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西北工业大学明德学院本科毕业设计论文毕业设计（论文）外文文献翻译 题 目：四旋翼无人机位置和姿态跟踪控制系 别 自动化系 专 业 自动化 班 级 191002 学生姓名 张 川 学 号 103613 指导教师 聂 聪 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Matlab46/ 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。即使状态变量和的光滑曲线表明，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。参考文献1 RaffoGV,OrtegaMG,RubioFR.Anintegralpredictive/nonlinear Hcontrol 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Control EngPract2011;19(10):1195207.29 Xu R,zgner.Slidingmodecontrolofaclassofunderactuatedsystems. Automatica2008;44:23341.30 YuX,ManZ.Fastterminalsliding-modecontrol design fornonlinear dynamical systems.IEEETransCircSystFundam TheoryAppl2002;49(2): 2614.20Research ArticlePosition and attitude tracking control for a quadrotor UAVJing-Jing Xiongn, En-Hui ZhengCollege of Mechanical and Electrical Engineering, China Jiliang University, Hangzhou 310018, PR Chinaa r t i c l e i n f oArticle history:Received 3 November 2013Received in revised form29 December 2013Accepted 16 January 2014This paper was recommended forpublication by Jeff Pieper.Keywords:Quadrotor UAVUnderactuatedNovel robust TSMCSMCSynthesis controla b s t r a c tA synthesis control method is proposed to perform the position and attitude tracking control of thedynamical model of a small quadrotor unmanned aerial vehicle (UAV), where the dynamical model isunderactuated, highly-coupled and nonlinear. Firstly, the dynamical model is divided into a fully actuatedsubsystem and an underactuated subsystem. Secondly, a controller of the fully actuated subsystem isdesigned through a novel robust terminal sliding mode control (TSMC) algorithm, which is utilized toguarantee all state variables converge to their desired values in short time, the convergence time is sosmall that the state variables are acted as time invariants in the underactuated subsystem, and, acontroller of the underactuated subsystem is designed via sliding mode control (SMC), in addition, thestabilities of the subsystems are demonstrated by Lyapunov theory, respectively. Lastly, in order todemonstrate the robustness of the proposed control method, the aerodynamic forces and moments andair drag taken as external disturbances are taken into account, the obtained simulation results show thatthe synthesis control method has good performance in terms of position and attitude tracking whenfaced with external disturbances.& 2014 ISA. Published by Elsevier Ltd. All rights reserved.1. IntroductionThe quadrotor unmanned aerial vehicles (UAVs) are being usedin several typical missions, such as search and rescue missions,surveillance, inspection, mapping, aerial cinematography and lawenforcement 15.Considering that the dynamical model of the quadrotor is anunderactuated, 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 researchersmuch attention, has been a useful and efficient control algorithmfor handling systems with large uncertainties, time varying prop-erties, nonlinearities, and bounded external disturbances 6. Theapproach is based on defining exponentially stable sliding surfacesas a function of tracking errors and using Lyapunov theory toguarantee all state trajectories reach these surfaces in finite-time,and, since these surfaces are asymptotically stable, the statetrajectories slides along these surfaces till they reach the origin7. But, in order to obtain fast tracking error convergence, thedesired poles must be chosen far from the origin on the left half ofs-plane, simultaneously, this will, in turn, increase the gain of thecontroller, which is undesirable considering the actuator satura-tion in practical systems 8,9.Replacing the conventional linear sliding surface with the non-linear terminal sliding surface, the faster tracking error convergenceis to obtain through terminal sliding mode control (TSMC). Terminalsliding mode has been shown to be effective for providing fasterconvergence than the linear hyperplane-based sliding mode aroundthe equilibrium point 8,10,11. TSMC was proposed for uncertaindynamic systems with pure-feedback form in 12. In 13, a robustadaptive TSMC technique was developed for n-link rigid roboticmanipulators with uncertain dynamics. A global non-singular TSMCfor rigid manipulators was presented in 14. Finite-time control ofthe robot system was studied through both state feedback anddynamic output feedback control in 15. A continuous finite-timecontrol scheme for rigid robotic manipulators using a new form ofterminal sliding modes was proposed in 16. For the sake ofachieving finite-time tracking control for the rotor position in theaxial direction of a nonlinear thrust active magnetic bearing system,the robust non-singular TSMC was given in 17. However, theconventional TSMC methods are not the best in the convergencetime, the primary reason is that the convergence speed of thenonlinear sliding mode is slower than the linear sliding mode whenthe state variables are close to the equilibrium points. In 18, a novelTSMC scheme was developed using a function augmented slidinghyperplane for the guarantee that the tracking error converges tozero in finite-time, and was proposed for the uncertain single-inputand single-output (SISO) nonlinear system with unknown externaldisturbance. In the most of existing research results, the uncertainexternal disturbances are not taken into account these nonlinearsystems. In order to further demonstrate the robustness of novelContents lists available at ScienceDirectjournal homepage: /locate/isatransISA Transactions0019-0578/$-see front matter & 2014 ISA. Published by Elsevier Ltd. All rights reserved./10.1016/j.isatra.2014.01.004nCorresponding author.E-mail addresses: jjxiong357 (J.-J. Xiong),ehzheng (E.-H. Zheng).Please cite this article as: Xiong J-J, Zheng E-H. Position and attitude tracking control for a quadrotor UAV. ISA Transactions (2014), http:/10.1016/j.isatra.2014.01.004iISA Transactions () TSMC, the external disturbances are considered into the nonlinearsystems and are applied to the controller design.In this work, we combine two components in the proposedcontrol: a novel robust TSMC component for high accuracytracking performance in the fully actuated subsystem, and a SMCcomponent that handles the external disturbances in the under-actuated subsystem.Even though many classical, higher order and extended SMCstrategies, have been developed for the flight controller design forthe quadrotor UAV (see for instance 1923, and the list is notexhaustive), and, these strategies in the papers 1923 wereutilized to dictate a necessity to compensate for the externaldisturbances, in addition, the other control methods, such asproportionalintegraldifferential (PID) control 24,25, backstep-ping control 26,27, switching model predictive attitude control28, etc., have been proposed for the flight controller design, mostof the aforementioned control strategies have been proposed inorder to make the quadrotor stable in finite-time and the stabili-zation time of the aircraft may be too long to reflect theperformance of them. In addition, the stabilization time is essen-tial significance for the quadrotor UAV to quickly recover fromsome unexpected disturbances. For the sake of decreasing thetime, a synthesis control method based on the novel robust TSMCand SMC algorithms is applied to the dynamical model of thequadrotor UAV. The synthesis control method is proposed toguarantee all system state variables converge to their desiredvalues in short time. Furthermore, the convergence time of thestate variables are predicted via the equations derived by the novelrobust TSMC, this is demonstrated by the following sections.The organization of this work is arranged as follows. Section 2presents the dynamical model of a small quadrotor UAV. Thesynthesis control method is detailedly introduced in Section 3. InSection 4, simulation results are performed to highlight the overallvalidity and the effectiveness of the designed controllers. InSection 5, a discussion, which is based on different synthesiscontrol schemes, is presented to emphasize the performance ofthe proposed synthesis control method in this work, followed bythe concluding remarks in Section 6.2. Quadrotor modelIn order to describe the motion situations of the quadrotor modelclearly, the position coordinate is to choose. The model of thequadrotor 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 theposition 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,rrepresents the quadrotors angular velocity in the body-frame. mdenotes the total mass. g represents the acceleration of gravity. ldenotes the distance from the center of each rotor to the center ofgravity.The orientation of the quadrotor is given by the rotation matrixR:E-B, where R depends on the three Euler angles ,0, whichrepresent the roll, the pitch and the yaw, respectively. AndA?=2;=2;A?=2;=2;A?;.The transformation matrix from ,0to p,q,r0is given bypqr264375 10? sin0cossincos0? sincoscos264375_266437751The dynamical model of the quadrotor can be described by thefollowing equations 5,24,29:x 1mcossincos sinsinu1?K1_xmy 1mcossinsin? sincosu1?K2_ymz 1mcoscosu1?g?K3_zm_Iy?IzIxJrIx_rlIxu2?K4lIx_ _Iz?IxIy?JrIy_rlIyu3?K5lIy_Ix?IyIz1Izu4?K6Iz_8:2where Kidenote the drag coefficients and positive constants;r1?23?4;i; stand for the angular speed of thepropeller i Ix,Iy,Izrepresent the inertias of the quadrotor;Jrdenotesthe inertia of the propeller;u1denotes the total thrust on the bodyin the z-axis;u2and u3represent the roll and pitch inputs,respectively;u4denotes a yawing moment.u1 F1F2F3F4;u2 ?F2F4; u3 ?F1F3; u4 d?F1F2F3F4=b;,where Fi b2idenote the thrust generated by four rotors and areconsidered as the real control inputs to the dynamical system, bdenotes the lift coefficient;d denotes the force to moment scalingfactor.3. Synthesis controlCompared with the brushless motor, the propeller is very light,we ignore the moment of inertia caused by the propeller. Eq. (2) isdivided into two parts:z#u1coscosm?g1Izu42435?K3m_z_Ix?IyIz?K6Iz_24353xy#u1mcossinsin? cos#cossinsin#?K1m_x?K2m_y2435#l=Ix00l=Iy#u2u3#_Iy?IzIx?K4lIx_Iz?IxIy?K5lIy_24358:4where Eq. (3) denotes the fully actuated subsystem (FAS), Eq. (4)denotes the underactuated subsystem (UAS). For the FAS, a novelrobust TSMC is used to guarantee its state variables converge totheir desired values in short time, then the state variables areregarded as time invariants, therefore, the UAS gets simplified. Forthe UAS, a sliding mode control approach is utilized. The specialsynthesis control scheme is introduced in the following sections.Fig. 1. Quadrotor UAV.J.-J. Xiong, E.-H. Zheng / ISA Transactions () 2Please cite this article as: Xiong J-J, Zheng E-H. Position and attitude tracking control for a quadrotor UAV. ISA Transactions (2014), http:/10.1016/j.isatra.2014.01.004i3.1. A novel robust TSMC for FASConsidering the symmetry of a rigid-body quadrotor, therefore,we get IxIyLet x1 z?0and x2 _z _?0. The fully actuatedsubsystem is written by_x1 x2;_x2 f1g1u1d1(5where f1 ?g 0?0;g1 coscos=m 0;0 1=Iz?;u1 u1u4?0and d1 ?K3_z=m?K6_=Iz?0:To develop the tracking control, the sliding manifolds aredefined as 18,30s2_s11s11s1m01=n016as4_s32s32s3m02=n026bwhere s1 zd?z; s3d?, Zdanddare the desired values ofstate variables Z and, respectively. In addition, the coefficients1;2;1;2 are positive, m01;m02;n01;n02are positive odd integerswith m01on01and m02on02.Let s20 and s40. The convergence time is calculated asfollows:ts1n011n01?m01ln1s10?n01?m01=n0111 !7ats3n022n02?m02ln2s30?n02?m02=n0222 !7bIn accordance with Eq. (5) and the time derivative of s2and s4, wehave_s2zd?u1mcoscosgK3m_z1_s11ddtsm01=n0118a_s4 d?1Izu4K6Iz_2_s32ddtsm02=n0238bThe controllers are designed byu1mcoscoszdg1_s11m01n01sm01?n01=n011_s11s21sm1=n12?9au4 Izd2_s32m02n02sm02?n02=n023_s32s42sm2=n24?9bwhere1,2,1, and2are positive,m1, n1, m2, and n2are positiveodd integers with m1on1and m2on2:Under the controllers, the state trajectories reach the areas(1,2) of the sliding surfaces s20 and s40 along_s2 ?1s2?01sm1=n12and_s4 ?2s4?02sm2=n24in finite-time, respectively. Thetime is defined ast01rn11n1?m1ln1s10?n1?m1=n11110at02rn22n2?m2ln2s30?n2?m2=n22210bwhere011?K3_z=m=jsm1=n12j;1 L1=jsm1=n12j1;L1 jK3_z=mjmax;140;1 fjs2jrL1=1m1=n1g022?K6_=Iz=jsm2=n24j;2 L2=jsm2=n24j2L2 jK6_=Izjmax;240;2 fjs4jrL2=2m2=n2gProof 1. In order to illustrate the subsystem is stable, here, weonly choose the state variable z as an example and Lyapunovtheory is applied.Considering the Lyapunov function candidateV1 s22=2Invoking Eqs. (8a) and (9a) the time derivative of V1is derived_V1 s2_s2 s2?1s2?1sm1=n12K3_z=m ?1s22?01sm1n1=n12Considering that (m1n1) is positive even integer, thats,_V1r0: The state trajectories of the subsystem converge to thedesired equilibrium points in finite-time. Therefore, the subsystemis asymptotically stable.3.2. A SMC approach for UASIn this section, the details about sliding mode control of aclassofunderactuatedsystemsarefoundin29.LetQ u1mcossinsin? cos#,andy1 Q?1x y?0;y2 Q?1_x_y?0;y3 ?0;y4 _?0. The underactuated subsystem is written ina cascaded form_y1 y2;_y2 f2d2;_y3 y4;_y4 f3g2u2d3:11According to Eqs. (9a) and (9b) we can select the appropriateparameters to guarantee the control law u1and yaw angleconvergetothedesired/referencevaluesinshorttime.Thats,_u1 0,becomes time invariant, then_ 0, Q is timeinvariant matrix and non-singular because u1is the total thrustand nonzero to overcome the gravity. As a resultf2 cossinsin?0;d2 Q?1diagK1=m K2=m?Qy2;f3 0;g2 diagl=Ixl=Iy?;u2 u2u3?0;d3 diag?lK4=Ix?lK5=Iy?y4Define the tracking error equationse1 yd1?y1;e2_e1_yd1?y2;e3_e2yd1?f2;e4_e3:yd1?f2y1y2f2y2f2f2y3y4?8:12where the vector yd1denotes the desired value vector.The sliding manifolds are designed ass c1e1c2e2c3e3e413where the constants ci40.By making_s ?Msgns?s, we getu2f2y3g2?1c1e2c2e3c3e4:y01d?ddtf2y1y2hi?ddtf2y2f2hi?ddtf2y3hiy4?f2y3f3d3Msgnss8:9=;14whereM c2d2c32d2jjE1jj23d4jjyjj2;1Zf2=y1;2Zf2=y2;3Zf2=y3;E1 e1e2e3?0;y y1y2y3y4?0and40;40;jjd2jjod2jjE1jj2;d2 maxK1=mK2=mjjd3jjod4jjyjj2;d4 maxlK4=IxlK5=Iy:Accordingtof2y3? sinsin?coscoscos0?;and0 of2=y3cos2coso2?, and , therefore, f2=y3is invertible.J.-J. Xiong, E.-H. Zheng / ISA Transactions () 3Please cite this article as: Xiong J-J, Zheng E-H. Position and attitude tracking control for a quadrotor UAV. ISA Transactions (2014), http:/10.1016/j.isatra.2014.01.004iProof 2. The stability of the subsystem is illustrated by Lyapunovtheory as follows.Consider the Lyapunov function candidate:V 12sTsInvoking Eqs. (13) and (14), the time derivative of V is_V sT_s sTc1_e1c2_e2c3_e3_e4? sT?Msgnsc2d2c3f2y2d2f2y3d3?o ?sTs?M?c2d2c32d2jjE1jj2?3d4jjxjj2jjsjj1 ?sTs?jjsjj1r0:Therefore, under the controllers, the subsystem state trajec-tories can reach, and, thereafter, stay on the manifold S0 infinite-time.4. Simulation results and analysisIn this section, the dynamical model of the quadrotor UAV inEq. (2) is used to test the validity and efficiency of the proposedsynthesis control scheme when faced with external disturbances.The simulations of typical position and attitude tracking areperformed on Matlab 46/Simulink, which is equipped witha computer comprising of a DUO E7200 2.53 GHz CPU with 2 GB ofRAM and a 100 GB solid state disk drive. Moreover, the perfor-mance of the synthesis control is demonstrated through thecomparison with the control method in 29, which used a ratebounded PID controller and a sliding mode controller for the fullyactuated subsystem, and, a SMC approach for the underactuatedsubsystem.4.1. PID control and SMCIn this section, more details of the PID control and SMC methodfor a quadrotor UAV has been introduced, meanwhile, the simula-tion results and analysis, which verify the effectiveness of thesynthesis control scheme, can be found in 29. The approximate05101520253035404550036Time ( s )z ( m )05101520253035404550-202x ( m )referencereal05101520253035404550-101y ( m )Fig. 2. The positions (x,y,z), PID control and SMC.05101520253035404550-0.02500.025 ( rad )referencereal05101520253035404550-0.0600.06 ( rad )0510152025303540455000.51Time ( s ) ( rad )X: 25.48Y: 0.5003Fig. 3. The angles (,), PID control and SMC.0510152025303540455078910111213X: 39.54Y: 9.801Time ( s )u1 ( m/s2 )Fig. 4. The controller u1, PID control and SMC.Table 1Quadrotor model parameters.VariableValueUnitsm2.0kgIxIy1.25Ns2/radIz2.2Ns2/radK1K2K30.01Ns/mK4K5K60.012Ns/ml0.20mJr1Ns2/radb2Ns2d5N ms2g9.8m/s2Table 2Controller parameters.VariableValueVariableValue11231121m015m025n017n027m11m21n13n231102101L1=jsm1=n12j12L2=jsm2=n24j210.120.1c120c222c3810.1102032J.-J. Xiong, E.-H. Zheng / ISA Transactions () 4Please cite this article as: Xiong J-J, Zheng E-H. Position and attitude tracking control for a quadrotor UAV. ISA Transactions (2014), http:/10.1016/j.isatra.2014.01.004isimulation tests are shown in Figs. 24, however, the researchobjects are slightly changed to make obvious comparisons withthe following simulation tests.4.2. Novel robust TSMC and SMCIn this section, in order to justify the effectiveness of the pro-posed synthesis control method, the position and attitude trackingof the quadrotor have been performed.The initial position and angle values of the quadrotor for thesimulation tests are 0, 0, 0m and 0, 0, 0rad. In addition, thequadrotors model variables are listed in Table 1.The desired/reference position and angle values are used insimulation tests: xd1 m, yd1 m, Zd1 m,dd0 rad,d/6.28 rad. Besides, The controller parameters are listed in Table 2.The simulation results are shown in Figs. 510.The overall control scheme managed to effectively hold thequadrotors horizontal position and attitude in finite-time, asshown in Figs. 5 and 6. The finite-time convergence of the statevariables z andis clearly faster than the other state variables,therefore, it is safe to consider the pitch angleas time invariantafter 1.163 s. In addition, the altitude z reaches its reference valueunder the controller u1after 1.779 s, thus, it is reliable to considerthe controller u1as time invariant after 1.779 s. These verify thematrix Q is time invariant in short finite-time. The other statevariables reach their desired values after about 5 s. Even thoughthe smooth curves of the state variablesandshow that theyhave certain oscillation amplitudes, the amplitudes are varyingfrom ?0.05 rad and 0.05 rad. According to the initial conditions,parameters and desired/reference values, the convergence time ofthe state variables z andis essentially consistent with the valuescalculated by invoking Eqs. (7a) and (7b) and . This demonstratesthe effectiveness of the proposed synthesis control scheme.The linear and angular velocities, displayed in Figs. 7 and 8,respectively, exhibit the same behavior as the correspondingpositions and angles. Indeed, these state variables are driven totheir steady states as expected. This, once again, demonstrates theeffectiveness of the synthesis control scheme.The behavior of the sliding variables (s2,s4and s), shown inFig. 9, follows the expectations as all the variables converge totheir sliding surfaces. Furthermore, as desired, the finite-timeconvergence of s2and s4is obviously faster than the finite-timeconvergence of s. Similarly, this exhibits the same behavior asshown in Figs. 5 and 6.Seen from Fig. 10, it can be found that the four control inputvariables converge to their steady state values (19.66,0,0,0) afterseveral seconds, respectively. Besides, u1a0, this also verifies thematrix Q is time invariant in short finite-time. In spite of the highinitial values, u1and u4are almost no oscillation amplitudes. Thisalso denotes that the time derivative of u1trends to zero. Thus,compared with the equations in 29, Eq. (11) is greatly simplified.Finally, the robustness of the proposed overall control methodis demonstrated by considering the aerodynamic forces andmoments and air drag taken as the external disturbances intothe dynamical model of the quadrotor. Furthermore, these dis-turbance terms are also applied to the controller design. As aresult, the effects of these disturbance terms are invisible on all thestate variables, sliding variables, and controllers.5. DiscussionThe extensive simulation tests have been performed to evaluatethe different synthesis control schemes which are based onposition and attitude tracking of the quadrotor UAV. It can be0510152025303540012x ( m )X: 4.977Y: 0.9909referencereal0510152025303540012y ( m )X: 4.977Y: 0.99460510152025303540012Time ( s )z ( m )X: 1.779Y: 0.9932Fig. 5. The positions (x,y,z), novel robust TSMC and SMC.0510152025303540-0.0500.05 ( rad )referencereal0510152025303540-0.0500.05 ( rad )051015202530354000.51X: 1.163Y: 0.4992Time ( s ) ( rad )X: 21.01Y: 0.5002Fig. 6. The angles (,), novel robust TSMC and SMC.0510152025303540-0.4500.45u ( m/s )0510152025303540-0.4500.45v ( m/s )0510152025303540-1.401.4Time ( s )w ( m/s )Fig. 7. The linear velocities (u,v,w), novel robust TSMC and SMC.J.-J. Xiong, E.-H. Zheng / ISA Transactions () 5Please cite this article as: Xiong J-J, Zheng E-H. Position and attitude tracking control for a quadrotor UAV. ISA Transactions (2014), http:/10.1016/j.isatra.2014.01.004iclearly seen that while all the state variables converge to theirreference values as desired in finite-time, the convergence time isobviously different. It was shown that the synthesis controlmethod based on the novel robust TSMC and SMC is a morereliable and effective approach to perform the tracking control forthe quadrotor UAV.6. ConclusionThis work studies the position and attitude tracking control of asmall quadrotor UAV using the proposed control method based onthe aforementioned control algorithms. In order to further test theperformances of the designed controllers, the dynamical model ofthe quadrotor along with the controllers is simulated on Matlab/Simulink. The main conclusions are summarized as follows. (a) Thesix degrees of freedom converge to their desired/reference valuesin finite-time, respectively. (b) The convergence time of the statevariables z andis essentially consistent with the theoreticallycalculated values. (c) Compared to other variables, the pitch angleand the controller u1become time invariants in short time.(d) The four control input variables converge to their steady valuesin finite-time,u1and u4are almost no oscillation amplitudes. Allabove, the effectiveness and robustness of the proposed synthesiscontrol method have been demonstrated, and, the presentedsimulation results are promising at the position and attitudetracking control for the aircraft.AcknowledgmentThis work was partially supported by the National NaturalScience Foundation of China (60905034).References1 Raffo GV, Ortega MG, Rubio FR. An integral predictive/nonlinear H1 controlstructure for a quadrotor helicopter. Automatica 2010;46:2939.2 Derafa L, Benallegue A, Fridman L. Super twisting control algorithm for theattitude tracking of a four rotors UAV. J Frankl Inst 2012;349:68599.3 Luque-Vega L, Castillo-Toledo B, Loukianov AG. Robust block second ordersliding mode control for a quadrotor. J Frankl Inst 2012;349:71939.4 Garcia-Delgado L, Dzul A, Santibez V, Llama M. Quadrotors formationbansed on potential functions with obstacle avoidance. IET Control TheoryAppl 2012;6(12):1787802.5 Alexis K, Nikolakopoulos G, Tzes A. Model predictive quadrotor control:attitude, altitude and position experimental studies. IET Control Theory Appl2012;6(12):181227.6 Nekoukar V, Erfanian A. Adaptive fuzzy terminal sliding mode control for aclass of MIMO uncertain nonlinear systems. Fuzzy Sets Syst 2011;179:349.7 Ashrafiuon H, Scott-Erwin R. Sliding mode control of under-actuated multi-body systems and its application to shape change control. Int J Control2008;81(12):184958.0510152025303540-0.1500.15p ( rad/s )0510152025303540-0.100.1q ( rad/s )0510152025303540-1.201.2Time ( s )r ( rad/s )Fig. 8. The angular velocities (p,q,r), novel robust TSMC and SMC.051015202530354010203040u1 ( N )u2 ( N )u3 ( N )u4 ( N )X: 17.86Y: 19.660510152025303540-100100510152025303540-3030510152025303540-10010Time ( s )Fig. 10. The controllers (u1,u2,u3,u4), novel robust TSMC and SMC.0510152025303540-202s2s4sX: 0.8018Y: 0.0006840510152025303540-1.201.2X: 1.338Y: -0.00021290510152025303540-20020Time ( s )s(1)s(2)Fig. 9. The sliding variables (s2,s4,s), novel robust TSMC and SMC.J.-J. Xiong, E.-H. Zheng / ISA Transactions () 6Please cite this article as: Xiong J-J, Zheng E-H. Position and attitude tracking contro
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