傅里叶级数下连续小推力(译文)

1、Reduction of Low-Thrust Continuous Controlsfor Trajectory Dynamics前言摘要A novel method to evaluate the trajectory dynamics of low-thrust spacecraft is developed.The thrust vectorcomponents are represented as Fourier series in eccentric anomaly, and Gausss variational equations are averagedover one orb

2、it to define a set of secular equations.These secular equations are a function of only 14 of the thrustFourier coefficients, regardless of the order of the original Fourier series, and are sufficient to accurately determine alow-thrust spiral trajectory withsignificantly reduced computational requir

3、ements as compared with integration ofthe full Newtonian problem. This method has applications to low-thrust spacecraft targeting and optimal controlproblems.一种新的求解小推力轨道动力学的方法被开发出来了。推力向量的各坐标用偏近点角表示为傅里叶级数，并且将高斯变分方程平均到整个轨道上来提高久期方程组的精度。这些久期方程组仅仅是14个推力傅里叶系数的函数而忽略初始傅里叶级的次数，这些方程能够精确有效的解决小推力螺旋轨道问题，且与通常的牛顿问

4、题的积分方法相比能大大减少计算量。这一方法应用在小推力航天器的目标确定和最优控制问题中。I. Introduction 引言LOW-THRUST propulsion systems offer an efficient optionfor many interplanetaryand Earth-orbit missions. However,optimal control of these systems can pose a difficult designchallenge. Analytical or approximate solutions exist for severalsp

5、ecial cases of optimal low-thrust orbit-transfer problems, butthe generalcontinuous-thrust problem requires full numericalintegration of each initial condition and thrust profile. The trajectoryis highly sensitive to these variables; thus, the optimal control lawover tens or hundreds of orbits of a

6、spiral trajectory is often difficult todetermine.小推力推进系统为许多星际航行和地球轨道任务提供了一种高效的新选择。但是这种系统却对最优化控制提出了新的挑战。对于几种特殊的小推力下的轨道转移的最优控制问题已经出现了解析法和数值近似解的方法，但是对于一般的连续推力问题需要对每一个初始条件和推力值进行完全的数值积分。轨道参数的确定往往对这些变量很敏感，因此，（普通的）最优控制法则对于确定螺旋轨道的数十甚至上百个轨道是十分困难的。Analytical solutions have been developed for many special-case

7、transfers, such as the logarithmic spiral 13, Forbess spiral 1,the exponential sinusoid 4,5, the case of constant radial orcircumferential thrust 68, Markopouloss Keplerian thrustprograms 9, Lawdens spiral 10, and Bishop and Azimovsspiral 11. More recent solutions have used the calculus of variation

8、s12 or direct optimization methods 13 to determine optimallow-thrust control laws within certain constraints. Several methodsfor open-loop minimum-time transfers 1416 and optimal transfersusing Lyapunov feedback control 17,18 also exist. Averagingmethods incombination with other approaches have prov

9、en to beeffective at overcoming sensitivities to small variations in the initialorbit and thrust profile 1922, yet all solutions remain limited tocertain regions of the thrust and orbital parameter space.在一些特殊情形的轨道转移中，数值解法已经得到了挖掘，如对数螺旋（上升/下降），福布斯螺旋（上升/下降），指数正弦曲线，常值径向/切向推进，马氏开普勒推进，劳登螺旋，毕阿氏螺旋等。而最新的更多解

10、法则应用变量/变差的微积分或直接优化法来来确定特定约束条件下的小推力最优控制律。一些求解开环（轨道）的时间最优轨道转移和李雅普诺夫反馈最优轨道转移法也早就出现。实践证明，把平均法和其它方法结合的思路在解决对初始轨道/推力变量预测敏感的问题上有很有效的，但所有这些方法都局限于由推力和轨道参数形成的空间区域中。The focus of this study is a novel method to evaluate the effect oflow-thrust propulsion on spacecraft orbit dynamics with minimalconstraints. We

11、represent each component of the thrust accelerationas a Fourier series in eccentric anomaly and then average Gausssvariational equations over one orbit to define a set of secular equations.The equations are a function of only 14 of the thrust Fouriercoefficients, regardless of the order of the origi

12、nal Fourier series;thus, the full continuous control is reduced to a set of 14 parameters.本文研究的重点是提出一种评估/解决带有最少约束的航天器轨道动力学的小推力推进效应/问题。我们将推进加速度向量的各坐标用偏近点角表示为傅里叶级数，并且将高斯变分方程平均到整个轨道上来提高久期方程组的精度。忽略初始傅里叶级的次数，这些久期方程组仅仅是14个推力傅里叶级系的函数，因而整个连续控制问题的参数减少为14个。With the addition of a small correction term to elimi

13、nate offsets ofthe averaged trajectory due to initial conditions, the averaged secularequations are sufficient to determine a low-thrust trajectory with ahigh degree of accuracy. This is verified by comparison of theaveraged trajectory dynamics with the fully integrated Newtonianequations of motion

14、for a basic step-acceleration function and arandomly generated continuous-acceleration function. 我们用添加小的修正项的方法来消除由于初始条件造成的平均轨道的误差，这样，这个平均轨道就可以有足够的精度来解决小推力轨道（转移）问题。这个已经通过对比平均轨道动力法和基本分步加速度函数及随机连续加速度函数的牛顿运动方程全积分法证实。The method has certain limits of applicability. First, the thrustacceleration must be ab

15、le to be represented by a Fourier series, asis true for almost any physical system. The acceleration must beperiodic (not necessarily with a period of one orbit) with sufficientlylow magnitude that the orbit shape does not change significantlyfrom start to finish. The orbit may closely approach zero

16、 eccentricity,but the current method cannot tolerate exactly circular orbits.这个方法有其适用局限性。首先，跟其它的物理系统一样，加速度必须能够表示成为傅里叶级数。加速度必须具有足够小的周期来使轨道形状从开始到结束不发生大的改变。轨道的偏心率可以接近零，但目前的方法还不允许完全的圆轨道。The focus of the current paper is to introduce this methodology andpresent numerical justification of the result in so

17、me limited settings.Subsequent work will demonstrate the use of these secular equationsin solving orbit-transfer problems for low-thrust spacecraft and willstudy the results for near-circular orbits.本文的重点是引入这一新的方法并且数值的方法来验证在某些限制情形下的结果。后面的工作将论证和说明使用这种久期方程来解决小推力轨道转移问题并且近圆轨道情况下的结果。II. Variational Equat

18、ions 变分方程Weconsider a spacecraft of negligible mass in orbit about a centralbody, which isassumed to be a point mass. The spacecraft is subjectto a continuous thrust acceleration of potentially varying magnitudeand direction. The spacecraft trajectory can be described by theNewtonian equations of mo

19、tion:我们考虑一个对于其环绕的中心体质量可以忽略不计的在轨航天器，它可以认为是一个质点。航天器受一个连续可以大范围改变大小和方向的推力的作用。其轨道可以用牛顿运动方程来表述： （1） （2）where r is the position vector, v is the velocity vector, and is thestandard gravitational parameter of the central body. The thrustaccelerationvector F can be resolved along the radial, normal, andcircu

20、mferential directions:式中的r为位置矢量，v为速度矢量，为中心体的引力常数。推力加速度矢量F可以沿着径向、法向和切向分解： （3）where and. The Newtonianequations can be decomposed into the Lagrange planetary equations,which describe the time rate of change of the classical orbit elementsof a body subject to the , , and perturbations. The Gaussianform

21、 of the Lagrange planetary equations is presented next 23:式中：，。牛顿方程可以分解为拉格朗日行星方程，拉格朗日行星方程把经典的轨道根数随时间的变化率描述为受，和支配的扰动方程。高斯形式的行星方程可以表示为：where a is the semimajor axis, e is the eccentricity, i is the inclination,is the longitude of the ascending node, is the argumentof periapsis, is the true anomaly, E

22、is the eccentric anomaly, and is the mean longitude. The mean anomaly is thedifference of the mean longitude and the longitude of periapsis:这里式中的a为半长轴，e为偏心率，i为轨道倾角，为升交点赤经，为近地点辐角，为真近点角，E为偏近点角，而为真黄经。平近点角是平黄经和近心点经度的微分： （10）In the modeling and simulation of low-thrust spacecraft orbits,both the Newtonia

23、n equations and the Gauss equations provideidentical results. The Gauss equations are often preferred for clearvisualization of the orbit over time.在小推力航天器的轨道建模与模拟中，牛顿方程和高斯方程可以得到相同的结果。但在轨道时间历程的可视化中，高斯方程要好一些。III. Fourier Series Expansion of Control Law 控制律的傅里叶展开According to Fouriers theorem, any piec

24、ewise-smooth functionwith a finite number of jump discontinuities on the interval（0，L）can be represented by a Fourier series that converges to theperiodic extension of the function itself 24:根据傅里叶理论，任何一个在（0，L）上只存在有限个第一类间断点的分段光滑的函数都可以表示为一系列无限逼近与函数本身的傅里叶级数： （11）When jump discontinuities exist, the Fou

25、rier series converges to theaverage of the two limits. For an interval of interest, theFourier coefficients are found by当跳跃间断点存在时，傅里叶级数将逼近于左右极限的平均值。对于长度为的区间，傅里叶系数可以由下式给出： （12） （13） （14）Nearly all physical systems meet the conditions of piecewisesmoothness and jump discontinuities; thus, this represe

26、ntation can beapplied to almost any general low-thrust spacecraft control law.Given an arbitrary acceleration vector F, each component can berepresented as a Fourier series over an arbitrary time interval. TheFourier series can be expanded in time or in a time-varying orbitalparameter, such as true

27、anomaly, eccentric anomaly, or meananomaly. Let represent this arbitrary parameter:几乎所有的物理系统都可以满足分段光滑和有限个第一类间断点这个条件；因此，这种表示方法可以应用于任何一种普通的小推力航天器的控制律中。对于给定的任意的加速度矢量F，其各个分量都可以表示为在任意时间间隔上的傅里叶级数。傅里叶级数可以按时间展开，也可以按随时间变化的轨道参数展开，如真点近点角，偏心率或平近点角。以来表示这一任意的轨道参数： (15) (16) (17)The acceleration function is thus d

28、efined by the coefficients and, where the superscripts Dandindicate the accelerationdirection and the expansion variable, respectively.因此加速度方程就可以由系数和确定，这里的上标D和分别代表加速度的方向和展开变量。IV. Averaged Variational Equations 平均变分方程We begin our first-order averaging analysis by assuming anacceleration vector that i

29、s to be specified over one orbit period() with a sufficiently low magnitude that the size and shape ofthe orbit does not change significantly over one revolution.Therefore, we may average the Gauss equations over one orbit periodwith respect to mean anomaly to find equations for the mean orbitelemen

30、ts:现在开始一阶平均化分析，先假设一个加速度矢量,它可以在一个轨道周期()中表示，这个加速度矢量足够小，以使轨道的在一圈中的改变不是很明显。因此，我们可以将高斯方程用真近点角在一个轨道周期上平均来得到平均轨道参数方程。 （18）whererepresents any orbit element. Ourperiodic-orbitassumption slightly simplifies the Fourier series and coefficientdefinitions:这里的代表任意的轨道参数，上面的对以为周期的轨道的假设，使得傅里叶级数及其系数的描述得到稍许简化： (19) (

31、20) (21) （22） （23） （24）At this point, the choice of orbital parameter for the thrustaccelerationvector components Fourier series expansion becomessignificant. If the acceleration components are expanded as Fourierseries in true anomaly and the independent parameter for theaveraging is likewise shift

32、ed to true anomaly, the resulting secularequations become quite lengthy and complex. For example, theequation for the semimajor axis becomes：由此可见，在对推力加速度矢量的傅里叶展开时，所使用的轨道参数的选择是很重要的。如果加速度分量和其它用于平均化分析的参数都按真近点角展开成傅里叶级数的话，则由此得出的久期方程将变得冗长而复杂。例如，半长轴方程将变成： （25） Note that the denominator can be expanded as a

33、 cosineseries in its own right:我们注意到分母可以展开成余弦级数： （26）Thus, in the true anomaly expansion, each averaged equationcontains integrals of products of the sine and cosine series. Resolvedinto secular equations, each equation will contain the full Fourierseries for each relevant acceleration direction. Ev

34、en morecomplicated results are found if the expansion and averaging arecarried out in mean anomaly.因此，在真近点角的展开中，每一个平均方程中都含有正/余弦乘积的所有项目。当方程转化为久期方程后，每一个方程都含有各个方向的加速度的所有傅里叶级数。如果展开和平均化是以平近点角进行的，那么会得到更加复杂的结果。However, if the acceleration vector components are expandedas Fourier series in eccentric anomaly

35、 and the averaging iscarried out with eccentric anomaly as the independent parameter,the problematic denominators are eliminated, as：但是，如果加速度向量的每一个分量都是按偏近点角展开的并且，平均化的各独立参数也是按偏近点角进行的，那么上述有问题的分母便可以消去，： （27）Applying this, the averaged Gauss equations with respect toeccentric anomaly can be stated as应用此

36、式，关于偏近点角的高斯方程就可以表述为 （28） （29） （30） （31） （32） （33）Substituting the Fourier series for the thrust vector componentsinto the averaged Gauss equations, we encounter the orthogonalityconditions：用傅里叶级数代替推力分量得到平均化的高斯方程，我们便遇到正交条件： （34） （35）This orthogonality eliminates all but the zeroth-, first-, and/orsec

37、ond-order coefficients of each thrust-acceleration Fourier series.Thus, the average rates of change of the orbital elements a, e, i, ,and are only dependent on the 14 Fourier coefficients：，and ,regardless of the order of the original thrust Fourier series.Henceforth, we will drop the superscript E,

38、as all thrust-accelerationFourier series will beexpanded in eccentric anomaly:这个正交性消掉了推力加速度傅里叶展开级数中除了第一，第二和/或第三项系数外的所有系数。因此，不考虑初始推力加速度傅里叶级数的次/阶数，轨道根数的平均变化率a, e, i, , , 和只与这14个傅里叶系数有关：，和。由于所有的推力加速度分量都是以偏近点角展开的，所以在以后的叙述中，我们将省略上标E： （36） （37） （38） （39） （40） （41）The assumption of a thrust-acceleration ve

39、ctor specified over onlyone orbit period is not strictly necessary; the same averaging methodcan be used with acceleration functions specified over arbitrarylengths, simply by substituting Eqs. (1217) and averaging the Gaussequations over the full interval. An example of this is presented inSec. V.

40、In general, when , the zeroth, (m/2)th, and mthcoefficients will remain, with fractional indices required in theoriginal Fourier series when m is not an even integer. However, theaveraging assumption may become less valid for aperiodic controllaws of long duration, for which the orbit changes signif

41、icantly fromstart to finish.我们不一定要严格按照对推力加速度矢量只能在一个轨道周期中表示的假设，这样的平均化分析的方法可以在任意长度的轨道段中进行，只需把方程(1217)和高斯方程在整个区间上替换就行。第下面的第五部分中将给出这样的一个例子。通常的，当时，方程中只有第零次，(m/2)次和m次傅里叶系数，如果m不是偶数，那么将会出现分数指数的形式。但是如果是这样的话，这种平均化假设在解决轨道始末两个状态改变量明显的控制律问题中有效性会大打折扣。Equations (3641) contain singularities in the case of zeroeccen

42、tricity or inclination, due to singularities in the Gaussequations. To avoid numerical difficulties when evaluatingtrajectories that closely approach circular or equatorial orbits, wesubstitute the variables.由于高斯方程存在奇点的问题，方程(3641)在偏心率为零和轨道倾角为零时也会出现奇点。为了能够解决在评估近圆和近赤道轨道时出现的数值计算问题，我们把变量作如下替换： （42） （43）

43、 （44） （45）The averaged differential equations for the new variables canbe derived using the preceding approach. These substitutions areeffective for near-zero eccentricity and inclination; however, we findinaccuracies when integrating trajectories that pass through exactlycircular orbits, indicating

44、 that our averaging analysis must be reconsideredfor this case. A deeper analysis of these singularities willbe carried out in the future.含有上述新变量的平均化的微分方程也可以用先前的方法推导得出。这一变量替换可以有效的解决近圆和近赤道轨道问题；但是我们发现即使这种替换变量的方程在解决完全圆轨道时，也会出再不准确，这就说明，针对于这种情况，我们前面所述的平均化分析要重新考虑。对于这种奇异点的情况，将来还要做更深入的研究和分析。V. Agreement wit

45、h Newtonian Equations 与牛顿方程的相容/一致性In the following, we present several examples to indicate theveracity of our analytical result. These simulations and comparisonsare to provide additional insight into this result. Additional numericalverifications of these results were made but are not reported her

46、e.To verify the averaged secular equations, we first consider a simplecontrol law: a step-acceleration function in the circumferentialdirection only, with two burns and coast arcs, as shown in Fig. 1.The Fourier series for this step function is determined byEqs. (2224)：接下来我们用几个例子来说明上述分析结果的正确性。我们将用这些

47、模拟和对比来更加深入的洞察和理解上述分析结果。我们做了更多的数值验证，但不在这里一一举出。为了验证平均化的久期方程的正确性，我们先考虑一个切向分步加速度模式下的控制律问题，在机动过程中，包含有两个点火和两个滑行弧，过程如下图1所示。分段函数下的傅里叶级数由方程(2224)给出：Fig. 1 Step circumferential acceleration. 图1 分段切向加速度 （46） （47） （48） （49）Figure 2 compares this Fourier series, numerically evaluated up toorder 1000, to the seri

48、es of only the five terms that appear in theaveraged secular equations. Clearly, there is considerable variationbetween these two representations of the periodic step-accelerationcontrol law. (All dimensions herein are normalized to a standardgravitational parameter; thus, figure units are not state

49、d.)图2中，我们对久期方程中的1000项傅里叶级数和5项傅里叶级数的数值评估结果的精度作了对比。很明显，这种对周期性分段推力的控制律用不同的项数逼近（在精度上）会有很大的不同。（所有的量纲均规范化为标准的重力参数；因此在图中没有标注单位）Fig. 2 Fourier series for step circumferential acceleration.图2 分段切向加速度的傅里叶级数表示（对比）Fig. 3 Osculating orbital elements of spacecraft subject to stepcircumferential acceleration.图3 受分

50、段切向推力（加速度）的航天器的吻切轨道根数Another case of interest is acceleration with constant magnitudebut varying direction, as on a spacecraft with one gimbaled thruster.As a simple example of this case, we consider an acceleration thatoscillates sinusoidally between the radial and circumferentialdirections, as sho

51、wn in Fig. 4. The Fourier series for the componentsof this acceleration vector are immediate:, , and, and all other coefficients are zero.我们所关注的另一种情况是航天器上载有可以改变推进方向的推进器，使航天器受到常值连续可变方向推力（加速度）。这种情况下的一个简单的例子就是加速度方向在径向和法向间按正弦规律变化，如图4所示。这种情形下的加速度矢量分量的傅里叶表达式中的系数便为：, ,和，而其它所有的系数均为零。Figure 5 shows the traje

52、ctory resulting from this acceleration,as determined by both the Newtonian equations and the averagedsecular equations. Again, there is close agreement between the twomethods.由牛顿方程和平均久期方程算得的在这种加速度作用下的轨道如图5所示。同样，这两种方法之间也存在相容/一致性。Next, we consider a more complex control law. Figures 68describe the tra

53、jectory of an example system for which the thrustaccelerationFourier coefficients were randomly selected up toorder 10 within the range of -2.5e- 7 to 2.5e -7. Figure 6compares the time histories of the osculating orbital elements asdetermined by both methods. We note that there is precise agreement

54、between the Newtonian equations and the averaged secularequations for the first several orbits. In the later orbits, there is somedrift, most noticeable in mean anomaly and argument of periapsis,due to higher-order effects not captured in the averaging process andto mismatch between the average init

55、ial conditions and the secularinitial conditions. Correction of this drift will be addressed in Sec. VI.下面，我们考虑一个更为复杂的控制律。图68所描述的实例中，加速度傅里叶展开系数是随机选取的，达到10阶精度，且范围在-2.5e- 7 , 2.5e -7。图6将两种方法算得的吻切轨道根数时程进行了对比。我注意到，在初始的几圈轨道中，牛顿方程和久期方程解具有极好的一致性。但在靠后几圈轨道中，解发生了一些漂移，其中平近点角和近地点辐角的变化最为明显，这是由于更高阶的影响因素没有在平均化分析中考虑到，且平均化的初始条件和久期方程的初始条件并不完全匹配/相容。我们将在本文第六部分中对这些漂移进行修正。Figure 7 compares one component of this acceleration vector overthree orbits with the first five terms