Efficient Orbit Propagation of Orbital Elements Using Modified Chebyshev Picard Iteration Method

This paper focuses on propagating perturbed two-body motion using orbital elements combined with a novel integration technique.While previous studies show that Modified Chebyshev Picard Iteration(MCPI)is a powerful tool used to propagate position and velocity,the present results show that using orbital elements to propagate the state vector reduces the number of MCPI iterations and nodes required,which is especially useful for reducing the computation time when including computationally-intensive calculations such as Spherical Harmonic gravity,and it also converges for>5.5x as many revolutions using a single segment when compared with cartesian propagation.Results for the Classical Orbital Elements and the Modified Equinoctial Orbital Elements(the latter provides singularity-free solutions)show that state propagation using these variables is inherently well-suited to the propagation method chosen.Additional benefits are achieved using a segmentation scheme,while future expansion to the two-point boundary value problem is expected to increase the domain of convergence compared with the cartesian case.MCPI is an iterative numerical method used to solve linear and nonlinear,ordinary differential equations(ODEs).It is a fusion of orthogonal Chebyshev function approximation with Picard iteration that approximates a long-arc trajectory at every iteration.Previous studies have shown that it outperforms the state of the practice numerical integrators of ODEs in a serial computing environment;since MCPI is inherently massively parallelizable,this capability is expected to increase the computational efficiency of the method presented.

Short-Term Orbit Prediction with J₂and Mean Orbital Elements

An analytical theory for calculating perturbations of the orbital elements of a satellite due to J2 to accuracy up to fourth power in eccentricity is developed. It is observed that there is significant improvement in all the orbital elements with the present theory over second-order theory. The theory is used for computing the mean orbital elements, which are found to be more accurate than provided by Bhatnagar and taqvi’s theory (up to second power in eccentricity). Mean elements have a large number of practical applications.

Biomechanical analysis of an absorbable material for treating fractures of the inferior orbital wall

AIM:To investigate the biomechanical properties and practical application of absorbable materials in orbital fracture repair.METHODS:The three-dimensional(3D)model of orbital blowout fractures was reconstructed using Mimics21.0 software.The repair guide plate model for inferior orbital wall fracture was designed using 3-matic13.0 and Geomagic wrap 21.0 software.The finite element model of orbital blowout fracture and absorbable repair plate was established using 3-matic13.0 and ANSYS Workbench 21.0 software.The mechanical response of absorbable plates,with thicknesses of 0.6 and 1.2 mm,was modeled after their placement in the orbit.Two patients with inferior orbital wall fractures volunteered to receive single-layer and double-layer absorbable plates combined with 3D printing technology to facilitate surgical treatment of orbital wall fractures.RESULTS:The finite element models of orbital blowout fracture and absorbable plate were successfully established.Finite element analysis(FEA)showed that when the Young’s modulus of the absorbable plate decreases to 3.15 MPa,the repair material with a thickness of 0.6 mm was influenced by the gravitational forces of the orbital contents,resulting in a maximum total deformation of approximately 3.3 mm.Conversely,when the absorbable plate was 1.2 mm thick,the overall maximum total deformation was around 0.4 mm.The half-year follow-up results of the clinical cases confirmed that the absorbable plate with a thickness of 1.2 mm had smaller maximum total deformation and better clinical efficacy.CONCLUSION:The biomechanical analysis observations in this study are largely consistent with the clinical situation.The use of double-layer absorbable plates in conjunction with 3D printing technology is recommended to support surgical treatment of infraorbital wall blowout fractures.

Orbital Elements Ephemerides and Interfaces Design of LEO Satellites

Low earth orbit satellites,with unique advantages,are prosperous types of navigation augmentation satellites for the GNSS satellites constellations.The broadcast ephemeris element needs to be designed as an important index of the augmented LEOs.The GPS ephemerides of 16/18 elements cannot be directly applied to the LEOs because of the poor fitting accuracies in along-track positional component.Besides,the ill-conditioned problem of the normal-matrix exists in fitting algorithm due to the small eccentricity of the LEO orbits.Based on the nonsingular orbital elements,5 sets of ephemerides with element numbers from 16 to 19 were designed respectively by adding or modifying orbital elements magnifying the along-track and radial positional components.The fitting experiments based on the LEO of 300 to 1500 km altitudes show that the fitting UREs of the proposed 16/17/18/18*/19-element ephemerides are better than 10/6/4/5/2.5 cm,respectively.According to the dynamical range of the fitting elements,the interfaces were designed for the 5 sets of ephemerides.The effects of data truncation on fitting UREs are at millimeter level.The total bits are 329/343/376/379/396,respectively.29/15 bits are saved for the 16/17-element ephemerides compared with the GPS16 ephemeris,while 64/61/41 bits can be saved for the 18/18*/19-element ephemerides compared with the GPS18 elements ephemeris.

Linearized relative motion equations through orbital element differences for general Keplerian orbits

A new formulation of the orbital element-based relative motion equations is developed for general Keplerian orbits.This new solution is derived by performing a Taylor expansion on the Cartesian coordinates in the rotating frame with respect to the orbital elements.The resulted solution is expressed in terms of two different sets of orbital elements.The first one is the classical orbital elements and the second one is the nonsingular orbital elements.Among of them,however,the semi-latus rectum and true anomaly are used due to their generality,rather than the semi-major axis and mean anomaly that are used in most references.This specific selection for orbital elements yields a new solution that is universally applicable to elliptic,parabolic and hyperbolic orbits.It is shown that the new orbital element-based relative motion equations are equivalent to the Tschauner–Hempel equations.A linear map between the initial orbital element differences and the integration constants associated with the solution of the Tschauner–Hempel equations is constructed.Finally,the presented solution is validated through comparison with a high-fidelity numerical orbit propagator.The numerical results demonstrate that the new solution is computationally effective;and the result is able to match the accuracy that is required for linear propagation of spacecraft relative motion over a broad range of Keplerian orbits.

Calculation of Orbital Elements for Released Tethered Satellite

The tethered satellite system has a great potential and one of its very useful applications is momentum transfer. Raising a payload by deploying it upward from an orbitor on a long tether and then releasing it represents a rather important possible application with significant fael economy. This paper presents a dynamic model set up for a two body tethered satellite system and two control laws of deployment used to simulate the deployment of the system, gives calculation formulas for six orbital elements of two sub satellites and discusses calculation examples.

Study on relative orbital configuration in satellite formation flying

In this paper, the relative orbital configurations of satellites in formation flying with non-perturbation and J2 perturbation are studied, and an orbital elements method is proposed to obtain the relative orbital configurations of satellites in formation. Firstly, under the condition of non-perturbation, we obtain many shapes of relative orbital configurations when the semi-major axes of satellites are equal. These shapes can be lines, ellipses or distorted closed curves. Secondly, on the basis of the analysis of J2 effect on relative orbital configurations, we find out that J2 effect can induce two kinds of changes of relative orbital configurations. They are distortion and drifting, respectively. In addition, when J2 perturbation is concerned, we also find that the semi-major axes of the leading and following satellites should not be the same exactly in order to decrease the J2 effect. The relationship of relative orbital elements and J2 effect is obtained through simulations. Finally, the minimum relation perturbation conditions are established in order to reduce the influence of the J2 effect. The results show that the minimum relation perturbation conditions can reduce the J2 effect significantly when the orbital element differences are small enough, and they can become rules for the design of satellite formation flying.

Three-dimensional finite element analysis of hyperbaric oxygen therapy on the optic nerve from a damaged orbit

A middle-aged male patient with a right orbital comminuted fracture underwent computer tomography scanning, and a three-dimensional finite element model of the eyes and relevant tissues was established. Optic nerve stress in a hyperbaric oxygen environment was simulated and analyzed by changing the elastic modulus and external pressure of the skull at the damage side. Results showed that stress maximized at the contact site of the optic nerve and the eyeball in the damaged and intact eye orbits. Optic nerve stress at the damaged orbit significantly increased; however, stress in the intact orbit only slightly changed with decreased elastic modulus of the skull while external pressure remained unchanged. Maximum optic nerve stress increased in the damaged and intact side, along with increased external pressure, while elastic modulus remained unchanged. These experimental findings suggested that the optic nerve was compressed under hyperbaric oxygen and optic nerve stress was greater in the damaged orbit than in the intact orbit.

Solution set on the natural satellite formation orbits under first-order earth's non-spherical perturbation

Using the reference orbital element approach, the precise governing equations for the relative motion of formation flight are formulated. A number of ideal formations with respect to an elliptic orbit can be designed based on the relative motion analysis from the equations. The features of the oscillating reference orbital elements are studied by using the perturbation theory. The changes in the relative orbit under perturbation are divided into three categories, termed scale enlargement, drift and distortion respectively. By properly choosing the initial mean orbital elements for the leader and follower satellites, the deviations from originally regular formation orbit caused by the perturbation can be suppressed. Thereby the natural formation is set up. It behaves either like non-disturbed or need little control to maintain. The presented reference orbital element approach highlights the kinematics properties of the relative motion and is convenient to incorporate the results of perturbation analysis on orbital elements. This method of formation design has advantages over other methods in seeking natural formation and in initializing formation.

Modified filter for mean elements estimation with state jumping

To investigate the real-time mean orbital elements(MOEs)estimation problem under the influence of state jumping caused by non-fatal spacecraft collision or protective orbit trans-fer,a modified augmented square-root unscented Kalman filter(MASUKF)is proposed.The MASUKF is composed of sigma points calculation,time update,modified state jumping detec-tion,and measurement update.Compared with the filters used in the existing literature on MOEs estimation,it has three main characteristics.Firstly,the state vector is augmented from six to nine by the added thrust acceleration terms,which makes the fil-ter additionally give the state-jumping-thrust-acceleration esti-mation.Secondly,the normalized innovation is used for state jumping detection to set detection threshold concisely and make the filter detect various state jumping with low latency.Thirdly,when sate jumping is detected,the covariance matrix inflation will be done,and then an extra time update process will be con-ducted at this time instance before measurement update.In this way,the relatively large estimation error at the detection moment can significantly decrease.Finally,typical simulations are per-formed to illustrated the effectiveness of the method.

Artificial Sun synchronous frozen orbit control scheme design based on J_2 perturbation

Sun synchronous orbit and frozen orbit formed due to J 2 perturbation have very strict constraints on orbital parameters,which have restricted the application a lot.In this paper,several control strategies were illustrated to realize Sun synchronous frozen orbit with arbitrary orbital elements using continuous low-thrust.Firstly,according to mean element method,the averaged rate of change of the orbital elements,originating from disturbing constant accelerations over one orbital period,was derived from Gauss＇ variation of parameters equations.Then,we proposed that binormal acceleration could be used to realize Sun synchronous orbit,and radial or transverse acceleration could be adopted to eliminate the rotation of the argument of the perigee.Finally,amending methods on the control strategies mentioned above were presented to eliminate the residual secular growth.Simulation results showed that the control strategies illustrated in this paper could realize Sun synchronous frozen orbit with arbitrary orbital elements,and can save much more energy than the schemes presented in previous studies,and have no side effect on other orbital parameters＇ secular motion.

An Orbit Determination Using SGP4 Propagator and Doppler Shifts for CubeSats

The two line elements(TLEs),released by the North American Aerospace Defense Command(NORAD),are chosen for CubeSats' mission operators.Unfortunately,they have errors and other accompanied problems,which cause large deviations in the in-track component.When a TLE value is available at a certain epoch,the dominant error is the angular error.It is proposed to correct the angular error by solving-for the mean argument of latitude at the desired epoch.A batch least squares technique and range rate measurements are used for the correction process.With the assistance of satellite tool kit(STK)software and Matlab,a simulation to verify the orbit determination(OD)technique is implemented.This paper provides an angular correction low cost OD method and presents a complete analysis for various test cases.This approach maintains high accuracy in cross-track and radial and makes great improvement in in-track at the same time,but it is exclusive for circular orbits.When it is applied to an elliptical orbit,the error will be unacceptable.Therefore,the angular error is corrected using the longitude of periapsis which totally mitigates the error at the epoch under consideration.For inclinations less than 20 o,the mean longitude is preferred for the angular correction as it provides more accuracy compared with the mean argument of latitude.

Reducing Influence of Gravity Model Error in Precise Orbit Determination of Low Earth Orbit Satellites

Based on the orbit integration and orbit fitting method, the influence of the characters of the gravity model, with different precisions, on the movement of low Earth orbit satellites was studied. The way and the effect of absorbing the influence of gravity model error on CHAMP and GRACE satellite orbits, using linear and periodical empirical acceleration models and the so-called "pseudo-stochastic pulses" model, were also analyzed.

Impulsive orbit control for spacecraft around asteroid

An impulse feedback control law to change the mean orbit elements of spacecraft around asteroid is presented. First, the mean orbit elements are transferred to the osculating orbit elements at the burning time. Then, the feedback control law based on Gauss’s perturbation equations of motion is given. And the impulse control for targeting from the higher circulation orbit to the specified periapsis is developed. Finally, the numerical simulation is performed and the simulation results show that the presented impulse control law is effective.

An improved velocity determination method based on GOCE kinematic orbit

A new velocity determination algorithm with combination of remove and restore method, outliers detection method and Chebyshev fitting method with redundant observations is proposed. An optimal selection of number of sampling points is given. The result shows that, when the number of sampling points is 19, the three-dimension （3D） interpolation precision of velocity is superior to 0.1 mm/s, which is above 3 times better than that of Chebyshev fitting method with redundant observations and far better than those of the conventional interpolation methods.

TLE orbit determination using simplex method

Two-Line Element(TLE)datasets are the only orbital data source of Earth-orbiting space objects for many civil users for their research and applications.The datasets have uneven qualities that may affect the reliability of the propagated positions of space objects using a single TLE.The least squares approach to use multiple TLEs also suffers from the poor quality of some TLEs,and reliable error information cannot be available.This paper proposes a simplex algorithm to estimate an optimal TLE from multiple TLEs and obtain the uncertainty of each element.It is a derivative-free technique that can deal with various orbit types.Experiments have demonstrated that using the TLE estimated from the simplex method is more reliable,stable,and effective than those from the batch least squares method.As an application example,the optimal TLE and its uncertainty are used for predicting the fallen area,keeping the actual fallen site in the prediction areas.

井下带式输送机智能巡检机器人结构的设计

为解决带式输送机采用传统人工检测故障效率低等问题,针对问题设计出一种适用于井下输送机故障自动巡检机器人系统,区别于其他带式输送机巡检机器人。该巡检机器人可以实现大坡度和大转弯半径的运动,首先利用ADAMS分析现在主流的巡检机器人模型,分析其优劣性;利用SOLIDWORKS软件建立巡检机器人的三维模型,提出了摩擦驱动与齿轮齿条驱动的两用机器人模型;再利用ANSYS软件对轨道和机架进行了有限元分析,仿真结果表明巡检轨道与机器人本体的应力小于许用应力,能够平稳地检测输送机的故障;最后再利用ADAMS软件对巡检机器人进行动力学分析,得出巡检机器人可以进行无障碍的水平和爬坡运动。

高阶轨道角动量传输光纤设计及传输特性研究(内封底文章)

为解决一般轨道角动量传输光纤传输轨道角动量模式数量少、质量差的问题,提出了一种基于正六边形空气孔排列的新型光子晶体光纤结构。该光纤引入了空气填充率高的矩形空气孔以及采用高折射率材料填充环形传输区域,能够有效提高环形传输区域和包层间的折射率差,且正六边形排列空气孔有利于提高模间有效折射率差。经过结构优化得到最优光纤结构,有限元法的分析结果表明,最优结构下,该光纤在常用波段S+C+L+U波段上能够支持142种轨道角动量模式的传输,最高阶数达到36阶。且所提出光纤具有良好的传输特性,本征模式的最高限制性损耗为10−9 dB/m量级,与典型轨道角动量传输光子晶体光纤相比至少降低了一个数量级;最大有效模场面积能够达到206.18μm^(2),最小非线性系数低至0.397 W^(−1)∙km^(−1);色散平坦且最小色散变化低至1.4578 ps/(nm∙km);所有本征模式纯度均在93.4%~96.8%范围内。且该光纤具有较好的制备可行性,对制造精度要求不高。因此,该光纤在基于轨道角动量光纤的复用系统中具有广阔的应用前景,为提高通信容量提供了一种有效手段。

航空圆柱直齿轮热摆辗成形工艺分析

为研究航空圆柱直齿轮热摆辗成形过程中金属变形规律和齿形充填特点,利用DEFORM-3D软件建立了圆柱直齿轮热摆辗成形三维刚塑性有限元模型,模拟了其热摆辗成形过程。结果表明,圆柱直齿轮热摆辗成形过程中,坯料变形区分为主动变形区和被动变形区,并随着摆头的运动而呈周期性变化;随着变形的进行,主动变形区金属流动方向从径向转变为轴向充填模具型腔,齿形从上端到下端逐渐成形,坯料呈蘑菇状;齿根区域等效应变与等效应力值最大,齿顶温度高于齿根温度;轴向载荷随塑性变形的进行逐渐增大,最大值约为2500 kN,径向载荷呈周期性变化,径向载荷远小于轴向载荷。开展了圆柱直齿轮热摆辗工艺实验,实验结果与模拟结果吻合。

有限元方法在眼眶生物力学中的应用进展

有限元方法(FEM)是力学研究中常用的数学方法,将物体划分为离散且相互作用的有限单元。在医学研究中,有限元分析(FEA)可模拟难以开展的生物力学实验。眼眶手术极具挑战性且具有陡峭的学习曲线,给眼科医生带来了巨大挑战。FEM可模拟分析眼眶组织的力学特性,为眼眶相关疾病的诊断和治疗提供了新的方法。随着技术的发展,FEM在眼眶疾病的诊疗中愈发成熟,并成为眼眶生物力学研究的热门领域。本文综述了眼眶FEM的最新进展,包括建立眼眶FEA模型、模拟眼眶结构以及在眼眶相关疾病中的应用情况。此外,还讨论了FEM的局限性和未来的研究方向。眼眶FEA作为一种辅助诊疗数字化工具,将随着技术的发展逐渐释放其在眼眶疾病诊疗方面的潜力。