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Modeling and Sliding Mode Control of a Quadrotor

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  Modeling and Sliding Mode Control of a QuadrotorUnmanned Aerial Vehicle Nour BEN AMMAR, Soufiene BOUALL`EGUE and Joseph HAGG`EGE Research Laboratory in Automatic Control (LA.R.A), National Engineering School of Tunis (ENIT)University of Tunis El Manar, BP 37, Le Belv´ed`ere, 1002 Tunis, TunisiaE-mails: nourelhouda.benammar@enit.rnu.tn, soufiene.bouallegue@issig.rnu.tn, joseph.haggege@enit.rnu.tn  Abstract —In this paper, a detailed mathematical model fora Quadrotor Vertical Take-Off and Landing (VTOL) type of Unmanned Aerial Vehicles (UAVs) is firstly established for thenonlinear attitude and position control. All aerodynamic forcesand moments of the studied Quadrotor UAV are describedwithin an inertial frame. The dynamic model is obtained usingthe Newton-Euler formalism. A nonlinear Sliding Mode Con-trol (SMC) approach is then designed for this vehicle in order tostabilize its vertical flight dynamics. The tracking of an helicaldesired trajectory is investigated for the SMC-controlled Quadrotorcraft. Demonstrative numerical simulation are carried outin order to demonstrate the effectiveness of the proposed controlapproach.  Index Terms —VTOL aircraft, Quadrotor UAV, modeling, flightdynamics, sliding mode control, attitude and position stabiliza-tion, Lyapunov theory, path tracking. I. I NTRODUCTION An Unmanned Aerial Vehicle (UAV) refers to a flyingmachine without an on-board human pilot [1], [2], [3], [4],[5]. These vehicles are being increasingly used in many civildomains, especially for surveillance, environmental researches,security, rescue and traffic monitoring.Researchers have led to different designs for this type of aircrafts. A Quadrotor UAV is one of the Vertical Take-Off andLanding (VTOL) designs which are proven to have promisingflying concepts due to their high maneuverability. The complexmechanical structure of the Quadrotor, its strongly nonlinearand coupled dynamics, its multiple inputs-outputs and theobservation difficulty of its states allowed this VTOL aircraftto be a popular topic of research in the field of robotics andnonlinear control theory. So, modeling and control of this kindof nonlinearl systems became increasingly difficult and hardtasks in the practical design and prototyping framework.Several linear control approaches, such as PID,LinearQuadratic Regulator (LQR) and Linear Quadratic Gaussian(LQG), have been proposed in the literature and applied forattitude stabilization and/or altitude tracking of Quadrotors[6], [7]. However, these methods can impose limitations onapplication of Quadrotors for extended flight regions, i.e.aggressive maneuvers, where the system is no longer linear.Moreover, the stability of the closed-loop system can only beachieved for small regions around the equilibrium point, whichare extremely hard to compute. In addition, the performanceson tracking trajectories of these control laws are not satisfac-tory enough comparing with other more advanced methods.To overcome this problem, nonlinear control alternatives, suchas the feedback linearization [8], SMC [9], [10], [11] andBackstepping [13] approaches are recently used in the VTOLaircrafts control framework. An integral predictive/nonlinear H ∞  strategy has been also proposed and applied by G.V.Raffo et al. in [12]. In this paper, a nonlinear SMC approachis proposed for the attitude stabilization for a Quadrotor. Roll,pitch and yaw dynamics are separately controlled thanks toLyapunov-based designed SMC controllers. A nonlinear modelof the studied UAV is firstly established using the Newton-Euler formulation.The remainder of this paper is organized as follows. SectionII presents the flight dynamics modeling of the Quadrotor UAVbased on the well known Newton-Euler approach. Section IIIis devoted to design a nonlinear SMC approach for the UAVflight stabilization and path tracking. All numerical simulationresults, obtained for modeling and control, are presented anddiscussed in Section IV. Section V concludes this paper.II. M ODELING OF THE  Q UADROTOR  UAVDesign and analysis of control systems are usually startedby carefully considering mathematical models of physicalsystems. In this section, a complete dynamical model of thestudied Quadrotor UAV is established using the Newton-Eulerformalism.  A. System description and aerodynamic forces A Quadrotor is an UAV with four rotors that are controlledindependently. The movement of the Quadrotor results fromchanges in the speed of the rotors. The structure of Quadrotorin this paper is assumed to be rigid and symmetrical. Thecenter of gravity and the body fixed frame srcin are coincided.The propellers are rigid and the thrust and drag forces areproportional to the square of propeller’s speed. The studiedQuadrotor rotorcraft is detailed with their body- and inertial-frames       (       )  and       (           )  respec-tively, as shown in Fig. 1.Let consider the following model partitions naturally intotranslational and rotational coordinates [1], [3], [4], [5]:        ∈ R 3      ϕ  ∈ R 3 (1)where          denotes the position vector of the centerof mass of the Quadrotor relative to the fixed inertial frame, ISSN: 2356-5608 Proceedings of Engineering & Technology (PET) pp. 834-840 Copyright IPCO-2016 3rd International Conference on Automation, Control, Engineering and Computer Science (ACECS'16)  Fig. 1. Mechanical structure of the Quadrotor and related frames.     ϕ   denotes the attitude of the Quadrotor given bythe Euler angles  ϕ ,    and   .We note that,  ϕ  is the roll angle around the   -axis,    is thepitch angle around the   -axis and    are the roll angle aroundthe   -axis. All those angles are bounded as follows: −     ϕ     (2) −          (3) −       (4)Each motor M   (  =1, 2, 3 and 4) of the Quadrotor producesthe force which is proportional to the square of the angularspeed. Known that the motors are supposedly turning only in afixed direction, the produced force      is always positive. Thefront and rear motors (M1 and M3) rotate counter-clockwise,while the left and right motors (M2 and M4) rotate clockwise.As given in [1], [5], [2], the gyroscopic effects and theaerodynamic torques tend to cancel in trimmed flight becausethe mechanical design of the Quadrotor. The total thrust     isthe sum of individual thrusts of each motor. Let denote by   the total mass of the Quadrotor and    the acceleration of thegravity.The orientation of the Quadrotor is given by the rotationmatrix            →       which depends on the three Euler angles  ϕ   and defined by the following equation:   ϕ     ϕ −  ϕ    ϕ ϕ     ϕ − ϕ −  ϕ ϕ  (5)where           and          .During its flight, the Quadrotor is subjected to externalforces like the gusts of wind, gravity, viscous friction andothers self generated such as the thrust and drag forces. Inaddition, external torques are provided mainly by the trustof rotors and the drag on the body and propellers. Momentsgenerated by gyroscopic effects of motors are also noted.The trust force generated by the     rotor of the Quadrotoris given by:              2  2      2   (6)where    is the air density,    and    are the radius and thesection of the propeller respectively,       is the aerodynamicthrust coefficient.The aerodynamic drag torque, caused by the drag force atthe propeller of the     rotor and opposed the motor torque,is defined as follows:             2  2      2   (7)where      is the aerodynamic drag coefficient.The pitch torque is a function of the difference (   3 −   1 ),the roll torque is proportional to the term (   4 −   2 ) and theyaw one is the sum of all reactions torques generated by thefour rotors and due to the shaft acceleration and propellerdrag. All these pitching, rolling and yawing torques are definedrespectively as follows:           3 −   1   (8)   ϕ        4 −   2   (9)           1 −   2      3 −   4   (10)where    is a constant coefficient and    denotes the distancefrom the center of each rotor to the center of gravity.Two gyroscopic effects torques, due to the motion of thepropellers and the Quadrotor body, are additively provided.These moments are given respectively by:      4   =1 Ω ∧ [          −   +1   ]   (11)      Ω ∧   Ω  (12)where  Ω  is the vector of the angular velocity in the fixedearth frame and                    is the inertia matrix of the Quadrotor,     ,      and      denote the inertias of the   -axis,  -axis and   -axis of the Quadrotor, respectively,      denotesthe   -axis inertia of the propellers’ rotors.The Quadrotor is controlled by independently varying thespeed of the four rotors. Hence, these control inputs are definedas follows:   1  2  3  4       ϕ                −      −          −    −    21  22  23  24  (13)where       and       are two parameters depending on theair density, the geometry and the lift and drag coefficients of the propeller as given in Eq. (6) and Eq. (7), and   1  2  3  4  arethe angular speeds of the four rotors, respectively.From Eq. (13), it can be observed that the input   1  denotesthe total thrust force on the Quadrotor body around the   -axis, the inputs   2  and   3  represent the roll and pitch torques,respectively. The input   4  represents the yawing torque.   B. Modeling with Newton-Euler formalism While using the Newton-Euler formalism for modeling, theNewton’s laws lead to the following motion equations of theQuadrotor: {                           Ω      −    −    −    (14)where          ϕ       4 ∑  =1       denotes the totalthrust force of the four rotors,           1  2  3       is the air drag force which resists to the Quadrotor motion,             is the gravity force,         ϕ          represents the total rolling, pitching and yawing torques,     and      are the gyroscopic torques and          4  5  6    ϕ 2     2     2    is the torque resulting from theaerodynamic frictions.Substituting the position vector and the forces expressionsintoEq. (14), we have the following translational dynamics of the Quadrotor:          ϕ    ϕ   1 −   1             ϕ   1 −   2          ϕ 1 −  −   3     (15)From the second part of Eq. (14), and while substitutingeach moment by its expression, we deduce the followingrotational dynamics of the rotorcraft:   ϕ       −             −        Ω     −   4     ϕ 2       2         −         ϕ    −        Ω    ϕ −   5      2       3         −            ϕ −   6      2       4 (16)where   1  2  6  are the drag coefficients and positive constant, Ω      1 −  2     3 −  4  is the overall residual rotor angularvelocity.Taking       ϕ   ϕ                   as state vec-tor, the following state-space representation of the studiedQuadrotor is obtained as follows:                 1     2   2     1  4  6     3 Ω   4     2  22     1  2   3     4   4     4  2  6     6 Ω   2     5  24     2  3   5     6   6     7  2  4     8  26     3  4   7     8   8     9  8    1    ϕ    ϕ   1   9     10   10     10  10    1    ϕ − ϕ   1   11     12   12     11  12    ϕ   1 −  (17)where:  1       −         2   −  4      3   −         4       −          5   −  5      6   −         7       −          8   −  6      9   −  1    10   −  2    11   −  3    1         2         3       III. S LIDING  M ODE  C ONTROL OF THE  Q UADROTOR  A. Basic concepts of SMC  The SMC is a type of Variable Structure Control (VSC).Its basic idea is to attract the system states towards a surface,called sliding surface, suitably chosen and design a stabilizingcontrol law that keeps the system states on such a surface. Forthe choice of the sliding surface shape, the general form of Eq. (18) was proposed by Stoline and Li in [13]:                 − 1      (18)where    denotes the variable control (state),       is thetracking error defined as          −    ,     is a positiveconstant that interprets the dynamics of the surface and     isthe relative degree of the sliding mode controller.Condition, called attractiveness is the condition under whichthe state trajectory will reach the sliding surface. There aretwo types of conditions of access to the sliding surface. Inthis paper, we will use the Lyapunov based approach. Itconsists to make a positive scalar function, given by Eq. (19)and called Lyapunov candidate function, for the system statevariables and then choose the control law that will decreasethis function:                             (19)In this case, the Lyapunov function can be chosen as:               2 (20)The derivative of this above function is negative when thefollowing expression is checked:                 (21)The purpose is to force the system state trajectories toreach the sliding surface and stay on it despite the presenceof uncertainty. The sliding control law contains two terms asfollows:                      (22)where         denotes the equivalent control which is a way todetermine the behaviour of the system when an ideal slidingregime is established. it is calculated from the followinginvariance condition of the surface: {                   (23)  and         is a discontinuous function calculated by checkingthe condition of the attractiveness. It is useful to compensatethe uncertainties of the model and often defined as follows:        −          (24)where     is a positive control parameter and       is thesign operator.  B. SMC controllers design for the Quadrotor  For the attitude control, we use the rotational motion modelgiven by Eq. (16). The translational dynamics model of Eq. (15) is used to design the Quadrotor position controller.Let also consider the state vector given by Eq. (17).We begin by defining the tracking errors which representthe difference between the set-point and current values of thestate:  {    +1             −            (25)The sliding surfaces are chosen based on the tracking errorssuch as:     ϕ     2     1  1        4     2  3        6     3  5        8     4  7        10     5  9        12     6  11 (26)Let consider for the roll dynamics SMC design the follow-ing Lyapunov function:        ϕ      2 ϕ  (27)While referring to Eq. (19) and Eq. (21), we deduce theexpression of the derivative roll surface given as:    ϕ   −   1     ϕ   (28)By changing    2  with its expression and referring to theabove equations, the control law   2  is given by:  2     1   −  1  4  6 −  3 Ω   4 −  2  22   1  −  1   1 −   1     ϕ    (29)While following exactly the same steps as the roll controllerdesign, the control inputs   3  and   4 , responsible of generatingthe pitch and yaw rotations respectively, are calculated asfollows:  3     2   −  4  2  6 −  6 Ω   2 −  5  24   3  −  2   3 −   2         (30)  4     3   −  7  2  4 −  8  26   5  −  3   5 −   3         (31)Using the same method, we deduced the control laws   1 ,    and     for the stabilization of    ,    and    positions of theQuadrotor, respectively. These control inputs are computed asfollows:  1    ϕ   −  11  12     11  −  6   11 −   6           (32) TABLE IQ UADROTOR MODEL PARAMETERS .Parameters Values and unitsLift coefficient  b  2.984 e-05 N.s 2 /rad 2 Drag coefficient  d  3.30 e-07  N.s 2 /rad 2 Mass  m  0.5  kg Arm length  l  50  cm Motor inertia  J  r  2.8385 e-05  N.m/rad/s 2 Quadrotor inertia  J diag  (0 . 005 , 0 . 005 , 0 . 010) aerodynamic friction coeffs.  κ 1 , 2 , 3  0.3729translational drag coeffs.  κ 4 , 5 , 6  5.56 e-04acceleartion of the gravity  g  9.81  m/s 2       1  −  9  8     7  −  4   7 −   4        (33)       1  −  9  10     9  −  5   9 −   5        (34)IV. S IMULATION  R ESULTS  A ND  D ISCUSSION In this section, the proposed SMC approach for the Quadro-tor attitude stabilization is implemented in order to verify hisvalidity and efficiency. For the simulation, we use the physicalparameters of Table I . The initial position and angle valuesare set as          and         .Even though the reference angle were changed in everymoment, the proposed control scheme managed to effectivelyhold the quadrotor’s attitude in finite-time, as shown in Fig. 2and Fig. 3 for the attitude dynamics control, and in Fig. 4 andFig. 5 for the position dynamics tracking. In Fig. 6, we presentthe helical trajectory tracking of the Quadrotor. It is shown thateven though the quadrotor’s attitude and position are affectedby the abruptly changed reference angles, the designed SMCcontrollers are able to drive all these state variables back to thenew reference angle and position within seconds. Moreover,the aerodynamic forces and moments are taken into accountin the controllers design. Those demonstrate the robustness of the proposed control strategy and its effectiveness.  Time (sec)0 10 20 30 40 60    r   o    l    l   a   n   g    l   e    (   r   a    d    ) -101 φ Desired  Time (sec)0 10 20 30 40 60    p    i   t   c    h   a   n   g    l   e    (   r   a    d    ) -505 θ Desired  Time (sec)0 10 20 30 40 60     Y   a   w    a   n   g    l   e    (   r   a    d    ) -202 ψ Desired 505050 Fig. 2. SMC- based results for the attitude tracking of the Quadrotor.
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