Lissajous Orbits at the Collinear Libration Points in the Restricted Three-Body Problem with Oblateness

In the present work, the collinear equilibrium points of the restricted three-body problem are studied under the effect of oblateness of the bigger primary using an analytical and numerical approach. The periodic orbits around these points are investigated for the Earth-Moon system. The Lissajous orbits and the phase spaces are obtained under the effect of oblateness.

Port-Hamiltonian Based Control of the Sun-Earth 3D Circular Restricted Three-Body Problem: Stabilization of the <i>L</i>₁Lagrange Point

In this paper, we use Port-Hamiltonian framework to stabilize the Lagrange points in the Sun-Earth three-dimensional Circular Restricted Three-Body Problem (CRTBP). Through rewriting the CRTBP into Port-Hamiltonian framework, we are allowed to design the feedback controller through energy-shaping and dissipation injection. The closed-loop Hamiltonian is a candidate of the Lyapunov function to establish nonlinear stability of the designed equilibrium, which enlarges the application region of feedback controller compared with that based on linearized dynamics. Results show that the Port-Hamiltonian approach allows us to successfully stabilize the Lagrange points, where the Linear Quadratic Regulator (LQR) may fail. The feedback system based on Port-Hamiltonian approach is also robust against white noise in the inputs.

Artificial Equilibrium Points in the Low-Thrust Restricted Three-Body Problem When Both the Primaries Are Oblate Spheroids

This paper studies the existence and stability of the artificial equilibrium points (AEPs) in the low-thrust restricted three-body problem when both the primaries are oblate spheroids. The artificial equilibrium points (AEPs) are generated by canceling the gravitational and centrifugal forces with continuous low-thrust at a non-equilibrium point. Some graphical investigations are shown for the effects of the relative parameters which characterized the locations of the AEPs. Also, the numerical values of AEPs have been calculated. The positions of these AEPs will depend not only also on magnitude and directions of low-thrust acceleration. The linear stability of the AEPs has been investigated. We have determined the stability regions in the xy, xz and yz-planes and studied the effect of oblateness parameters A1(0A1?and ?A2(0A2<1) on the motion of the spacecraft. We have found that the stability regions reduce around both the primaries for the increasing values of oblateness of the primaries. Finally, we have plotted the zero velocity curves to determine the possible regions of motion of the spacecraft.

A New Critical Value Concerning the Genealogy of Long Period Families at L_4 in the Restricted Three-body Problem

We found another critical mass ratio value -↑μ between μ4 and μ5 concerning the genealogy of the long period family around the equilateral equilibrium point L4 in the restricted three-body problem. This value has not been pointed out before. We used numerical computations to show how the long period family evolves around this critical value. The case is similar to that of the critical values between μ2 and μ4, with slight difference in evolution details.

A Special Case of Variational Formulation for Two-Point Boundary Value Problem in L2(Ω)

We consider the nonlinear boundary value problems for elliptic partial differential equations and using a maximum principle for this problem we show uniqueness and continuous dependence on data. We use the strong version of the maximum principle to prove that all solutions of two-point BVP are positives and we also show a numerical example by applying finite difference method for a two-point BVP in one dimension based on discrete version of the maximum principle.

Restricted Three Body Problem with Stokes Drag Effect

The existence and stability of stationary solutions of the restricted three body problem under the effect of the dissipative force, Stokes drag, are investigated. It is observed that there exist two non collinear stationary solutions. Further, it is also found that these stationary solutions are unstable for all values of the parameters.

Halo Orbits in the Photo-Gravitational Restricted Three-Body Problem

We study of halo orbits in the circular restricted three-body problem (CRTBP) with both the primaries as sources of radiation. The positioning of the triangular equilibrium points is discussed in a rotating coordinate system.

Oblateness Effect of Saturn on Halo Orbits of L1 and L2 in Saturn-Satellites Restricted Three-Body Problem

The Circular Restricted Three-Body Problem (CRTBP) with more massive primary as an oblate spheroid with its equatorial plane coincident with the plane of motion of the primaries is considered to generate the halo orbits around L1 and L2 for the seven satellites (Mimas, Enceladus, Tethys, Dione, Rhea, Titan and Iapetus) of Saturn in the frame work of CRTBP. It is found that the oblateness effect of Saturn on the halo orbits of the satellites closer to Saturn has significant effect compared to the satellites away from it. The halo orbits L1 and L2 are found to move towards Saturn with oblateness.

Improved semi-analytical computation of center manifolds near collinear libration points

In the framework of the circular restricted three-body problem, the center manifolds associated with collinear libration points contain all the bounded orbits moving around these points. Semianalytical computation of the center manifolds and the associated canonical transformation are valuable tools for exploring the design space of libration point missions. This paper deals with the refinement of reduction to the center manifold procedure. In order to reduce the amount of calculation needed and avoid repetitive computation of the Poisson bracket, a modified method is presented. By using a polynomial optimization technique, the coordinate transformation is conducted more efficiently. In addition, an alternative way to do the canonical coordinate transformation is discussed, which complements the classical approach. Numerical simulation confirms that more accurate and efficient numerical exploration of the center manifold is made possible by using the refined method.

A new control law based on characteristic exponent assignment for libration point orbit maintenance

A new method is developed for stabilizing motion on collinear libration point orbits using the formalism of the circular restricted three body problem. Linearization about the collinear libration point orbits yields an unstable linear parameter-varying system with periodic coefficients. Given the variational equations, an innovative control law based on characteristic exponent assignment is introduced for libration point orbit maintenance. A numerical simulation choosing the Richardson＇s third order approximation for a halo orbit as a nominal orbit is conducted, and the results demonstrate the effectiveness of this control law.

Solar Radiation Pressure Effects on Stability of Periodic Orbits in Restricted Four-Body Problem

In this work, the Hamiltonian of the four-body problem is considered under the effects of solar radiation pressure. The equations of motion of the infinitesimal body are obtained in the Hamiltonian canonical form. The libration points and the corresponding Jacobi constants are obtained with different values of the solar radiation pressure coefficient. The motion and its stability about each point are studied. A family of periodic orbits under the effects of the gravitational forces of the primaries and the solar radiation pressure are obtained depending on the pure numerical method. This purpose is applied to the Sun-Earth-Moon-Space craft system, and the results obtained are in a good agreement with the previous work such as (Kumari and Papadouris, 2013).

Multi-modes control method for spacecraft formation establishment and reconfiguration near the libration points

This paper studies Multi-modes control method for libration points formation establishment and reconfiguration. Firstly, relations between optimal impulse control and Floquet modes are investigated. Method of generating modes is proposed. Characteristics of the mode coefficients stimulated at different time are also given. Studies show that coefficients of controlled modes can be classified into four types, and formation establishment and reeonfiguration can be achieved by multi-impulse control with the presented method of generating modes. Then, since libration points formation is generally unstable, mutli-modes keeping control method which can stabilize five Floquet modes simultaneously is proposed. Finally, simulation on formation establishment and reconfiguration are carried out by using method of generating modes and mutli-modes keeping control method. Results show that the proposed control method is effective and practical.

Halo Orbits around Sun-Earth L1 in Photogravitational Restricted Three-Body Problem with Oblateness of Smaller Primary

This paper deals with generation of halo orbits in the three-dimensional photogravitational restricted three-body problem, where the more massive primary is considered as the source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. Both the terms due to oblateness of the smaller primary are considered. Numerical as well as analytical solutions are obtained around the Lagrangian point L1, which lies between the primaries, of the Sun-Earth system. A comparison with the real time flight data of SOHO mission is made. Inclusion of oblateness of the smaller primary can improve the accuracy. Due to the effect of radiation pressure and oblateness, the size and the orbital period of the halo orbit around L1 are found to increase.

STUDIES OF MELNIKOV METHOD AND TRANSVERSAL HOMOCLINIC ORBITS IN THE CIRCULAR PLANAR RESTRICTED THREE－BODY PROBLEM

Non-Hamiltonian systems containing degenerate fixed points obtained from twodegrees of freedom near-integrable Hamiltonian systems through non-canonicaltransformations are dealt with in this paper. Two criteria .for determining theexistence of transversal homoclinic and heteroclinic orbits are presented. By exploitingthese criteria the existence of the transversal homoclinic orbits and so, of thetransversal homoclinic tangle .phenomenon in the near-integrable circular planarrestricted three-body problem with sufficiently small mass ratio of the two primaries isproven. Under some assumptions, the existence of the transversal heleroclinic orbits isproven. The global qualitative phase diagram is also illustrated.

Existence of Equilibrium Points in the R3BP with Variable Mass When the Smaller Primary is an Oblate Spheroid

The paper deals with the existence of equilibrium points in the restricted three-body problem when the smaller primary is an oblate spheroid and the infinitesimal body is of variable mass. Following the method of small parameters;the co-ordinates of collinear equilibrium points have been calculated, whereas the co-ordinates of triangular equilibrium points are established by classical method. On studying the surface of zero-velocity curves, it is found that the mass reduction factor has very minor effect on the location of the equilibrium points;whereas the oblateness parameter of the smaller primary has a significant role on the existence of equilibrium points.

Effect of Perturbations in Coriolis and Centrifugal Forces on the Non-Linear Stability of <i>L</i>₄in the Photogravitational Restricted Three Body Problem

Effect of perturbations in Coriolis and centrifugal forces on the non-linear stability of the libration point L4 in the restricted three body problem is studied when both the primaries are axis symmetric bodies (triaxial rigid bodies) and the bigger primary is a source of radiation. Moser’s conditions are utilized in this study by employing the iterative scheme of Henrard for transforming the Hamiltonian to the Birkhoff’s normal form with the help of double D’Alembert’s series. It is found that L4 is stable for all mass ratios in the range of linear stability except for the three mass ratios μc1, μc2 and μc3, which depend upon the perturbations ε1 and ε1 in the Coriolis and centrifugal forces respectively and the parameters A1,A2,A3 and A4 which depend upon the semi-axes a1,b1,c1;a2,b2,c2 of the triaxial rigid bodies and p, the radiation parameter.

肌筋膜触发点手法治疗张口受限患者的效果分析

目的:探讨应用肌筋膜触发点手法治疗张口受限患者的有效性。方法:选取2022年3月至2023年12月在我院口腔颌面外科及修复科就诊的张口受限患者42例,均采用肌筋膜触发点手法治疗先后对颞肌、咬肌、翼内肌、翼外肌和舌骨上肌群进行治疗,评价指标为最大主动张口度和主观舒适度评分。结果:干预前的最大主动张口度均值为3~26(16.07±5.93)mm,张口受限以Ⅱ级为主;干预后的最大主动张口度均值为6~30(20.64±5.91)mm,张口受限以Ⅰ级为主,干预前后两者差异有统计学意义(t=-9.677,P<0.001)。其中,除5例有头颈部放疗病史的患者张口度未改善外,其余张口受限患者的张口度均有不同程度的改善;患者对手法治疗的主观舒适度评价分为60~100(82.43±14.40)。结论:肌筋膜触发点手法治疗能一定程度改善非放疗引起的张口受限,在改善与肌肉痉挛有关的张口受限方面疗效较好,治疗舒适度佳、安全、经济,可作为张口受限患者的保守治疗手段或其他治疗方法的辅助治疗,值得在临床推广。

地月历表误差对共线平动点周期轨道的误差影响分析

基于圆形限制性三体问题下地月平动点的3阶轨道解析解,分析不同初值条件下的轨道演化特性,并基于误差传播理论研究地月历表误差对地月共线平动点周期轨道在不同初值及初值误差时的影响。结果表明:1)轨道误差与地月历表误差在同一个量级,当地月历表误差均值不为0时,其峰值大约为地月历表误差的2.5倍,轨道误差随时间变化的无量纲周期约为轨道无量纲周期的0.5倍;2)在同一条轨道的不同位置,地月历表误差影响存在较大差异,平面Lyapunov轨道、Halo轨道和垂直Lyapunov轨道误差的峰值点分别位于或接近X向最大值与最小值处、Z向最大值与最小值处和z=0处。

一种面向限制性三体问题的轨道-姿态运动模态构建方法

针对平动点附近的具有强非线性和耦合性的轨道-姿态动力学机理分析,以及如何利用其动力学特性完成相关任务操作等问题,提出了基于Floquet理论的轨道-姿态运动模态构建方法,并探讨了所构建模态的典型应用。首先,在平动点轨道-姿态周期解的基础上,对轨道-姿态状态转移矩阵进行分块,根据分块矩阵特征值特点,建立了姿态模态和姿轨耦合模态,与轨道模态共同构成完整的轨道-姿态运动模态。然后,对模态特性进行了相空间分析。最后,以平动点相对轨道-姿态运动解析、平动点轨道-姿态稳定性控制和平动点轨道附近姿轨编队设计为应用案例,相应数值仿真表明了所提出模态的正确性,以及在典型航天任务设计与控制中的应用价值。

共线平动点中心流形上的轨道转移问题

圆形限制性三体问题共线平动点附近的平动点轨道由于其独特的动力学特性,在深空探测任务中有着重要价值,这些轨道间的轨道转移问题值得进行系统性研究.针对平动点轨道的计算与延拓,提出了一种基于数值的系统性计算平动点轨道的方法以及状态伴随法的轨道稳定维持策略.在此基础上,通过对大量平动点轨道不变流形以及平动点相空间中心流形的研究,设计了一套通过脉冲机动实现平动点轨道间轨道转移的系统性解决方案.该方法充分利用平动点动力学特性,在仿真验证中证实了方案的有效性,为平动点轨道转移研究提供了新的思路.