Graphical generation of periodic orbits of tschauner-hempel equations
Graphical generation of periodic orbits of tschauner-hempel equations
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DOI：
10.2514/1.56326
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发表时间：
2012-08
影响因子：
2.6
通讯作者：
M. Bando;A. Ichikawa
中科院分区：
文献类型：
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作者：
M. Bando;A. Ichikawa
在圆轨道上，卫星（跟随者）相对于参考卫星（引导者）的相对运动由自治非线性微分方程描述。原点处的线性化方程被称为Hill-Clohessy-Wiltshire（HCW）方程[1]。平面内运动和平面外运动是独立的。后者是简单的正弦运动。前者在Clohessy-Wiltshire条件下也有周期解。周期解由正弦函数给出，相对轨道为椭圆。这些轨道由代表尺寸和相位的两个参数参数来参数化。由于这种简单的性质，相对周期轨道用于编队飞行[2]。非线性相对动力学的周期轨道也可以用随动系统在惯性系中的椭圆轨道及其倾角的参数来表征，并明确给出了周期轨道的初始条件[3]。如果参考轨道是椭圆轨道，则相对运动方程中包含了轨道的真近点和半径，它们是周期函数。在原点处的线性化运动方程被称为Tschauner-Hempel（TH）方程。平面内运动和平面外运动保持独立。然而，在这种情况下，状态转移矩阵的推导不是立即的。在早期的研究[4-6]中，真实异常被用作自变量，因为这简化了所得方程，并且可以获得它们的转移矩阵。使用这种表示法[7，8]已经广泛地研究了Renaissance问题。平面外运动是正弦运动，平面内运动具有周期解[2，9]。[9]中给出了周期解的条件，它推广了CW条件. TH方程的周期轨道也用于编队飞行[2，9，10]。然而，它们比HCW方程的椭圆更复杂，并且它们的形状随着偏心率而变化。其中一些是非常扭曲的。因此，如何设计相对轨道并不直观清楚。文[11]给出了以时间为自变量的相对动力学的直接研究，文[12]给出了状态转移矩阵在偏心率下的级数展开式。[13]中给出了状态转移矩阵的封闭形式，并给出了椭圆交会的一个简单历史，将非线性相对动力学的周期轨道作为时间函数的特征推广到椭圆情形，并给出了周期轨道的初始条件[14]。在[15]中得到了基于轨道根数的相对动力学的一般解，并确定了解所在的相对运动不变流形。本文讨论TH方程平面内运动的周期解的生成问题。为此目的，使用[16]的状态转移矩阵，其作为真实异常的函数给出。解由四个常数参数化，并由此得出周期解的条件。在HCW方程的情况下，尺寸和相位参数被引入。本说明的主要贡献概述如下。首先，所有周期解所在的区域由两个椭圆确定，椭圆的大小取决于领导轨道的偏心率。给出了一种在给定大小和相位参数的情况下，确定周期轨道在给定真近点处起始点的图解方法。改变真异常的值，就产生了整个周期轨道族。区域和周期轨道的时域版本通过乘以先导轨道的半径来确定。这说明了周期轨道的形状如何随偏心率而变化，以及为什么某些轨道的形状会扭曲。当偏心率趋于零时，周期运动区域收缩为一个椭圆，这是HCW方程的周期轨道。size参数给出了leader和follower之间的距离，而phase决定了follower的位置。与[15]一样，可以计算到领导者的最大和最小距离。因此，我们的方法是有用的，设计相对轨道的大小和位置的规格。本说明的结构安排如下。第二节回顾了沿着椭圆轨道的相对运动方程.第三节给出了TH方程周期轨道的几何作图方法。最后，第四节给出结论。
T HE relative motion of a satellite (follower) with respect to the reference satellite (leader) in a circular orbit is described by autonomous nonlinear differential equations. The linearized equations at the origin are known as Hill–Clohessy–Wiltshire (HCW) equations [1]. The in-plane motion and the out-of plane motion are independent. The latter is a simple sinusoidal motion. The former has also periodic solutions under the so-called Clohessy–Wiltshire condition. Periodic solutions are given by sinusoidal functions, and the relative orbits are ellipses. These orbits are parametrized by two parameters representing size and phase. Because of this simple nature, relative periodic orbits are used for formation flying [2]. Periodic orbits of the nonlinear relative dynamics are also characterized by the parameters of the follower’s elliptic orbit and its inclination angle in an inertial reference frame, and initial conditions for periodic orbits are explicitly given [3]. If the reference orbit is elliptic, the equations of relative motion involve the true anomaly and the radius of the orbit, which are periodic functions. The linearized equations of motion at the origin are known as Tschauner–Hempel (TH) equations. The in-plane motion and the out-of plane motion remain independent. However, the derivation of the state transition matrix in this case is not immediate. In earlier studies [4–6], the true anomaly is used as independent variable, because this simplifies the resulting equations and the transition matrix can be obtained for them. Rendezvous problems have been extensively studied using this representation [7,8]. The out-of-plane motion is a sinusoidal motion, and the inplane motion has periodic solutions [2,9]. The condition, which generalizes the CW condition, for periodic solutions is given in [9]. Periodic orbits of the TH equations are also used for formation flying [2,9,10]. However, they are more complicated compared to ellipses of the HCW equations, and their shape changes with eccentricity. Some of them are very much distorted. Hence, it is not intuitively clear how to design relative orbits. The direct study of the relative dynamics using time as independent variable is given in [11], and a series expansion of the state transition matrix in the eccentricity is obtained in [12]. A closed form of the state transition matrix is recently derived in [13],where a brief history of elliptic rendezvous is also found. The characterization of periodic orbits of the nonlinear relative dynamics as functions of time is extended to the elliptic case, and initial conditions for periodic orbits are explicitly given [14]. The general solution of the relative dynamics based on the orbital elements is obtained in [15], and the relative motion invariant manifold, where the solution lies, is determined. This technical note is concernedwith the generation of all periodic solutions of the in-plane motion of the TH equations. For this purpose, the state transition matrix of [16], which is given as a function of the true anomaly, is used. The solution is parametrized by four constants, and the condition for periodic solution follows from this. As in the case of the HCWequations, size and phase parameters are introduced. The main contributions of this note are summarized as follows. First, the region where all periodic solutions lie is determined by two ellipses, whose size depends on the eccentricity of the leader orbit. Given size and phase parameters, a graphical method to determine the initial point of a periodic orbit at a given true anomaly is proposed. Varying the value of the true anomaly, the whole family of periodic orbits is generated. The time domain versions of the region and periodic orbits are determined by multiplying the radius of the leader orbit. This clarifies how the shape of a periodic orbit changes with eccentricity and why the shapes of some orbits are distorted. As the eccentricity goes to zero, the region of periodic motion shrinks to a single ellipse, which is a periodic orbit of the HCW equations. The size parameter gives the distance between the leader and the follower, and the phase determines the position of the follower. As in [15], the maximum and minimum distances to the leader can be calculated. Hence, our method is useful to design relative orbits with size and position specifications. This note is organized as follows. Section II reviews the equations of relativemotion along an elliptic orbit. Section III gives a geometric method to draw periodic orbits of the TH equations. Finally, Section IV gives conclusions.