Modelling of Opposed Lateral and Longitudinal Tilting Dual-Fan Unmanned Aerial Vehicle

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1 Modelling of Opposed Lateral and Longitudinal Tilting Dual-Fan Unmanned Aerial Vehicle N. Amiri A. Ramirez-Serrano R. Davies Electrical Engineering Department, University of Calgary, Canada ( Mechanical and Manufacturing Engineering Department, University of Calgary, Canada, ( Electrical Engineering Department, University of Calgary, Canada, ( Abstract: In this paper, first an overview and explanation of the Oblique Active Tilting (OAT) and Opposed Lateral Tilting (OLT) control concepts are presented. A complete dynamic model of vertical take-off and landing (VTOL) unmanned aerial vehicles (UAVs) having lateral and longitudinal rotor tilting mechanism is developed using a Newton-Euler formulation for double axis OAT mechanism (doat). Then, a theoretical analysis of OAT vehicle control response and stability for pitch, roll and yaw motion is described and simulated using the derived dynamic model. The aim of the aspects presented in this paper is to fully enable the highly maneuverable characteristics of UAV possessing doat which renders them to navigate in highly confined environments by performing semi-acrobatic maneuvers in hover lateral and forward flight. Keywords: Autonomous Unmanned aerial vehicle, Modeling, Vertical Take-off and Landing. 1. INTRODUCTION UAVs are becoming more useful everyday due to advancements in aerodynamics, propulsion, computers, and sensor technologies that allow aircrafts to have capabilities that were not available until recently. As UAV roles become more diversified, there is a continuous need to adapt to performing multiple tasks efficiently with a single airframe. This is especially important in VTOL airframes that can perform a greater number of diverse tasks. The Ability of small or medium air vehicles to access and operate in confined and obstructed environments is a key concern for the developments that are required in air mobility to enable aerial transportation systems and UAV complex mission execution. Satisfying this condition necessitates VTOL aerial vehicles to be more compact for a given payload, but on the other hand, maintaining effective vehicle control becomes increasingly difficult as their size reduces. When the available moment arms decrease in length, the proper control of the vehicle requires larger forces, which conventional control devices (e.g. ailerons) can no longer provide. Therefore, to obtain a compact UAV with effective control, the vehicle should be able to provide moments that do not depend on its dimensions. One type of control device which does not rely on moment arms is a gyroscope. It directly generates the large moments required to change the attitudes of satellites and space stations within short time periods Lim et al. (2004). In this direction, the author in Gress (2002) found that, by utilizing the vehicle s lift-fans themselves as control This work is supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. moment gyroscopes, they would be the basis for a powerful control system which would possess minimal weight and do not depend on the vehicle geometry or scale. Furthermore, with the helicopters limitations in both close environments and forward speed, development of alternate VTOL air vehicles has been considered by many researchers, see Bouabdallah (2007). Therefore, a combination of VTOL capability with efficient, high-speed cruise flight plus high maneuverable characteristics in tilt-rotor aircrafts have the potential to revolutionize UAVs. A good example of this type of VTOL aerial vehicles is VTOLs having lateral and longitudinal rotor tilting mechanism that gives them the unique ability to maneuver in confined spaces. A complete dynamic model of this type of UAVs is provided in details in this paper. This entirely new system, which uses only the dual lift-fans themselves for control, has been developed recently Gress (2007). It utilizes the inherent gyroscopic properties and driving torques of the fans for vehicle pitch control, and it eliminates the need for external control elements or lift devices. The system enables agile and compact VTOL air vehicles by generating pure and extensive moments rather than just forces. Fig. 1 shows the prototype of the VTOL UAV used to do simulations of predicted pitch, yaw and roll motions by considering the propellers tilt angles as inputs to the complete model of this type of vehicle that is presented in this paper. This prototype is called evader. The remainder of this paper is organized as follows: In Section II a lift-fan OAT mechanism and its capability of providing the three required moments(pitch, roll and yaw) are described. The proposed model is derived in Section Copyright by the International Federation of Automatic Control (IFAC) 2054

2 Fig. 1. Prototype of Dual-fan VTOL air vehicle having lateral and longitudinal tilting rotors. III. The results of simulating the model for lateral and longitudinal angles using the propellers speeds as input signals, plus considering the orientation angles as outputs, and also a simple control of pitch angle are shown in Section V. Finally, the main findings of this paper and the remaining works are summarized in Section VI. Fig.2.Fanstiltedlongitudinally90 forhighspeedforward flight. 2. LIFT-FAN OAT MECHANISM In this mechanism, unlike the full-scale tilt rotors, propellers can tilt in two directions providing also stability and control in hover. The required lift and control momentsareobtainedasfollowwithnoneedofanyhelicopter type cyclic controls. The roll movement is obtained by differential propeller speeds. The yaw angle can be controlled through differential longitudinal tilting. The gyroscopic moments issued from opposed lateral tilting, together with the torque generated by the collective longitudinal tilting allow for a significant pitching moment. In the following subsections, there is a brief description of how this vehicle works and how the essential moments are achieved. 2.1 Special Characteristics of OAT OAT composes a differential or opposed lateral tilting element for generating gyroscopic and fan-torque pitching moments, and the collective longitudinal tilting component for producing them from thrust vectors, as well as for controlling horizontal motion. This mechanism can contribute more than just stability and control in the conventional sense. Using the dual-axis version makes it possible to have an independent control of all six axes Gress (2003). High Speed Flight: Transition to high speed flight or airplane mode is achieved by tilting the fans longitudinally 90 degrees Fig. 2, during which longitudinal stability is maintained by lateral tilting and by the horizontal stabilizer at the rear of the aircraft. Because VTOL air vehicles do not require runways, their lifting surface-areas do not need be as large as those of a conventional airplane. There would be no need for conventional control surfaces (except the horizontal stabilizer) and associated dual control system, thereby reducing weight, complexity, and cost. And, because the entire wing-halves (fan shrouds) tilt, and differential longitudinal tilting of the fans generates a gyroscopic rolling moment (whether in hover or airplane mode), roll rates of the vehicle will be substantially higher than those using a conventional wing with ailerons. Gyroscopic pitch moments: Tilting both spinning fans simultaneously towards or away from one another laterally Fig. 3. a) Oppositely spinning disks tilted equally towards one another generating gyroscopic moment τ gyro, b) The whole system rotated about y axis to a new attitude orientation. produces gyroscopic moments perpendicular to their tilt axes at right angles direction. This moment τ gyro changes the vehicle s attitude as can be seen in Fig. 3, and this is the moment used to initiate control and dynamically stabilize the pitch attitude of our compact VTOL air vehicle. Returning the spinning discs to their neutral orientation will stop rotation of the vehicle in the case of space vehicles, where it will rest at the new attitude. In aerial vehicles, however, there are aerodynamic forces which tend to terminate or limit rotation of the vehicle without returning the fans to neutral. Fan-torque pitch moments: Using lift-fans as CMGs for air vehicle pitch control, there is another pitching moment associated with the fans lateral tilting which is a fan-torque pitching moment. Unlike the gyroscopic moment, a fantorque will remain after the tilting has stopped. Without this moment the fans would have to be tilting continuously to generate gyroscopic moments in order to reach a desired pitch angle or to compensate for a pitch disturbance. With this fans net torques providing a static pitching moment, these aerial vehicles have the potential to remain level in hover despite any pitch imbalances. So they have the ability to pitch while stationary, a particularly advantageous feature allowing direct target-pointing and VTOL from sharply inclined surfaces. Till now only tandem-rotor helicopters can make the pitch hover stationary property, see Chunhua et al. (2004). Thrust-vectoring pitch moments: The fan net torque may be insufficient to provide the static restoration. Therefore, to improve the vehicle s static stability in all instances, 2055

3 Fig. 4. Schematic of VTOL aerial vehicle with dual-axis OAT mechanism. an additional pitch control moment is obtained by collectively tilting the fans in the longitudinal direction while simultaneously tilting them laterally. These improvements all derive from the resulting characteristic of non-vertical thrust vector, which also provides more direct horizontal motion control. Therefore, the fans tilting for full and proper pitch control of the UAV will be in an oblique direction. Hence the name of this control method in either of its two executions is single-axis or dual-axis OAT. model. In this section, the translational and the rotational dynamic equations of the tilt-rotor aerial vehicle are presented. In this modeling, a general form of this vehicle is considered with doat ability, in which each of its ducts can have different lateral and longitudinal angles and different propeller speeds that have not been considered in previous works, see Kendoul et al.(2006) and Gress(2007). The equations of motion for a rigid body subject to body force F tot R 3 and torque τ R 3 applied at the center of mass are given by Newton-Euler equations with respect to the body coordinate frame (B) (see Fig. 5) and can be written as { m v B +ω mv B = F tot I ω +ω Iω = τ where v B R 3 is the body velocity vector, ω R 3 is the body angular velocity vector, m R specifies the mass, and I R 3 is an inertia matrix. 2.2 Overview of soat and doat Single-Axis Oblique Active Tilting (soat): In the simplest method, called single-axis OAT or soat, the fans or propellers tilt about a fixed and oblique horizontal axis, and the corresponding tilt path lies along a vertical plane oriented at a fixed angle α from the longitudinal direction Fig. 5. The tilt angle β is measured along the tilt-path plane, and is zero when the propeller spin axis is vertical. soat provides full, helicopter-like pitch control of the vehicle. Moreover, it also improves stability and control in yaw and roll either by reducing their high degree of coupling intuitively associated with dual-fan rotorcrafts or by taking advantage of that coupling. This distinct superiority, together with its simplicity, makes soat an exceptional choice of control method for small UAVs. Dual-Axis Oblique Active Tilting (doat): There is much more to be gained by taking full advantage of the dual-axis OAT capability including the potential for better control response for independent 6-axis control, vertical takeoffs and landings from severely sloped terrain, remaining perfectly level in hover, remaining stationary while pitching and yawing to track a target, and extreme maneuvering in three dimensional space, see Gress (2003). The capabilities of doat are still an open area of research and exploration. To investigate these capabilities and verify the characteristics of this control mechanism, in this paper a full model of the dual-fan VTOL aerial vehicle with lateral and longitudinal tilting rotors is derived in this paper which represents a general dynamic model for this kind of vehicle and can be used to explore the features of both soat and doat. 3. LATERAL AND LONGITUDINAL ROTOR TILTING VTOL MODELING The performance of the UAV controller will be dependent on the availability of a sufficiently accurate vehicle Fig. 5. Schematic of soat equipped VTOL aerial vehicle. 3.1 Translational Dynamics In this subsection the Cartesian equations of motion for the VTOLs vehicle having lateral and longitudinal tiltrotors are defined. Aerodynamic forces and moments are derived using a combination of momentum and blade element theory Leishman (2006). The VTOL has two motors with propellers. The direction of the thrust can be redirected by tilting the propellers laterally and longitudinally. A voltage applied to each motor results in a net torque being applied to the rotor shaft, Q i, which results in a thrust, T i. In other words, a propeller produces thrust by pushing air in a direction perpendicular to its plane of rotation. Whether the airflow is in the direction of the angular velocity vector or opposite depends on the shape of the propeller. Forward velocity causes a drag force on the rotor that acts opposite to the direction of travel, D i. The thrust and drag can be defined as T i = 1 2 C TρAr 2 ω 2 i,d i = 1 2 C DρAr 2 ω 2 i (1) where A is the blade area, ρ is the density of air, r is the radius of the blade, ω i is the angular velocity of the propellers, and C T 0 and C D 0 are aerodynamic coefficients depending on the blade geometry and the fluid density of the medium which is air in this case. T i (i = 1,2) represents the thrust force produced by left and right propellers. At hover, it can be assumed that the thrust and drag are proportional to the square of the propellers rotation speed. Thus the thrust and drag forces are given by T i = CT ω 2 i,d i = CD ω 2 i (2) 2056

4 The ground effect is not considered in the equations of this paper, but it is important to have in mind to consider ground and wall effects in maneuvers in confined spaces which is part of the future work of this research. The rotation matrices are comprised of ducts pitch (α i ) and roll (β i ) manipulation and transform the thrust vectors to the force vectors applied to the CG, as it can be seen in (3) below [ ] [ ] Cαi 0 S αi R y (α i ) = 0 1 0,R x (β i ) = 0 C βi S βi S αi 0 C αi 0 S βi C βi [ ] C αi 0 S αi R xy (β,α) i = R x (β i )R y (α i ) = S βi S αi C βi S βi C αi (3) C βi S αi S βi C βi C αi Where C α = Cos(α) and S α = Sin(α). When the thrust vector is multiplied by the rotation matrices the forces applied to the CG are represented by (4). F CG = R xy (β,α) 1 T 1 +R xy (β,α) 2 T 2 R xy (β,α) 1 D 1 R xy (β,α) 2 D 2 (4) It is very important to consider the vehicle s orientation when calculating the Cartesian equations of motion. Similar to most aerial vehicles, this type of tilt-rotor UAV can control its Cartesian position with its attitude. The Cartesian equations of motion can be derived by multiplying the force vector (F CG ) by the rotation matrix (R zyx ) to give the force vector applied to the inertial frame (E). The rotation matrix used in our development is in the form R zyx, with respect to the right-hand convention R z (ψ), R y (θ), and R x (φ). The total force F E tot acting on the vehicle s center of gravity is the sum of the lifting and dragging forces F E CG created by the rotors, the gravity F g and the aerodynamic forces F E a which is considered as a disturbance, namely F E tot = F E CG +F g +F E a = R yxz F CG +(0,0, mg) E +F E a The aerodynamic lift and drag forces may be considered as external disturbance in vertical flight mode, but need to be taken into account in dynamic modeling for horizontal flight. There is also the friction forces on the vehicle body in horizontal motion which has been considered in the term F E a in (5). 3.2 Rotational Dynamics In this subsection all the major torques acting on the vehicle in order to derive the angular acceleration equations of motion are presented. Equation (6), outlines the Euler rigid body motion equations for the vehicle s principle angular acceleration pitch( θ), roll ( φ), and yaw ( ψ) considering the vehicle s principle axis inertia (I xx, I yy and I zz ) and the sum of torques (τ). I xx φ+(izz I yy ) ψ θ = τ x I yy θ +(Ixx I zz ) φ ψ = τ y I zz ψ +(Iyy I xx ) θ φ = (6) τ z (5) It has been identified that there are four major torques acting on the vehicle. Gyroscopic moments (A i,i = 1,2): One of the primary torques acting on the vehicle is the gyroscopic torques created when tilting the ducts. Forcing propellers to perform laterally in opposite directions will create gyroscopic moments (τ i gyro,i = 1,2) which are perpendicular to their respectivespinandtiltaxes.theyarecreatedaboutaperpendicular axis from the orthogonal axis of rotation of the propeller and the orthogonal tilt axis. It is worth to note that each duct creates its own torques about its principle axis, where ω i will be positive or negative depending on the direction of rotation, and each duct is capable of pitch (α i ) and roll (β i ) motions. For example, with the propeller spinning clockwise and positive tilt rotation velocity for α i, there is a reactionary torque created perpendicular to the orthogonal axis of the spinning propeller and the tilt axis of β i. These moments are defined by the cross product of the kinetic moments (I r ω r ) of the propellers and the tilt velocity vector. In order to express these gyroscopic moments in the fixed body frame with respect to the vehicle s CG, the above equations should be multiplied by the rotational matrices R xy (βα) 1 and R xy (βα) 2, as below: τ gyro = R xy (β,α) 1 A 1 +R xy (β,α) 2 A 2 A i = I r β i ω r I r α i ω r 0 Propeller torques (Q i,i = 1,2): As the blades rotate, they are subject to drag forces which produce torques around the aerodynamic venter O. These moments act in opposite direction relative to ω. Q 1 = (0,0, Q 1 ) T,Q 2 = (0,0,Q 2 ) T The positive quantities Q i can be written as a function of propeller speeds: Q i = C Q ωi 2,C Q > 0. These torques can be written in B as Q = [ 2 R(β,α) i Q i = i=1 S α1 Q 1 +S α2 Q 2 S β1 C α1 Q 1 S β2 C α2 Q 2 C β1 C α1 Q 1 +C β2 C α2 Q 2 Thrust vectoring moment: These torques are derived based on the thrust vector T i and the translational displacement of the ducts and the vehicle s CG, represented as a vector d i. The torques are the cross-product of the thrust vector, with respect to the vehicle s CG, and the displacement vector d i which can be defined in B as d 1 = (0, l,h) T and d 2 = (0,l,h) T. ] (7) τ thrust = [R xy (β,α) 1 T 1 ] d 1 +[R xy (βα) 2 T 2 ] d 2 (8) Reactionary torques (P i,i = 1,2): The reactionary torques are comprised of the counter torques experienced bytheductwithrespecttothevehicle scgandthetilting rotations of the ducts. τ react = R xy (β,α) 1 P 1 +R xy (β,α) 2 P 2 P i = I r β i I r α i

5 Finally, the complete expression of the torque vector, with respect to CG of the vehicle and expressed in B is: τ = τ gyro +τ thrust +τ prop τ react (9) 3.3 Equations of Motion τ = [τ x τ y τ z ] T Table 1. Parameters parameter value parameter value m 8 kg I r kg.m 2 g 9.81 m.s 1 I x kg.m 2 l 0.4 m I y kg.m 2 h 0.08 m I z kg.m 2 Finally replacing all torque expressions in (6) the equations of motion can be written as: I xx φ = ψ θ(i yy I zz )+C α1 I r β 1 ω r +C α2 I r β 2 ω r S α1 Q 1 +S α2 Q 2 h[s β1 C α1 T 1 +S β2 C α2 T 2 ] +l[c β1 C α1 T 1 C β2 C α2 T 2 ]+C α1 I r β1 +C α2 I r β2 I yy θ = φ ψ(i zz I xx )+S β1 S α1 I r β 1 ω r +C β1 I r α 1 ω r +S β2 S α2 I r β 2 ω r +C β2 I r α 2 ω r +S β1 C α1 Q 1 S β2 C α2 Q 2 h[s α1 T 1 +S α2 T 2 ]+S β1 S α1 I r β1 (10) +C β1 I r α 1 +S β2 S α2 I r β2 +C β2 I r α 2 I zz ψ = θ φ(ixx I yy ) C β1 S α1 I r β 1 ω r +S β1 I r α 1 ω r C β2 S α2 J r β 2 ω r +S β2 I r α 2 ω r C β1 C α1 Q 1 +C β2 C α2 Q 2 +l[ S α1 T 1 +S α2 T 2 ] C β1 S α1 I r β1 +S β1 I r α 1 C β2 S α2 I r β2 +S β2 I r α 2 And from (5) the full expression of translational dynamic equations is defined as: mẍ = C ψ C θ F x +(C ψ S θ S φ C φ S ψ )F y +(C ψ S θ C φ +S ψ S φ )F z mÿ = S ψ C θ F x +(S ψ S θ S φ +C ψ C φ )F y (11) +(S ψ S θ C φ S φ C ψ )F z m z = S θ F x +S φ C θ F y +C θ C φ F z mg 4. SIMULATION The vehicle equations of motion in pitch without any aerodynamic external moments is the sum of the pitch axis elements of all the moments described in the previous section, and from (10) can be written as 1 2 I xx θ = cos(α)i r βωr htβcos(α)+cos(α)i r β (12) In this equation it is assumed that the vehicle propellers are tilted with the same lateral and longitudinal angles (α 1 = α 2,β 1 = β 2 ). The propellers speeds are also considered constant and the same for both rotors (ω 1 = ω 2 )butindifferentdirections.i xx isassumedconstantand independent of the propeller tilt angle β. It is a nonlinear function of β. Thereby, to make the pitch motion equation a linear function of propeller tilt angle, it is assumed to get small values. Therefore, in (12) cos(β) is approximated by 1 and sin(β) by β. We are looking for the pitch response of the vehicle to a control input. In this case, the lateral tilt angle β is considered as a control input with the function β = 1 2 β m(1 cosat),0 t π/a Considering this input in (12), the results of the vehicle s pitchresponsearedepictedinfig.6andfig.7fordifferent Fig. 6. The vehicle pitch response to the tilt angles inputs for different lateral angles and α = 15. Fig. 7. The vehicle pitch response to the tilt angles inputs for different longitudinal angles and β = 15. lateral and longitudinal angles, where the initial conditions are set at zero (θ(0) = 0, θ(0) = 0). Moreover, in Fig. 6 the effect of having gyroscopic moment is depicted as we can get 100 degree of pitch motion by changing β. Table (1) shows the parameters of the vehicle that have been used in simulations of this paper. The equations of motion for rolling and yawing with the same assumptions stated above for pitch motion are derived from (10) as: 1 2 I yy φ = βsin(α)i r βωr +I r αω r htsin(α) (13) 1 2 I zz ψ = sin(α)i r βωr +βi r αω r (14) The roll and yaw response to the tilt angle inputs are shown in Fig. 8. Fig. 9 represents the predicted pitch angle by changing ω. In the above figures the presented simulations are for 0 t π/a. Vehicle pitch response with a simple proportional controller: The P controller is in the form of β = k p θ wherek p isthecontrollergain.figure 10showsthevehicle pitch response to P controller in presence of a constant disturbance signal for different propeller speeds. As it can 2058

6 Fig. 8. Pitch, roll and yaw responses of a tilting fan VTOL vehicle by adding inputs as α = 15 and β = 0.5β m (1 cosat). Fig. 9. Predicted pitch response for different propeller speeds. Fig. 10. Feedback proportional control of the vehicle pitch angle with k p = 1 for different propeller speeds. be seen in this figure, the system is stable and convergent toθ m = 15.InFig. 11,thepitchresponsewithPcontroller is shown for different α angles. In all these simulations it is assumed that the propellers speeds are enough to make the vehicle stay in hover. 5. CONCLUSION A description of Lift-fan OAT mechanism is presented, which has the potential to provide a combination of VTOL capability with efficient, high-speed cruise flight plus high maneuverable characteristics in tilt-rotor aircrafts. A complete dynamic model of an example of this type of VTOL aerial vehicles which is VTOLs having lateral and longitudinal rotor tilting mechanism, is discussed here. Some simulation scenarios were made to explore the ability of our model and therefore the capabilities of the vehicle Fig. 11. Feedback proportional control of the vehicle pitch angle with k p = 1 for different longitudinal angles. itself. There are more advantages associated with OAT than just stability and control in the conventional sense which have not been explored yet. Examining all these properties by applying proper choice of controller to verify the soat and doat capabilities are what we are looking for in future works. The equations of motion of this design are highly coupled and nonlinear, so future work would be a challenging objective. REFERENCES S. Bouabdallah design and control of quadrotors with application to autonomous flying. PhD thesis, EPFL university, K. B. Lim, and J. Y. Shin, and D. D. Moerder, and E. G. Cooper. A new approach to attitude stability and control for low airspeed vehicles. Collection of Technical Papers - AIAA Guidance, Navigation, and Control Conference, 2: , Y. Guo, G. Ma, and C. Li. Steering law design to control moment gyroscopes for near minimum time attitude maneuver. 5th IEEE Conference on Industrial Electronics and Applications (ICIEA), F. Kendoul, and I. Fantoni, and R. Lozano. Modeling and control of a small autonomous aircraft having two tilting rotors. IEEE Transactions on Robotics, 22: , G. R. Gress. Using dual propellers as gyroscopes for tiltprop hover control. American Institute of Aeronautics and Astronautics, G. R. Gress. A Dual-Fan VTOL Aircraft Using Opposed Lateral Tilting for Pitch Control. American Helicopter Society, 59th Annual Forum, May, 2003, Phoenix, Arizona. H. Chunhua, Z. Jihong, H. Jinchun, and S. Zengqi. Output tracking of an unmanned tandem helicopter based on dynamic extension method. Fifth World Congress on Intelligent Control and Automation,WCICA 2004., 6: , G. R. Gress. Lift Fans as Gyroscopes for Controlling Compact VTOL Air Vehicles: Overview and Development Status of Oblique Active Tilting. American Helicopter Society 63rd Annual Forum, J. G. Leishman. Principles of Helicopter Aerodynamics. Cambridge University Press, Second Edition,