MiG-21 modeling and simulation

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This work provides MiG-21 modeling as an unmanned aerial vehicle by the full nonlinear equations of motion: Force Equations and Moment Equation; Kinematic equations and Navigation equations. By this model, the responses of UAV to different control effects are surveyed in simulation, which evaluates the UAV''s ability to control when it performs complex motions. Research MIG-21 MODELING AND SIMULATION Le Ngoc Lan1*, Nguyen Vu2, Hoang Minh Dac1, Pham Thi Phuong Anh1, Le Thanh Ngoc3 Abstract: This work provides MiG-21 modeling as an unmanned aerial vehicle by the full nonlinear equations of motion: Force Equations and Moment Equation; Kinematic equations and Navigation equations By this model, the responses of UAV to different control effects are surveyed in simulation, which evaluates the UAV's ability to control when it performs complex motions This is the basic background for the synthesis of UAV-MiG-21 control systems to follow complex orbits Keywords: Nonlinear equations of motion, Simulation in simulink, Modeling of MiG-21 NOMENCLATURE Fi (Oi xi yi zi) Fb (Ob xb yb zb) FS (OS xS yS zS) Fw (Ow xw yw zw) , , ψ p, q, r u, v, w Inertia reference frame Body-fixed frame Stability frame Wind frame Roll, Pitch, Yaw angles Roll, Pitch, Yaw rates Body frame velocity of aircraft in the x, y, z-body directions  Flight path angle  Flight path heading angle α Angle of Attack (AOA) β Side-slip angle (or Angle of Side-slip: AOS)  Air density S Wing area total b, c Wing span, Wing chord Cx, Cy, Cz (CD, CY, CL) Drag, Side and Lift force aerodynamic coefficients Cl, Cm, Cn X, Y, Z axis aerodynamic moment coefficients NED North, East, and Down components of n frame vector INTRODUCTION Dynamic modeling is an important step in the development and the control of a system as UAV or autonomous aircraft In fact the model allows the designers to analyze the system possibilities and its behavior under various conditions Especially for UAV it is highly appreciable to have a closed to real model for simulation and controller tuning before implementing For describing the motion of unmanned aerial vehicles, the usual model of lateral motion, longitudinal motion and stabilization of bank angle are used [8], [9] They are linear models with approximate parameters defined in the standard conditions However, in case of complex maneuvers, they cannot be separated into independent motions, the development of dynamic model for simulation based on full nonlinear equations is required, especially for type of UAV built from fighter aircraft Journal of Military Science and Technology, Special Issue, No.57A, 11 - 2018 35 Electronics and Automation In recent years, the military has missions that need to use MiG-21 to automatically perform missions In order to synthesize the control system for MiG21 aircraft, it is necessary to build a dynamic modeling for the MiG-21 and to correct the model parameters (including trigonometric parameters) To solve this problem, the full nonlinear model for the MiG-21 aircraft is used The modeling for MiG-21 by applying standard method for deriving the full nonlinear equations of motion: the Force and Moment Equations, the Kinematic and Navigation Equations in order are presented bellow COORDINATE SYSTEMS AND UAV ORIENTATION AND POSITION CALCULATION DIAGRAM In this paper uses the coordinate system follows ([1], [2]): Inertial Frame (NED), Tangent plane frame (NED), Vehicle-carried vertical frame, Body-fixed frame, Stability frame, and Wind frame This section presents the mathematical relationship between the rotational motion of the axes in moving of aircraft and the solving in the building of models 2.1 Axis conventions and direction cosine matrix - The relationship between the vehicle-carried vertical and body frame: The Euler angles (, , and ) define the orientation of the body fixed frame with respect to the vehicle-carried vertical frame The rotations required to transform the vehicle-carried vertical frame to the body-fixed frame The heading angle  is a rotation about the zV axis into a new frame (designated (x1, y1, z1)); the pitch angle  is a rotation about the y1 axis in to the (x2, y2, z2); the roll angle  is a rotation about the x2 (or xB) axis into the body-fixed axis system x2, xb yV x1    y1, y2 O  xV yb   zb z2 zV, z1 Figure Relationship between vehicle-carried vertical and body-fixed frames These rotations are described by: LBV (DCM) = L.L.L 36 (1) L N Lan, …, L T Ngoc, “MiG-21 modeling and simulation.” Research cos cos  cos sin  sin   =  sin  sin  cos  cos  sin sin  sin  sin  cos  cos sin  cos   cos  sin  cos  sin  sin  (2) cos  cos  cos  sin  sin  sin  cos - Relationship between the body-fixed, wind and stability frame: The relationship between the body-fixed, wind, and stability frame are show in figure All three frames have their origin at the center of gravity of the aircraft (plane) The x axis in the wind frame (xW) is aligned with the velocity vector of the aircraft The side-slip angle β and angle of attack α define the orientation of the wind with respect to the body axes O yw v u α w yb,ys β zb,zw xb zb Vacosβ Vasinβ xs Va xw Figure Relationship of body-fixed, stability and wind frames The total transformation from the body frame to the wind frame is given by: LWB (α,β) = LSB(α).Lws(β) =  cos  cos    sin   cos  sin    sin  cos  cos   sin  sin   sin     cos   (3) The definitions of the body-fixed frame are: u = Vacosαcosβ v = Vasinβ w = Vasinαcosβ (4) The total velocity V, angle of attack α, and side-slip angle β, can be expressed in terms of these body-fixed axis velocities as: Va = |Va| = (u2 + v2 + w2)1/2 α = tan-1(w/u) β = sin-1(v/Va) (5) Including the impact of the wind - The relationship between wind frame and tangent plane frame (geographic system): The flight path coordinate system relates the velocity vector of the aircraft with respect to the Earth to the geographic system via two angles: the flight path angle γ and heading angle χ The heading angle χ is measured from North to the projection of Va in the local tangent plane and the flight path angle γ takes vertically up to Va Journal of Military Science and Technology, Special Issue, No.57A, 11 - 2018 37 Electronics and Automation The transformation matrix is given by:  cos  cos   LVG    sin   sin  cos   cos  sin  cos  sin  cos   sin     cos   (6) 2.2 UAV orientation and position calculation algorithm diagram There are two commonly used methods of expressing the orientation of one three axis coordinate system with respect to another The two methods are Euler angles and Quaternions The Euler angle method, which is the conventional designation relating a moving-axis system to a fixed-axis system, is used frequently in aircraft simulations Its strengths lie in a relatively simple mechanization in digital computer simulation of aircraft dynamics Another beneficial aspect of this technique is that the Euler angle rates and the Euler angles have an easily interpreted physical significance The negative attribute to the Euler angle coordinate transformation method is the mathematical singularity that exists when the pitch angle θ approaches 900 In order to resolve the ambiguity resulting from the singularity in the Euler angle representation of rotations about the three axes, a four-parameter system - the quaternion system - was developed Solving trigonometric equations in moving of aircraft steps according to the following diagram, figure 3 MIG-21 MOTION EQUATIONS First step of design a controller of aircraft (UAV) is to derive the dynamic model This section will present on standard method for deriving the full nonlinear equations of motion of fixed wing aircraft (MiG-21 etc…) and applying the solving trigonometric equations in moving of aircraft 3.1 Aircraft forces and moments Newton’s second law of motion is used to determine the effect of the net force and moment on a body It states that the summation of all external forces acting on a body is equal to the time rate of change of the linear momentum of the body, and the summation of the external moments acting on the body is equal to the time rate of change of the angular momentum The summary of the equations consists of three forces and moments equations are expressed in matrix form in Equations to 14 The gravitational force acting on the airplane acts through the center of gravity (CG) of the airplane Because the body axis system is fixed to the center of gravity, the gravitational force will not produce any moments It will contribute to the external force acting on the airplane, however, and have components along the respective body axis The gravity components along the x, y, and z axes are expressed in Eq 11 The thrust force due to propulsion can have components that act along each of the body axis In addition, the propulsive forces can create moments if the thrust does not act through the CG (MiG-21 has the thrust act through the CG so propulsion couldn’t the moment) The propulsive forces and moments acting along 38 L N Lan, …, L T Ngoc, “MiG-21 modeling and simulation.” Research the body axis system are denoted in Eq and Eq 13 respectively - Figure Steps to model the aircraft (MiG-21) applying solving trigonometric equations in moving Force Equations: Newton’s second law:   Fx   ax   Fx    a     F    F    y M  y  y  Aircarft    Fz   az   Fz  Pr op  Aero   F Where the Faero are given by:  x  Fy   Fz  Aero and MiG-21 has: 1   V a SC x  1    V a2 SC y  2   V SC  z   a   Fx   Fx  P F    0     y  Fz  Pr op   Pr op   (7) (8) (9) P – the propeller thrust force Journal of Military Science and Technology, Special Issue, No.57A, 11 - 2018 39 Electronics and Automation  u   rv  qw   g x   a x           v    pw  ru    g y    a y   w   qu  pv   g   a       z  z (10)  gx      g sin         g  L  y BV     g sin  cos   g   g   g cos  cos    z  z   where: (11) By integrating the Eq 10, one can determine the velocities of aircraft (u, v, w), after that angle of attack α, and side-slip angle β is given by Eq - Moment Equations: MCG = (rAero - rCG).FAero + MAero + (rProp-rCG).FProp + MProp (12)    J x J z  J xz2 C1  C5  ( J y  J z ) J z  J xz2 C2   Jz  Jx Jy C6  M Aero J xz Jy C7  Jz  ( J  J y ) J x  J xz2 C8  x  ( J x  J y  J z ) J xz  Jy C3  1   Va SbCl   M roll        M pitch    Va2 ScCm    M  2  yaw   V SbC   a n 2   p   (C1.r  C2 p ).q  C3 M roll  C4 M yaw      2  q    C5 q  C6 ( p  q )  C7 M pitch   r   (C p  C r ).q  C M  C M  roll yaw     C4  J xz  C9  Jx  (13) (14) The angular velocities of aircraft (p, q, r) can determine by integrating the Eq 14 3.2 Kinematic equations The orientation of the airplane can be described by three consecutive rotations whose order is important The angular rotations are called the Euler angles Euler method: The relationship between the angular velocities in the body frame (p, q and r) and the Euler rates (  ,  , and  ) is shown:     sin  tan     cos             sin  / cos  cos  tan    p     sin  . q  cos  / cos    r  (15) By integrating theses equations, one can determine the Euler angles (, , and ) Quaternion method: Another way, the kinematic equations are using the Euler-Rodrigues quaternions This type of implementation is considered superior to the simple Euler angle equations, since quaternion equations are linear and the solution does not 40 L N Lan, …, L T Ngoc, “MiG-21 modeling and simulation.” Research exhibit gimbal lock singularity The equations, which are presented below, are described in more detail in [2]  q    p  q  r   q0       r  q   qx   q x   p  q    q  r p   qy  y      r  q  q  p   q z    z (16) By integrating theses equations, one can determine quaternions As shown in [2], the equations for computing the Euler angles are: 2 2     a tan 2[ 2( q0 q x  q y q z ), ( q0  q z  q x  q y )]      a sin[ 2( q0 q y  q x q z )]         a tan 2[ 2( q q  q q ), ( q  q  q  q )]  z x y x y z     (17) 3.3 Navigation equations The state variables pn, pe, and pd are inertial frame position quantities, where as the velocities u, v, and w are body frame quantities The relationship between the position and velocities in the inertial frame is given by:  p n  VN  u     1      p  V  DCM  e  E v  p  V   w  d  D   (Navigation equations) (18) where DCM = LBV could give by Eq cos cos  p n       p e    sin  sin  cos  cos  sin  p   cos  sin  cos  sin  sin  d  cos sin sin  sin  sin  cos  cos cos  sin  sin  sin  cos  sin    u    sin  cos . v  cos  cos   w  (19) By integrating theses equations, one can determine the position of aircraft (pn, pe, pd) Relationship between the velocities in tangent plane frame (geographic system) and velocities in wind frame is given by: VN  V   cos  cos  cos  sin    1  a   cos   VE   LVG      sin  V     sin  cos  sin  cos   D     sin     cos   1 Va      0   (20) The Flight path angle and Ground speed heading angle are given by:  = asin(VD/Va)  = atan(VE/VN) The Equations 10, 14, 15, 18 are full nonlinear equations of motion 3.4 MiG-21 nonlinear model implementation in simulink Aircraft Orientation and Position equations; Aircraft Forces and Moments equations; and the relationship equations of spatial coordinates in space (trigonometric equations) described in Section 2, can now be implemented in Simulink environment Atmosphere Model is used on the Standard Atmosphere (Compute standard atmosphere pressure, temperature, density and speed of sound as functions of MSL altitude) Earth Model is built base on The WGS-84 Earth model (provides local Earth radii and gravity as a function of position) Aerodynamic, Journal of Military Science and Technology, Special Issue, No.57A, 11 - 2018 41 Electronics and Automation Propulsion, and Inertia coefficients are used on the MiG MiG-21 21 aircraft database Figure 4, shows the general structure of MiG MiG-21 21 Simulink nonlinear model Figure MiG-21 MiG 21 Simulink model general sstructure tructure tructure SIMULATION RESULTS AND DISCUSSION After getting the model; some checks of the MiG MiG-21 21 longitudinal dynamics responses to (elevator) and Lateral dynamics responses to (aileron, rudder) deflections of non non-linear near models are illustrated in ffigures igures 55-7 The response responsess of MiG MiG-21 21 when aircraft performs maneuver are illustrated in figures igures 8, Initial conditions for simulation: Maircraft craft=8890kg, Va=244.9m/s, α=2.91950, =2.91950, =2.91950, Elevator = =-2.952260, 2.952260, Quaternion parameters = [0.9997, 0, 0.02547, 0], Engine thrust=13170N, Altitude=5000m 4.1 MiG-21 MiG 21 mo model del responses to control inputs MiG-21 MiG 21 dynamics response responses to In Init itiall conditions conditions are illustrated in ffigure igure 55 The aircraft is stabilized at an initial altitude (5000m) with deflection of elevator is 2.952660 In this graph shows shows when controls elevator generate generatess moment Mpitch, this will cause the aircraft to pitch to up up Pitch rate q varies around to the steady state at 0,, angle of attach goes to the steady state state This Th s shows that the system is controllable and stable in longitudinal motion motion Keep the initial conditions for simulation, examines aircraft response to a unit (11 ) 1s aileron step input and a unit rudder step input input The response of the aircraft dynamics a unit aileron step input is shown in fig igure ure Both the Roll and S Spiral piral modes appear convergent characteristics in this check The R Roll oll mode converges relatively quickly quickly,, whereas the S Spiral piral converges very slowly indeed The R Roll oll mode is most clearly seen in the roll rate response p where it determines rise at the zero seconds and recovery when the pulse is removed at 1s Spiral mode characteristic is rather more subtle and is most easily seen in roll 42 L N Lan, …, L T Ngoc, “MiG “MiG 21 21 m modeling odeling and simulation imulation imulation.” ” Research attitude response  where it determines the longer term convergence to zero Figure MiG-21 MiG 21 response to -2.95266 2.952660 elevator input input Figure MiG-21 MiG 21 response to 10 aileron step input input Journal of Military Science and Technology, Special Issue, No No.57A 57A, 11 - 2018 2018 43 Electronics and Automation The response of the aircraft dynamics a unit rudder step input is shown in figure It very clear that the response is dominated by the oscillatory Dutch Roll mode (the Roll mode and Spiral mode are not discernible in the response) Figure MiG-21 response to 10 rudder step input Through the system's response (see figure 7), it is shown that when the angle of bank is small, the calculation of α=-, =- has a small error, so it is still appropriate The responses of MiG-21 Lateral dynamics show that this model is controllable and stable in Lateral motion 4.2 MiG-21 model in basic maneuvers Keep the initial conditions for this simulation - Turn with constant bank angle (450) The response of MiG-21 when MiG-21 performs turn circle with constant bank angle =450 is shown in the figure The time spent flying the aircraft from ψ =00 to ψ =3600 is 170s According to the MiG-21Bis Pilot’s Flight Operating 0.64 *VAvg 0.64 * (244.9  254.8)   160s Instructions, the maneuvering time is tan  * tan 45 44 L N Lan, …, L T Ngoc, “MiG-21 modeling and simulation.” Research That corresponds to the maneuverability of the MiG-21 Figure MiG-21 response to when performs turn circle at bank angle =450 - Helical roll with constant aileron angle (-70) Figure MiG-21 response to when MiG-21 performs roll 0-3600(helical roll) MiG-21 response to when MiG-21 performs helical roll is shown in the figure Journal of Military Science and Technology, Special Issue, No.57A, 11 - 2018 45 Electronics and Automation This can by controls aileron =-70 in 9,4s The total time taken to make the maneuver helical roll (from =00 to =3600) is 9,4s That corresponds to the maneuverability of the MiG-21 In addition, the graph shows that when the plane performs maneuver at a small roll angle, approximation methods α =  - and β = ψ - have relatively small deviations (they will be equal when the roll angle  =0); but when maneuvering with large roll angle, the difference between them increases In addition, the responses are shown in figure 8, shows that these models with approximate parameters defined in the standard conditions are no longer true Surveying of the MiG-21 model has responses in accordance with MiG-21 aircraft documentation CONCLUSION The objective this paper is achieved by applying standard method for deriving the full nonlinear equations of motion and methods solving trigonometric equations in moving of fixed wing aircraft for the MiG-21 The results of the mathematical model are used to build and survey the model in Simulink The responses of MiG-21 dynamics show that this model is controllable and stable Surveying of the MiG-21 model when the aircraft performs maneuver has responses correspond to the maneuverability of the MiG-21 The model is efficient and suitable for designing an accurate controller for various phases of flights (including complex maneuvers) in Simulink The built model makes the synthesis of control system for unmanned MiG-21 aircraft for performance complicate fly mission become more advantageous REFERENCES [1] Yasimina Bestaoui Sebbane Smart Autonomous Aircraft-Flight Control and Planning for UAV CRC Press, Taylor & Francis Group ISBN-13: 978-14822-9916-8, 2016 [2] Philips W.F Hailey C.E and Gebert G.A Review of Attitude Representations Used for Aircraft Kinematics Journal of Aircraft ISBN-38: 718-737, 2001 [3] M.V.Cook Flight dynamics principles Butterworth-Heinemann ISBN: 978-0-7506-6927-6, 2007 [4] A Elsayed Ahmed, A Hafez, A N Ouda, H Eldin Hussein Ahmed, H Mohamed Abd-Elkader Modeling of a Small Unmanned Aerial Vehicle World Academy of Science, Engineering and Technology International Journal of Aerospace and Mechanical Engineering 2015 [5] Mark Peter, Michael A.Konyak The Engineering Analysis and Design of Aircraft Dynamics Model For the FAA Target Generation Facility Air Traffic Engineering Co., LLC 2012 [6] Sổ tay phi công MiG-21BiS, Quân chủng Không quân, 1980 [7] Kỹ thuật lái dẫn đường máy bay MiG-21BiS, Quân chủng Không quân, 1987 [8] Đinh Văn Tuân Xây dựng phương pháp tổng hợp hệ thống điều khiển cho 46 L N Lan, …, L T Ngoc, “MiG-21 modeling and simulation.” Research khí cụ bay hành trình có thiết bị dẫn đường qn tính, 2009, Luận án tốt nghiệp [9] Vũ Hồng Quang Tổng hợp hệ thống điều khiển chuyển động cạnh khoang cho máy bay khơng người lái, 2008, Luận án tốt nghiệp TĨM TẮT XÂY DỰNG MƠ HÌNH VÀ MƠ PHỎNG MiG-21 Bài báo đề cập đến việc xây dựng mơ hình động học đầy đủ máy bay MiG21 thiết bị bay không người lái sở sử dụng hệ phương trình động học phi tuyến đầy đủ: Các phương trình lực mơment, phương trình động hình học vị trí Bằng mơ hình động học xây dựng được, đáp ứng UAV tác động điều khiển khác nghiên cứu thông qua phương pháp mơ phỏng, qua đánh giá khả điều khiển UAV thực quỹ đạo bay phức tạp Đây sở ban đầu để tổng hợp hệ điều khiển UAV-MiG-21 theo quỹ đạo phức tạp Từ khóa: Các phương trình chuyển động phi tuyến, Simulink, Mơ hình động học máy bay MiG-21 Received 2nd July 2018 Revised 22 th September 2018 Accepted 17 th October 2018 Author affiliations: Academy of Military Science and Technology) Department of Military Science; Air Defense and Air Force Technical Institute *Email: lengoclan12@gmail.com Journal of Military Science and Technology, Special Issue, No.57A, 11 - 2018 47 ... through the CG (MiG-21 has the thrust act through the CG so propulsion couldn’t the moment) The propulsive forces and moments acting along 38 L N Lan, …, L T Ngoc, MiG-21 modeling and simulation. ”... is tan  * tan 45 44 L N Lan, …, L T Ngoc, MiG-21 modeling and simulation. ” Research That corresponds to the maneuverability of the MiG-21 Figure MiG-21 response to when performs turn circle... dynamic modeling for the MiG-21 and to correct the model parameters (including trigonometric parameters) To solve this problem, the full nonlinear model for the MiG-21 aircraft is used The modeling
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