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.一種新的求解小推力軌道動力學的方法被開發(fā)出來了。推力向量的各坐標用偏近點角表示為傅里葉級數(shù)，并且將高斯變分方程平均到整個軌道上來提高久期方程組的精度。這些久期方程組僅僅是14個推力傅里葉系數(shù)的函數(shù)而忽略初始傅里葉級的次數(shù)，這些方程能夠精確有效的解決小推力螺旋軌道問題，且與通常的牛頓問

4、題的積分方法相比能大大減少計算量。這一方法應用在小推力航天器的目標確定和最優(yōu)控制問題中。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.小推力推進系統(tǒng)為許多星際航行和地球軌道任務提供了一種高效的新選擇。但是這種系統(tǒng)卻對最優(yōu)化控制提出了新的挑戰(zhàn)。對于幾種特殊的小推力下的軌道轉(zhuǎn)移的最優(yōu)控制問題已經(jīng)出現(xiàn)了解析法和數(shù)值近似解的方法，但是對于一般的連續(xù)推力問題需要對每一個初始條件和推力值進行完全的數(shù)值積分。軌道參數(shù)的確定往往對這些變量很敏感，因此，（普通的）最優(yōu)控制法則對于確定螺旋軌道的數(shù)十甚至上百個軌道是十分困難的。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.在一些特殊情形的軌道轉(zhuǎn)移中，數(shù)值解法已經(jīng)得到了挖掘，如對數(shù)螺旋（上升/下降），福布斯螺旋（上升/下降），指數(shù)正弦曲線，常值徑向/切向推進，馬氏開普勒推進，勞登螺旋，畢阿氏螺旋等。而最新的更多解

10、法則應用變量/變差的微積分或直接優(yōu)化法來來確定特定約束條件下的小推力最優(yōu)控制律。一些求解開環(huán)（軌道）的時間最優(yōu)軌道轉(zhuǎn)移和李雅普諾夫反饋最優(yōu)軌道轉(zhuǎn)移法也早就出現(xiàn)。實踐證明，把平均法和其它方法結(jié)合的思路在解決對初始軌道/推力變量預測敏感的問題上有很有效的，但所有這些方法都局限于由推力和軌道參數(shù)形成的空間區(qū)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.本文研究的重點是提出一種評估/解決帶有最少約束的航天器軌道動力學的小推力推進效應/問題。我們將推進加速度向量的各坐標用偏近點角表示為傅里葉級數(shù)，并且將高斯變分方程平均到整個軌道上來提高久期方程組的精度。忽略初始傅里葉級的次數(shù)，這些久期方程組僅僅是14個推力傅里葉級系的函數(shù)，因而整個連續(xù)控制問題的參數(shù)減少為14個。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. 我們用添加小的修正項的方法來消除由于初始條件造成的平均軌道的誤差，這樣，這個平均軌道就可以有足夠的精度來解決小推力軌道（轉(zhuǎn)移）問題。這個已經(jīng)通過對比平均軌道動力法和基本分步加速度函數(shù)及隨機連續(xù)加速度函數(shù)的牛頓運動方程全積分法證實。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.這個方法有其適用局限性。首先，跟其它的物理系統(tǒng)一樣，加速度必須能夠表示成為傅里葉級數(shù)。加速度必須具有足夠小的周期來使軌道形狀從開始到結(jié)束不發(fā)生大的改變。軌道的偏心率可以接近零，但目前的方法還不允許完全的圓軌道。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.本文的重點是引入這一新的方法并且數(shù)值的方法來驗證在某些限制情形下的結(jié)果。后面的工作將論證和說明使用這種久期方程來解決小推力軌道轉(zhuǎn)移問題并且近圓軌道情況下的結(jié)果。II. Variational Equat

18、ions 變分方程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:我們考慮一個對于其環(huán)繞的中心體質(zhì)量可以忽略不計的在軌航天器，它可以認為是一個質(zhì)點。航天器受一個連續(xù)可以大范圍改變大小和方向的推力的作用。其軌道可以用牛頓運動方程來表述： （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為位置矢量，v為速度矢量，為中心體的引力常數(shù)。推力加速度矢量F可以沿著徑向、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：，。牛頓方程可以分解為拉格朗日行星方程，拉格朗日行星方程把經(jīng)典的軌道根數(shù)隨時間的變化率描述為受，和支配的擾動方程。高斯形式的行星方程可以表示為：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:這里式中的a為半長軸，e為偏心率，i為軌道傾角，為升交點赤經(jīng)，為近地點輻角，為真近點角，E為偏近點角，而為真黃經(jīng)。平近點角是平黃經(jīng)和近心點經(jīng)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.在小推力航天器的軌道建模與模擬中，牛頓方程和高斯方程可以得到相同的結(jié)果。但在軌道時間歷程的可視化中，高斯方程要好一些。III. Fourier Series Expansion of Control Law 控制律的傅里葉展開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:根據(jù)傅里葉理論，任何一個在（0，L）上只存在有限個第一類間斷點的分段光滑的函數(shù)都可以表示為一系列無限逼近與函數(shù)本身的傅里葉級數(shù)： （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當跳躍間斷點存在時，傅里葉級數(shù)將逼近于左右極限的平均值。對于長度為的區(qū)間，傅里葉系數(shù)可以由下式給出： （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:幾乎所有的物理系統(tǒng)都可以滿足分段光滑和有限個第一類間斷點這個條件；因此，這種表示方法可以應用于任何一種普通的小推力航天器的控制律中。對于給定的任意的加速度矢量F，其各個分量都可以表示為在任意時間間隔上的傅里葉級數(shù)。傅里葉級數(shù)可以按時間展開，也可以按隨時間變化的軌道參數(shù)展開，如真點近點角，偏心率或平近點角。以來表示這一任意的軌道參數(shù)： (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數(shù)和確定，這里的上標D和分別代表加速度的方向和展開變量。IV. Averaged Variational Equations 平均變分方程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:現(xiàn)在開始一階平均化分析，先假設一個加速度矢量,它可以在一個軌道周期()中表示，這個加速度矢量足夠小，以使軌道的在一圈中的改變不是很明顯。因此，我們可以將高斯方程用真近點角在一個軌道周期上平均來得到平均軌道參數(shù)方程。 （18）whererepresents any orbit element. Ourperiodic-orbitassumption slightly simplifies the Fourier series and coefficientdefinitions:這里的代表任意的軌道參數(shù)，上面的對以為周期的軌道的假設，使得傅里葉級數(shù)及其系數(shù)的描述得到稍許簡化： (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：由此可見，在對推力加速度矢量的傅里葉展開時，所使用的軌道參數(shù)的選擇是很重要的。如果加速度分量和其它用于平均化分析的參數(shù)都按真近點角展開成傅里葉級數(shù)的話，則由此得出的久期方程將變得冗長而復雜。例如，半長軸方程將變成： （25） Note that the denominator can be expanded as a

33、 cosineseries in its own right:我們注意到分母可以展開成余弦級數(shù)： （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，在真近點角的展開中，每一個平均方程中都含有正/余弦乘積的所有項目。當方程轉(zhuǎn)化為久期方程后，每一個方程都含有各個方向的加速度的所有傅里葉級數(shù)。如果展開和平均化是以平近點角進行的，那么會得到更加復雜的結(jié)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個分量都是按偏近點角展開的并且，平均化的各獨立參數(shù)也是按偏近點角進行的，那么上述有問題的分母便可以消去，： （27）Applying this, the averaged Gauss equations with respect toeccentric anomaly can be stated as應用此

36、式，關(guān)于偏近點角的高斯方程就可以表述為 （28） （29） （30） （31） （32） （33）Substituting the Fourier series for the thrust vector componentsinto the averaged Gauss equations, we encounter the orthogonalityconditions：用傅里葉級數(shù)代替推力分量得到平均化的高斯方程，我們便遇到正交條件： （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:這個正交性消掉了推力加速度傅里葉展開級數(shù)中除了第一，第二和/或第三項系數(shù)外的所有系數(shù)。因此，不考慮初始推力加速度傅里葉級數(shù)的次/階數(shù)，軌道根數(shù)的平均變化率a, e, i, , , 和只與這14個傅里葉系數(shù)有關(guān)：，和。由于所有的推力加速度分量都是以偏近點角展開的，所以在以后的敘述中，我們將省略上標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.我們不一定要嚴格按照對推力加速度矢量只能在一個軌道周期中表示的假設，這樣的平均化分析的方法可以在任意長度的軌道段中進行，只需把方程(1217)和高斯方程在整個區(qū)間上替換就行。第下面的第五部分中將給出這樣的一個例子。通常的，當時，方程中只有第零次，(m/2)次和m次傅里葉系數(shù)，如果m不是偶數(shù)，那么將會出現(xiàn)分數(shù)指數(shù)的形式。但是如果是這樣的話，這種平均化假設在解決軌道始末兩個狀態(tài)改變量明顯的控制律問題中有效性會大打折扣。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點的問題，方程(3641)在偏心率為零和軌道傾角為零時也會出現(xiàn)奇點。為了能夠解決在評估近圓和近赤道軌道時出現(xiàn)的數(shù)值計算問題，我們把變量作如下替換： （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.含有上述新變量的平均化的微分方程也可以用先前的方法推導得出。這一變量替換可以有效的解決近圓和近赤道軌道問題；但是我們發(fā)現(xiàn)即使這種替換變量的方程在解決完全圓軌道時，也會出再不準確，這就說明，針對于這種情況，我們前面所述的平均化分析要重新考慮。對于這種奇異點的情況，將來還要做更深入的研究和分析。V. Agreement wit

45、h Newtonian Equations 與牛頓方程的相容/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)：接下來我們用幾個例子來說明上述分析結(jié)果的正確性。我們將用這些

47、模擬和對比來更加深入的洞察和理解上述分析結(jié)果。我們做了更多的數(shù)值驗證，但不在這里一一舉出。為了驗證平均化的久期方程的正確性，我們先考慮一個切向分步加速度模式下的控制律問題，在機動過程中，包含有兩個點火和兩個滑行弧，過程如下圖1所示。分段函數(shù)下的傅里葉級數(shù)由方程(2224)給出：Fig. 1 Step circumferential acceleration. 圖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.)圖2中，我們對久期方程中的1000項傅里葉級數(shù)和5項傅里葉級數(shù)的數(shù)值評估結(jié)果的精度作了對比。很明顯，這種對周期性分段推力的控制律用不同的項數(shù)逼近（在精度上）會有很大的不同。（所有的量綱均規(guī)范化為標準的重力參數(shù)；因此在圖中沒有標注單位）Fig. 2 Fourier series for step circumferential acceleration.圖2 分段切向加速度的傅里葉級數(shù)表示（對比）Fig. 3 Osculating orbital elements of spacecraft subject to stepcircumferential acceleration.圖3 受分

50、段切向推力（加速度）的航天器的吻切軌道根數(shù)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.我們所關(guān)注的另一種情況是航天器上載有可以改變推進方向的推進器，使航天器受到常值連續(xù)可變方向推力（加速度）。這種情況下的一個簡單的例子就是加速度方向在徑向和法向間按正弦規(guī)律變化，如圖4所示。這種情形下的加速度矢量分量的傅里葉表達式中的系數(shù)便為：, ,和，而其它所有的系數(shù)均為零。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.由牛頓方程和平均久期方程算得的在這種加速度作用下的軌道如圖5所示。同樣，這兩種方法之間也存在相容/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，我們考慮一個更為復雜的控制律。圖68所描述的實例中，加速度傅里葉展開系數(shù)是隨機選取的，達到10階精度，且范圍在-2.5e- 7 , 2.5e -7。圖6將兩種方法算得的吻切軌道根數(shù)時程進行了對比。我注意到，在初始的幾圈軌道中，牛頓方程和久期方程解具有極好的一致性。但在靠后幾圈軌道中，解發(fā)生了一些漂移，其中平近點角和近地點輻角的變化最為明顯，這是由于更高階的影響因素沒有在平均化分析中考慮到，且平均化的初始條件和久期方程的初始條件并不完全匹配/相容。我們將在本文第六部分中對這些漂移進行修正。Figure 7 compares one component of this acceleration vector overthree orbits with the first five terms