﻿ 悬停状态下小型无人直升机飞行动力学模型辨识<sup>*</sup>
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Identification of flight dynamics models of a small-scale unmanned helicopter in hover condition
WU Meiliwen, CHEN Ming, WANG Fang
School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China
Received: 2018-06-21; Accepted: 2018-09-19; Published online: 2018-10-17 15:16
Corresponding author. CHEN Ming, E-mail:chenming@buaa.edu.cn
Abstract: In order to better study the hover dynamics characteristics of small-scale unmanned helicopter, the in-depth dynamics model analysis of linear system identification and nonlinear modeling was conducted in this paper on an 8.1 kg electric helicopter with 3-axis gyro augmentation. In the linear system identification procedure, frequency-domain identification method was adopted. Double systems were obtained by using command signals from both before and after the gyro part. In the nonlinear modeling procedure, body dynamics, rotor dynamics, and tail rotor dynamics were modeled separately. The tail rotor dynamics utilized 3-stage identification method to extract the base model, the gyro model, and the overall model. A nonlinear-linear combined modification method was decided for improving the models' performance. The results show that the 13-state high-order model has higher simulation accuracy compared with the 11-state model. The flight data of the helicopter's base model for dual system linear model has high quality in high-frequency domain, and the maximum frequency is 30 rad/s. Apart from the flapping equation parameters and tail rotor parameters, the combined modification method got 7 parameters of the nonlinear mathematical model (NMM) corrected, which fits the experimental hover data effectively.
Keywords: frequency response     small-scale unmanned helicopter     3-axis gyro     nonlinear analysis     system identification

1) 总是希望获得简单却又精确的动力学模型。为了获得这个精确的动力学模型，最先可以想到的就是通过首要物理原则进行非线性建模。但是小型无人直升机由于低雷诺数的特点，详细的非线性建模无法保证在全部飞行包线内获得准确的估计。而且小型无人直升机不如全尺寸直升机数据丰富[10]，建模具有挑战性。因此获取简单而局部精确的线性模型通常是较好的解决办法。多数研究获得的模型阶次不大于13阶[3]，其中包含刚体机体动力学、旋翼挥舞动力学和稳定杆动力学等。以往的研究对于携带伺服小翼的机械增稳结构进行了动力学探讨，但是研究没有涉及携带三轴陀螺仪的控制增稳模型。

2) 总是通过借助实验数据来提取该飞行状态下精确的模型。在这个过程中，系统辨识方法占据重要的地位。系统辨识分为频域辨识方法和时域辨识方法。时域辨识方法有最大似然估计和子空间方法等，这些方法在不同的领域各有优点，如子空间方法在处理多输入多输出问题上具有优势[11-12]，而最大似然法[13]应用于非线性数学模型(Nonlinear Mathematical Model，NMM)有优势。频域辨识方法则通过对比拟合飞行数据在频域上的信息来获取最优的模型。在频域领域测量噪声和过程噪声是没有估计误差的[14]，这让闭环辨识问题变得简单，而且频域信息的分析更能帮助如鲁棒控制方法的设计[15-16]。CIFER软件是公认的具有高可靠性的频域辨识工具[1, 2, 14, 17]

1 非线性飞行动力学建模

 图 1 JR700直升机系统 Fig. 1 JR700 helicopter system

 参数 JR700 Raptor 50[8] X-cell[3] HeLion[7] 质量/kg 8.1 4.8 8.15 9.75 旋翼实度 0.052 0.05 0.05 0.055 桨叶转动惯量/(kg·m2) 0.04 0.035 0.02 0.055 主旋翼直径/cm 153.4 134.37 152.4 141 旋翼转速/(rad·s-1) 178 191 167 193.73

1.1 刚性机体动力学

1.1.1 直接测量法

1.1.2 地面物理实验

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1.2 旋翼动力学

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1.3 尾桨动力学

 图 2 尾桨动力学结构 Fig. 2 Tail rotor dynamics configuration

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2 双系统线性模型辨识和模型修正方法 2.1 双系统线性模型辨识

 图 3 信号系统示意图 Fig. 3 Signal system illustration

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2.2 非线性-线性模型结合修正方法

Model A和Model B状态变量分别为

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Model A状态空间模型为

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1) ZwZδcol

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2) MaLb

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3) XuYv

x方向的受力由2个部分组成：主旋翼受力和机体受力。

x方向速度导数为

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y方向受力由3方面决定，则速度导数为

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3 辨识结果分析与实验验证 3.1 模型结构验证与分析

 参数 Model A Model B 数值 CR(Cramer-Rao)/% 敏感度/% 数值 CR(Cramer-Rao)/% 敏感度/% τf 0.078 41 10.01 0.895 9 0.055 95 8.32 1.066 Ma 195.10 11.54 2.248 440.4 6.59 2.656 Lb 224.12 20.05 2.010 740.2 5.765 2.521 Ba 0.275 6 10.44 2.241 0.252 2 17.47 8.066 Ab 0.275 6 0.252 2 Aδlat 0.034 64 11.88 2.444 0.083 12 9.017 2.165 Aδlon 0.386 3 8.710 1.693 0.446 2 8.122 1.962 Bδlat 0.407 4 9.563 1.538 0.508 9 7.983 1.859 Bδlon -0.042 43 14.75 3.415 -0.093 63 9.941 2.373 价值函数 107.092 29.686 5

1) 如表 2所示，相对比较Model B来说，Model A的价值函数过高，说明辨识结果拟合实验数据的效果一般。Model A的价值参数很难降低，一种假设是因为Model A的模型结构并不合适，无法匹配真实的动力学模型，所以辨识精度有限。在Model A中，动力学模型视直升机和三轴陀螺仪部件为一整体，意味着辨识出来的各项导数是考虑陀螺仪之后的有偏移的数值。因为陀螺仪有自己的角速率反馈控制，所以全机模型的阶次应当比单纯直升机挥舞模型的阶次要高。挥舞模型引入的是一阶模型，如考虑增加陀螺仪的一阶反馈，整体挥舞模型应为二阶。

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2) 实验中飞行数据的频域信息展现在图 4图 5中(实线)，由图中可以看出，Model B的相干值参数要高于Model A，特别是在高频领域。在Model A中，专业操纵手飞行的辨识数据，在高频区域，相干值参数0.6以上的频率最高为13 rad/s，而在Model B中，高质量频域数据可以达到30 rad/s并保持较高的相干值参数。飞行数据的质量，不管是主轴响应还是轴外响应，比文献[8]中都要好。也就是说，三轴陀螺仪在其中起到了增加飞行数据有效带宽的重要作用。飞行数据频率范围越宽，会包含更多的动力学信息，那么也会让辨识的模型更加贴近真实情况。同样，飞行数据中越高的高频特性也会让高阶模型的辨识变得有意义，因为高阶模型的细节信息主要存在于高频区域，应用高阶模型来辨识可以获得更高的高频贴合度，使模型在更宽的频域范围内更加贴近于真实的动力学特性。因此结合问题一的解决方案，这里引入二阶挥舞方程，来同时解决2个问题。忽略陀螺仪的真实反馈参数，而用二阶挥舞方程参数来代替陀螺仪在Model A中引起的二阶效应。配置二阶挥舞项b1= ，应用文献[26]中的挥舞方程，并在悬停状态下，忽略直升机桨叶的锥度角效应，形成以下的6自由度耦合纵横向状态方程：

 图 4 Model A 13阶高阶模型频域辨识结果与悬停实验数据对比 Fig. 4 Comparison of frequency-domain identification results of 13-state high-order model with experimental hover data for Model A
 图 5 Model B 13阶高阶模型频域辨识结果与悬停实验数据对比 Fig. 5 Comparison of frequency-domain identification results of 13-state high-order model with experimental hover data for Model B
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3.2 双系统高阶模型辨识结果

 参数 Model A Model B 数值 CR/% 敏感度/% 数值 CR/% 敏感度/% τf 0.071 30 7.425 0.681 8 0.055 60 8.204 0.856 0 Xu -0.060 20c -0.060 20c Xa -9.8a -9.8a Yv -0.142 0c -0.142 0c Yb 9.8a 9.8a Zw -1.714 12.09 5.473 -1.714 12.67 5.795 Mu 0.037 10a 0.045 80a Mv -0.001 62a -0.004 42a Ma 191.3 6.422 1.714 448.4 7.465 3.006 Lu -0.002 21a -0.002 81a Lv -0.102 2a -0.158 0a Lb 237.2 6.753 1.735 740.9 5.905 2.530 Nr 0.409 3 19.85 5.892 -1.084 28.2 13.47 Kxr -403.70 21.23 1.779 Kxx -20.47 15.36 1.965 Ba -0.426 6 4.567 1.425 0.208 6 20.03 9.650 Ab -0.426 6 0.208 6 Aδlat 0.025 47 7.875 2.553 -0.032 70 24.62 5.610 Aδlon -0.343 2 6.999 1.430 -0.456 8 7.78 1.78 Bδlat 0.391 3 7.242 1.094 0.504 6 8.011 1.655 Bδlon -0.021 31 23.06 7.520 -0.053 30 18.46 4.161 Zδcol -45.87 4.251 1.925 -45.87 4.253 1.944 Nδcol -10.58 19.93 9.765 0b Nδped 176.01 11.47 1.874 75.28 2.844 3.778 价值函数 49.098 9 39.907 8 注：上标a表示理论值；b表示由模型结构考虑移除的参数；c表示单独辨识参数并在模型中固定不变。

 图 6 Model A 13阶高阶模型时域仿真结果与悬停实验数据对比 Fig. 6 Comparison of time-domain simulation results of 13-state high-order model with experimental hover data for Model A
 图 7 Model B 13阶高阶模型时域仿真结果与悬停实验数据对比 Fig. 7 Comparison of time-domain simulation results of 13-state high-order model with experimental hover data for Model B

 参数 数值 a0 5.75 Kcol 0.293 2 Kβ 160.57 Ixx 0.396 Iyy 0.653 ΔXu -0.041 ΔYv -0.048 5

 参数 JR700 13阶 Raptor 50 X-cell HeLion Model A Model B τf 0.071 30 0.055 60 0.043 0.052 τs+τf 0.127 0.272 0.299 Lb 237.2 740.9 735.5 320 583.5 Ma 191.3 448.4 228 204 265.3 Aδlat 0.025 47 -0.032 70 0.008 9a 0 0 Aδlon -0.343 2 -0.456 8 -0.242 2a -0.53a -0.42a Bδlat 0.391 3 0.504 6 0.031 5a 0.42a 0.4a Bδlon -0.021 31 -0.053 30 -0.011 2a 0 0 注：上标a表示原始数值[3, 7-8]按本文单位标准换算。

 模态 13阶Model A 13阶Model B 俯仰耦合模态 [0.673 7, 14.666 0] [0.426 1, 21.621 4] 滚转耦合模态 [0.285 9, 14.593 5] [0.328 8, 26.942 2] 航向模态 [0.504 5, 19.883 2] (-1.084 0) 垂向模态 (-1.713 6) (-1.713 6) 纵向速度模态 [0.257 8, 0.139 7] [0.357 6, 0.192 2] 横向速度模态 [0.240 8, 0.256 6] [0.239 0, 0.131 9]

4 结论

1) 在辨识实验中同时记录陀螺仪之前和之后的操纵信号，分别作为操纵输入，进行频域系统辨识分析，可以同时得到双系统线性模型。通过数据得知，一阶挥舞方程不适用于Model A动力学特性，陀螺仪的反馈作用使挥舞动力学的高阶效应更加明显。而Model B的飞行数据具有更高质量的高频数据，利用Model B飞行数据辨识直升机基底模型有利于提高模型高频区域的精确度。

2) 尾桨动力学的辨识应用三阶段辨识法。第1阶段为基底模型，可以通过航向辨识实验得到；第2阶段为陀螺仪模型，应用PI反馈假设，可以通过地面扫频实验得到；第3阶段为整体增稳模型。三阶段辨识法获得辨识结果有很高的频域辨识精度，时域验证中基底模型和整体模型均能很好地仿真拟合实验数据。

3) 除了线性模型以外，同时进行非线性建模，并利用非线性-线性模型结合修正方法提高了线性模型和NMM的精度。除了挥舞动力学方程及航向动力学以外，NMM修正7个非线性参数，总体来讲修正后NMM在悬停状态仿真效果表现优异。

4) 在频域辨识过程中，增加二阶挥舞项的13阶次双系统线性模型相比11阶次双系统线性模型有更好的辨识效果，辨识结果价值函数更低，模型可靠性更高，高阶频域贴合程度更好。模型具有2个主要模态，俯仰耦合模态和滚转耦合模态，符合同种类型直升机特点。相比Model A和Model B机动灵活性更高。

 [1] METTLER B, TISCHLER M B, KANADE T. System identification modeling of a small-scale unmanned rotorcraft for flight control design[J]. Journal of the American Helicopter Society, 2002, 47(1): 50-63. DOI:10.4050/JAHS.47.50 [2] 杨帆, 熊笑, 陈宗基, 等. 超小型直升机动力学模型的建模、辨识与验证[J]. 北京航空航天大学学报, 2010, 36(8): 913-917. YANG F, XIONG X, CHEN Z J, et al. Modeling, system identification and validation of small rotorcraft-based unmanned aerial vehicle[J]. Journal of Beijing University of Aeronautics and Astronautics, 2010, 36(8): 913-917. (in Chinese) [3] METTLER B. Identification modeling and characteristics of miniature rotorcraft[M]. New York: Kluwer Academic Publishers, 2002: 53-90. [4] GAVRILETS V, METTLER B, FERON E.Nonlinear model for a small-size acrobatic helicopter[C]//AIAA Guidance, Navigation, and Control Conference and Exhibit.Reston: AIAA: 2001: 4333. [5] GAVRILETS V, MARTINOS I, METTLER B, et al.Control logic for automated aerobatic flight of a miniature helicopter[C]//AIAA Guidance, Navigation, and Control Conference and Exhibit.Reston: AIAA, 2002: 4834. [6] KHALIGH S P, FAHIMI F, ROBERT K C. A system identification strategy for nonlinear model of small-scale unmanned helicopters[J]. Journal of the American Helicopter Society, 2016, 61(4): 1-13. [7] CAI G, CHEN B M, LEE T H, et al.Comprehensive nonlinear modeling of an unmanned-aerial-vehicle helicopter[C]//AIAA Guidance, Navigation, and Control Conference and Exhibit.Reston: AIAA, 2008: 7414. [8] BHANDARI S, COLGREN R. High-order dynamics models of a small UAV helicopter using analytical and parameter identification techniques[J]. Journal of the American Helicopter Society, 2015, 60(2): 1-10. [9] GRAUER J, CONROY J, HUBBARD J, et al. System identification of a miniature helicopter[J]. Journal of Aircraft, 2009, 46(4): 1260-1269. DOI:10.2514/1.40561 [10] KAPILTA T T, DRISCOLL J T, DIFTLER M A, et al.Helicopter simulation development by correlation with frequency sweep flight test data[C]//American Helicopter Society 45th Annual Forum Proceedings.Boston, MA: American Helicopter Society (AHS), 1989: 631-643. [11] PANIZZA P, RICCARDI F, LOVERA M. Black-box and grey-box identification of the attitude dynamics for a variable-pitch quadrotor[J]. IFAC-PapersOnLine, 2015, 48(9): 61-66. DOI:10.1016/j.ifacol.2015.08.060 [12] BERGAMASCO M, LOVERA M.Rotorcraft system identification: An integrated time-frequency domain approach[C]//Advances in Aerospace Guidance, Navigation and Control.Berlin: Springer, 2013: 161-181. [13] DE MENDONCA C B. Flight vehicle system identification:A time-domain methodology[J]. Journal of Guidance, Control, and Dynamics, 2016, 39(3): 737-738. [14] TISCHLER M B, REMPLE R K. Aircraft and rotorcraft system identification[M]. Reston: AIAA, 2006: 321-491. [15] LA CIVITA M, PAPAGEORGIOU G, MESSNER W C, et al. Design and flight testing of an H∞ controller for a robotic helicopter[J]. Journal of Guidance, Control, and Dynamics, 2006, 29(2): 485-494. [16] 李继广, 陈欣, 李亚娟, 等. 飞翼无人机机动飞行非线性鲁棒控制方法[J]. 北京航空航天大学学报, 2018, 44(1): 89-98. LI J G, CHEN X, LI Y J, et al. Nonlinear robust control method for maneuver flight of flying wing UAV[J]. Journal of Beijing University of Aeronautics and Astronautics, 2018, 44(1): 89-98. (in Chinese) [17] TISCHLER M, CAUFFMAN M.Comprehensive identification from frequency responses, Vol.2: User's manual: NASA CP 10150[R].Washington, D.C.: NASA, 1994. [18] WU W. Identification method for helicopter flight dynamics modeling with rotor degrees of freedom[J]. Chinese Journal of Aeronautics, 2014, 27(6): 1363-1372. DOI:10.1016/j.cja.2014.10.002 [19] LA CIVITA M.Integrated modeling and robust control for full-envelope flight of robotic helicopters[D].Ann Arbor, MI: Carnegie Mellon University, 2002: 21-98. [20] TANG S, ZHENG Z, QIAN S, et al. Nonlinear system identification of a small-scale unmanned helicopter[J]. Control Engineering Practice, 2014, 25: 1-15. DOI:10.1016/j.conengprac.2013.12.004 [21] PADFIELD G D. Helicopter flight dynamics[M]. Oxford: Blackwell Publishing, 2008: 87-184. [22] HEFFLEY R K, MNICH M A.Minimum-complexity helicopter simulation math model: NASA-11665[R].Washington, D.C.: NASA, 1988. [23] CAI G, CHEN B M, LEE T H. Unmanned rotorcraft systems[M]. Berlin: Springer, 2011: 70-111. [24] LEISHMAN G J. Principles of helicopter aerodynamics[M]. Cambridge: Cambridge University Press, 2006: 115-169. [25] CAI G, CHEN B M, PENG K, et al. Modeling and control of the yaw channel of a UAV helicopter[J]. IEEE Transactions on Industrial Electronics, 2008, 55(9): 3426-3434. DOI:10.1109/TIE.2008.926780 [26] CHEN R T N.Effects of primary rotor parameters on flapping dynamics: NASA-1431[R].Washington, D.C.: NASA, 1980.

#### 文章信息

WU Meiliwen, CHEN Ming, WANG Fang

Identification of flight dynamics models of a small-scale unmanned helicopter in hover condition

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(3): 546-559
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0384