无人机外文翻译-四旋翼无人机位置和姿态跟踪控制【中文4860字】【PDF+中文WORD】
无人机外文翻译-四旋翼无人机位置和姿态跟踪控制【中文4860字】【PDF+中文WORD】,中文4860字,PDF+中文WORD,无人机,外文,翻译,四旋翼,位置,姿态,跟踪,控制,中文,4860,PDF,WORD
【中文4860字】
四旋翼无人机位置和姿态跟踪控制
摘要: 一个综合控制方法是提出要执行的位置和姿态跟踪小型四旋翼的动力学模型无人机（UAV），那里的动力学模型是欠驱动控制，高度耦合非线性的。首先，动力学模型分为全面启动子系统和欠驱动子系统；其次，全面启动子系统的控制器通过一种新的强大的终端滑模控制（台积电）的算法，这是用来保证所有状态变量在短时间内收敛到自己想要的值，收敛时间是如此之小，状态变量担任时间不变量的欠驱动子系统，另外，在欠驱动子系统的控制器通过滑模控制（SMC）设计。此外，该子系统的稳定性都证明了Lyapunov理论；最后，为了证明所提出的控制方法的鲁棒性，空气动力学的力和力矩，并作为外部扰动空气阻力考虑在内，得到的仿真结果表明，合成控制方法的立场和态度方面都有不错的表现当遇到外部干扰跟踪。
关键词：四旋翼无人机，欠驱动，新颖的鲁棒台积电，SMC，综合控制
1.介绍
四旋翼无人飞行器（UAV）正被用于一些典型的任务，如搜索和救援任务，监督，检查，测绘，航空摄影和法律的强制执行。
考虑到旋翼的动力学模型是一个欠驱动，高度耦合的和非线性的系统，很多控制策略，已经开发了一类相似的系统。其中，滑模控制，这已引起研究人员的
瞩目，一直是一个有用的和有效的控制算法，处理系统具有较大不确定性，随时间变化的特性，非线性和有界外部干扰。该方法是基于定义指数稳定的滑动面
作为机能缺失跟踪误差sandusing李亚普诺夫理论的 ，保证所有的状态轨迹在有限时间到达这些表面，另外，这些表面是渐近稳定，状态轨迹滑动沿着这些表面，直到他们到达原点。但是，为了获得快速跟踪误差收敛，期望的极点必须远离原点选择上的左半部分s平面，同时，这将反过来增加了控制器的增益，这是不可取的考虑，在实际系统中的致动器饱和。
与非取代了传统的线性滑动面线性终端滑动面，更快的跟踪误差收敛是获得通过终端滑模控制，终端的滑动模式已被证明是有效的，用于提供更快收敛比围绕平衡点的线性超平面型滑模。台积电提出了不确定动态系统与纯料回分钟。一个鲁棒自适应台积电技术被用于正刚性连接的机械手具有不确定动态发展。一个全球性的非奇异台积电刚性机械臂正在呈现。机器人系统的有限时间控制是通过两个状态反馈和动态输出反馈控制研究。使用终端的滑动模式的一种新形式的刚性机械手的连续有限时间控制方案被建议。为了实现有限时间跟踪控制中的转子位置的非线性推力主动磁轴承系统的轴向，强劲的非奇异台积电被赋予。然而，传统的台积方法不是最好的收敛时间，主要的原因是非线性滑模的收敛速度比时的状态变量是接近平衡点的线性滑动模式慢。使用增强功能的滑动一个新的计划，台积电开发超平面对跟踪误差收敛到零的有限时间，提出了不确定性的单输入和单输出（SISO）非线性系统具有未知外部干扰的保证。在大多数现有的研究成果，在不确定的外部干扰都没有考虑这些非线性系统。为了进一步展示的新颖TSMC的鲁棒性，外部干扰被认为是进入非线性系统和被施加到所述控制器的设计。
图1 四旋翼无人机
在这项工作中，我们结合两部分组成控制，对于高精度的新颖的鲁棒台积电组件在完全致动子系统的跟踪性能以及一个SMC 组件处理在欠驱动子系统的外部干扰。
尽管许多经典，高阶和SMC扩展策略，已经开发了飞行控制器设计的四旋翼无人机，在报纸上这些策略被用来决定一个必要补偿外部干扰，此外，其他的控制方法，如比例-积分-微分（PID）控制，回步平控制，开关模型预测姿态控制等等。已经提出了用于在飞行控制器的设计，上述控制策略，已经提出了为使旋翼稳定在有限时间和空气工艺的稳定时间可能太长，以反映他们的表现，稳定时间为四旋翼无人机，快速从一些意想不到的干扰中恢复至关重要的意义。为了减少时间，基于新颖的鲁棒TSMC和SMC算法合成控制方法被应用到的动态模型四旋翼无人机。合成控制方法，提出以保证所有的系统状态变量在短时间内收敛到他们的期望值。此外，状态变量的收敛时间进行了预测通过由新颖的鲁棒台积电得出的方程式，这表现在以下几个部分。
这项工作的组织安排如下：第2节提出了一个小的四旋翼无人机的动力学模型。合成控制方法是在第3节详细的介绍。在第4节，仿真结果分析，以突出整体有效性和所设计的控制器的有效性。第5节的讨论，这是基于不同的合成控制方案，提出了强调表现在这项工作中提出的综合控制方法，其次是结束语在第6部分。
2. 旋翼模型
为了描述的旋翼模型的运动情况，显然，位置坐标是选择。旋翼是建立在这一工作由主体框架B和接地E型如图呈现。让矢量表示旋翼的重心的位置和向量表示其在地球帧的线速度。向量表示旋翼的角速度在主体框架，表示的总质量。表示重力加速度。表示从每个转子的中心至重心的距离。
在旋翼的方向是由旋转矩阵R给定：,其中R取决于三个欧拉角，这代表了翻滚，俯仰。且，，。
从变换矩阵到被给出
(1)
在旋翼的动力学模型可以由以下方程来描述
(2)
式中，Ki表示阻力系数和正的常数，，静置螺旋桨的角速度，，，代表旋翼的转动惯量，表示螺旋桨的转动惯量，表示总瑟斯顿体在轴；和表示的侧倾和俯仰的输入；表示偏航力矩。，，，。其中表示由四个转子所产生的推力和被认为是真正的控制输入到动力系统，表示升力系数;表示的力，力矩的比例因子。
3.综合控制
与无刷电机相比，螺旋桨是很轻的，我们忽略的转动惯量所引起的螺旋桨。式（2）是 分为两部分：
(3)
(4)
其中公式（3）表示完全致动子系统（FAS），式（4）表示的欠驱动子系统（UAS）。对于FAS，一个新颖的鲁棒TSMC用于保证其状态变量在短时间内收敛到其所需的值，然后，状态变量被视为时间不变性，因此，UAS得到简化。对于UAS，滑模控制方法利用。特别合成控制方案在以下几节介绍。
3.1一种新型强大的台积电FAS
考虑到一个刚体旋翼的对称性，然而，我们得到，和完全触动子系统写的是
(5)
为了开发跟踪控制，滑动歧管被定义为
(6)
当，，和是状态变量的期望值。此外，该系数是正的，是正奇数整数让和收敛时间的计算方法如下
(7)
根据公式（5）与S2和S4的时间导数，我们有
(8)
该控制器被设计
(9)
这里是积极的，也是正奇数整数且，根据控制器的状态轨迹到达的区域滑动表面，在有限时间内，时间被定义为
(10)
在
证明1为了说明该子系统是稳定的，在这里，我们只选择了状态变量，和 Lyapunov得以理论应用。
考虑到Lyapunov函数
调用方程（8）和（9）V1的时间导数导出
考虑到为正偶数而且。该子系统的状态轨迹在有限时间收敛到期望的平衡点，因此，子系统是渐近稳定的。
3.2 SMC的方式为无人机
在本节中，左右推拉的一类欠驱动系统的模式控器的细节被发现在SMC方法的无人机系统。
，，
在欠驱动子系统是写在一个级联的形式
(11)
根据公式（9），我们可以选择合适的参数，以保证控制律和偏航角ψ收敛到期望的/参考值在很短的时间。时，不随时间变化，然后，是不随时间变化和非奇异的矩阵，作为其结果是总推力和非零克服重力。
确定跟踪误差方程
(12)
其中所述载体来表示所希望的值的矢量
滑动歧管被设计成
(13)
其中常数
由于
，
可以得到
(14)
图2 位置，PID控制和SMC
图3 坐标，PID控制和SMC
证明2该子系统的稳定性由李雅普诺夫说明理论如下：
考虑Lyapunov函数
调用（13）和（14），V的时间导数是
因此，在控制器，子系统的状态轨迹可以达到，此后，在有限时间保持。
图4 控制器，PID控制和SMC
表1 旋翼模型参数
表2 控制器参数
4.仿真结果与分析
在本节中，式中的四旋翼无人机的动力学模型，当遇到外部干扰，式（2）用于测试所提出的合成控制方案的有效性和效率。典型的位置和姿态跟踪的仿真在Matlab7.1.0.246/ Simulink中进行的，其配备了包括DUO E72002.53 GHz的CPU与2GB的内存和100GB的固态硬盘驱动器的计算机。此外，该合成控制的性能通过被证实的与控制方法相比，它使用一个速率控制方法相比，有界的PID控制器和滑模控制器的完全驱动子系统,一个SMC方法的欠驱动子系统。
4.1 PID控制和SMC
在本节中，将PID控制和SMC方法的更多细节对于aquadrotor无人机已经出台,同时，仿真结果和分析，从而验证的有效性综合控制方案，可以发现
图5 坐标（X，Y，Z），新颖的鲁棒台积电和SMC
图6 角，新颖的鲁棒台积电和SMC
模拟测试显示在图2-4，然而，研究科目略有改变，使具有明显的比较以下模拟测试。
4.2 新颖的鲁棒台积电和SMC
在本节中，为了证明所提出的合成控制方法的有效性，已经进行了四旋翼的位置和姿态跟踪。
在旋翼的模拟测试的初始位置和角度的值是[0,0,0]和[0,0,0]。此外，该旋翼模型变量列于表1中。所需/参考位置和角度值在模拟测试中使用，此外，该控制器参数列于表2，仿真结果示于图5-10。
整体控制方案管理，以有效地保持在有限时间的四旋翼水平位置和姿态，图5和图6有所展示，状态变量z和ψ的有限时间收敛显然比其他状态变量更快，因此，它是安全的考虑俯仰角ψ为不随时间变化1.163s之后。此外，高度Z达到1.779s后，控制器U1根据其参考值，因此，它是可靠的后1.779秒到考虑控制器U1为不随时间变化。这些验证矩阵Q是时间不变的短有限时间。在其他变量后约5秒达到他们的期望值。即使状态变量φ和θ的光滑曲线表明，它们有一定的振荡幅度，该幅度是从-0.05弧度到0.05弧度不同。根据初始条件，参数和希望/参考值，状态变量z和ψ的收敛时间是通过调用方程计算值基本一致。这表明所提出的合成控制方案的有效性。
图7 线速度（U，V，W），新颖的鲁棒台积电和SMC
线速度和角速度，显示在图7和8，分别表现出相同的行为，相应的位置和角度，事实上，这些状态变量被驱赶到它们的稳定状态如预期。这再次证明了综合控制方案的有效性。
滑动变量（S2，S4和s），示于图9，如下的期望，因为所有的变量收敛到其滑动面。此外，如需要，为S2和S4的有限时间收敛明显大于s的有限时间收敛速度更快。同样的，这表现出相同的行为，在图中所示5和6。
由图可见，如图10所示，可以发现，该四个控制输入变量几秒钟后，分别收敛于稳态值。此外，这也验证了矩阵是不随时间变化在短期有限的时间，尽管有较高的初始值，和几乎没有振动的振幅。这也表示其中趋势的时间导数为零。因此， 在方程的方程组进行比较，（11）被大大简化。
最后，提出了在所有的控制方法的鲁棒性证明通过考虑气动力和力矩，并作为外部干扰到旋翼的动力学模型空气阻力。而且，这些干扰术语也适用于在控制器的设计。其结果是，这些扰动方面的影响是不可见的所有状态变量，滑动变量，和控制器。
5.讨论
广泛的模拟测试已经完成，评估不同的合成控制计划，是基于四旋翼无人机的位置和姿态的跟踪，它可以beclearly看出，虽然根据需要在有限时间所有的状态变量收敛到他们的参考值时，收敛时间显然是不同的。结果表明，基于该新颖的鲁棒TSMC和SMC的合成控制方法是一种更可靠和更有效的方法来执行跟踪控制的旋翼UAV。
6.结论
这项工作研究的立场和态度使用基于上述控制算法所提出的控制方法跟踪一个小四旋翼无人机的控制权，为了进一步测试设计的控制器的性能，随着控制器的四旋翼的动力学模型进行仿真基于Matlab / Simulink的。主要结论概述如下。 （a）本六个自由度在有限时间分别收敛到其期望/参考值。 （b）该状态变量z和ψ的收敛时间是与理论计算值（C）基本上是一致的相对于其他变量，俯仰角ψ和控制器成为时间变量在很短的时间（D）四个控制输入变量收敛到在有限时间稳定的价值观， 和几乎没有振动振幅。所有上述情况，提出了合成控制方法的有效性和鲁棒性已被证实，并且，所呈现的模拟结果是有希望的位置和姿态跟踪控制的飞机。
致谢
本工作部分由中国国家自然科学基金科学基金（60905034）提供。
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ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Research Article
Position and attitude tracking control for a quadrotor UAV
Jing-Jing Xiong n, En-Hui Zheng
College of Mechanical and Electrical Engineering, China Jiliang University, Hangzhou 310018, PR China
Please cite this article as: Xiong J-J, Zheng E-H. Position and attitude tracking control for a quadrotor UAV. ISA Transactions (2014), http:
//dx.doi.org/10.1016/j.isatra.2014.01.004i
a r t i c l e i n f o
Article history:
Received 3 November 2013 Received in revised form 29 December 2013
Accepted 16 January 2014
This paper was recommended for publication by Jeff Pieper.
Keywords: Quadrotor UAV Underactuated Novel robust TSMC SMC
Synthesis control
a b s t r a c t
A synthesis control method is proposed to perform the position and attitude tracking control of the dynamical model of a small quadrotor unmanned aerial vehicle (UAV), where the dynamical model is underactuated, highly-coupled and nonlinear. Firstly, the dynamical model is divided into a fully actuated subsystem and an underactuated subsystem. Secondly, a controller of the fully actuated subsystem is designed through a novel robust terminal sliding mode control (TSMC) algorithm, which is utilized to guarantee all state variables converge to their desired values in short time, the convergence time is so small that the state variables are acted as time invariants in the underactuated subsystem, and, a controller of the underactuated subsystem is designed via sliding mode control (SMC), in addition, the stabilities of the subsystems are demonstrated by Lyapunov theory, respectively. Lastly, in order to demonstrate the robustness of the proposed control method, the aerodynamic forces and moments and air drag taken as external disturbances are taken into account, the obtained simulation results show that the synthesis control method has good performance in terms of position and attitude tracking when faced with external disturbances.
& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
The quadrotor unmanned aerial vehicles (UAVs) are being used in several typical missions, such as search and rescue missions, surveillance, inspection, mapping, aerial cinematography and law enforcement [1–5].
Considering that the dynamical model of the quadrotor is an underactuated, highly-coupled and nonlinear system, many con- trol strategies have been developed for a class of similar systems. Among them, sliding mode control, which has drawn researchers' much attention, has been a useful and efficient control algorithm for handling systems with large uncertainties, time varying prop- erties, nonlinearities, and bounded external disturbances [6]. The approach is based on defining exponentially stable sliding surfaces as a function of tracking errors and using Lyapunov theory to guarantee all state trajectories reach these surfaces in finite-time, and, since these surfaces are asymptotically stable, the state trajectories slides along these surfaces till they reach the origin [7]. But, in order to obtain fast tracking error convergence, the desired poles must be chosen far from the origin on the left half of s-plane, simultaneously, this will, in turn, increase the gain of the controller, which is undesirable considering the actuator satura- tion in practical systems [8,9].
n Corresponding author.
E-mail addresses: jjxiong357@gmail.com (J.-J. Xiong), ehzheng@cjlu.edu.cn (E.-H. Zheng).
Replacing the conventional linear sliding surface with the non- linear terminal sliding surface, the faster tracking error convergence is to obtain through terminal sliding mode control (TSMC). Terminal sliding mode has been shown to be effective for providing faster convergence than the linear hyperplane-based sliding mode around the equilibrium point [8,10,11]. TSMC was proposed for uncertain dynamic systems with pure-feedback form in [12]. In [13], a robust adaptive TSMC technique was developed for n-link rigid robotic manipulators with uncertain dynamics. A global non-singular TSMC for rigid manipulators was presented in [14]. Finite-time control of the robot system was studied through both state feedback and dynamic output feedback control in [15]. A continuous finite-time control scheme for rigid robotic manipulators using a new form of terminal sliding modes was proposed in [16]. For the sake of achieving finite-time tracking control for the rotor position in the axial direction of a nonlinear thrust active magnetic bearing system, the robust non-singular TSMC was given in [17]. However, the conventional TSMC methods are not the best in the convergence time, the primary reason is that the convergence speed of the nonlinear sliding mode is slower than the linear sliding mode when the state variables are close to the equilibrium points. In [18], a novel TSMC scheme was developed using a function augmented sliding hyperplane for the guarantee that the tracking error converges to zero in finite-time, and was proposed for the uncertain single-input and single-output (SISO) nonlinear system with unknown external disturbance. In the most of existing research results, the uncertain external disturbances are not taken into account these nonlinear systems. In order to further demonstrate the robustness of novel
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quadrotor is set up in this work by the body-frame B and the earth- frame E as presented in Fig. 1. Let the vector [x,y,z]' denotes the position of the center of the gravity of the quadrotor and the vector [u, v,w]' denotes its linear velocity in the earth-frame. The vector [p,q,r]' represents the quadrotor's angular velocity in the body-frame. m denotes the total mass. g represents the acceleration of gravity. l denotes the distance from the center of each rotor to the center of gravity.
The orientation of the quadrotor is given by the rotation matrix R:E-B, where R depends on the three Euler angles [ϕ,θ,ψ]0 , which represent the roll, the pitch and the yaw, respectively. And ϕAð π=2; π=2Þ; θ Að π=2; π=2Þ; ψ Að π; πÞ.
The transformation matrix from [ϕ,θ,ψ]0 to [p,q,r]0 is given by
2 p 3
2 1 0 sin θ
32 ϕ_ 3
6 q 7
6 0 cos ϕ sin ϕ cos θ 76 θ_ 7
4 5 ¼ 4
56 7
ð1Þ
r 0 sin ϕ cos ϕ cos θ
4 ψ_ 5
Fig. 1. Quadrotor UAV.
The dynamical model of the quadrotor can be described by the following equations [5,24,29]:
8 x€ ¼ 1 ð cos ϕ sin θ cos ψ þ sin ϕ sin ψ Þu1 K1 x_
m m
>
TSMC, the external disturbances are considered into the nonlinear
> y€ ¼ 1
K2 y_
1
> mð cos ϕ sin θ sin ψ sin ϕ cos ψ Þu m
>
systems and are applied to the controller design.
> z€ ¼ 1
K3 z_
<> mð cos ϕ cos θÞu1 g m
In this work, we combine two components in the proposed
ϕ€ _
Iy Iz
Jr θ_ Ω l
K4 lϕ_
ð2Þ
control: a novel robust TSMC component for high accuracy
> ¼ θψ_
>
Ix þ Ix
r þ Ixu2 Ix
tracking performance in the fully actuated subsystem, and a SMC
> θ€ ¼ ψ_ ϕ_ Iz Ix Jr ϕ_ Ωr þ l u3 K5 lθ_
component that handles the external disturbances in the under-
>
>
> ψ€
Iy Iy
_ _ Ix Iy 1
Iy Iy
K6
actuated subsystem.
Even though many classical, higher order and extended SMC
:> ¼ ϕθ
Iz þ Izu4 Iz ψ_
strategies, have been developed for the flight controller design for
where Ki denote the drag coefficients and positive constants;
Ωr ¼ Ω1 Ω2 þ Ω3 Ω4; Ωi ; stand for the angular speed of the
the quadrotor UAV (see for instance [19–23], and the list is not
exhaustive), and, these strategies in the papers [19–23] were
propeller i Ix,Iy,Iz
represent the inertias of the quadrotor;Jr denotes
the inertia of the propeller;u1 denotes the total thrust on the body
utilized to dictate a necessity to compensate for the external
disturbances, in addition, the other control methods, such as
in the z-axis;u2
and u3
represent the roll and pitch inputs,
respectively;u4 denotes a yawing moment.u1 ¼ ðF1 þ F2 þ F3 þ F4Þ;
proportional–integral–differential (PID) control [24,25], backstep-
ping control [26,27], switching model predictive attitude control
u2 ¼ð F2
þF4Þ; u3
2
¼ð F1
þ F3Þ; u4
¼ dð F1
þ F2
þ F3
þ F4Þ=b;,
[28], etc., have been proposed for the flight controller design, most of the aforementioned control strategies have been proposed in order to make the quadrotor stable in finite-time and the stabili- zation time of the aircraft may be too long to reflect the performance of them. In addition, the stabilization time is essen- tial significance for the quadrotor UAV to quickly recover from some unexpected disturbances. For the sake of decreasing the time, a synthesis control method based on the novel robust TSMC and SMC algorithms is applied to the dynamical model of the quadrotor UAV. The synthesis control method is proposed to guarantee all system state variables converge to their desired
where Fi ¼ bΩi denote the thrust generated by four rotors and are considered as the real control inputs to the dynamical system, b
denotes the lift coefficient;d denotes the force to moment scaling factor.
3. Synthesis control
Compared with the brushless motor, the propeller is very light, we ignore the moment of inertia caused by the propeller. Eq. (2) is divided into two parts:
values in short time. Furthermore, the convergence time of the state variables are predicted via the equations derived by the novel
" z€ #
2 u1 cos ϕ cos θ 3 2
g
m
K3 3
z_
Þ
m 3
robust TSMC, this is demonstrated by the following sections.
ψ€ ¼ 4 1
5þ4 ϕ_ θ_ Ix Iy K6 5 ð
The organization of this work is arranged as follows. Section 2 presents the dynamical model of a small quadrotor UAV. The
8 " x€ #
>
Izu4
" cos ψ sin ψ
u1
Iz Iz ψ_
#" cos ϕ sin θ #
x_
2 K1 3
— m
synthesis control method is detailedly introduced in Section 3. In
Section 4, simulation results are performed to highlight the overall
>> y€ ¼ m
>
<
sin ψ
cos ψ
sin ϕ
þ4 K2 5
y_
— m
validity and the effectiveness of the designed controllers. In
> " ϕ€ #
" l=Ix 0
#" u #
2 θ_ ψ_
Iy Iz I
K4 l _ 3
ϕ
I
ð4Þ
Section 5, a discussion, which is based on different synthesis >
2 x x
control schemes, is presented to emphasize the performance of
> θ€ ¼
0 l=Iy
u3 þ4 ψ_ ϕ_ Iz Ix
K5 lθ_ 5
>:
the proposed synthesis control method in this work, followed by
Iy Iy
the concluding remarks in Section 6.
2. Quadrotor model
In order to describe the motion situations of the quadrotor model clearly, the position coordinate is to choose. The model of the
where Eq. (3) denotes the fully actuated subsystem (FAS), Eq. (4) denotes the underactuated subsystem (UAS). For the FAS, a novel robust TSMC is used to guarantee its state variables converge to their desired values in short time, then the state variables are regarded as time invariants, therefore, the UAS gets simplified. For the UAS, a sliding mode control approach is utilized. The special synthesis control scheme is introduced in the following sections.
J.-J. Xiong, E.-H. Zheng / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
7
3.1. A novel robust TSMC for FAS
Considering the symmetry of a rigid-body quadrotor, therefore, we get Ix ¼ Iy Let x1 ¼ ½zψ ]0 and x2 ¼ ¼ ½z_ψ_ ]0 . The fully actuated
Considering the Lyapunov function candidate
2
V 1 ¼ s2=2
η sm1 =n1 _
Invoking Eqs. (8a) and (9a) the time derivative of V1 is derived
subsystem is written by
V_ 1 ¼ s2s_2 ¼
s2ð ε1s2
1 2 þ K3z=mÞ
( x_ 1 ¼ x2;
2 0 m1 þ n1 Þ=n1
x_ 2 ¼ f 1 þ g1u1 þ d1
ð5Þ
¼ ε1s2
η1sð
2
where f 1 ¼½ g 0]0 ; g1 ¼½ cos ϕ cos θ=m 0; 0 1=Iz ]; u1 ¼ ½u1 u4]0
and d1 ¼½ K3z_=m K6ψ_ =Iz ]0 :
To develop the tracking control, the sliding manifolds are defined as [18,30]
s2 ¼ s_1 þω1s1 þξ1s1m1 =n1 ð6aÞ
Considering that (m1 þn1) is positive even integer, that’s,
V_ 1 r0: The state trajectories of the subsystem converge to the desired equilibrium points in finite-time. Therefore, the subsystem
is asymptotically stable.
3.2. A SMC approach for UAS
0 0
s4 ¼ s_3 þω2s3 þξ2s3m2 =n2 ð6bÞ
In this section, the details about sliding mode control of a
0 0
class of underactuated systems are found in [29]. Let
where s1 ¼ zd z; s3 ¼ ψ d ψ, Zd and ψd are the desired values of
state variables Z and ψ, respectively. In addition, the coefficients
" cos ψ sin ψ
Q ¼ u1
#
, and y1 ¼ Q ½x y] ; y2 ¼ Q
½x_ y_ ] ;
ðω1; ω2; ξ1; ξ2Þ are positive, m0 ; m0 ; n0 ; n0 are positive odd integers
m sin ψ cos ψ
1 0 1 0
with m0 o n0
and m0 o n0 .
1 2 1 2
y ¼ ½ϕ θ]0 ; y
¼ ½ϕ_ θ_ ]0 . The underactuated subsystem is written in
1 1 2 2 3 4
Let s2 ¼ 0 and s4 ¼ 0. The convergence time is calculated as
a cascaded form
follows:
1 m1 Þ=n1 !
y_ 1
¼ y2;
ξ
n0 ω1½s1ð0Þ]ðn0 0 0 þ y_ 2 ¼ f 2 þd2;
1 1
ts1 ¼ ω n0 0 ln ð7aÞ _
1ð 1 m1Þ
n0
ξ1
ω2½s3ð0Þ]ðn2 m2 Þ=n2 þξ !
y3 ¼ y4;
y_ 4 ¼ f 3 þg2u2 þ d3: ð11Þ
0 0 0
2 2
According to Eqs. (9a) and (9b) we can select the appropriate
ts3 ¼ ω
ln
2
m0
2ðn0 2Þ ξ2
ð7bÞ
parameters to guarantee the control law u1 and yaw angle ψ
In accordance with Eq. (5) and the time derivative of s2 and s4, we have
converge to the desired/reference values in short time. That’s,u_ 1 ¼ 0,ψ becomes time invariant, then ψ_ ¼ 0, Q is time invariant matrix and non-singular because u1 is the total thrust
u1 K3 d m0 =n0
s_2 ¼ z€d cos ϕ cos θ þ g þ z_ þω1s_1 þξ1 s 1 1 ð8aÞ
m m dt 1
and nonzero to overcome the gravity. As a result
2 2 3 2
f ¼½ cos ϕ sin θ sin ϕ]0 ; d2 ¼ Q 1diag½K1=m K2=m]Qy ; f ¼ 0; g
1 K6 d m0 =n0
2
2
s_4 ¼ ψ€ d I u4 þ I
ψ_ þω2s_3 þξ2dts
ð8bÞ
¼ diag½l=Ix l=Iy]; u
¼ ½u
u ]0 ; d
¼ diag½ lK =I lK =I ]y
3 2 2 3 3
z z
4 x 5 y 4
The controllers are designed by
Define the tracking error equations
m m0 ðm0 n0 Þ=n0
m =n
8 e1 ¼ yd y1;
cos ϕ cos θ
1 1
1
1
u1 ¼ z€d þ g þω1s_1 þξ1
1s 1
n0 1
s_1 þε1s2 þη1s2
ð9aÞ > 1
>
< 1
1 > e2 ¼ e_ 1 ¼ y_ d
d
y2;
ð12Þ
>
／ m0
m0 n0 Þ=n0
m2 =n2 ＼
e3 ¼ e_ 2 ¼ y€ 1 f 2;
u4 ¼ Iz
ψ€ d þω2s_3 þξ2 2sð 2
2 2 s_3 þε2s4 þη2s
ð9bÞ >
n0 3
4 > :::d
’ ∂f 2
∂f 2
∂f 2
2 > e4 ¼ e_ 3 ¼ y1
∂y y2 þ∂y f 2 þ∂y y4
:
where ε1,ε2,η1, and η2 are positive,m1, n1, m2, and n2 are positive
1 2 3
d
odd integers with m1 on1 and m2 on2:
Under the controllers, the state trajectories reach the areas (Δ1,Δ2) of the sliding surfaces s2 ¼ 0 and s4 ¼ 0 along s_2 ¼ ε1s2
where the vector y1 denotes the desired value vector.
The sliding manifolds are designed as
s ¼ c1e1 þc2e2 þc3e3 þe4 ð13Þ
η0 m1 =n1
0 m2 =n2
1s2 and s_4 ¼ ε2s4 η2s4 in finite-time, respectively. The time is defined as
where the constants ci 40.
By making s_ ¼ MsgnðsÞ λs, we get
n1
ε1½s1ð0Þ]ðn1 m1 Þ=n1 þδ1
::: h i
t0 ln
ð10aÞ
8 c1e2 þc2e3 þc3e4 þ y0
∂f 2 y 9
1 rε n m δ
d d
<
> 1 y 2 >
1ð 1 1Þ 1
u2 ¼
∂f 2 g
∂y 2
1>>
d h ∂f 2 i
f
dt ∂y2 2
d h∂f 2 i
y
dt ∂y3 4
dt ∂ 1 >
>
=
ð14Þ
3
t0
2
> >
n2 ε2½s3ð0Þ]ðn2 m2 Þ=n2 þδ2
> ∂f >
2 rε
2ðn2
ln
m2Þ δ2
ð10bÞ
>: ∂y3 ðf 3 þd3Þþ MsgnðsÞþλs >;
where
η0
m1 =n1
m1 =n1
where
M ¼ ðc2d2 þc3β d2ÞjjE1jj2 þβ d4jjξðyÞjj2 þρ;
1 ¼ η1 þð K3z_=mÞ=js2 j; η1 ¼ L1=js2 jþδ1; 2 3
L1 ¼ jK3z_=mjmax; δ1 40; Δ1 ¼ fjs2jrðL1=η1Þ
m1 =n1
g
β1 Z∂f 2=∂y1; β2 Z∂f 2=∂y2; β3 Z∂f 2=∂y3;
η0 m2 =n2
m2 =n2
E1 ¼ ½e1e2e3] ; ξðyÞ¼ ½y1y2y3y4] and λ 40;
2 ¼ η2 þð K6ψ_ =Iz Þ=js4 j; η2 ¼ L2=js4 jþδ2
ρ 0 d
0
o d E
0
d max K mK m
m2 =n2
4 ; jj 2jj
2jj 1jj2; 2 ¼
ð 1=
2= Þ
L2 ¼ jK6ψ_ =Iz jmax; δ2 40; Δ2 ¼ fjs4jrðL2=η2Þ
g
jjd3jjo d4jjξðyÞjj2; d4 ¼ maxðlK4=IxlK5=IyÞ:
Proof 1. In order to illustrate the subsystem is stable, here, we
According to∂f 2 ¼ ［ sin ϕ sin θ
cos ϕ cos θ cos ϕ0];
only choose the state variable z as an example and Lyapunov
and 0 o ｜｜∂f 2=∂y3｜｜ ¼ ｜ cos 2ϕ cos θo2, and , therefore, ∂f 2=
theory is applied.
｜｜
∂y3
∂y3 is invertible.
｜｜ ｜
reference real
2
x ( m )
0
-2
0 5 10 15 20 25 30 35 40 45 50
1
y ( m )
0
13
X: 39.54
Y: 9.801
12
u ( m/s 2 )
11
10
1
9
8
7
0 5 10 15 20 25 30 35 40 45 50
Time ( s )
Fig. 4. The controller u1, PID control and SMC.
-1
0 5 10 15 20 25 30 35 40 45 50
6
Table 1
Quadrotor model parameters.
z ( m )
Variable
Value
Units
m
2.0
kg
Ix ¼ Iy
1.25
Ns2/rad
Iz
2.2
Ns2/rad
K1 ¼ K2 ¼ K3
0.01
Ns/m
K ¼ K ¼ K
0.012
Ns/m
l
0.20
m
Jr
1
Ns2/rad
b
2
Ns2
d
5
N ms2
g
9.8
m/s2
3
0
0 5 10 15 20 25 30 35 40 45 50
Time ( s )
Fig. 2. The positions (x,y,z), PID control and SMC.
4 5 6
reference real
0.025
f ( rad )
0
-0.025
.06
Variable
Value
Variable
Value
ω1 ξ1
1
1
ω2 ξ2
3
1
m0
5
0
5
0.06
n1
0
7
n2
7
5
10
15
20
25 30
35
40
45
50
m1
1
m2
1
1
n1
3
n2
3
ε1
10
ε2
10
X: 25.48
Y: 0.5003
η1
L = sm1 =n1
η2
L2=j
m2 =n2
0.5
δ1
0.1
δ2
0.1
c1
20
c2
22
c3
8
ρ
1
0
λ
0.1
β1
0
5
10
15
20
25 30
Time ( s )
35
40
45
50
β2
0
β3
2
0
0 5 10 15 20 25 30 35 40 45 50
Table 2
Controller parameters.
q ( rad )
0
1 m2
0
-
0
y ( rad )
1 j 2 jþδ1 s4 jþδ2
0
Fig. 3. The angles (ϕ,θ,ψ), PID control and SMC.
Proof 2. The stability of the subsystem is illustrated by Lyapunov theory as follows.
Consider the Lyapunov function candidate:
V 1sT s
¼ 2
Invoking Eqs. (13) and (14), the time derivative of V is
V_ ¼ sT s_ ¼ sT ½c1e_ 1 þc2e_ 2 þc3e_ 3 þ e_ 4]
4. Simulation results and analysis
In this section, the dynamical model of the quadrotor UAV in Eq. (2) is used to test the validity and efficiency of the proposed synthesis control scheme when faced with external disturbances. The simulations of typical position and attitude tracking are performed on Matlab 7.1.0.246/Simulink, which is equipped with a computer comprising of a DUO E7200 2.53 GHz CPU with 2 GB of RAM and a 100 GB solid state disk drive. Moreover, the perfor- mance of the synthesis control is demonstrated through the comparison with the control method in [29], which used a rate
／
¼ sT MsgnðsÞþc2d2 þc3
∂f 2 d
2
∂y 2 þ
∂f 2 d ＼
3
∂y3
b
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