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Design of Low Altitude Long Endurance Solar-Powered UAV Using Genetic Algorithm
基于遗传算法的低空长航时太阳能无人机设计

Abu Bakar 1, Li Ke 1,*, Haobo Liu 1, Ziqi Xu 1 and Dongsheng Wen 1,2,*
阿布·巴卡尔 1、李克 1，*、刘浩波 1、徐子琪 1、温东生 1、2、*

1 National Key Laboratory of Human Machine and Environment Engineering,
1人体机与环境工程国家重点实验室，

School of Aeronautical Science and Engineering, Beihang University, Beijing 100191, China; abu_bakar_ist@hotmail.com (A.B.); liuhaobo@buaa.edu.cn (H.L.); 15652583677@buaa.edu.cn (Z.X.)
北京航空航天大学航空科学与工程学院，中国北京100191;abu_bakar_ist@hotmail.com （A.B.）;liuhaobo@buaa.edu.cn （H.L.）;15652583677@buaa.edu.cn （Z.X.）

2 School of Chemical and Process Engineering, University of Leeds, Leeds LS2 9JT, UK
2利兹大学化学与工艺工程学院，英国 Leeds LS2 9JT

* Correspondence: like@buaa.edu.cn (L.K.); d.wen@buaa.edu.cn (D.W.)
*通信方式：like@buaa.edu.cn （L.K.）;d.wen@buaa.edu.cn （D.W.）

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Citation: Bakar, A.; Ke, L.; Liu, H.; Xu, Z.; Wen, D. Design of Low Altitude Long Endurance
引自：Bakar， A.;柯，L.;刘 H.;徐 Z.;温 D. 低空长续航设计

Solar-Powered UAV Using Genetic Algorithm. Aerospace 2021, 8, 228. https://doi.org/10.3390/ aerospace8080228
使用遗传算法的太阳能无人机。航空航天 2021， 8， 228。https://doi.org/10.3390/ 航空航天8080228

Academic Editor: Lakshmi N Sankar
学术编辑：Lakshmi N Sankar

Received: 10 June 2021
收稿日期： 2021-06-10

Accepted: 4 August 2021
录用日期： 2021-08-04

Published: 16 August 2021
发布日期：2021 年 8 月 16 日

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Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
版权所有：© 2021 年归作者所有。被许可人 MDPI，瑞士巴塞尔。本文是根据知识共享署名 （CC BY） 许可证 （https:// creativecommons.org/licenses/by/ 4.0/） 的条款和条件分发的开放获取文章。

Abstract: This paper presents a novel framework for the design of a low altitude long endurance solar-powered UAV for multiple-day flight. The genetic algorithm is used to optimize wing airfoil using CST parameterization, along with wing, horizontal and vertical tail geometry. The mass estimation model presented in this paper is based on structural layout, design and available materials used in the fabrication of similar UAVs. This model also caters for additional weight due to the change in wing airfoil. The configuration is optimized for a user-defined static margin, thereby incorporating static stability in the optimization. Longitudinal and lateral control systems are developed for the optimized configuration using the inner–outer loop strategy with an LQR and PID controller, respectively. A six degree-of-freedom nonlinear simulation is performed for the validation of the proposed control scheme. The results of nonlinear simulations are in good agreement with static analysis, validating the complete design process.
摘要： 本文提出了一种新的多日飞行低空长航时太阳能无人机设计框架。遗传算法用于使用 CST 参数化以及机翼、水平和垂直尾翼几何形状来优化机翼。本文提出的质量估计模型基于结构布局、设计和用于制造类似无人机的可用材料。由于机翼翼型的变化，该模型还迎合了额外的重量。该配置针对用户定义的静态裕量进行了优化，从而在优化中加入了静态稳定性。纵向和横向控制系统是为优化配置而开发的，分别使用带有 LQR 和 PID 控制器的内-外循环策略。执行了六自由度非线性仿真以验证所提出的控制方案。非线性仿真的结果与静态分析非常一致，验证了完整的设计过程。

Keywords: solar-powered UAV (unmanned aerial vehicle); genetic algorithm; optimization; LQR (linear quadratic regulator); PID (proportional integral derivative)
关键词：太阳能无人机（无人驾驶飞行器）;遗传算法;优化;LQR（线性二次稳压器）;PID （比例积分导数）

1.Introduction
1.引言

At present, aviation focuses on economic, fuel-efficient, and long-endurance systems that incur low operating costs. The growing demand for airborne operations has resulted in the adoption of unmanned aircraft that can remain in the air for significantly longer times than aircraft with pilots on board. The UAV design has been developed extensively to attain these objectives. At present, UAVs can have an endurance of more than a day [1,2]. Engines that consume fuel are not the solution to the ever-growing demands for airborne time. The endurance of an aircraft must satisfy the mission requirement, which can be a few hours (broadcasting a soccer match), a few days (border patrolling, search and rescue missions), or even years (continuous surveillance, communication relay, and meteorological investigations). Exceptional endurance of months or years is possible with a solar-powered aircraft. Photovoltaic cells mounted on wings can be used to capture solar energy during the daytime. A part of this energy can be used directly to run the propulsion system and onboard electronics, and the remaining part can be stored in onboard batteries for use during the night. If sufficient solar energy can be stored in onboard batteries during the day to last the succeeding night, we can design an aircraft that can fly for years.
目前，航空业专注于经济、省油和长航时、运营成本低的系统。对空中运营的需求不断增长，导致无人驾驶飞机的采用，与机上飞行员的飞机相比，无人驾驶飞机可以在空中停留的时间要长得多。为了实现这些目标，无人机设计已经得到了广泛的发展。目前，无人机的续航时间可以超过一天 [1,2]。消耗燃料的发动机并不能解决不断增长的航空时间需求。飞机的续航能力必须满足任务要求，可以是几个小时（转播足球比赛）、几天（边境巡逻、搜救任务），甚至几年（连续监视、通信中继和气象调查）。太阳能飞机可以实现数月或数年的卓越续航能力。安装在机翼上的光伏电池可用于在白天捕获太阳能。这些能量的一部分可以直接用于驱动推进系统和机载电子设备，其余部分可以储存在机载电池中，以备夜间使用。如果机载电池中可以在白天储存足够的太阳能以持续下一个夜晚，我们就可以设计出一架可以飞行多年的飞机。

Theoretically, these solar-powered aircraft can fly forever, and their endurance is limited only by the reliability of the subsystems.
从理论上讲，这些太阳能飞机可以永远飞行，它们的续航能力仅受子系统可靠性的限制。

Solar airplanes can be divided into two categories: high altitude long endurance and low altitude long endurance aircraft. Zephyr [3] and Solara [4] are examples of high altitude long endurance solar aircraft. Designed by QinetiQ, Zephyr achieved three world records in July 2010. The UAV was launched for flight trials on 9 July 2010, and stayed aloft for 14 nights (336 h 22 min) at an altitude of 70,740 ft (21,561 m) above the US Army’s Yuma
太阳能飞机可分为两大类：高空长航时飞机和低空长航时飞机。Zephyr [3] 和 Solara [4] 是高海拔长航时太阳能飞机的例子。由 QinetiQ 设计的 Zephyr 在 2010 年 7 月创下了三项世界纪录。该无人机于 2010 年 7 月 9 日发射进行飞行试航，在美国陆军尤马上空 70,740 英尺（21,561 米）的高度高空停留了 14 晚（336 小时 22 分钟）

Aerospace 2021, 8, 228. https://doi.org/10.3390/aerospace8080228 https://www.mdpi.com/journal/aerospace
航空航天 2021， 8， 228.https://doi.org/10.3390/aerospace8080228https://www.mdpi.com/journal/aerospace

Proving Ground in Arizona. Sky-Sailor [5], Solong [6], and AtlantikSolar [7] are examples of low altitude long endurance solar UAVs. These solar UAVs have demonstrated the potential for perpetual flight and the advantages of solar technology. The present work is also about the design and optimization of fixed-wing low altitude long endurance solar-powered UAVs.
亚利桑那州的试验场。Sky-Sailor [5]、Solong [6] 和 AtlantikSolar [7] 是低空长航时太阳能无人机的例子。这些太阳能无人机展示了永久飞行的潜力和太阳能技术的优势。目前的工作还涉及固定翼低空长航时太阳能无人机的设计和优化。

The design process of solar airplanes was not published in the early stages of solar flights [5]. During the past decade, many authors published literature explaining the design process of solar UAVs. However, most of these designs were limited to the theoretical level and did not reach the proof-of-concept phase [8–14]. Certain designs were manufactured and successfully flown to validate different flight parameters, although these were not tested for a full day–night flight [15–21]. Very few designs have demonstrated perpetual flight capability [5–7].
太阳能飞机的设计过程在太阳能飞行的早期阶段没有公布 [5]。在过去的十年中，许多作者发表了解释太阳能无人机设计过程的文献。然而，这些设计大多局限于理论层面，没有达到概念验证阶段[8–14]。制造并成功飞行了一些设计来验证不同的飞行参数，尽管这些设计没有针对完整的昼夜飞行进行测试[15–21]。很少有设计表现出永久飞行能力[5–7]。

As far as airfoil selection is concerned, most of the literature compared few airfoils and selected the airfoil with the best lift-to-drag ratio and minimum moment coefficient. The authors in [22] studied 150 airfoils and selected E178 airfoil due to its smooth upper surface. Similarly, [23] discussed MH114, NACA4412, and SD7073 and choose MH114 because of the higher lift-to-drag ratio and large thickness at the trailing edge to facilitate any avionics installation. A similar procedure was adopted in [10,14]. Matter et al. [24] developed an
就翼型选择而言，大多数文献都比较了少数翼型，并选择了具有最佳升阻比和最小力矩系数的翼型。[22] 的作者研究了 150 个翼型，并选择了 E178 翼型，因为它的上表面光滑。同样，[23] 讨论了 MH114、NACA4412 和 SD7073，并选择了 MH114，因为它具有更高的升阻比和后缘的大厚度，以方便任何航空电子设备安装。[10,14] 也采用了类似的程序。Matter等[24]开发了一种

Xfoil-Matlab interface to optimize solar UAV airfoil at a Reynolds number of 2.0 × 105. The optimized airfoil showed an increase of 13% in the lift-to-drag ratio. The authors in [25]
Xfoil-Matlab 接口，用于优化雷诺数为 2.0 × 105 的太阳能无人机翼型。优化后的翼型显示升阻比增加了 13%。[25] 的作者

also used the genetic algorithm to optimize solar-powered UAV airfoil to maximize the radiation incidence and lift coefficient and minimize the drag by lift ratio. Betancourth et al. [9] selected E212 and the upper surface was considered as a polyline connected by several short line sections. Thereby, it was possible to arrange photovoltaic cells without considerable deformation [26]. In the design of Sky-Sailor [5], W. Engel designed a special airfoil named WE3.55-9.3. The design procedure and objectives were not published. In the proposed framework, the airfoil design is integrated into solar UAV design. Airfoil is also considered a design variable along with wingspan, chord, horizontal and vertical tail span, chord and axial location of the tail assembly. Hence, it is possible to design airfoil considering the overall performance of the UAV. Secondly, if the airfoil changes, rib and wing skin will also change, consequently changing the weight of the UAV. The proposed framework also caters for such changes.
还使用遗传算法优化太阳能无人机翼型，以最大化辐射入射和升力系数，并最小化升力比。Betancourth等[9]选择了E212，上表面被认为是一条由几条短线段连接的折线。因此，可以在不发生明显变形的情况下布置光伏电池[26]。在 Sky-Sailor [5] 的设计中，W. Engel 设计了一种名为 WE3.55-9.3 的特殊翼型。设计程序和目标未公布。在拟议的框架中，翼型设计被集成到太阳能无人机设计中。机翼也被认为是一个设计变量，以及尾翼组件的翼展、弦、水平和垂直尾翼、弦和轴向位置。因此，可以设计考虑到无人机整体性能的翼型。其次，如果翼型发生变化，肋骨和机翼蒙皮也会发生变化，从而改变无人机的重量。拟议的框架也迎合了这些变化。

The accurate prediction of the structural mass of a solar UAV is very important. Several mass estimation models are discussed in the literature. A real and simplified airplane structure was dimensioned and actual shear force and bending moment were calculated for the wing and fuselage for the given load case in [11,27]. This method showed deviation from actual data. Noth [5] concluded that models based on the data of HALSOL and twin-boom aircraft [28,29] could not be used for small land-launched solar UAVs. He collected data of over 400 models and sailplanes and ranked solar UAVs in the top 5% in terms of build quality. This model is widely used to predict the structure mass of solar UAVs [10,15,16,19,30–32]. As presented in Section 2.3.3, this model also failed to predict the structural mass accurately. The estimation of the structural mass of solar UAVs is significantly influenced by the materials, structural design and layout (double spare, single spare, D-box). In the present study, the UAV dimensions, structural layout, design, and density of materials are used to predict the structural mass. The proposed model is validated with an existing solar UAV with high accuracy.
准确预测太阳能无人机的结构质量非常重要。文献中讨论了几种质量估计模型。在 [11,27] 中，在给定的载荷工况下，对真实和简化的飞机结构进行了尺寸标注，并计算了机翼和机身的实际剪力和弯矩。这种方法与实际数据存在偏差。Noth [5] 得出结论，基于 HALSOL 和双臂飞机 [28\u201229] 数据的模型不能用于小型陆基太阳能无人机。他收集了 400 多个模型和滑翔机的数据，并将太阳能无人机在制造质量方面排在前 5%。该模型被广泛用于预测太阳能无人机的结构质量 [10,15,16,19,30–32]。如第 2.3.3 节所述，该模型也未能准确预测结构质量。太阳能无人机结构质量的估计受材料、结构设计和布局（双备用、单备用、D-box）的显着影响。在本研究中，无人机尺寸、结构布局、设计和材料密度用于预测结构质量。所提出的模型与现有的太阳能无人机进行了高精度的验证。

Naturally, the solar UAV configuration must also be stable. Static stability is often measured in terms of the static margin. The literature discussed above provides no guide- line about the static margin for solar UAVs. In the proposed framework, the solar UAV is optimized for a user-specified static margin. The optimization is performed for cruise conditions. The Matlab genetic algorithm is used for optimization. Only static stability is incorporated in the optimization framework. To ensure dynamic stability, the dynamic response of the optimized solar-powered UAV is studied. In addition, longitudinal and lateral controls are developed for the optimized configuration. An inner–outer loop control
当然，太阳能无人机的配置也必须稳定。静态稳定性通常根据静态裕量来衡量。上面讨论的文献没有提供关于太阳能无人机静态裕量的指南。在所提出的框架中，太阳能无人机针对用户指定的静态裕量进行了优化。针对巡航条件执行优化。Matlab 遗传算法用于优化。优化框架中仅包含静态稳定性。为保证动态稳定性，对优化后的太阳能无人机的动态响应进行了研究。此外，还为优化配置开发了纵向和横向控制。内外循环控制

strategy is used with an LQR and PID controller, respectively. Finally, 6-DOF nonlinear simulation is performed in Matlab Simulink to validate the design process and control system.
策略分别与 LQR 和 PID 控制器一起使用。最后，在 Matlab Simulink 中进行 6-DOF 非线性仿真，以验证设计过程和控制系统。

The remainder of this paper is organized as follows: The design methodology is presented in Section 2, including mass estimation, airfoil parameterization, and stability. The optimization framework is discussed in Section 3. The design of the solar UAV and the optimization results are discussed in Section 4. The dynamic analysis, linear control system design and nonlinear 6-DOF results of the optimized configuration are presented in Section 5. Section 6 presents a few conclusions and discussions, followed by the References section.
本文的其余部分组织如下：设计方法在第 2 节中介绍，包括质量估计、翼型参数化和稳定性。优化框架在 Section 3 中讨论。太阳能无人机的设计和优化结果在第 4 节中讨论。优化配置的动力学分析、线性控制系统设计和非线性 6-DOF 结果在第 5 节中介绍。第 6 节提出了一些结论和讨论，然后是参考文献部分。

2.Design Methodology
2.Design 方法

2.1.Power Required for Level Flight
2.1.平飞所需的功率

During steady level flight, the lift and propeller thrust is equal to the weight and drag, respectively:
在稳定水平飞行期间，升力和螺旋桨推力分别等于重量和阻力：

W = mg = C ρ SV2 (1)
W = 毫克 = C ρ SV2（1）

2

and
和

T = C ρ SV2

2

(2)

where W is weight, T is thrust, CL is lift coefficient, CD is drag coefficient, S is surface area, V is the velocity, m is mass and g is gravitational constant. We can determine the velocity from Equation (1):
其中 W 是重量，T 是推力，CL 是升力系数，CD 是阻力系数，S 是表面积，V 是速度，m 是质量，g 是引力常数。我们可以从方程 （1） 中确定速度：

V =  2mg CLρS
V =2毫克 CLρS

Using Equations (2) and (3), the power required for level flight, PLevel, is given by:
使用公式 （2） 和 （3），水平飞行所需的功率 PLevel 由下式给出：

(3)

P TV
P电视台

CD s(mg)3 s 2
CD s（毫克）3 s 2

(4)

Level =
级别 =

= 3/2 ρ

L

We replace the surface area in Equation (4) with the aspect ratio, AR, to obtain the following equation:
我们将方程 （4） 中的表面积替换为纵横比 AR，得到以下方程：

P  CD s 2ARg3 m3/2

(5)

2.2.Daily Energy Requirement
2.2.每日能量需求

level = 3/2 ρ
水平 =3/2ρ

L

The efficiency of the motor ηmot, controller ηclrt, gearbox ηgrb, and propeller ηplr must be considered to calculate the power consumption. The power required for the payload Ppld and avionics Pav is generally known. If the voltage is to be reduced for the payload and avionics, the efficiency of the step-down converter ηbec must also be considered. Therefore, the total power consumption Ptot is calculated using the equation given in Noth [5].
在计算功耗时，必须考虑电机 ηmot、控制器 ηclrt、变速箱 ηgrb 和螺旋桨 ηplr 的效率。有效载荷 Ppld 和航空电子设备 Pav 所需的功率是众所周知的。如果要降低有效载荷和航空电子设备的电压，还必须考虑降压转换器 ηbec 的效率。因此，总功耗 Ptot 是使用 Noth [5] 中给出的方程计算的。

Ptot =  1  PLevel + 1 Pav + Ppld (6)
Ptot = 1 PLevel + 1 Pav + PPLD（6）

Considering the efficiency of charging ηchrg and discharging ηdchrg of the batteries, the daily energy consumption Eelec tot is given by:
考虑到电池的 ηchrg 充电效率和放电 ηdchrg 的效率，每日能耗 Eelec tot 由下式给出：

Eelec tot = Ptot Tday + Tnight  ! (7) where Tday and Tnight are duration of day and night, respectively.
Eelec tot = Ptot Tday + Tnight ！（7） 其中 Tday 和 Tnight 分别是白天和黑夜的持续时间。

2.3.Mass Prediction Model
2.3.质量预测模型

It is highly important to accurately estimate the mass of a solar UAV because the power required for level flight is directly proportional to it. The mass prediction model in this study is adopted from Noth [5]. The exception is that rather than estimating the structural mass statistically, a new model based on structural layout, design and available materials is proposed.
准确估计太阳能无人机的质量非常重要，因为水平飞行所需的功率与它成正比。本研究中的质量预测模型采用自 Noth [5]。例外的是，它不是统计地估计结构质量，而是提出了一种基于结构布局、设计和可用材料的新模型。

2.3.1.Fixed Masses
2.3.1.固定质量

The fixed mass includes the masses of the payload and avionics. These masses are known and do not vary with the aircraft’s dimensions:
固定质量包括有效载荷和航空电子设备的质量。这些质量是已知的，并且不随飞机的尺寸而变化：

mf ixed = mav + mpld (8)
MF ixed = MAV + MPLD（8）

where mf ixed is fixed mass, mpld is payload mass and mav is avionics mass.
其中 MF ixed 是固定质量，MPLD 是有效载荷质量，MAV 是航空电子质量。

2.3.2.Battery Mass
2.3.2.电池质量

Depending on the mission requirements, the battery adds significant mass to the system. The battery mass is directly proportional to the product of the power required and the duration of the night and is inversely proportional to its energy density [5].
根据任务要求，电池会大大增加系统的质量。电池质量与所需功率和夜间持续时间的乘积成正比，与其能量密度成反比 [5]。

Tnight bat ηdchrgkbat tot

where mbat is battery mass and kbat is energy density of battery.
其中 MBAT 是电池质量，KBAT 是电池的能量密度。

2.3.3.Structural Mass
2.3.3.结构质量

The most significant challenge in solar UAV design is the estimation of the structural weight. An overestimation of the weight would result in selecting a heavier battery, which would take a long time to charge and increase the weight. An underestimation of the weight would result in selecting a lighter battery, which would be insufficient to last a night. The weight estimation methods discussed in Section 1 are based on statistical data of different manned sailplanes, unmanned radio-controlled models, NASA high-altitude prototypes, and twin-boom configurations. These models are derived from their geometric parameters such as wing area, wing aspect ratio and wingspan, rather than the structural layout, design and materials used. The structural weights of the four solar UAVs estimated using different models are presented in Table 1. It is important to mention here that, for solar UAV design studies discussed in the preceding section, structural weight was quoted for very few prototypes.
太阳能无人机设计中最大的挑战是结构重量的估计。高估重量会导致选择较重的电池，这将需要很长时间来充电并增加重量。低估重量会导致选择较轻的电池，这不足以持续一夜。第 1 节中讨论的重量估计方法基于不同载人滑翔机、无人无线电控制模型、NASA 高空原型机和双臂配置的统计数据。这些模型源自其几何参数，如机翼面积、机翼纵横比和翼展，而不是所使用的结构布局、设计和材料。表 1 列出了使用不同模型估计的四种太阳能无人机的结构重量。这里值得一提的是，对于上一节讨论的太阳能无人机设计研究，很少有原型机引用了结构重量。

Table 1. Comparison of structural weight estimation models.
表 1.结构重量估计模型的比较。

Weight in kg
重量 （kg）

UAV b (m) AR Noth [5] Stender [28] Rizzo [29] Leutenegger [11] Actual
无人机b （m）AR诺斯 [5]斯特德 [28]里佐 [29]洛伊滕艾格 [11]实际

Sky-Sailor 3.2 12.9 0.87 2.466 7.04 0.51 0.748
天空水手3.212.90.872.4667.040.510.748

SunSailor 4.2 13.15 2.016 3.743 10.056 0.85 1.70

AltantikSolar 5.69 18.7 4.733 5.381 14.95 - 1.52
AltantikSolar5.6918.74.7335.38114.95-1.52

Zephyr 18 11.6 189.43 37.461 67.91 51.0 20.0
西风1811.6189.4337.46167.9151.020.0

The most common wing structural layouts are double circular spare [16], single circular spare [7,18] and D-box [5,20,21]. Even for a specific structural layout, there can be many variations in the numbers of ribs, stringers, the diameter and thickness of spares and the thickness of ribs and skin. However, for a given span and aspect ratio, with different structural layouts, any mass prediction models presented in Table 1 will give the same prediction. In the proposed design process, weight estimation is based on the specific
最常见的机翼结构布局是双圆形备用 [16]、单圆形备用 [7,18] 和 D-box [5,20,21]。即使对于特定的结构布局，肋条、纵梁的数量、备件的直径和厚度以及肋骨和蒙皮的厚度也可能存在许多变化。然而，对于给定的跨度和纵横比，具有不同的结构布局，表 1 中给出的任何质量预测模型都将给出相同的预测。在拟议的设计过程中，权重估计基于特定的

structural design, layout, respective dimensions and actual material used in the fabrication of solar UAVs.
结构设计、布局、各自的尺寸和用于制造太阳能无人机的实际材料。

The wing structural layout selected for the current study is double circular spares with ribs and stringers. The reasons for this selection are the simplicity and practical experience in fabricating this layout. The selection of the material is also very important for solar UAV design. The materials should be lightweight but adequately strong enough to withstand flight loads. For the current study, the materials considered are 3 k carbon fiber, light density balsa wood and plywood. Several prototypes such as MARAAL [16], Sky-Sailor [5], AtlantikSolar [7], SunSailor [33] and LEEUAV [20] also used carbon fiber and balsa wood as primary materials.
为当前研究选择的机翼结构布局是带有肋和纵梁的双圆形备件。选择此布局的原因是构建此布局的简单性和实践经验。材料的选择对于太阳能无人机的设计也非常重要。材料应重量轻，但强度足以承受飞行载荷。对于当前的研究，考虑的材料是 3 k 碳纤维、轻质轻木和胶合板。一些原型，如 MARAAL [16]、Sky-Sailor [5]、AtlantikSolar [7]、SunSailor [33] 和 LEEUAV [20] 也使用碳纤维和轻木作为主要材料。

There is no study conducted for structure design. The structure design is adopted from a reference UAV available at the UAV design lab, Beihang University, Beijing. The reference UAV is also a solar-powered UAV with a total weight of 7.5 kg. This UAV has been extensively tested in various experimental flights with a cruise speed up to 15 m/s. The reference UAV has a front and rear spare with a thickness of 1 mm and diameter of 20 mm and 12 mm, respectively. Ribs are made from balsa wood with a thickness of 2 mm. Four stringers run along the span. The stringer at the trailing edge is made from balsa wood. The other three stringers are made from plywood. Wing surface is also balsa wood of 1 mm thickness. The reference UAV also has a third shorter spare adjacent to the front spare. The purpose of this spare is to provide extra stiffness and install batteries if required. The horizontal and vertical tail design is similar to the wing, except they have only one spare with a thickness and diameter of 1 mm and 12 mm, respectively. The tail boom is also a carbon fiber tube with a thickness and diameter of 1 mm and 20 mm, respectively. The fuselage is made from a carbon fiber sheet of 1 mm thickness. To estimate the length of the fuselage, similar solar UAVs are studied (Table 2). From the literature discussed in Section 1, the fuselage length of only three prototypes was quoted. From the geometric data of these three prototypes, it is concluded that the fuselage length could be assumed to be 21% of the wingspan.
没有进行结构设计的研究。结构设计采用了北京航空航天大学无人机设计实验室提供的参考无人机。参考无人机也是一款太阳能无人机，总重量为 7.5 公斤。这款无人机已在各种实验飞行中进行了广泛测试，巡航速度高达 15 m/s。参考无人机有一个前后备用件，厚度为 1 毫米，直径分别为 20 毫米和 12 毫米。肋骨由厚度为 2 毫米的轻木制成。四根纵梁沿着跨度运行。后缘的纵梁由轻木制成。其他三个纵梁由胶合板制成。机翼表面也是 1 毫米厚的轻木。参考无人机在前备件附近还有第三个较短的备件。此备件的目的是提供额外的刚度并在需要时安装电池。水平和垂直尾翼设计与机翼相似，只是它们只有一个备用机翼，厚度和直径分别为 1 毫米和 12 毫米。尾梁也是一根碳纤维管，厚度和直径分别为 1 毫米和 20 毫米。机身由 1 毫米厚的碳纤维板制成。为了估计机身的长度，研究了类似的太阳能无人机（表 2）。从第 1 节讨论的文献中，仅引用了三个原型的机身长度。从这三个原型机的几何数据中可以得出结论，可以假设机身长度为翼展的 21%。

Table 2. Ratio of fuselage length to wingspan.
表 2.机身长度与翼展的比率。

Solar UAV Wingspan (m) L. Fuselage (m) Ratio
太阳能无人机翼展 （m）L. 机身 （m）比率

Sky-Sailor [5] 3.2 0.82 0.256
天空水手 [5]3.20.820.256

AtlantikSolar [7] 5.7 0.81 0.142
亚特兰蒂克太阳能 [7]5.70.810.142

SunSailor [33] 4.2 0.94 0.224
太阳水手 [33]4.20.940.224

  average 0.207
平均0.207

The fuselage mass is determined based on its length and the average diameter of the reference solar UAV. A heat sink plastic film is used for the wing and tail surfaces in conjunction with balsa wood to save weight. The user can input the percentage coverage of balsa wood (100% implies that the entire wing and tail surface would be of balsa wood, i.e., plastic film would not be used).
机身质量是根据其长度和参考太阳能无人机的平均直径确定的。机翼和尾翼表面使用散热器塑料薄膜和轻木以减轻重量。用户可以输入轻木的覆盖率百分比（100% 意味着整个机翼和尾部表面都是轻木，即不使用塑料薄膜）。

The shape of the rib is identical to that of the airfoil and varies as the airfoil undergoes modifications during the optimization process. The user can change the number of holes and their respective locations. The diameters of these holes are input as percentages of the airfoil thickness at the respective locations. This prevents misleading results when the input diameter value is larger than the airfoil thickness at that particular location. The wing airfoil of the reference UAV is NACA 6409, and its tail airfoil is NACA 0009.
加强筋的形状与翼型的形状相同，并且随着翼型在优化过程中的修改而变化。用户可以更改孔的数量及其各自的位置。这些孔的直径以相应位置的翼型厚度的百分比形式输入。这可以防止当输入直径值大于该特定位置的翼型厚度时产生误导性结果。参考无人机的机翼翼型是 NACA 6409，它的尾翼翼是 NACA 0009。

The reference UAV is disassembled for measurements. Mass is calculated using geometric dimension, structural layout, design and material densities. Material densities considered for carbon fiber, balsa wood and plywood are 1600 kg/m3, 90 kg/m3 and 260 kg/m3, respectively. The structural components of the reference UAV are shown in Figure 1.
将参考无人机拆卸进行测量。质量是使用几何尺寸、结构布局、设计和材料密度计算的。碳纤维、轻木和胶合板考虑的材料密度分别为 1600 kg/m3、90 kg/m3 和 260 kg/m3。参考无人机的结构部件如图 1 所示。

Figure 1. Structural components of solar UAV: (a) Upper wing, (b) Lower wing, (c) Fuselage,
图 1.太阳能无人机的结构部件：（a） 上机翼，（b） 下机翼，（c） 机身，

(d) Wing rib.
（d） 翼肋。

The calculated and actual masses are compared in Table 3. The total estimated mass of the wing also contains the mass of stringers.
表 3 中比较了计算质量和实际质量。机翼的总估计质量也包含纵梁的质量。

Table 3. Comparison of calculated and actual structural mass.
表 3.计算结构质量与实际结构质量的比较。

0.3 m
0.3 米

Radius Radius
半径半径

0.9 m
0.9 米

Radius
半径

No of ribs Ribs/Unit 51
肋骨数量肋骨/单位51

Tail
尾巴

Balsa Coverage Upper Wing 100%
轻木覆盖上翼100%

Span 1 m
跨度1 m

Chord 0.22 m
弦0.22 m

Height 0.4 m
高度0.4 m

Chord 0.25 m
弦0.25 m

Thickness 0.001 m
厚度0.001 m

Radius 0.006 m
半径0.006 m

Balsa Coverage 50%
轻木覆盖率50%

Length 0.73 m
长度0.73 m

0.17 0.163
0,170,163

Tail Boom
尾梁

Fuselage
机身

Thickness 0.001 m
厚度0.001 m

Radius 0.01 m
半径0.01 m

Thickness 0.001 m
厚度0.001 m

Length 1.2 m
长度1.2 m

Diameter 0.065 m
直径0.065 m

0.068 0.0735
0,0680,0735 元

0.388 0.391
0.3880.391 元

Total 1.946 1.9875
合计1.9461.9875

The individual components and total structural mass of the reference UAV and the calculated mass are very close. The advantage of using this type of method is that we can accurately predict the mass given the available materials, structural layout and design. This method is very flexible and can be extended to D-box structural layout. The design of the wing and tail ribs is shown in Figure 2. The modeled airfoils exactly replicate the actual airfoils.
参考无人机的各个组件和总结构质量与计算的质量非常接近。使用这种方法的优点是，我们可以在给定可用材料、结构布局和设计的情况下准确预测质量。这种方法非常灵活，可以扩展到 D-box 结构布局。机翼和尾肋的设计如图 2 所示。建模的翼型完全复制了实际的翼型。

Figure 2. Rib design for wing and tail airfoil.
图 2.机翼和尾翼翼的肋条设计。

2.3.4.Solar Cells Mass
2.3.4.太阳能电池质量

To estimate the mass of the solar cell, the required solar cell area is computed using the following equation:
为了估计太阳能电池的质量，使用以下公式计算所需的太阳能电池面积：

Asc =  π  1 + Tnight 1  !Ptot (10)
Asc = π 1 + 骑士1 ！PTOT系列（10）

where Asc is the area of the solar cells, ηsc is the efficiency of the solar cells, ηcbr is the efficiency of the camber, ηmppt is the efficiency of the Maximum Power Point Tracker (MPPT), ηwthr is the margin factor for clouds and Imax is the maximum irradiation. The mass of the solar cells can be calculated by considering the encapsulation and mass density of the solar cell.
其中 Asc 是太阳能电池的面积，ηsc 是太阳能电池的效率，ηcbr 是外倾角的效率，ηmppt 是最大功率点跟踪器 （MPPT） 的效率，ηwthr 是云的裕量因子，Imax 是最大辐照度。太阳能电池的质量可以通过考虑太阳能电池的封装和质量密度来计算。

msc = Asc(ksc + kenc) (11)
毫秒 = Asc（ksc + kenc）（11）

where msc is the mass of solar cells, and ksc and kenc are solar cells and encapsulation mass density, respectively.
其中 MSC 是太阳能电池的质量，KSC 和 KENC 分别是太阳能电池和封装质量密度。

2.3.5.Maximum Power Point Tracker Mass
2.3.5.最大功率点跟踪器质量

To extract the maximum power from the solar cell, we must track the optimal working point on its current-to-voltage curve using MPPT. This point shifts continuously because of the irradiance conditions. The mass of the MPPT is calculated using:
为了从太阳能电池中提取最大功率，我们必须使用 MPPT 跟踪其电流-电压曲线上的最佳工作点。由于辐照度条件，该点不断移动。MPPT 的质量计算公式为：

mmppt = kmppt Imaxηscηcbrηmppt Asc (12) where mmppt is the mass of MPPT and kmppt is the mass-to-power ratio of MPPT.
mmppt = kmppt Imaxηscηcbrηmppt Asc（12），其中 mmppt 是 MPPT 的质量，kmppt 是 MPPT 的质量功率比。

2.3.6.Propulsion System Mass
2.3.6.推进系统质量

The propulsion system consists of an electric motor, controller, electronics, gearbox, and propeller. Based on the statistical analysis from Noth [5], the mass of the entire propulsion system is given by:
推进系统由电动机、控制器、电子设备、变速箱和螺旋桨组成。根据 Noth [5] 的统计分析，整个推进系统的质量由下式给出：

mprop = kprop Plevel (13)
mprop = kprop 普利夫 （13）

where mprop and kprop are the mass and mass-to-power ratio of the propulsion system, respectively.
其中 mprop 和 kprop 分别是推进系统的质量和质功率比。

2.4.Stability
2.4.稳定性

Stability is generally defined as the capability of an aircraft to return to its equilibrium state after any imbalance, gust, and control input. There are two types of stability: static and dynamic. For an airplane to be statically stable, the sign of the CMα curve must be negative. A statically stable plane tends to return to its equilibrium position. A statically unstable plane continues to increase the orientation after disturbance. Meanwhile, a statically neutral plane regains its position, implying that the net force or moment acting on the aircraft in the new orientation is zero. Static stability is generally measured in terms of the static margin, which is defined as follows:
稳定性通常定义为飞机在任何不平衡、阵风和控制输入后恢复到平衡状态的能力。稳定性有两种类型：静态稳定性和动态稳定性。要使飞机保持静态稳定，CMα 曲线的符号必须为负。静态稳定的平面往往会返回到其平衡位置。静态不稳定的平面在干扰后继续增加取向。同时，静态中性平面重新获得其位置，这意味着作用在新方向上的飞机的合力或力矩为零。静态稳定性通常根据静态裕度来衡量，静态裕度定义如下：

SM = XNP − XCG × 100% (14)
SM = XNP − XCG × 100% （14）

Lre f

Lre f is the reference length, XCG is the center of gravity and XNP is the neutral point. In the proposed framework, the solar UAV is optimized for a user-specified static margin at the equilibrium position. As the geometry of the UAV is changed during optimization, the neutral point will also change. One methodology is to fix XCG and calculate the static margin using XNP and Lre f . In this case, an additional constraint is required to fulfill the static margin requirement. In the current study, the required XCG is calculated using XNP, Lre f and the static margin. The aerodynamic moment is then shifted to the required XCG for analysis. For the design of a solar-powered UAV, a static margin of 15% is selected.
Lre f 是参考长度，XCG 是重心，XNP 是中性点。在所提出的框架中，太阳能无人机在平衡位置针对用户指定的静态裕量进行了优化。由于无人机的几何形状在优化过程中发生变化，中性点也会发生变化。一种方法是修复 XCG 并使用 XNP 和 Lre f 计算静态裕量。在这种情况下，需要额外的约束来满足 static margin 要求。在当前研究中，所需的 XCG 是使用 XNP、Lre f 和静态裕度计算的。然后将空气动力学力矩转移到所需的 XCG 进行分析。对于太阳能无人机的设计，选择了 15% 的静态裕量。

2.5.Airfoil Parameterization
2.5.翼型参数化

The airfoil must be parameterized to modify it during optimization. Several methods for airfoil parameterization are available in the literature (Hicks–Henne bump function [34], CST (class function/shape function transformation) [35], and PARSEC [36]). In the present study, CST is used for airfoil parametrization. This method can define a wide range of airfoils and can be extended to other shapes such as squares and circles. It is defined as:
必须对翼型进行参数化，才能在优化期间对其进行修改。文献中提供了几种翼型参数化方法（Hicks-Henne 碰撞函数 [34]、CST（类函数/形状函数变换）[35] 和 PARSEC [36]）。在本研究中，CST 用于翼型参数化。此方法可以定义各种翼型，并且可以扩展到其他形状，例如正方形和圆形。它被定义为：

Zupper = CN1(x)·Supper(x) + x·∆Zupper (15)
Zupper = CN1（x）·晚餐（x） + x·∆Zupper（15）

Zlower = CN1(x)·Slower(x) + x·∆Zlower (16) The class function CN1(x) is defined as:
Zlower = CN1（x）·Slower（x） + x·∆Zlower（16） 类函数 CN1（x） 定义为：

CN1(x) = xN1·(1 − x)N2 (17)
CN1（x） = xN1·（1 − x）注 2（17）

where ∆Z defines the trailing edge thickness and x ∈ [0, 1]. For airfoil shapes, N1 = 0.5 and N2 = 1. S(x) is defined as a linear combination of Bernstein polynomials and is given by:
其中 ∆Z 定义后缘厚度，x ∈ [0， 1]。对于翼型，N1 = 0.5 和 N2 = 1。S（x） 定义为 Bernstein 多项式的线性组合，由下式给出：

where
哪里

n

S(x) = ∑ aibi,n(x) (18)
S（x） = ∑ aibi，n（x）（18）

i=0

bi,n(x) = n xi(1 − x)n−i (19)
bi，n（x） = n 习（1 − x）n−i（19）

ai is the Bernstein coefficient and n is the degree of the polynomial. The quality of the CST airfoil is significantly dependent on the degree of the polynomial. In Figure 3, CST airfoils for polynomials of order zero, two, and four for NACA 63–137 are plotted in conjunction with the original airfoil. It is evident that the polynomial of order four provides a very good representation of the original airfoil. Hence, a fourth-order polynomial is used in the present optimization framework. It provides ten airfoil parameters: five each for the lower and upper portions.
ai 是伯恩斯坦系数，n 是多项式的次数。CST 翼型的质量在很大程度上取决于多项式的次数。在图 3 中，NACA 63-137 的零、二和四阶多项式的 CST 翼型与原始翼型一起绘制。很明显，四阶多项式很好地表示了原始机翼。因此，在本优化框架中使用了四阶多项式。它提供 10 个翼型参数：下部和上部各 5 个。

Figure 3. CST representation of NACA 63–137 airfoils.
图 3.NACA 63-137 翼型的 CST 表示。

3.Optimization Framework
3.优化框架

The optimization framework is modeled in Matlab using the genetic algorithm (GA) with selection, mutation and crossover functions. Seventeen design variables are con- sidered. The first 10 design variables are the CST parameters for the airfoil. The lower and upper bounds for the airfoils are based on 15 airfoils recommended for the design of solar UAVs in different publications. These airfoils are parameterized using CST. For each parameter, the highest and lowest of the values of all the airfoils are used as the upper and lower bounds, respectively (Figure 4).
优化框架在 Matlab 中使用具有选择、突变和交叉函数的遗传算法 （GA） 进行建模。考虑了 17 个设计变量。前 10 个设计变量是翼型的 CST 参数。翼型的下限和上限基于不同出版物中推荐用于太阳能无人机设计的 15 个翼型。这些翼型使用 CST 进行参数化。对于每个参数，所有翼型的值的最高值和最低值分别用作上限和下限（图 4）。

0.4

0.3

0.2

0.1

0

-0.1

-0.2

CST parameters for lower portion of 15 airfoils
15 个翼型下部的 CST 参数

1 1.5 2 2.5 3 3.5 4 4.5 5

Design Variable
设计变量

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

CST parameters for upper portion of 15 airfoils
15 个翼型上部的 CST 参数

1 1.5 2 2.5 3 3.5 4 4.5 5

Design Variable
设计变量

Figure 4. CST parameter for 15 airfoils recommended for solar UAVs.
图 4.推荐用于太阳能无人机的 15 个翼型的 CST 参数。

This results in very wide design space and the possibility of the production of non- feasible and unsmooth airfoils. The airfoils produced during optimization are passed through a series of assessments (maximum thickness, camber, curvature of the lower portion, and quality of the trailing edge) to ensure a feasible and smooth airfoil. The details of the design variables are listed in Table 4. There is no geometric and aerodynamic twist in the wing. The wing planform is a rectangle without taper and sweep. The dihedral angle of 7◦ is used from 45% of the wing semi-span for this study. The tail is a simple T-shape design with the horizontal tail mounted at the top of the vertical tail. The fuselage is not considered in the optimization, as recommended by the developer of Xflr5 (Xflr5 GUI). However, the fuselage and tail boom’s skin friction drag and form drag are considered in the optimization and calculated using W.H. Mason’s FRICTman code [37].
这导致了非常宽的设计空间，并有可能生产出不可行和不光滑的翼型。优化过程中生产的翼型经过一系列评估（最大厚度、外倾角、下部曲率和后缘质量），以确保翼型的可行性和平滑度。表 4 列出了设计变量的详细信息。机翼没有几何和空气动力学扭曲。机翼平面是一个没有锥度和后掠的矩形。在本研究中，7◦ 的二面角从机翼半跨度的 45% 开始使用。尾巴是一个简单的 T 形设计，水平尾巴安装在垂直尾巴的顶部。优化中不考虑机身，正如 Xflr5 （Xflr5 GUI） 的开发人员所建议的那样。然而，在优化中考虑了机身和尾梁的蒙皮摩擦阻力和形状阻力，并使用W.H. Mason的FRICTman代码[37]进行计算。

During optimization, Xflr5 is used to obtain aerodynamic coefficients. Xflr5 is an analysis tool for airfoils based on Xfoil’s airfoil analysis and also includes wing design and analysis capabilities using the lifting line theory, vortex lattice method and 3D panel model. The validation of Xfoil with experimental data of E387 airfoil at Reynolds number
在优化过程中，Xflr5 用于获得空气动力学系数。Xflr5 是基于 Xfoil 翼型分析的翼型分析工具，还包括使用升力线理论、涡流晶格法和 3D 面板模型的机翼设计和分析功能。用雷诺数的 E387 翼型实验数据验证 Xfoil

2.0 × 105 is shown in Figure 5. The results are in good agreement in the linear region. As solar UAVs fly at a lower angle of attack, Xflr5 is a good choice for preliminary design.
图 5 中显示了 2.0 × 105。结果在线性区域中非常一致。由于太阳能无人机以较低的迎角飞行，Xflr5 是初步设计的不错选择。

Table 4. Design variables of solar UAV.
表 4.太阳能无人机的设计变量

Axial location of tail (Origin at wing airfoil leading edge)
尾翼的轴向位置（原点在机翼翼型前缘）

1.3–1.5 m
1.3 - 1.5 米

14 Horizontal tail chord 0.2–0.31 m
14水平尾弦0.2–0.31 m

15 Vertical tail chord 0.2–0.3 m
15垂直尾弦0.2–0.3 m

16 Horizontal tail span 0.9–1.1 m
16水平尾跨0.9–1.1 m

17 Vertical tail height 0.4–0.5 m
17垂直尾部高度0.4–0.5 m

1.6

0.06

0.05

0.04

0.03

0.02

0.01

0

EXP
经验

XFoil
XFoil 公司

-5 0 0.02 0.04 0.06

Drag Coefficient
阻力系数

-5 0 5 10 15

Alpha
阿尔法

Figure 5. Comparison of Xflr5 and experimental data, E387 at Re = 2.0 × 105.
图 5.Xflr5 与实验数据的比较，Re = 2.0 × 105 时的 E387。

The analysis is performed in the following sequence:
分析按以下顺序执行：

1.Matlab GA provides 17 design variables.
1.Matlab GA 提供了 17 个设计变量。

2.The first 10 variables are used for airfoil generation using CST. The quality of the CST airfoil is assessed. When the airfoil is not smooth, the iteration is terminated, the objective function is assigned a value of 200, and the next set of design variables is considered. When the airfoil is smooth, we proceed to Step 3.
2.前 10 个变量用于使用 CST 生成翼型。评估 CST 翼型的质量。当翼型不平滑时，迭代将终止，为目标函数分配值 200，并考虑下一组设计变量。当翼型光滑时，我们继续进行第 3 步。

3.The CST airfoil and other geometry parameters are used to perform aerodynamic analysis in Xflr5. A VB script is written that calls the Xflr5.exe, writes all the variables in the respective fields, performs analysis, and writes the output file.
3.CST 翼型和其他几何参数用于在 Xflr5 中进行空气动力学分析。编写一个 VB 脚本，该脚本调用 Xflr5.exe，写入相应字段中的所有变量，执行分析，并写入输出文件。

4.Matlab reads the Xflr5 output file. This file contains all the aerodynamic forces and moment coefficients. The pitching moment coefficient is transferred to the required
4.Matlab 读取 Xflr5 输出文件。该文件包含所有空气动力和力矩系数。俯仰力矩系数被转移到所需的

C.G. location calculated from the static and neutral points using Equation (14).
使用公式 （14） 从静态点和中性点计算的 C.G. 位置。

5.The trim (zero pitching moment) angle of attack and the corresponding lift and drag coefficients are calculated.
5.计算纵倾（零俯仰力矩）攻角以及相应的升力和阻力系数。

6.The structural mass of the solar UAV is calculated (Section 2.3.3).
6.计算太阳能无人机的结构质量（第 2.3.3 节）。

7.The total mass is calculated by solving a cubic equation as presented in [5].
7.总质量是通过求解 [5] 中提出的三次方程来计算的。

8.Equations (1), (6) and (9)–(13). If the required solar cell area is greater than the wing area, the iteration is terminated, the objective function is assigned a value of 200, and the next set of design variables is considered.
8.方程 （1）、（6） 和 （9）–（13）。如果所需的太阳能电池面积大于机翼面积，则终止迭代，为目标函数分配值 200，并考虑下一组设计变量。

9.If the required solar cell area is less than the wing area, the objective function is calculated as follows:
9.如果所需的太阳能电池面积小于机翼面积，则目标函数计算如下：

Obj = abs(Li f t − Weight) + Ptot (20)
目标 = abs（Li f t − 重量） + Ptot（20）

The first term ensures that the lift produced is equal to the weight, whereas the second term helps to decrease mass and increase C3/2/CD. The number of generations is set to
第一项确保产生的升力等于重量，而第二项有助于减少质量并增加 C3/2/CD。代数设置为

100 with a population size of 25.
100 个，人口规模为 25。

4.Design of Solar UAV
太阳能无人机 4.Design

The solar UAV is designed to fly over a specific location on a specific day of the year. Given these specifications, tnight and tday are retrieved. The designed UAV is oriented to fly for tnight hours. The next step is to add robustness to the design [7].
太阳能无人机旨在在一年中的特定日期飞越特定位置。根据这些规范，将检索 tnight 和 tday。设计的无人机面向夜间飞行。下一步是增加设计的稳健性 [7]。

The first consideration is to have multiple-day endurance, tDate. This implies a window of a specific number of days or months for flight operation. To achieve this, the additional night duration, owing to date change, must be considered:
首先要考虑的是具有多天的耐久性 tDate。这意味着航班运行的特定天数或月数。为此，必须考虑由于日期变更而增加的夜间持续时间：

tDate = tmax

—

tmin
天明

(21)

exc
exc （极）

night
晚上

night
晚上

Meteorological factors such as clouds and water fog in the early morning and evening,
清晨和傍晚的云、水雾等气象因素，

tWeather , and aggressive flight conditions during the night, tPlevel , such as gusts, may result
t天气 ，以及夜间激进的飞行条件，可能会导致 tPlevel ，例如阵风

exc exc
excexc （非）

in additional power consumption. The total demand surplus time is:
在额外的功耗中。总需求剩余时间为：

treq = SUM tDate + tWeather + tPlevel (22)
treq = 总和tDate + tWeather + tPlevel（22）

This design is not fully robust. This is because the energy stored in the batteries starts to drain when the available solar power is less than the power required for level flight, which can occur more than 1 h before sunset. To illustrate this, a solar UAV is designed to perform a full-night operation in Beijing or Tianjin (40◦ N). The optimum time to perform a full night operation is 21 June, when tday = 14.8 h and tnight = 9.2 h. In this design (referred to as Design-1), treq = zero. This implies that we assume ideal conditions during flight. The input parameters for the design process are listed in Table 5.
此设计并非完全稳健。这是因为当可用的太阳能小于平飞所需的功率时，电池中存储的能量开始耗尽，这可能发生在日落前 1 小时以上。为了说明这一点，太阳能无人机被设计为在北京或天津 （40◦ N） 进行全天运行。进行整夜手术的最佳时间是 6 月 21 日，此时 tday = 14.8 小时，tnight = 9.2 小时。在此设计（称为 Design-1）中， treq = 零。这意味着我们在飞行过程中假设了理想的条件。表 5 列出了设计过程的输入参数。

Table 5. Input parameters for solar UAV design.
表 5.太阳能无人机设计的输入参数。

Inputs Description Value
InputsDescription值

g Gravitational constant 9.81 m/s2
g万有引力常数 9.81 m/s2

Alt Altitude 700 m
AltAltitude700 米

rho Density 1.15 kg/m3
rho密度1.15 kg/m3

Vcruise Cruise velocity 8.5 m/s
Vcruise巡航速度8.5 m/s

S.M. Static margin 15%
S.M.静态保证金15%

ηmctrl Efficiency of motor control 0.9
电机控制效率0.9

ηmotor Efficiency of motor 0.85
η电机效率0.85

ηgrbox Efficiency of gearbox 0.97
变速箱 ηgrbox效率0.97

ηprop Efficiency of propeller 0.80
η螺旋桨效率0.80

mavi Mass of avionic 0.5 kg
mavi航空电子质量0.5 kg

mpld Mass of payload 0.1 kg
mpld有效载荷质量0.1 kg

Pavi Power for avionics 5 W
用于航空电子设备的 Pavi Power 5 W

Ppld Power for payload 0.5 W
有效载荷的 Ppld 功率 0.5 W

ksc Mass density of solar cells 0.33 kg/m2
ksc太阳能电池质量密度0.33 kg/m2

kenc Mass density of encapsulation 0.26 kg/m2
kenc封装质量密度0.26 kg/m2

ηsc Efficiency of solar cells 0.19
太阳能电池的 ηsc 效率 0.19

kprop Mass-to-power ratio 0.008
kprop质量功率比0.008

ηMPPT Efficiency of MPPT 0.95
ηMPPT效率 MPPT0.95

ηchrg Efficiency of charging 0.95
ηchrg充电效率0.95

ηdchrg Efficiency of discharging 0.95
ηdchrg放电效率0.95

kbat Energy density of battery 240 Wh/kg
kbat电池能量密度240 Wh/kg

The energy simulation for Design-1 is shown in Figure 6a. It is evident that although the battery is designed for tnight = 9.2 h, it is insufficient for a continuous 24 h flight and is drained completely before sunrise. This is because the battery starts to supply the energy when the available solar power is less than the power required to maintain level flight. In this case, it is approximately 1.4 h earlier than sunset. In the next design (referred to as Design-2), an additional time of 1.4 h (denoted as tEnight) is included. The energy simulation for Design-2 is presented in Figure 6b. This design demonstrates the capability for continuous 24 h flight. The battery shows a minimum capacity of 10.8 Wh at sunrise, which corresponds to less than half an hour of flight. Although Design-2 displays
Design-1 的能量模拟如图 6a 所示。很明显，虽然电池是为 night = 9.2 h 设计的，但它不足以连续 24 h 飞行，并且在日出前完全耗尽。这是因为当可用的太阳能功率小于保持水平飞行所需的功率时，电池开始提供能量。在这种情况下，它比日落早大约 1.4 小时。在下一个设计（称为 Design-2）中，包括 1.4 小时的额外时间（表示为 tEnight）。Design-2 的能量模拟如图 6b 所示。此设计展示了连续 24 小时飞行的能力。电池在日出时显示的最小容量为 10.8 Wh，相当于不到半小时的飞行。虽然 Design-2 显示

potential, it is an ideal design. To add robustness, Equation (22) is incorporated (referred to as Design-3). The associated parameters are as follows:
潜力，这是一个理想的设计。为了提高稳健性，我们加入了公式 （22）（称为 Design-3）。关联参数如下：

tEnight = 1.4 h, tclouds = kcc f × tmax , tPlevel = 2.4 h, tdate = tmax
tEnight = 1.4 h，tclouds = kcc f × tmax，tPlevel = 2.4 h，tdate = tmax

—

tmin
天明

exc
exc （极）

night
晚上

exc
exc （极）

exc
exc （极）

night
晚上

night
晚上

kcc f is the cloud or fog thickness factor [38] and is considered to be 0.2.
KCC F 是云或雾的厚度因子 [38]，被认为是 0.2。

(a) (b)
（一）（二）

Figure 6. Energy simulation: (a) tnight = 9.2 h, (b) tnight = 9.2 h, tEnight = 1.4 h.
图 6.能量模拟：（a） 夜 = 9.2 小时，（b） 夜 = 9.2 小时，t夜 = 1.4 小时。

For a solar UAV that can perform multi-day operation during a three month window
对于可以在三个月窗口内执行多天运行的太阳能无人机

(1 May to 30 July), tmax = 10.3 h, tmin = 9.2 h, and tdate = 1.1 h. Hence, for Design-3,
（5 月 1 日至 7 月 30 日），tmax = 10.3 h，tmin = 9.2 h，tdate = 1.1 h。因此，对于 Design-3，

treq = 5.56 h. The total required time of flight for batteries is given by:
treq = 5.56 小时。电池所需的总飞行时间由下式给出：

treq = tmin + t

+ treq
+ 特雷克

(23)

tot
小

night
晚上

Enight
夜

exc
exc （极）

An important parameter of a battery is the state of charge, SOC. It is defined as the current battery capacity to the rated battery capacity. The over-discharging of a battery can attenuate its usable capacity. In addition, the minimum SOC may be limited to 5–10% to increase the battery life cycle [39]. A minimum SOC of 10% is imposed in Design-3. The output parameters for these three designs are listed in Table 6.
电池的一个重要参数是充电状态 SOC。它被定义为当前电池容量与额定电池容量之比。电池的过度放电会削弱其可用容量。此外，最低 SOC 可以限制为 5-10%，以延长电池寿命 [39]。Design-3 的最低 SOC 为 10%。表 6 列出了这三种设计的输出参数。

Table 6. Output parameters.
表 6.Output 参数。

Output Design-1 Design-2 Design-3
输出设计-1设计-2设计-3

Span (m) 5.1629 5.68533 5.83775
跨度 （m）5.16295.685335.83775

Wing chord (m) 0.2882 0.30711 0.30061
翼弦（米）0.28820.307110.30061

Axial location of tail (m) 1.3041 1.46781 1.49272
尾部轴向位置 （m）1.30411.467811.49272

Horizontal tail chord (m) 0.2002 0.25906 0.21497
水平尾弦 （m）0.20020.259060.21497

Vertical tail chord (m) 0.2453 0.21441 0.25869
垂直尾弦 （m）0.24530.214410.25869

Horizontal tail span (m) 0.9256 0.90948 0.92595
水平尾翼跨距 （m）0.92560.909480.92595

Vertical tail span (m) 0.4 0.44728 0.44781
垂直尾翼跨距（米）0.40.447280.44781

ma f (kg) 1.7136 2.1108 1.9774
马 F （kg）1.71362.11081.9774

mBatt (kg) 1.1967 1.5229 3.4294
米瓦特 （kg）1.19671.52293.4294

mProp (kg) 0.1696 0.1969 0.2807
米螺旋桨 （kg）0.16960.19690.2807

msc (kg) 0.498 0.5553 0.7312
磁质（kg）0.4980.55530.7312

mMPPT (kg) 0.0474 0.0529 0.0696
mMPPT （kg）0.04740.05290.0696

mTOTAL (kg) 4.2253 5.0387 7.0883
百万合计 （kg）4.22535.03877.0883

PLevel (W) 29.65 33.06 43.57
栎高比（W）29.6533.0643.57

The increase in the mass of the battery and total mass of the UAV owing to the additional time of flight is highly significant. Design-1 has a battery mass of 1.2 kg and
由于飞行时间的增加，电池质量和无人机总质量的增加非常显着。Design-1 的电池质量为 1.2 kg，

a total mass of 4.22 kg. With the addition of tEnight of 1.4 h, Design-2 has a battery mass of 1.52 kg and total mass of 5.04 kg. The further addition of a treq of 5.56 h and minimum SOC of 10% increases the battery mass to 3.4 kg and total mass to 7.1 kg. The convergence of the objective function is presented in Figure 7. The mean and best values of the objective function for the last 20 generations are almost equal. This implies that all the configurations in the generation are identical. One design iteration requires 45 s on a personal i5 laptop with 16 GB RAM and 2.43 GHz processor. A total of 100 generations with 25 population sizes may require more than one day.
总质量为 4.22 kg。增加 1.4 小时的 tEnight 后，Design-2 的电池质量为 1.52 kg，总质量为 5.04 kg。进一步增加 5.56 小时的 treq 和 10% 的最小 SOC 将电池质量增加到 3.4 kg，总质量增加到 7.1 kg。目标函数的收敛性如图 7 所示。过去 20 代目标函数的平均值和最佳值几乎相等。这意味着生成中的所有配置都是相同的。在配备 16 GB RAM 和 2.43 GHz 处理器的个人 i5 笔记本电脑上，一次设计迭代需要 45 秒。总共 100 代和 25 个种群大小可能需要一天以上的时间。

Figure 7. Convergence history of objective function, Design-3.
图 7.目标函数的收敛历史，设计 3。

The lift, drag, and pitching moment coefficients and the lift-to-drag ratio of the optimized configuration are shown in Figure 8a. The trim angle of attack is 3.1◦, where the pitching moment is zero. For this trim angle of attack, the lift coefficient is 0.96, and the lift-to-drag ratio is 28.4 (which is nearly the maximum for this design). The Xflr5 model showing panels’ density and control surfaces is also shown in Figure 8b.
优化配置的升力、阻力和俯仰力矩系数以及升阻比如图 8a 所示。纵倾攻角为 3.1◦，其中俯仰力矩为零。对于这个纵倾攻角，升力系数为 0.96，升阻比为 28.4（这几乎是此设计的最大值）。显示面板密度和控制表面的 Xflr5 模型也显示在图 8b 中。

(a) (b)
（一）（二）

Figure 8. (a) Lift, drag, moment, and L/D of Design-3, (b) Xflr5 model.
图 8.（a） Design-3 的升力、阻力、力矩和 L/D，（b） Xflr5 模型。

The optimized airfoil, shown in Figure 9, has a thickness of 9.7% at 25% and a maxi- mum camber of 4.6% at 43.5% of the chord. The airfoil is highly smooth, and the trailing edge quality is good.
如图 9 所示，优化的翼型在 25% 处的厚度为 9.7%，在弦的 43.5% 处的最大外倾角为 4.6%。翼型高度光滑，后缘质量好。

For Design-3, the wing incidence is set at 2◦. As the steady-state alpha for Design-3
对于 Design-3，机翼入射设置为 2◦。作为 Design-3 的稳态 Alpha

is 3.1◦, the wing airfoil encounters a local incidence of 5.1◦ with a lift coefficient of 1.07. The Reynolds number for a cruise speed of 8.5 m/s is 1.65 × 105. Fifteen different airfoils suggested for solar UAVs in the literature are analyzed in Xflr5 at a Reynolds number of
为 3.1◦，则机翼翼型遇到的局部入射为 5.1◦，升力系数为 1.07。巡航速度为 8.5 m/s 的雷诺数为 1.65 × 105。文献中建议的 15 种不同的翼型在 Xflr5 中以雷诺数

1.65 × 105. cl3/2/cd for these airfoils for a lift coefficient of approximately 1.07 are plotted in Figure 10. The performance of optimized airfoil is higher than most of the airfoils and very close to Sky-Sailor [5] airfoil. This plot clearly shows the importance of airfoil selection
1.65 × 105.这些翼型的升力系数约为 1.07 的 Cl3/2/cd 如图 10 所示。优化翼型的性能高于大多数翼型，非常接近 Sky-Sailor [5] 翼型。这个图清楚地显示了翼型选择的重要性

with regard to the design lift coefficient and drag coefficient. These airfoils can be used for this lift coefficient, but a few airfoils evidently operate at higher cl3/2/cd.
关于设计升力系数和阻力系数。这些翼型可用于此升力系数，但少数翼型显然在更高的 cl3/2/cd 下运行。

Figure 9. Optimized airfoil for Design-3.
图 9.为 Design-3 优化了翼型。

Figure 10. Performance comparison of airfoils.
图 10.翼型的性能比较。

The energy simulation for 21 June is presented in Figure 11a. The SOC is also shown in black color. The solar UAV takes off at 07:00 with a completely drained battery (SOC = 0). The available solar power is higher than the required power. The batteries become com- pletely charged (SOC = 1) within 6 h. The available solar energy can be used to attain an altitude or increase speed. At approximately 19:00, the available solar power is less than that required, and the batteries start to supply energy. At 20:00, solar power becomes unavailable, and the UAV operates completely on batteries. The next morning, when the solar power is higher than required, the batteries show a remaining capacity of 335 Wh. This corresponds to an SOC of 0.41.
图 11a 显示了 6 月 21 日的能量模拟。SOC 也以黑色显示。太阳能无人机于 07：00 起飞，电池完全耗尽 （SOC = 0）。可用的太阳能功率高于所需的功率。电池在 1 小时内充满电 （SOC = 6）。可用的太阳能可用于达到高度或提高速度。大约 19：00，可用的太阳能功率低于所需功率，电池开始供电。20：00，太阳能变得不可用，无人机完全依靠电池运行。第二天早上，当太阳能功率高于所需时，电池的剩余容量为 335 Wh。这对应于 0.41 的 SOC。

(a) (b)
（一）（二）

Figure 11. Design-3, energy simulation: (a) 21 June and (b) 1 May.
图 11.设计 3，能量模拟：（a） 6 月 21 日和 （b） 5 月 1 日。

A similar simulation for 1 May is also presented in Figure 11b. Owing to the lower solar irradiation on 1 May, the battery takes a longer time to charge fully. More battery is consumed to perform full-night flight because of the increased night hours. The next morning, when the available solar power is equal to the required power, the battery shows a minimum SOC of 0.37. It is concluded that Design-3 has the potential to perform continuous flight operations. However, it is important to note that the charge margin time [7] is reduced owing to the date change.
图 11b 中还显示了 5 月 1 日的类似模拟。由于 5 月 1 日的太阳辐射较低，电池需要更长的时间才能充满电。由于夜间时间增加，执行全天飞行会消耗更多电池。第二天早上，当可用太阳能功率等于所需功率时，电池显示的最小 SOC 为 0.37。结论是 Design-3 具有执行连续飞行操作的潜力。但是，需要注意的是，由于日期更改，费用保证金时间 [7] 会缩短。

5.Dynamic Stability and Control System Design
5.动态稳定与控制系统设计

In this section, the response of the optimized solar-powered UAV (Design-3) is dis- cussed by applying the linearized equation of motion. These equations are based on small perturbations from the trim conditions. Additionally, these equations are decou- pled. That is, the perturbations in longitudinal forces and moments depend neither on the lateral/directional perturbations nor the lateral/directional control inputs, and the perturbations in lateral/directional forces and moments depend neither on the longitudinal perturbations nor the longitudinal control inputs. Altitude hold and bank to turn control (BTT) systems are developed. The proposed control laws are then implemented in the nonlinear 6-DOF Matlab block for validation.
在本节中，通过应用线性运动方程来讨论优化的太阳能无人机 （Design-3） 的响应。这些方程基于修剪条件的小扰动。此外，这些方程式也被 消解了。也就是说，纵向力和力矩的扰动既不取决于横向/方向扰动，也不取决于横向/方向控制输入，横向/方向力和力矩的扰动既不取决于纵向扰动也不取决于纵向控制输入。开发了高度保持和坡度转弯控制 （BTT） 系统。然后将所提出的控制律在非线性 6-DOF Matlab 模块中实现以进行验证。

The first step is to estimate the moments of inertia. As the proposed optimization framework provides only the C.G. location for the desired static margin, a reasonable mass distribution that provides the required C.G. location is assumed. This distribution is passed on to Xflr5 to calculate the moments of inertia [40]. For Design-3, the neutral point and C.G. location are 0.72 m and 0.675 m, respectively (origin at fuselage nose). This yields a static margin of 15%. The estimated mass distribution is shown in Figure 12.
第一步是估计惯性矩。由于所提出的优化框架仅提供所需静态裕量的 C.G. 位置，因此假设提供所需 C.G. 位置的合理质量分布。这个分布被传递给 Xflr5 来计算惯性矩 [40]。对于 Design-3，中性点和 CG 位置分别为 0.72 m 和 0.675 m（原点为机头）。这产生了 15% 的静态利润率。估计的质量分布如图 12 所示。

Figure 12. Mass distribution for required C.G. location, Design-3.
图 12.所需 CG 位置的质量分配，设计 3。

5.1.Attitude and Altitude Control
5.1.姿态和高度控制

The linearized small disturbance longitudinal rigid body equations of motion in the
线性化小扰动纵向刚体运动方程

state-space form are [41]:
state-space 形式是 [41]： .

xlong = Alongxlong + Blongηlong (24)
xlong = Alongxlong + Blongηlong（24）

where
哪里

A =
答=

long
长 

Xu Xw 0 −g0cosθ0
徐旭0−g0cosθ0

Zu Zw u0 + Zq −g0sinθ0
祖Zwu0 + Zq−g0sinθ0

(25)

Mu + M . Zu Mw + M . Zw Mq + u0 M . −M . g0sinθ0
Mu + M .祖MW + M .ZwMq + u0 M .−M . g0sinθ0

0 0 1 0
00 10

and
和

Xδe XδT

 Zδe ZδT 

Blong = M

+ M . Z
+ M .Z

M + M . Z
M+ M .Z 系列

 (26)

The longitudinal state variable vector, xlong, is given by:
纵向状态变量向量 xlong 由下式给出：

xlong = u w q θ T (27)
xlong =uwqθT（27）

where u, w, q, and θ are the forward velocity, vertical velocity, pitch rate, and pitch angle, respectively. A longitudinal control vector, ηlong, is given by:
其中 u、w、q 和 θ 分别是前进速度、垂直速度、俯仰速率和俯仰角。纵向控制向量 ηlong 由下式给出：

ηlong = δe δT T (28)
ηlong =δeδTT（28）

δe and δT are perturbations from the trim in the elevator and throttle settings, respec- tively. Xflr5 is used to calculate the stability and control derivatives. Xflr5 predictions can be used for preliminary design analysis. These predictions can be improved using high fidelity tools or wind tunnel testing. The definitions and relationships between the dimen- sional stability derivatives for longitudinal dynamics and dimensionless derivatives of aerodynamic coefficients are presented in [40,41]. The longitudinal derivatives for Design-3 are listed in Table 7. XFLR5 does not compute derivatives with respect to w. . Hence, these are assumed to be zero.
δe 和 δT 是升降舵和油门设置中配平的扰动，具体来说是。Xflr5 用于计算稳定性和控制导数。Xflr5 预测可用于初步设计分析。这些预测可以使用高保真工具或风洞测试来改进。纵向动力学的维稳定性导数与空气动力学系数的无量纲导数之间的定义和关系见[40,41]。表 7 列出了 Design-3 的纵向导数。XFLR5 不计算关于 w 的导数。因此，这些值假定为零。

Table 7. Longitudinal derivatives of aerodynamic coefficients.
表 7.空气动力学系数的纵向导数。

Derivatives Value
衍生品价值

CXu −0.051647

CXα 0.6825

Czu 0.00659

CLα 5.7669

CLq 7.5009
接线盒 7.5009

CMu −0.00684

CMα −1.424

CMq −21.431

The resulting plant matrix is given by:
得到的植物基质由下式给出：

A =
答=

long
长 

 (29)

The eigenvalues of the longitudinal plant matrix A are −11.13 + 4.68i, −11.13 − 4.68i,
纵向植物矩阵 A 的特征值为 −11.13 + 4.68i、−11.13 − 4.68i、

−0.068 + 0.91i, and −0.068 − 0.91i, which corresponds to a short period and the phugoid mode. The short-period mode has a relatively short time period and is generally damped
−0.068 + 0.91i 和 −0.068 − 0.91i，对应于短周期和泡状模式。短周期模式的时间段相对较短，并且通常是阻尼的

substantially. The phugoid mode has a significantly longer time period and is damped marginally. The properties of the short period and phugoid mode of the present system are given in Table 8.
大幅。phugoid 模式的时间段明显更长，并且略微阻尼。表 8 给出了当前系统的短周期和 phugoid 模式的特性。

Table 8. Properties of short period and phugoid mode for Design-3.
表 8.Design-3 的短周期和 phugoid 模式的属性。

Mode Dumping Ratio ζ Natural Frequency ω [s−1] Period T [s]
固有频率 ω ζ模式转储比[s−1]周期 T [s]

Short-Period Mode 0.923 12.07 1.32
短周期模式0.92312.071.32

Phugoid Mode 0.074 0.91 6.93
Phugoid 模式0.0740.916.93

In the attitude and altitude control system design, it is assumed that the forward velocity is controlled by a separate controller and maintained constant at 8.5 m/s. The engine control and actuator dynamics are omitted for simplicity. Control surfaces are modeled as simple trailing edge flaps with x and y hinge position at 70% of chord and 50%
在姿态和高度控制系统设计中，假设前进速度由单独的控制器控制并保持在 8.5 m/s 的恒定值。为简单起见，省略了发动机控制和执行器动力学。控制表面被建模为简单的后缘襟翼，x 和 y 铰链位置位于弦的 70% 和 50% 处

of thickness, respectively. The control surface sizing can be refined using high fidelity tools. The resulting state-space model is expressed as:
的厚度。可以使用高保真工具优化控制表面尺寸。生成的状态空间模型表示为：

∆w.
∆ w.

.

−7.0192 7.1385 0

−3.206

∆q =

−5.4389 −15.30 0

0 1 0

+

−59.29

0

[θδe] (30)
[θδe]（30）

Figure 13a shows the longitudinal open-loop step response. The vertical velocity and pitch rate have non-zero values. The pitch angle continues to decrease dramatically. Therefore, a controller must be designed for the desired pitch angle response. In this study, a linear quadratic regulator (LQR) controller is used to determine the gains for the feedback loop. The goal is to identify a control vector η(t) that drives the state from a specified initial state x(t) to a desired final state xd t f such that a specified performance index (J)
图 13a 显示了纵向开环阶跃响应。垂直速度和音高速率具有非零值。俯仰角继续急剧减小。因此，必须针对所需的俯仰角响应设计控制器。在本研究中，线性二次稳压器 （LQR） 控制器用于确定反馈环路的增益。目标是确定一个控制向量 η（t），该向量将状态从指定的初始状态 x（t） 驱动到所需的最终状态 xd t f，以便指定的性能指数 （J）

is minimized:
最小化：

where g is specified as
其中 g 指定为

J = t f g(x(τ), η(τ), τ)dτ (31)
J =t f g（x（τ）， η（τ）， τ）dτ（31）

t

g = xTQx + ηT Rη (32)
g = xTQx + ηT Rη（32）

where Q is a positive semi-definite weight matrix and R is a positive-definite weight matrix. Q and R are selected to provide the optimal response as desired by the designer. For this design:
其中 Q 是正半定权重矩阵，R 是正定权重矩阵。选择 Q 和 R 以提供设计人员所需的最佳响应。对于此设计：

Q =

0 0 0

0 0 0

0 0 1

and R = [1] (33)
和 R = [1]（33）

The corresponding gain vector is given by:
相应的增益矢量由下式给出：

Klong = 0.0271 −0.0482 −1.0 (34)
Klong = 0.0271−0.0482−1.0（34）

(a) (b)
（一）（二）

Figure 13. Longitudinal step response: (a) Open Loop, (b) Closed Loop.
图 13.纵向阶跃响应：（a） 开环，（b） 闭环。

The feedback response for the attitude-hold control system in response to a 1◦ step input is shown in Figure 13b. The rise and settling time of the pitch angle are 0.784 s and 1.48 s, respectively, with zero steady-state error. As the designed solar UAV must demonstrate exceptional endurance at a constant altitude, an altitude-hold control system is required. To design this system, we introduce a vertical height equation [41] in the
姿态保持控制系统响应 1◦ 步长输入的反馈响应如图 13b 所示。俯仰角的上升和稳定时间分别为 0.784 s 和 1.48 s，稳态误差为零。由于设计的太阳能无人机必须在恒定高度下表现出卓越的续航能力，因此需要一个高度保持控制系统。为了设计这个系统，我们在

current state-space model.
当前状态空间模型。

.

∆h = u0(∆θ − ∆α) (35)
∆h = u0（∆θ − ∆α）（35）

The resulting state space model is given as:
生成的状态空间模型如下：

α −7.019 0.8398 0 0

  −0.377

∆q −46.23 −15.305 0 0

=

+

−59.29

[∆δ ] (36)
[∆δ ]（36）

∆θ 0 1

∆h −8.5 0

0 0 0 e

8.5 0 0
8.5 00 元

The corresponding LQR gains are [0.2308 − 0.0482 − 1.000], which correspond to the state alpha, pitch rate, and pitch angle, respectively. This system is the same as Equation (30), except vertical velocity is replaced by alpha. A PID controller is suggested in the outer loop that would generate a theta command resulting from the difference between the target height and current height. The parameters of the PID controller are P = 0.07025, I = 0.0003309, D = −0.0300, and N = 0.5372. The response to the 1 m step input is shown in Figure 14. The linear pitch angle accurately matches the pitch angle command, and the target height is achieved with zero steady-state error. The maximum elevator deflection to produce desired pitch angle is 1◦.
相应的 LQR 增益为 [0.2308 − 0.0482 − 1.000]，分别对应于状态 alpha、俯仰速率和俯仰角。该系统与方程 （30） 相同，只是垂直速度被 alpha 取代。建议在外部循环中使用 PID 控制器，该控制器将根据目标高度和当前高度之间的差异生成 theta 命令。PID 控制器的参数为 P = 0.07025、I = 0.0003309、D = −0.0300 和 N = 0.5372。对 1 m 步长输入的响应如图 14 所示。线性俯仰角与俯仰角命令精确匹配，并且目标高度为零稳态误差。产生所需俯仰角的最大电梯偏转为 1◦。

Figure 14. Longitudinal outer-loop step response.
图 14.纵向外环阶跃响应。

5.2.Heading Control
5.2.航向控制

The linearized small disturbance lateral rigid body equations of motion in state-space
状态空间中线性化小扰动横向刚体运动方程

form are given by [41]:
形式由 [41] 给出： .

xlat = Alatxlat + Blatηlat (37)
xlat = 阿拉特拉特 + 布拉特 （37）

where
哪里

Yv Yp g0 cos Θ0 Yr − u0

Yδr Yδa

Alat =
阿拉特 =

Lv Lp 0 Lr

0 1 0 0
0100 0

Nv Np 0 Nr

and Blat =
和 Blat =

Lδr Lδa

0 0

Nδr Nδa

 (38)

Lateral state vector xlat and input control vector ηlat are given by:
横向状态向量 xlat 和输入控制向量 ηlat 由下式给出：

xlat = v p φ r T , ηlat = δr δa T (39)
xlat = vpφr T ， ηlat = δrδa T（39）

where v, p, φ and r are the lateral velocity, roll rate, roll angle and yaw rate, respectively. δr and δa are the perturbations from trim in the rudder and aileron. It is also convenient to replace lateral velocity with sideslip angle, β, using the following relationship.
其中 v、p、φ 和 r 分别是横向速度、滚转率、滚转角和偏航率。ΔR 和 ΔA 是方向舵和副翼中 TRIM 的扰动。使用以下关系式将横向速度替换为侧滑角 β 也很方便。

β ∼= tan β = v
β ∼= 棕褐色 β = v

V0
V0 系列

(40)

The resulting plant matrix is given by [42]:
得到的植物基质由下式给出 [42]：

Alat =
阿拉特 =

Yv Yp/V0 (g0 cos Θ0)/V0 (Yr − u0)/V0
YvYp/V0（g0 cos Θ0）/V0（Yr − u0）/V0

LvV0 Lp 0 Lr

0 1 0 0
0100 0

NvV0 Np 0 Nr

 (41)

The dynamics of the resulting system are the same as those of the original system except that the state vector is now β p φ r T. For heading angle control, both skid to turn (STT) and bank to turn can be implemented. In STT, rudders are the primary
结果系统的动力学与原始系统的动力学相同，只是状态向量现在β p φ r T。对于航向角控制，可以实现滑移转弯 （STT） 和倾斜转弯。在 STT 中，方向舵是主要的

control surface. Rudders are used to create sideslip and aircraft turns in the direction of
控制表面。方向舵用于产生侧滑和飞机转向

rudder deflection. Ailerons can be used to encounter roll produced due to yaw and to keep the wings level if desired. In BTT, ailerons are the primary control surface. Ailerons are deflected to bank the aircraft into a turn. In the current study, a BTT maneuver is modeled. Lateral aerodynamic derivatives estimated using Xflr5 are listed in Table 9.
方向舵偏转。副翼可用于应对由于偏航而产生的滚动，并在需要时保持机翼水平。在 BTT 中，副翼是主要的控制面。副翼偏转，使飞机倾斜转弯。在目前的研究中，对 BTT 机动进行了建模。表 9 列出了使用 Xflr5 估计的横向空气动力学导数。

Table 9. Lateral derivatives of aerodynamic coefficients.
表 9.空气动力学系数的横向导数。

Derivatives Value
衍生品价值

CYβ −0.2339
碳β−0.2339

CY p −0.2156
CY p-0.2156

CYr 0.1841
塞浦路斯0.1841

Cl β −0.1193

Cl p −0.7429
Cl p-0.7429

Clr 0.2709

Cnβ 0.0277
碳β0.0277

Cn p −0.1431
中国 p−0.1431

Cnr −0.0231
CNR - 0.0231

The lateral dynamics are complex as compared to longitudinal dynamics, involv- ing one force and two moments. Eigenvalues of lateral plant matrix Alat are −26.09,
与纵向动力学相比，横向动力学很复杂，涉及一个力和两个力矩。横向植物基质 Alat 的特征值为 −26.09，

−1.47 + 2.12i, −1.47 − 2.12i and 0.116, which corresponds to roll subsidence, Dutch roll
−1.47 + 2.12i、−1.47 − 2.12i 和 0.116，对应于滚动沉降、荷兰滚动

and spiral mode, respectively. The positive value of the spiral mode indicates that it is
和 spiral 模式。螺旋模式的正值表示它是

unstable. Before designing a linear BTT control system, we must first add the BTT equation to our system. The BTT equation is given by:
稳定。在设计线性 BTT 控制系统之前，我们必须首先将 BTT 方程添加到我们的系统中。BTT 方程由下式给出：

r ≈ g φ (42)
R ≈ 克φ（42）

V0
V0 系列

It states that any non-zero roll angle will induce a yaw rate. This equation can
它指出，任何非零滚动角都会引起偏航率。这个方程式可以

.

be incorporated into state-space using ψ = r. After augmenting the BTT equation in
使用 ψ = R 合并到状态空间中。在

Equation (41), the plant matrix and control matrix are given by:
方程 （41），植物矩阵和控制矩阵由下式给出：

−0.2831 −0.0896 1.1541 −0.9235 0

−13.0276 −27.8583 0 10.1602 0

0.1696

50.0247

Alat =
阿拉特 =

0 1.0000 0 0 0
01.000000 00

2.7068 −4.8078 0 −0.7767 0

0 0 1.1529 0 0

and Blat =
和 Blat =

0

−0.0064

0

(43)

Like longitudinal control, LQR is used to model the BTT maneuver. Q and R matrix
与纵向控制一样，LQR 用于模拟 BTT 机动。Q 和 R 矩阵

are:
是：

Q =

10 0 0 0 0
100 00 0 0

0 80 0 0 0
08000 0 0

0 0 30 0 0
0030 0 0 0

0 0 0 5 0
000 50

0 0 0 0 8
000 08

and R = [1] (44)
和 R = [1]（44）

Weights in the Q matrix are selected to obtain the optimum response for all the states. The corresponding LQR gains are Klat= [1.4579 8.4355 11.8906 −0.9540 2.8284]. The poles of the closed loop system (Alat − BlatKlat) are −448.2, −1.03 + 1.6i, 1.03 − 1.6i, −0.45 + 0.28i and −0.45 − 0.28i. All the roots are now negative; hence, the system is now stable. The
选择 Q 矩阵中的权重以获得所有状态的最佳响应。相应的 LQR 增益为 Klat= [1.4579， 8.4355， 11.8906 −0.9540， 2.8284]。闭环系统 （Alat − BlatKlat） 的极点为 −448.2、−1.03 + 1.6i、1.03 − 1.6i、−0.45 + 0.28i 和 −0.45 − 0.28i。现在所有的根都是负的;因此，该系统现在是稳定的。这

response to a 110◦ turn is shown in Figure 15. The yaw rate is limited to 2◦/s.
对 110◦ 转弯的响应如图 15 所示。偏航角速率限制为 2◦/s。

Figure 15. Heading angle control response for 110◦ reference.
图 15.航向角控制响应为 110◦ 参考。

5.3.Six-DOF Non-Linear Simulation
5.3.六自由度非线性仿真

The proposed control strategy is validated in the 6-DOF Matlab Simulink model. The aerodynamic model used for this simulation is presented below.
所提出的控制策略在 6-DOF Matlab Simulink 模型中进行了验证。用于该模拟的空气动力学模型如下所示。

L = CLqS where CL = CL
L = CLqS，其中 CL = CL

c

+ CL α + CL q + CL δe

0 α 2V q δe

D = CDqS where CD = CD0 + CDα α + CDδe δe
D = CDqS，其中 CD = CD0 + CDα α + CDδe δe

MA = CmqSc where Cm = Cm
MA = CmqSc，其中 Cm = Cm

c

+ Cm α + Cm q + Cm δe
+ 厘米 α +厘米 q + 厘米 δe

0 α 2V q δe

LA = ClqSb where Cl = Cl
LA = ClqSb，其中 Cl = Cl

b   b
哈必 斯

+ Cl β + Cl p + Cl r + Cl δa + Cl δr

0 β 2V p 2V r δa δr
0β2Vp2Vrδa δr

NA = CnqSb where Cn = Cn
NA = CnqSb，其中 Cn = Cn

+ Cn β + b Cn p + b Cn r + Cn δa + Cn δr

0 β 2V p 2V r δa δr
0β2Vp2Vrδa δr

Y = CYqS where CY = CY
Y = CYqS，其中 CY = CY

b   b
哈必 斯

+ CY β + CY p + CY r + CY

δa + CY δr

0 β 2V p 2V r δa δr
0β2Vp2Vrδa δr

where L, D, MA, LA, NA, Y, q and c are lift, drag, pitching moment, rolling moment, yawing moment, side force, dynamic pressure and mean aerodynamic chord, respectively. CL0 , CD0 , Cm0 , Cl0 , Cn0 and CY0 are lift, drag, pitching, rolling, yawing and side force coefficient at α = δe = β = δa = δr = 0, respectively. The final trajectory is to climb to an altitude of
其中 L、D、MA、LA、NA、Y、q 和 c 分别是升力、阻力、俯仰力矩、滚动力矩、偏航力矩、侧向力、动压和平均空气动力学弦。CL0 、 CD0 、 Cm0 、 Cl0 、 Cn0 和 CY0 分别是 α = δe = β = δa = δr = 0 处的升力、阻力、俯仰、滚动、偏航和侧向力系数。最后的轨迹是爬升到

350 m at the rate of 2 m/s, after being hand-launched from an initial altitude of 0 m. After climbing to 350 m, there is a small cruise segment. At t = 250 s, the UAV starts a climbing turn, climbing with a rate of 1 m/s and turning with a yaw rate of 2◦/s. After climbing to 700 m, the UAV holds this altitude and continues to fly in a circular path with the same yaw rate. As noted in Figure 15, the BTT maneuver also produces a sideslip angle. In the 6-DOF simulation, the rudder is used to encounter the sideslip angle using a proportional controller, often known as a coordinated turn. Throughout the trajectory, the velocity of
从 0 m 的初始高度手动发射后，以 2 m/s 的速率飞行 350 m。爬升到 350 m 后，有一个小的游轮部分。在 t = 250 s 时，无人机开始爬升转弯，以 1 m/s 的速率爬升，以 2◦/s 的偏航速率转弯。爬升到 700 m 后，无人机保持这个高度，并以相同的偏航率继续沿圆形路径飞行。如图 15 所示，BTT 机动也会产生侧滑角。在 6-DOF 仿真中，方向舵用于使用比例控制器遇到侧滑角，通常称为协调转弯。在整个轨迹中，速度

8.5 m/s is maintained. To account for any expected sensor noises, the Matlab Simulink built-in band limited white noise model is used (Figure 16).
保持 8.5 m/s。为了考虑任何预期的传感器噪声，使用了 Matlab Simulink 内置的频带限制白噪声模型（图 16）。

Figure 16. Block diagram of current control system and implementation of noise.
图 16.电流控制系统和噪声实施的框图。

The angle of attack and sideslip angle are shown in Figure 17a. At the final cruise altitude of 700 m, alpha is around 3.2◦, which is consistent with static analysis and design. Sideslip is zero throughout the trajectory. In Figure 17b, x and y coordinates are plotted to visualize the circular motion. For the given yaw rate of 2◦/s, the UAV flies in a circular path with 480 m diameter.
攻角和侧滑角如图 17a 所示。在 700 m 的最终巡航高度时，alpha 约为 3.2◦，这与静态分析和设计一致。Sideslip 在整个轨迹中为零。在图 17b 中，绘制了 x 和 y 坐标以可视化圆周运动。对于给定的 2◦/s 偏航率，无人机沿直径为 480 m 的圆形路径飞行。

(a) (b)
（一）（二）

Figure 17. Simulation results: (a) Alpha and Beta, (b) Circular Motion.
图 17.仿真结果：（a） Alpha 和 Beta，（b） 圆周运动。

The height profile of the solar UAV with axial location is also shown in Figure 18a. After reaching the target height of 700 m, the UAV successfully maintains this altitude. Euler angles are also presented in Figure 18b. During climbing turn, the pitch angle is around 10◦. After achieving a target height, the pitch angle is reduced to 3.2◦, which is
具有轴向位置的太阳能无人机的高度剖面也如图 18a 所示。在达到 700 m 的目标高度后，无人机成功保持了这一高度。图 18b 中也显示了欧拉角。在爬坡转弯时，俯仰角约为 10◦。达到目标高度后，俯仰角减小到 3.2◦，即

equal to the angle of attack. Hence, the solar UAV is flying with zero flight path angle. For the yaw angle, the range of the plot is −180◦ to 180◦ (one complete circle). To elaborate, the trajectory of the solar UAV is shown in a 3D plot (Figure 19). Different segments of the
等于攻角。因此，太阳能无人机以零飞行路径角飞行。对于偏航角，绘图的范围是 −180◦ 到 180◦（一个完整的圆）。详细地说，太阳能无人机的轨迹显示在 3D 图中（图 19）。的不同部分

trajectory are shown with different colors.
轨迹以不同的颜色显示。

(a) (b)
（一）（二）

Figure 18. Simulation results: (a) Height Profile, (b) Euler Angles.
图 18.仿真结果：（a） 高度剖面，（b） 欧拉角。

Figure 19. Solar UAV trajectory in 3D.
图 19.3D 太阳能无人机轨迹。

6.Conclusions
6.结论

In this paper, the genetic algorithm is used to optimize a solar-powered UAV for a specific height, cruise speed and static margin. The wing airfoil is also considered as a design variable and a new structural mass estimation model is proposed. This model also caters for the change in structural mass with the change in wing airfoil during optimization. The objective is to design an airfoil for specified flight conditions, considering overall performance and weight estimation. To add robustness in the design, an extra battery is added as the system starts to consume the battery as soon as the required power is lower than the available solar power. The optimized configuration has a total mass of 7.08 kg with a battery mass of 3.4 kg. The optimized airfoil is very smooth, with a thickness and camber of 9.7% and 4.6%, respectively. To ensure control and handling, dynamic stability and control systems are also discussed using linearized equations of motion. The control laws are developed using an LQR and PID controller. The designed altitude and BTT controllers are implemented in a closed-loop nonlinear 6-DOF simulation. The results of the 6-DOF simulation are in good agreement with linear static analysis, validating the complete design process.
在本文中，遗传算法用于优化太阳能无人机的特定高度、巡航速度和静态裕度。机翼翼型也被考虑为一个设计变量，并提出了一种新的结构质量估计模型。该模型还考虑了优化过程中机翼翼型变化引起的结构质量变化。目标是为指定的飞行条件设计翼型，同时考虑整体性能和重量估计。为了提高设计的稳健性，当所需功率低于可用的太阳能功率时，系统就会开始消耗电池，从而添加额外的电池。优化后的配置总质量为 7.08 kg，电池质量为 3.4 kg。优化后的翼型非常光滑，厚度和外倾角分别为 9.7% 和 4.6%。为了确保控制和处理，还使用线性化运动方程讨论了动态稳定性和控制系统。控制律是使用 LQR 和 PID 控制器开发的。设计的高度和 BTT 控制器在闭环非线性 6-DOF 仿真中实现。6-DOF 仿真的结果与线性静态分析非常一致，验证了完整的设计过程。

Author Contributions: Conceptualization and methodology, A.B.; software, H.L.; validation, Z.X.; writing—review and editing, A.B.; supervision, L.K.; project administration, D.W.; funding acquisi- tion, L.K. All authors have read and agreed to the published version of the manuscript.
作者贡献： 概念化和方法论，A.B.;软件，H.L.;验证，Z.X.;写作——审查和编辑，AB;监督，L.K.;项目管理，DW;资金收购，L.K.所有作者均已阅读并同意手稿的已发表版本。

Funding: This work is supported by the Chinese National Natural Science Foundation (No. 61773039) and Aeronautical Science Foundation of China (No. 2017X51018).
资金： 这项工作由中国国家自然科学基金（No. 61773039）和中国航空科学基金（No. 2017X51018）支持。

Institutional Review Board Statement: Not applicable.
机构审查委员会声明：不适用。

Informed Consent Statement: Not applicable.
知情同意书：不适用。

Data Availability Statement: Not applicable.
数据可用性声明：不适用。

Conflicts of Interest: The authors declare no conflict of interest.
利益冲突：作者声明没有利益冲突。

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