Mini coaxial rocket-helicopter: aerodynamic modeling, hover control, and implementation

Transcription

1 Mini coaxial rocket-helicopter: aerodynamic modeling, hover control, and implementation E. S. Espinoza,2, O. Garcia, R. Lozano,3, and A. Malo Laboratoire Franco-Mexicain d Informatique et Automatique, LAFMIA UMI 375 CNRS-CINVESTAV Mexico {eespinoza,ogarcias,rlozano,alexmalo}@ctrl.cinvestav.mx 2 Universidad Politecnica de Pachuca, Hidalgo Mexico steed@upp.edu.mx 3 Laboratoire Heudiasyc, Université de Technologie de Compiègne, Compiègne, France rlozano@hds.utc.fr Abstract This paper presents the first stage of the development of a mini coaxial rocket-helicopter (MCR UAV) whose main characteristic is that it can be catapulted and then hover at a long distance from the launching site. A functional description of this mechanism is given and the aerodynamic model is obtained using the classical Newton-Euler equations. The aerodynamic effects of the vehicle are obtained by studying the wind analysis (Propeller Momentum Theory) in hover flight. An embedded system is built in order to validate the proposed aerodynamic prototype, and a classical nonlinear control law based on backstepping procedure for the complete system is implemented in order to test this vehicle in autonomous flight. Finally, simulation and practical results are presented for hover flight, and the future tasks of this project are listed. keywords: Mini coaxial rocket-helicopter, aerodynamic model, hover flight, and embedded system. INTRODUCTION In recent years, developments of mini UAVs (Unmanned Aerial Vehicles) have been growing thanks to operational and technological advances in electronics, computing science, and mechanics. Environmental monitoring, surveillance, and homeland security are civilian missions, whereas special warfare, battle group surveillance are common objectives for military missions. Mini coaxial helicopters are capable of taking off and landing in reduced, and complex spaces or cluttered environments. Due to their configuration, coaxial helicopters require mechanism such as swashplates, stabilizer bars, and tilt-rotors in order to control the direction of rotor thrust vector. The coaxial configuration reduces the reaction torque generated from single rotor configuration (conventional helicopters), and generates powerful thrust force for hover flight. In addition, coaxial helicopters are mechanically stable because the center of

2 gravity is below a pair of counter-rotating propellers. Many theoretical and practical developments about coaxial helicopters have been reported in the literature. In [2], authors present a robust control for a coaxial micro helicopter. Design and control of an indoor coaxial helicopter are discussed in []. A simplified model and backstepping control for a coaxial helicopter can be found in the reference [4]. Nevertheless, this paper presents a mini coaxial rocket-helicopter which possesses control surfaces (ailerons) to control the attitude flight. The complete aerodynamic flight model is obtained using the classical Newton-Euler equations, while a classical control law based on the backstepping procedure is used to achieve the global stability for the vehicle in hover mode. The main contribution of this paper is to present the Mini Coaxial Rocket-Helicopter (MCR UAV) which can be catapulted and then hover at a long distance from the launching site, see Figure, with the objective of reducing the energy consumption, increasing the range, and developing surveillance missions. This prototype is developed at LAFMIA CNRS-CINVESTAV Mexico in collaboration with the Institut Franco- Allemand de Recherches de Saint-Louis (ISL), Laboratoire Heudiasyc (Université de Technologie de Compiègne, France) and Centre de Recherche en Automatique de Nancy (CRAN-Nancy Université, France). The paper is organized as follows: section 2 introduces a functional description and the aerodynamic model of the vehicle. In section 3 the control technique based on the classical backstepping methodology and the stability analysis are presented. Simulation results of the closed-loop system and experimental results are shown in section 4 and section 5 respectively. Finally, section 6 gives the conclusions and future works of this project. Third task: Hover mode Rotor deployment Second task: Fin/aileron deployment Fourth task: Cruise mode First task: Launching Camera Observation Launching device Five task: Landing in a safe area Figure : Objective of the project. 2 MINI COAXIAL ROCKET-HELICOPTER In this section, the functional description of the vehicle is described, and the complete aerodynamic model is obtained using the classical Newton-Euler equations. 2

3 2. Description The mini coaxial rocket-helicopter (MCR UAV) is based on a couple of counter rotating brushless motors, and its main characteristic of this vehicle is that it is capable of being launched through an external dispositive (e.g. a rocket launching device, an air launcher, etc), and transforming itself into a coaxial helicopter at a long distance from the launching site. Once the vehicle reaches the objective (a place, a building, an uninhabited area, etc.), it performs hover flight, and can inspect the environment acquiring and transmitting information through a RF camera to a ground station. Concerning the functional description, the MCR vehicle possesses aerodynamic control surfaces (ailerons) which are manipulated to control the roll and pitch motion while the difference of the velocities of the two motors regulates the yaw motion. 2.2 Mathematical equations Consider an inertial fixed frame and a body frame fixed attached to the center of gravity of the helicopter denoted by I={x I, y I, z I } and B={x B, y B, z B }, respectively, see Figure 2, [3]. φ x B z B ψ B θ y B ξ z I y I I x I Figure 2: MCR UAV. Assume the generalized coordinates of the mini UAV as q = (x, y, z, ψ, θ, φ) T R 6, where ξ = (x, y, z) T R 3 represents the translation coordinates relative to the inertial frame, and η = (ψ, θ, φ) T R 3 describes the vector of the three Euler angles with rotations around z, y, x axes. These angles ψ, θ, and φ are called yaw, pitch and roll, respectively. Assume the translational velocity and the angular velocity in the body frame as ν = (u, v, w) T R 3 and Ω = (p, q, r) T R 3, respectively. The Newton-Euler equations of motion for a rigid object provide the dynamic model for the MCR UAV and are given by the following expressions [3], [4] ξ = V m V = RF Ṙ = R ˆΩ () I Ω = Ω IΩ + Γ (2) where F R 3 and Γ R 3 are the total force and torque acting on the vehicle, respectively. V = (ẋ, ẏ, ż) T R 3 is the translational velocity in the inertial frame, m R denotes the mass of the MCR UAV, I R 3 3 contains 3

4 the moments of inertia of the mini helicopter, and ˆΩ is a skew-symmetric matrix such that ˆΩa = Ω a. Thus, R represents the transformation matrix from the body frame to the inertial frame c θ c ψ s φ s θ c ψ c φ s ψ c φ s θ c ψ + s φ s ψ R = c θ s ψ s φ s θ s ψ + c φ c ψ c φ s θ s ψ s φ c ψ s θ s φ c θ c φ c θ where the shorthand notation of s a = sin, c a = cos, and t a = tan is used. For this matrix, the order of the rotations is considered as yaw, pitch and roll (ψ, θ, φ) [6]. T c T c z B z B x B Y yb L y B x B lrl r D D lrl r δ r δ p L r L p Figure 3: Schematic of the vehicle Forces The forces acting on the vehicle include those of the propulsion system F p, aerodynamic effects F a and weight F w [8]. These forces are described as following F = F x F y F z = F p + F a + F w Propulsion forces The thrust force is generated by the two motors and is described as F p = where T c is the thrust force of the two motors (T c = T + T 2 ). In this analysis, the thrust force is oriented parallel to the axis z B of the body frame, see Figure 3. Aerodynamic forces For the aerodynamic analysis, two rotation matrices B and W are used [4]. These rotation matrices represent the transformation of the aerodynamic forces from the body frame to stability frame B, and the stability frame to wind frame W, and are written as c α s α c β s β B =, W = s β c β s α c α T c 4

5 where α is the angle of attack and β are the sideslip angle. Thus, the aerodynamic forces in the body frame are written as L F a = B T W T Y D where L, Y, and D are the aerodynamic forces: lift, side force, and drag, respectively. Gravitational force The force due to the weight of the vehicle is described as F w = R T mg where g is the acceleration due to gravity Moments The moments acting on the mini helicopter are due to actuators (actuator moment Γ act, and gyroscopic moment Γ gyro ), and the aerodynamic effects Γ a [8]. These moments are obtained as following Γ = Γ L Γ M Γ N = Γ act + Γ gyro + Γ a with Γ act = τ φ τ θ τ ψ, Γ gyro = Γ a = q(i r ω r I r2 ω r2 ) p( I r ω r + I r2 ω r2 ) L M N, where τ φ = l r L r, τ θ = l r L p and τ ψ = τ M + τ M2 are the control inputs with l r that represents the distance from the center of mass to the forces L r and L p, see Figure 3. τ M and τ M2 denote the moments produced by the motors M and M 2. ω r and ω r2 denote the angular velocities of the rotors, I r and I r2 are the inertia moments of the propellers. L, M and N are the aerodynamic rolling, pitching and yawing moments, respectively [3], [5] Aerodynamic analysis in hover flight Since the stability, control and dynamics of the MCR vehicle are affected by the propulsion system, the wind produced by the propellers (Propeller Momentum Theory) is analyzed to obtain the behavior of the vehicle in hover mode. Figure 4 shows the vehicle submerged in the propeller slipstream [9], [], []. The thrust equation is described as T c = ṁ a v = ρav s (v ds v o ) (3) 5

6 T c v o v i v s = v o + v i v ds = v o + 2v i Figure 4: Vehicle submerged in the propeller slipstream. with v s = v o + v i v ds = v o + 2v i (4) v r = 2v i where A represents the area of the rotor disc, ṁ a is the mass flow rate, ρ denotes the air density, v o is the freestream velocity, v i represents the induced velocity and is directed opposite to the thrust, v s is the slipstream velocity, v ds denotes the far downstream velocity, and finally, v r is the resultant velocity in the propeller slipstream. Considering the hover case (α =, β = and v o = ), equations (3)-(4) give the induced velocity v h in hover flight as T c v i = v h = 2ρA Finally, the aerodynamic forces and moments are written as (5) D = 2 ρv2 rsc D L = 2 ρv2 rsbc l Y = 2 ρv2 rsc Y M = 2 ρv2 rs cc m L = 2 ρv2 rsc L N = 2 ρv2 rsbc n (6) where S represents the fin-aileron area, c is the fin-aileron chord, and b is the fin-aileron span. C D, C Y and C L are the aerodynamical non-dimensional coefficients of drag, sideforce and lift. C l, C m and C n are aerodynamical non-dimensional coefficients of the aerodynamic rolling, pitching and yawing moments. 6

7 2.2.4 Equations of motion The nonlinear model obtained by the Newton-Euler formulation is described as ẍ = Ax m c θc ψ + Ay m (s φs θ c ψ c φ s ψ ) + Az m (c φs θ c ψ + s φ s ψ ) ÿ = Ax m c θs ψ + Ay m (s φs θ s ψ + c φ c ψ ) + Az m (c φs θ s ψ s φ c ψ ) z = Ax φ = m θ ψ c θ s θ + Ay m s φc θ + Az m c φc θ g + θ φs θ c θ + I xx [Γ L + qr(i yy I zz )] + c φs θ c θ I zz [Γ N + pq(i xx I yy )] + s φs θ c θ I yy [Γ M pr(i xx I zz )] θ = φ ψc θ + c φ I yy [Γ M pr(i xx I zz )] ψ = + s φ θ φ c θ I zz [ Γ N pq(i xx I yy )] + θ ψs θ c θ + c φ c θ I zz [Γ N + pq(i xx I yy )] + s φ c θ I yy [Γ M pr(i xx I zz )] (7) where 3 CONTROL TECHNIQUE A x = L A y = Y A z = T c D Γ L = τ ϕ + q(i r ω r I r2 ω r2 ) + L Γ M = τ θ + p( I r ω r + I r2 ω r2 ) + M Γ N = τ ψ + N In this section, the control technique for the stabilization of the vehicle in hovering flight is presented using a classical backstepping procedure [2], [7]. For simplicity, the equations of motion (7) are separated into three subsystems. One subsystem describes the longitudinal motion (pitching, and translation in the x z plane), the second subsystem describes the lateral motion of the vehicle (rolling, and translation in the y z plane), and the last subsystem describes the directional motion (yawing, and translation in the x y plane), [4]. Since the yaw motion is mechanically stable using contra-rotating propellers, the gyroscopic moment Γ gyro will essentially be zero. Considering the condition φ =, and ψ =, the longitudinal subsystem is written as ẍ = Ax m c θ + Az m s θ z = Ax m s θ + Az m c θ g θ = Γ M Iyy (8) where A x = L, A z = T c D and Γ M = τ θ + M. Defining a change of variables u θ = τ θ + M (9) I yy I yy 7

8 Substituting, it yields ẍ = L m c θ + m T cs θ m Ds θ z = m T cc θ m Dc θ + L m s θ g () θ = u θ In order to stabilize the altitude of this vehicle, a linear control law is proposed as T c = m ( k z (z z d ) k z2 ż + D c θ m c θ L ) m s θ + g () it results ẍ = L m c θ L m t θs θ + gt θ z = k z (z z d ) k z2 ż (2) θ = u θ Taking a change of variables x = x; x 2 = ẋ; x 3 = θ; x 4 = θ, the state space representation is written as follows ẋ = x 2 ẋ 2 = L m c x 3 L s x3 + gt x3 ẋ 3 = θ (3) ẋ 4 = u θ In order to control the longitudinal subsystem (3), a nonlinear control law based on the backstepping procedure is proposed. First, let us define the error e as e = x x d (4) differentiating (4) defining the following positive function whose derivative is ė = ẋ ẋ d (5) V = k 2 e2 (6) V = k e ė = k e (ẋ ẋ d ) (7) and using ė = ( x 2 x d 2), it yields ( V = k e x2 x d ) 2 (8) Now, let us define x v 2 as the virtual control input, such that x v 2 = x d 2 e (9) then, substituting (9) in (8), it implies V = k e 2 + k e (x 2 x v 2) (2) Now, the error e 2 is defined as e 2 = x 2 x v 2 (2) 8

9 it results V = k e 2 + k e e 2 (22) Now, the following positive function is proposed as taking its derivative as V 2 = k 2 2 e2 2 (23) { L V 2 = k 2 e 2 ė 2 = k 2 e 2 (ẋ 2 ẋ v 2) = k 2 e 2 ( m c x 3 + L } ) s x3 gt x3 ẋ v 2 Defining the virtual control input as { L m c x 3 + L m t } v x 3 s x3 gt x3 { L m c x 3 + L } v s x3 gt x3 = δ v = ẋ v 2 + k e + e 2 (25) k 2 (24) and V 2 yields { L V 2 = k 2 e 2 2 k e e 2 + k 2 e 2 (δ v m c x 3 + L }) s x3 gt x3 (26) Defining the error e 3 then, V2 yields { L e 3 = δ v m c x 3 + L } s x3 gt x3 (27) V 2 = k 2 e 2 2 k e e 2 + k 2 e 2 e 3 (28) Now, proposing a positive function V 3 as whose derivative is given as V 3 = k 3 e 3 [ δ v V 3 = k 3 2 e2 3 (29) ( L g ( ) ] + t 2 ) x 3 x 4 Let us define the virtual control input as (( L g ( )) ) + t 2 v x 3 x4 such that (( L g ( ) ) + t 2 ) v x 3 x 4 = δ2 v = δ v + k 2 e 2 + e 3 (3) k 3 (3) then, V yields ( L V 3 = k 3 e 2 3 k 2 e 2 e 3 + k 3 e 3 [δ2 v g ( ) ] + t 2 ) x 3 x 4 (32) finally, defining the error e 4 as V 3 yields e 4 = δ v 2 ( L g ( ) + t 2 ) x 3 x 4 (33) V 3 = k 3 e 2 3 k 2 e 2 e 3 + k 3 e 3 e 4 (34) Proposing the last positive function as V 4 = k 4 2 e2 4 (35) 9

10 it results V 4 = k 4 e 4 ė 4 = k 4 e 4 ( δv 2 ( d u θ + d 2 x 2 4) ) (36) where d = L g ( + ) d 2 = L s x3 + L m c x 3 + 2L m c x 3 + 2L m s x 3 2gt x3 2gt 3 x 3 From (36), a control input u θ is proposed such that V 4 = k 4 e 2 4 k 3 e 4 e 3, it results u θ = ( δ 2 v + k ) 3 e 3 + e 4 h 2 x 2 4 h k 4 (37) where h = L g ( + ) h 2 = L s x3 + L m c x 3 + 2L m c x 3 + 2L m s x 3 2gt x3 2gt 3 x 3 now, a Lyapunov function is proposed as V = V + V 2 + V 3 + V 4 (38) and V yields V = k e 2 k 2 e 2 2 k 3 e 2 3 k 4 e 2 4 (39) Finally, the system (8) is stable in the origin [7]. In order to control the lateral subsystem, the control methodology presented to stabilize the longitudinal subsystem is employed. where A y = Y, A z = T c D and Γ L = τ φ + L. as ÿ = Ay m c φ Az m s φ φ = Γ L I xx (4) On the other hand, in order to stabilize the remaining directional subsystem, a linear control input is proposed τ ψ = I zz ( k ψ ψ k ψ2 ψ) N (4) which is substituted in and it yields ψ = τ ψ + N I zz (42) ψ = k ψ ψ k ψ2 ψ (43) Finally, the constants k ψ, k ψ2 are chosen such that the equation (43) is Hurwitz polynomial.

11 4 SIMULATION In this section, the simulation results of the longitudinal subsystem are shown. The closed-loop system is validated using the Matlab Simulink with the initial conditions x =.5 m, z =. m and θ =. rad and altitude desired z =.5 m. Figure 5 illustrates that the position response is stabilized, and the desired altitude is reached. The pitch angle and control law are shown in the Figure 6 in which the response of the pitch angle remains at zero Position x [m] Position z [m] time (s) time (s) Figure 5: Position x and z. Angle θ [rad] time (s) u θ [Nm] Figure 6: Angle θ and control input u θ. 5 PRACTICAL RESULTS This section presents the embedded system, and the practical results obtained in real-time during the hover flight. The airframe of the MCR UAV is built of carbon fibre and polystyrene foam sheet (Depron). This vehicle is powered by two brushless motors in coaxial configuration and the size of the propellers is 7 5 in, see Figure 7. The weight of the vehicle is 45 gr. The autopilot consists of a Digital Signal Controller (DSC TMS32F288), an inertial measurement unit (Microstrain 3DM-GX3-25 AHRS) and a communication device (XBee-PRO RF Module), see Figure 7. The microcontroller (DSC) processes the information from sensors, computes the control law, and then sends signals to actuators. The IMU provides the angles, and angular velocities. In order to observe the performance of the vehicle in real-time during the autonomous flight, a telemetry system is developed using the software LabVIEW and GSM ActiveX controls which illustrate the states variables such as attitude and position. In this stage of the project, the aerial vehicle is stabilized in hover flight. Figure 8 shows the performance in roll motion of the vehicle during the autonomous flight. The pitch angle and control are depicted in the Figure 9,

12 Figure 7: Mini coaxial rocket-helicopter and Autopilot. and the Figure illustrates the behavior of the yaw angle and control of this aerodynamic platform Roll control [µc signal] Roll angle [deg] Figure 8: Roll angle and control. 4 2 Pitch control [µc signal] Pitch angle [deg] Figure 9: Pitch angle and control..5 Yaw control [µc signal] Yaw angle [deg] Figure : Yaw angle and control. 6 CONCLUSIONS In this paper the hover flight of proposed aerodynamic prototype (called Mini Coaxial Rocket-Helicopter MCR) has been proved. For this purpose, the complete aerodynamic flight model of the vehicle, including the wind 2

13 effects, is obtained using the classical Newton-Euler equations. The stability of the vehicle in hover mode has been analyzed in order to validate the classical control technique based on the backstepping methodology. An embedded system has been developed to test the vehicle in real-time. Finally, simulation results which highlight the proposed nonlinear control law in closed-loop system and the experimental results obtained in real-time during the hover flight shown an effective performance of the developed system. The next stage of this project is to design and develop the device for launching the vehicle, to incorporate additional instrumentation including GPS navigation, velocity sensors and vision system for observation. REFERENCES [] S.Bouabdallah, R.Siegwart, and G. Caprari Design and Control of an Indoor Coaxial Helicopter, International Conference on Intelligent Robots and Systems 26, Beijing, China, 26. [2] P. Castillo, R. Lozano, and A. Dzul, Modelling and Control of Miniflying Machines, Springer-Verlag, London, July 25. [3] M. V. Cook, Flight Dynamics Principles, Butterworth-Heinemann, second edition, USA, 27. [4] A. Dzul, T. Hamel, and R Lozano, Modeling and Nonlinear Control for a Coaxial Helicopter, IEEE International Conference on Systems, Man and Cybernetics 22, Hammamet, Tunisia, 22. [5] B. Etkin, and L. Duff Reid, Dynamics of Flight: Stability and Control, John Wiley and Sons, 3rd Ed., 996. [6] H. Goldstein, C.P. Poole and J.L Safko, Classical Mechanics, Adison-Wesley, USA, 2nd Ed., 983. [7] H. Khalil, Nonlinear Systems, Prentice Hall, New York, 995. [8] R. Lozano, Unmanned Aerial Vehicles Embedded Control, John Wiley-ISTE Ltd, 2. [9] B.W. McCormick, Aerodynamics of V/STOL Flight, Dover Publications, 967, USA. [] B.W. McCormick, Aerodynamics, Aeronautics and Flight Mechanics, John Wiley and Sons, USA, 995. [] W.F. Phillips, Mechanics of flight, John Wiley and Sons, USA, 24. [2] D. Schafroth, C.Bermes, S.Bouabdallah, and R.Siegwart, Modeling, system identification and robust control of a coaxial micro helicopter, Control Engineering Practice, Elsevier, 2. [3] R. F. Stengel, Flight Dynamics, Princeton University Press, USA, 24. [4] B. L. Stevens and F. L. Lewis, Aircraft control and simulation, John Wiley and Sons, USA,