﻿ 小型无人机总体参数对动态滑翔能力的影响
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 哈尔滨工程大学学报  2020, Vol. 41 Issue (6): 817-823  DOI: 10.11990/jheu.201901024 0

### 引用本文

LIU Siqi, BAI Junqiang. Influence of small UAV parameters on dynamic soaring ability[J]. Journal of Harbin Engineering University, 2020, 41(6): 817-823. DOI: 10.11990/jheu.201901024.

### 文章历史

Influence of small UAV parameters on dynamic soaring ability
LIU Siqi , BAI Junqiang
School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China
Abstract: Dynamic soaring is a technology used to improve the endurance performance of small unmanned aerial vehicles (UAVs) by utilizing wind energy. The research on the influence of small UAVs' overall parameters on its soaring ability can reduce the energy consumption of UAVs during soaring. In this paper, a dynamic soaring trajectory optimization model of small UAVs is established based on the Runge-Kutta integration method. Considering the solution characteristics of trajectory recursive algorithms, further simplification is implemented to effectively reduce the number of variables and constraints in the optimization design. The Runge-Kutta integration optimization model is used to study the influence of some of the aircraft parameters on the dynamic soaring ability. The study results show that the improved Runge-Kutta integration optimization model can realize the objective of dynamic soaring trajectory optimization. Moreover, aircrafts with less weight, small wing load, large-enough maximum lift coefficient, and appropriate aspect ratio have a good dynamic soaring ability, and a large power law exponent provides a relatively favorable wind field for dynamic soaring.
Keywords: dynamic soaring    effortless flight    Runge-Kutta integration method    trajectory optimization    aircraft parameters    flying control

1 动态滑翔飞行建模 1.1 风场建模

 ${V_W} = {V_R}{\left( {\frac{h}{{{H_R}}}} \right)^p}$ (1)

 Download: 图 1 不同p值下的梯度风场模型 Fig. 1 Wind field model with different p

1.2 动态滑翔原理介绍

 $m\dot V = - D - mg{\rm{sin}}{\kern 1pt} {\kern 1pt} \gamma - m{\dot V_W}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi$ (2)

 $E = mgh + \frac{1}{2}m{V^2}$ (3)

 $\dot e = {\left( {\frac{E}{{mg}}} \right)^\prime } = \dot h + \frac{1}{g}V\dot V$ (4)

 ${\dot h = V{\rm{sin}}{\kern 1pt} {\kern 1pt} \gamma }$ (5)
 ${{{\dot V}_W} = \frac{{{\rm{d}}{V_W}}}{{{\rm{d}}h}}\dot h}$ (6)

 $\dot e = - \frac{{{\rm{d}}{V_W}}}{{{\rm{d}}h}}\frac{1}{g}{V^2}{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi - \frac{{DV}}{{mg}}$ (7)

1.3 飞行动力学建模

 Download: 图 3 动力学模型角度及坐标定义 Fig. 3 Forces acting on UAV and angles

 $\left\{ {\begin{array}{*{20}{l}} {m\dot V = - D - mg{\rm{sin}}{\kern 1pt} {\kern 1pt} \gamma - m{{\dot V}_W}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi }\\ {mV{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma \dot \psi = L{\rm{sin}}{\kern 1pt} {\kern 1pt} \mu - m{{\dot V}_W}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi }\\ {mV\dot \gamma = L{\rm{cos}}{\kern 1pt} {\kern 1pt} \mu + m{{\dot V}_W}{\rm{sin}}{\kern 1pt} {\kern 1pt} \gamma {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi - mg{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma }\\ {\dot h = V{\rm{sin}}{\kern 1pt} {\kern 1pt} \gamma }\\ {\dot x = V{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi + {V_W}}\\ {\dot y = V{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi } \end{array}} \right.$ (8)

 $\left\{ {\begin{array}{*{20}{l}} {L = \rho {S_w}{C_L}{V^2}/2}\\ {D = \rho {S_w}{C_D}{V^2}/2}\\ {{C_D} = {C_{D0}} + {K_D}C_L^2} \end{array}} \right.$ (9)

2 动态滑翔航迹优化方法 2.1 优化模型

 $\mathit{\boldsymbol{\dot x}}(t) = f(\mathit{\boldsymbol{x}}(t),\mathit{\boldsymbol{u}}(t))$ (10)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{X}}_0} = ({x_{1,1}}, \cdots ,{x_{6,1}}, \cdots ,{x_{1,M}}, \cdots ,{x_{6,M}},}\\ {{u_{1,1}},{u_{2,1}}, \cdots ,{u_{1,M}},{u_{2,M}},{t_f},{V_R}) = }\\ {(\mathit{\boldsymbol{x}}_1^{\rm{T}},\mathit{\boldsymbol{x}}_2^{\rm{T}}, \cdots ,\mathit{\boldsymbol{x}}_M^{\rm{T}},\mathit{\boldsymbol{u}}_1^{\rm{T}},\mathit{\boldsymbol{u}}_2^{\rm{T}}, \cdots ,\mathit{\boldsymbol{u}}_M^{\rm{T}},{t_f},{V_R})} \end{array}$ (11)

 $\left\{ {\begin{array}{*{20}{l}} {\Delta t = {t_f}/(M - 1)}\\ {{{\hat u}_j} = ({u_j} + {u_{j + 1}})/2}\\ {k_{i,j}^1 = {f_i}({x_j},{{\hat u}_j})}\\ {k_{i,j}^2 = {f_i}({x_j} + k_{i,j}^1\Delta t/2,{{\hat u}_j})}\\ {k_{i,j}^3 = {f_i}({x_j} + k_{i,j}^2\Delta t/2,{{\hat u}_j})}\\ {k_{i,j}^4 = {f_i}({x_j} + k_{i,j}^3\Delta t,{{\hat u}_j})} \end{array}} \right.$ (12)

 $\begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {C_{i + 6(j - 1)}} = {x_{i,j + 1}} - {x_{i,j}} - }\\ {(k_{i,j}^1 + 2k_{i,j}^2 + 2k_{i,j}^3 + k_{i,j}^4)\Delta t/6} \end{array}$ (13)

 ${x_{i,m + 1}} = {x_{i,m}} + (k_{i,m}^1 + 2k_{i,m}^2 + 2k_{i,m}^3 + k_{i,m}^4)\Delta t/6$ (14)

 $\begin{array}{*{20}{l}} {\mathit{\boldsymbol{X}} = ({x_{1,1}}, \cdots ,{x_{6,1}},{u_{1,1}},{u_{2,1}}, \cdots ,{u_{1,M}},{u_{2,M}},{t_f},{V_R}) = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (\mathit{\boldsymbol{x}}_1^{\rm{T}},\mathit{\boldsymbol{u}}_1^{\rm{T}},\mathit{\boldsymbol{u}}_2^{\rm{T}}, \cdots ,\mathit{\boldsymbol{u}}_M^{\rm{T}},{t_f},{V_R})} \end{array}$ (15)

 $\left\{ {\begin{array}{*{20}{l}} {V({t_f}) = V({t_0})}\\ {\lambda ({t_f}) = \lambda ({t_0})}\\ {h({t_f}) = h({t_0})} \end{array}} \right.$ (16)

 $|\psi ({t_f}) - \psi ({t_0})| \le \Delta {\psi _{{\rm{max}}}}$ (17)

 $h - \frac{1}{2}b|{\rm{sin}}(\mu )| \ge {h_{{\rm{min}}}}$ (18)

Sachs[5]的研究中则只采用了h≥0.5 m的最小高度约束，进行方法验证时采用和Sachs相同的高度约束条件，后续研究中则采用式(18)作为高度约束条件。

 ${\rm{min}}{\kern 1pt} {\kern 1pt} J(X) = {V_R}$ (19)

2.2 动态滑翔轨迹优化仿真验证

 Download: 图 4 动态滑翔最小代表速度对应航迹 Fig. 4 Optimal dynamic soaring cycle requiring minimum wind strength
 Download: 图 5 本文飞行器能量变化曲线 Fig. 5 Energy change of optimal dynamic soaring cycle
 Download: 图 6 动态滑翔对应控制及状态变量 Fig. 6 Variables of optimal dynamic soaring cycle

 $\left\{ {\begin{array}{*{20}{l}} {\Delta {E_{{\rm{ gain }}}} = - mg\int_0^t {\frac{{{\rm{d}}{V_W}}}{{{\rm{d}}h}}} \frac{1}{g}{V^2}{\rm{sin}}{\kern 1pt} {\kern 1pt} \gamma {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \psi {\rm{d}}t}\\ {\Delta {E_{{\rm{ lose }}}} = mg\int_0^t {\frac{{DV}}{{mg}}} {\rm{d}}t = \int_0^t D V{\rm{d}}t} \end{array}} \right.$ (20)

3 飞行器总体参数对动态滑翔能力的影响

3.1 飞行器重量

 Download: 图 7 飞行器重量对动态滑翔能力的影响 Fig. 7 Influence of UAV mass on dynamic soaring results

3.2 最大升力系数

 Download: 图 9 最大升力系数对动态滑翔能力的影响 Fig. 9 Influence of CLmax on dynamic soaring results
3.3 展弦比

1) 影响阻力系数中诱导阻力因子KD的大小：

 ${K_D} = \frac{1}{{\pi {\rm{AR}}}}$ (21)

2) 为保证飞行器与地面高度的约束不变，飞行器翼展固定的条件下，机翼面积随展弦比变化。按照公式，有：

 ${S_w} = \frac{{{b^2}}}{{{\rm{AR}}}}$ (22)

 Download: 图 10 展弦比对动态滑翔能力的影响 Fig. 10 Influence of aspect ratio on dynamic soaring results
3.4 风场强度变化指数

 Download: 图 11 风场强度变化指数对动态滑翔能力的影响 Fig. 11 Influence of wind power law exponent on dynamic soaring results
4 结论

1) 基于信天翁三自由度动力学模型，改进了Runge-Kutta积分方法构建了动态滑翔航迹优化方法模型，并结合轨迹的递推求解特点对该方法进行了进一步简化，有效减少了所需考虑的变量及约束个数。对该方法进行了验证，证明动态滑翔方法能够有效减少飞行器机械能损失，同时该优化模型能够为进一步研究飞行器的动态滑翔能力提供数据和方法的支持。

2) 利用改进型Runge-Kutta积分优化模型对飞行器部分总体参数对动态滑翔能力的影响进行了探究。通过对比不同总体参数下所需的最小风场代表速度来评估其对动态滑翔能力的影响。拥有较小重量、小翼载、足够大的最大升力系数、合适的展弦比的飞行器具有更好的机动能力，而机动性能较强的飞行器在逆风爬升、顺风俯冲及侧风机动中更有优势。这样的飞行器具有更好的动态滑翔能力。同时，风场强度变化指数更大的风场能够提供更大的侧风梯度，对动态滑翔相对更加有利。

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