Optimal Aircraft Routing in General Wind Fields

S. J. Bijlsma∗

摘要

AIRCRAFT navigation involves routing and guidance ofnairplanes as a part of flight planning. In this Note, optimalnaircraft routing refers to route determination for an airplane travelingnhorizontally between two given points so that the transit time isnminimized, assuming that the meteorological conditions (innparticular, the wind field) are fully known beforehand for thencomplete passage. Aircraft guidance refers to guiding the aircraftnalong this route.nThe determination of the minimal-time path for an airplane wasnfirst accomplished by Zermelo [1] in a lecture in Prague, which gavenrise to a number of publications in this field, including those in whichnthe analogy between optical and minimum-flight problems wasnpointed out [2]. The actual interest in the problem came after thenSecond World War, when regular long-distance flights came intonoperation. After some less successful attempts, a useful manualnmethod for the calculation of minimal-flight paths was developed bynintroducing time fronts analogous to wave fronts in geometricalnoptics [3].nThe introduction of the computer made this manual methodnobsolete. The obvious next step was to program appropriate methodsnsuch as applications of the calculus of variations and graph theory forncomputer application. Although the minimal-flight problem can benformulated very simply as a problem of the calculus of variations, thensolution must generally be obtained by iteration so that apart from thenfact that convergence problems may occur, the iteration processncould converge to a relative minimum instead of converging to annabsolute minimum. On the other hand, the graph method alwaysnyields an absolute minimum, but at the cost of large computation timenand memory space.nAs an ultimate application of the calculus of variations, annextension of the technique of neighboring optimal control wasnintroduced to compute near-optimal trajectories in general windnfields, starting from nominal solutions to the Zermelo problemnobtained with simple analytical or zero wind fields [4]. In [5], it wasnclaimed that the excellent performance of this approach in practice isnachieved because winds typically vary in a smooth manner and donnot contain many sharp nonlinearities or discontinuities. However,nthe wind does not generally vary in a smooth manner. Consequently,nthe choice of a nominal solution is not always obvious and neither isnthe convergence to a near-optimal trajectory, as we will see in thennext section.nInefficient routing may result in excess fuel burn and, accordingly,nin excess emissions. Therefore, combining the approaches of thencalculus of variations and graph theory, we propose a method thatnalways yields an optimal solution with moderate computationalneffort and memory space. The minimal-flight problem is a simplenexample of the control problem of Bolza [6] from the calculus ofnvariations. Integration of the model equations and varying the initialnheading yields a one-parameter family of solutions (extremals)nemanating from the point of departure and, under certain conditions,ncontinuous in their dependence on this parameter. New extremalsncan be inserted at a specific time if the distance between two adjacentnextremals becomes too large at that time. If this distance is chosen tonbe sufficiently small, the starting values of the new extremal can benobtained by linear interpolation between the corresponding values ofnits neighbors. In this way, a network is built up containing points thatncan be reached along minimal-time tracks after a certain number ofntime steps. The ultimate minimal-time track from origin tondestination is obtained by selecting that extremal, which ends closestnto the destination. This method was tested extensively during then1970s at the Royal Netherlands Meteorological Institute in manynpractical situations related to minimal-time ship routing [7].nIn the following section, the problem of Bolza is discussed innrelation to minimal-time aircraft routing for constant airspeed. Next,nthis discussion is generalized for airspeeds, which may depend onnposition, time, and heading, and the relation with minimal-time shipnrouting is elucidated. Conclusions are presented in the last section.

机译：飞机导航涉及飞机的选路和制导，这是飞行计划的一部分。在本注释中，最佳航空器航线是指在两个气象点（尤其是风场）事先已知并完全通过的气象条件下，确定在两个给定点之间水平飞行的飞机的航线，以最小化渡越时间。飞机制导是指沿着这条路线引导飞机。n飞机的最小时间路径的确定最初是由Zermelo [1]在布拉格的一次演讲中完成的，该领域的许多出版物引起了轰动，其中包括类似的出版物。没有指出光学和最小飞行问题之间的关系[2]。在第二次世界大战后，当常规的长途航班无人驾驶时，人们对该问题产生了真正的兴趣。经过一些较不成功的尝试后，通过引入类似于几何几何学中的波前的时间前沿[3]，开发了一种用于计算最小飞行路径的有用的手动方法。n计算机的引入使这种手动方法变得过时了。显而易见的下一步是对适当的方法进行编程，例如变分演算的应用和计算机应用的图论。尽管最小飞行问题可以很简单地表示为变化的演算问题，但是通常必须通过迭代来获得解，因此除了可能会出现收敛问题的事实之外，迭代过程还可以收敛到相对最小值而不是收敛到绝对值。最低。另一方面，图法总是提供绝对最小值，但要付出较大的计算时间和存储空间的代价。n作为微积分的最终应用，没有引入邻域最优控制技术的附件扩展来计算近最优轨迹。一般的风场，从简单解析风场或零风场获得的Zermelo问题的名义解开始[4]。在[5]中，人们宣称这种方法的出色性能在实践中无法实现，因为风通常以平滑的方式变化，并且不包含许多尖锐的非线性或不连续性。但是，风通常不会以平稳的方式变化。因此，如在下一节中将看到的，标称解的选择并不总是很明显，并且收敛到接近最佳的轨迹也不是那么容易。n低效的路线选择可能会导致过度的燃油消耗，从而导致过多的排放。因此，结合变分法和图论的方法，我们提出了一种总是产生具有适度的计算工作量和存储空间的最优解的方法。最小飞行问题是变量计算中Bolza [6]控制问题的简单示例。模型方程的积分和变化的初始航向产生了从出发点开始需要的一参数系列的解（极值），在某些条件下，它们对这个参数的依赖性不连续。如果两个相邻的末梢之间的距离在那时变得太大，则可以在特定的时间插入新的末梢。如果将此距离选择得足够小，则可以通过在邻居的对应值之间进行线性插值来获得新极值的起始值。这样，就建立了一个网络，其中包含在一定数量的时间步长之后沿最小时间轨迹可以到达的点。通过选择最接近目的地的极值，可以获得从起点到目的地的最终最小时间轨迹。 1970年代在荷兰皇家气象学院对这种方法进行了广泛的测试，涉及与最小时间航路有关的许多实际情况。[7]在下一节中，将讨论Bolza的问题与恒定空速的最小时间航空器的航行无关。接下来，将对空速进行一般性的讨论，这可能取决于位置，时间和航向，并阐明了与最短时间航行路线的关系。结论在最后一部分中给出。

Optimal Aircraft Routing in General Wind Fields

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