

Chaotic dynamical systems are characterized by the existence of a predictability horizon, connected to the notion of Lyapunov time, beyond which predictions of the state of the system are meaningless. In order to study the main features of orbit determination in the presence of chaos, Spoto and Milani (Celest Mech Dyn Astron 124:295–309, 2016) applied the classical least-squares fit and differential correction algorithm to determine a chaotic orbit and a dynamical parameter of a simple discrete system—Chirikov standard map (cf. Chirikov in Phys Rep 52:263, 1979)—with observations distributed beyond the predictability horizon. They found a time limit beyond which numerical calculations are affected by numerical instability: the computability horizon. In this article, we aim at pushing forward such inherent obstacle to numerical calculations in chaotic orbit determination by applying the classical and the constrained multi-arc method (cf. Alessi et al. in Mon Not R Astron Soc 423:2270–2278, 2012) to the same dynamical system. These strategies entail the determination of an orbit when observations are grouped in separate observed arcs. For each arc, a set of initial conditions is determined and, in the case of the constrained multi-arc method, all subsequent arcs are constrained to belong to the same trajectory. We show that the use of these techniques in place of the standard least-squares method has significant advantages: Not only can we perform accurate numerical calculations well beyond the computability horizon, in particular, the constrained multi-arc strategy improves considerably the determination of the dynamical parameter.

Periodic motion around libration points in the Elliptic Restricted Three-Body Problem

Periodicity of motion around the collinear libration point associated with the Elliptic Restricted Three-Body Problem is studied. A survey of periodic solutions in the Circular Restricted Three-Body Problem is presented considering both Sun–Earth and Earth–Moon systems. Halo, Lyapunov and Vertical families around L1, L2 and L3 points are investigated, and their orbital period ranges through the entire family are reported. Resonant motions within the orbit families in the circular problem are identified and selected as suitable initial guess to find periodic orbits in the elliptic problem, which are targeted using a differential correction algorithm. Periodic solutions found are cataloged depending on the number of revolutions around libration points. Geometry, dynamical behavior and stability properties of single-revolution orbits are shown, as well as double-, triple- and quadruple-revolution solutions.

Rational Minimax Approximation via Adaptive Barycentric Representations

Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation - precisely the case where rational functions outperform polynomials by a landslide. We show that far more robust algorithms than previously available can be developed by making use of rational barycentric representations whose support points are chosen in an adaptive fashion as the approximant is computed. Three variants of this barycentric strategy are all shown to be powerful: (1) a classical Remez algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares, and (3) a differential correction algorithm. Our preferred combination, implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and then switch to (1) for generically quadratic convergence. By such methods we can calculate approximations up to type (80, 80) of $|x|$ on $[-1, 1]$ in standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan, and Carpenter required 200-digit extended precision.

Robustness analysis method for orbit control

Satellite orbits require periodical maintenance due to the presence of perturbations. However, random errors caused by inaccurate orbit determination and thrust implementation may lead to failure of the orbit control strategy. Therefore, it is necessary to analyze the robustness of the orbit control methods. Feasible strategies which are tolerant to errors of a certain magnitude can be developed to perform reliable orbit control for the satellite. In this paper, first, the orbital dynamic model is formulated by Gauss’ form of the planetary equation using the mean orbit elements; the atmospheric drag and the Earth's non-spherical perturbations are taken into consideration in this model. Second, an impulsive control strategy employing the differential correction algorithm is developed to maintain the satellite trajectory parameters in given ranges. Finally, the robustness of the impulsive control method is analyzed through Monte Carlo simulations while taking orbit determination error and thrust error into account.

High-precision repeat-groundtrack orbit design and maintenance for Earth observation missions

The focus of this paper is the design and station keeping of repeat-groundtrack orbits for Sun-synchronous satellites. A method to compute the semimajor axis of the orbit is presented together with a station-keeping strategy to compensate for the perturbation due to the atmospheric drag. The results show that the nodal period converges gradually with the increase of the order used in the zonal perturbations up to $$J_{15}$$ . A differential correction algorithm is performed to obtain the nominal semimajor axis of the reference orbit from the inputs of the desired nodal period, eccentricity, inclination and argument of perigee. To keep the satellite in the proximity of the repeat-groundtrack condition, a practical orbit maintenance strategy is proposed in the presence of errors in the orbital measurements and control, as well as in the estimation of the semimajor axis decay rate. The performance of the maintenance strategy is assessed via the Monte Carlo simulation and the validation in a high fidelity model. Numerical simulations substantiate the validity of proposed mean-elements-based orbit maintenance strategy for repeat-groundtrack orbits.

Design of impulsive Earth-Moon Halo transfers: lunar proximity and direct options

Techniques associated with stable manifold and lunar flyby have been applied to the construction of optimal transfers to Earth-Moon $L_{1} /L_{2}$ libration point orbits. Compared with traditional design methods and to reduce maneuver cost, the design process presents a detailed analysis on the effect of lunar proximity with multiple constraints. An accurate and fast design strategy for seeking an insertion point and modifying the stable manifold to satisfy these constraints is proposed. Combined this strategy with the differential correction algorithm, the optimal transfer trajectory can be determined from a low-Earth orbit to a halo orbit around the $L_{1} /L_{2}$ libration point within a little computational time. Different amplitudes and insertion points of halo orbit in conjunction with various constraint conditions about lunar flyby are considered to deeply examine the efficiency and reliability of the design algorithm. Preliminary results indicate that the required mission cost has a significant correlation with lunar proximity constraints, and demonstrate that the method of constructing impulsive lunar halo transfer trajectories with multiple constraints is feasible.

Optical Short-Arc Association Hypothesis Gating via Angle-Rate Information

Given a short-arc optical observation with estimated angle rates, the admissible region is a compact region in the range/range-rate space defined such that all likely and relevant orbits are contained within it. An alternative boundary value problem formulation has recently been proposed where range/range hypotheses are generated with two angle measurements from two tracks as input. In this paper, angle-rate information is reintroduced as a means to eliminate hypotheses by bounding their constants of motion before a more computationally costly Lambert solver or differential correction algorithm is run.

Navigation Solution to Maneuver Spacecraft Relative to Spheres Centered on Cooperative Satellites

A differential correction algorithm is presented to deliver an impulsive maneuver to a satellite to place it within the overlapping spheres, with user-defined radii, centered around multiple cooperative satellites within a constrained time. The differential correction algorithm develops and utilizes the state transition matrix along with the equations of motion and multiple satellites’ state information to determine the optimum trajectory to achieve the desired results. The results from the algorithm are presented for prograde orbits. The results are presented in order to allow for mission design trade-offs, including satellite separation distance and required based on the satellites’ initial inclinations. The results also provide empirical methods to determine the miss distance and optimum solutions for the provided problem. This work ultimately provides the framework for applying the algorithm to a unique user defined responsive space scenario.

Navigation Solution to Maneuver a Spacecraft Relative to a Sphere Centered on a Cooperative Satellite

A differential correction algorithm is presented to deliver an impulsive maneuver to a satellite to place it within a sphere, with a user defined radius, centered around a non-maneuvering satellite within a constrained time. The differential correction algorithm develops and utilizes the State Transition Matrix along with the Equations of Motion and multiple satellite׳s state information to determine the optimum trajectory to achieve the desired results. The results from the differential correction algorithm are very accurate for prograde orbits, as presented. The results allow for orbit design trade-offs, including satellites׳ initial inclinations, semi-major axes, as well as the ballistic coefficients. The results also provide an empirical method to determine the optimum ΔV solution for the provided problem. Understanding that the minimum fuel solution lies with a semi-major axis ratio of 1, a very accurate empirical approximation is presented for semi-major axis ratio values less than and greater than 1. This work ultimately provides the generalized framework for applying the algorithm to a unique user defined maneuvering spacecraft scenario.

DGNSS RSIM을 위한 GPS/Galileo 의사거리 보정기법

본 논문에서는 위성항법시스템(GNSS)의 다양화에 따른 DGNSS 기준국(RSIM, Reference Station and Integrity Monitor)의 재구축을 위하여, 유럽연합(EU) 위성항법시스템인 Galileo의 E1 의사거리 보정정보 생성 알고리즘과 GPS/Galileo 시뮬레이션을 통한 성능검증에 대해 다룬다. 먼저 DGPS RSIM에서 DGNSS RSIM으로 전환을 위한 운영적 측면에서의 기술 및 메시지 표준과 사용자 방송 측면에서의 메시지 표준에 대해 살펴본다. 일반적으로 GNSS의 의사거리 보정을 위해서는 정확한 GNSS 위성위치와 사용자 위치를 알아야만 한다. 그러므로 Galileo 위성위치를 정확하게 계산하기 위해서, Galileo ICD 문건의 위성위치 계산식을 이용하여 사용자 수신기에서 제공하는 궤도력 정보를 기반으로 해당 위성 위치를 추정한다. 그리고 위성시계 옵셋과 사용자 수신기의 시각오차, GPS와 Galileo 위성의 시스템 타임 옵셋을 계산하여 GPS/Galileo 의사거리 보정정보를 생성한다. GPS/Galileo 시뮬레이터를 연동한 성능검증 플랫폼을 기반으로 GPS/Galileo 보정정보의 오차를 분석하고, 측위정확도를 분석하여 그 성능을 검증하였다. 국제기구(RTCM)에서 요구하는 기준국 운영을 위한 측위 성능을 충족할 수 있음을 확인하였다. In order to prepare for recapitalization of differential GNSS (DGNSS) reference station and integrity monitor (RSIM) due to GNSS diversification, this paper focuses on differential correction algorithm using GPS/Galileo pesudorange. The technical standards on operation and broadcast of DGNSS RSIM are described as operation of differential GPS (DGPS) RSIM for conversion of DGNSS RSIM. Usually, in order to get the differential corrections of GNSS pesudorange, the system must know the real positions of satellites and user. Therefore, for calculating the position of Galileo satellites correctly, using the equation for calculating the SV position in Galileo ICD (Interface Control Document), it estimates the SV position based on Ephemeris data obtained from user receiver, and calculates the clock offset of satellite and user receiver, system time offset between GPS and Galileo, then determines the pseudorange corrections of GPS/Galileo. Based on a platform for performance verification connected with GPS/Galileo integrated signal simulator, it compared the PRC (pseudorange correction) errors of GPS and Galileo, analyzed the position errors of DGPS, DGalileo, and DGPS/DGalileo respectively. The proposed method was evaluated according to PRC errors and position accuracy at the simulation platform. When using the DGPS/DGalileo corrections, this paper could confirm that the results met the performance requirements of the RTCM.

Navigation solution to maneuver a spacecraft relative to multiple satellites and ground locations

A dynamic differential correction algorithm is presented to deliver an impulsive maneuver to a satellite to place it within overlapping spheres, with user defined radii, centered around multiple non-maneuvering satellites and ground location׳s user defined line-of-sight within a constrained time. The solution is further constrained by elevation considerations for the ground locations. The differential correction algorithm develops and utilizes the state transition matrix, along with the equations of motion and multiple satellite׳s state information and ground location state information to determine the optimum trajectory to achieve the desired results. The results from the differential correction algorithm are very accurate for prograde orbits, as presented. The results allow for orbit design trade-offs, including the maneuvering satellite׳s initial inclination. The results deliver an analytical method to determine the optimum ΔV solution for the provided problem. This work ultimately establishes the generalized framework for applying the algorithm to a unique maneuvering spacecraft scenario which includes multiple satellites and ground locations.

Control of orientation for spacecraft formations in the vicinity of the Sun-Earth L2 libration point

Formation flying in the vicinity of the libration point is an important concept for space exploration and demands reliable and accurate techniques for the control of a spacecraft. On the basis of previous works, this paper addresses the problem of relative orientation control of spacecraft formation flying utilizing the framework of the circular restricted three-body problem (CR3BP) with the Sun and Earth as the primary gravitational bodies. Two specific tasks are accomplished in this study. First, the tangent targeting method (TTM), an efficient two-level differential correction algorithm, is exploited to control the Chief/Deputy architecture to maintain a prespecified orientation. The time spent within the orientation error corridor between successive maneuvers is maximized while the relative separation between the vehicles is held constant at each target point. The second task is to further optimize the maneuver intervals by dropping the constraint imposed on the relative vehicle separation. Numerical investigation indicates that the number of maneuvers can be significantly reduced and the length of time between successive maneuvers can be greatly increased by utilizing the TTM.

Equilibrium configurations of the tethered three-body formation system and their nonlinear dynamics

This paper considers nonlinear dynamics of tethered three-body formation system with their centre of mass staying on a circular orbit around the Earth, and applies the theory of space manifold dynamics to deal with the nonlinear dynamical behaviors of the equilibrium configurations of the system. Compared with the classical circular restricted three body system, sixteen equilibrium configurations are obtained globally from the geometry of pseudo-potential energy surface, four of which were omitted in the previous research. The periodic Lyapunov orbits and their invariant manifolds near the hyperbolic equilibria are presented, and an iteration procedure for identifying Lyapunov orbit is proposed based on the differential correction algorithm. The non-transversal intersections between invariant manifolds are addressed to generate homoclinic and heteroclinic trajectories between the Lyapunov orbits. (3,3)-and (2,1)-heteroclinic trajectories from the neighborhood of one collinear equilibrium to that of another one, and (3,6)- and (2,1)-homoclinic trajectories from and to the neighborhood of the same equilibrium, are obtained based on the Poincare mapping technique.

Trajectory correction maneuver design using an improved B-plane targeting method

A spacecraft for interplanetary mission is usually perturbed by some disturbance sources. The trajectory correction maneuver (TCM) is required to adjust this trajectory error, and the B-plane targeting method is widely used in this field. However, this B-plane targeting method is based on the differential correction algorithm, and a numerical Jacobian matrix is usually used for this algorithm. Therefore, our main goal in this paper is to suggest the improved B-plane targeting method to overcome the disadvantages of the conventional B-plane method which requires a numerical Jacobian matrix for the initial perturbation selection and iterations. For this improvement, an analytical Jacobian matrix is introduced instead of the numerical Jacobian matrix. Then, another B-plane approach that offers an analytical solution is suggested using the target eccentricity instead of the target time of closest approach (TCA). Using a modified Kepler's equation, the previous B-plane targeting approach can be replaced with the new method through the analytical solution.

The J 2 invariant relative configuration of spaceborne SAR interferometer for digital elevation measurement

A 3-craft formation configuration is proposed to perform the digital elevation model (DEM) for the distributed spaceborne interferometric synthetic aperture radar (InSAR), and it is optimized by the modified ant colony algorithm to have the best compatibility with J2 invariant orbits created by differential correction algorithm. The configuration has succeeded in assigning the across-track baseline to vary periodically and with its mean value equal to the optimal baseline determined by the relative height measurement accuracy. The required relationship between crafts’ magnitudes and phases is formulated for the general case of interferometry measure from non-orthographic and non-lateral view. The J2 invariant configurations created by differential correction algorithm are employed to investigate their compatibility with the required configuration. The colony algorithm is applied to search the optimal configuration holding the near-constant across-track baseline under the J2 perturbation, and the absolute height measurement accuracy is preferable as expected.

Identification of aerodynamic characteristics with discontinuities of the first kind from trajectory data for a free-flying object

A method is proposed for identifying the aerodynamic characteristics of a model the flow about which varies considerably in the course of motion. In this method, the trajectory of the model is split into sections with different types of the flow and then similar data from these sections are collected into groups and processed. The aerodynamic characteristics are identified by the method of successive approximation using a differential correction algorithm. The accuracy of the results is analyzed.

Research on precise design of the Mars probe ballistic trajectory

针对火星探测直接转移轨道精确求解问题, 提出了一种快速微分修正算法. 基于二体模型建立控制参数和目标参数偏差关系的数学模型, 并由此求解系统的偏导数矩阵; 采用B平面参数作为中间变量, 对算法进行两层迭代设计, 有效地减少了求解过程中的积分运算次数. 以2018年火星探测机会为例对算法进行了验证, 仿真结果表明, 利用圆锥曲线拼接法得到的轨道初值, 求解一条标准轨道只需6~9次积分迭代. 通过STK对计算结果进行了对比和验证.