Quad-Rotor UAV: High-Fidelity Modeling and Nonlinear PID Control

Transcription

1 AIAA Moeling an Simulation Technologies Conference - 5 August, Toronto, Ontario Canaa AIAA -86 Qua-Rotor UAV: High-ielity Moeling an Nonlinear PID Control Alaein Bani Milhim an Youmin Zhang Concoria University, Montreal, Quebec, HG M8, Canaa Camille-Alain Rabbath Defence Research an Development Canaa, Valcartier, Quebec, GJ X5, Canaa Qua-rotor helicopter is an Unmanne Aerial Vehicle (UAV), whose lift is generate by four rotors locate on the corner of X-shape. Due to simplicity of its ynamics an its ability to hover, quarotor helicopter becomes as a popular platform for UAV. Current esigns mostly consier a linear moel for controller esign. In this paper, we erivate nonlinear ynamic equations of the quarotor UAV for the hovering motion from basic Newton s secon law. Nonlinear Proportional Integral Derivative (PID) controller is propose to control the qua-rotor UAV. This controller uses the spee an the orientation of each propeller in hovering motion of the qua-rotor UAV. A number of trajectories are use to emonstrate the effectiveness of the esigne controller. Nomenclature M = mass of the vehicle x, y, z = x, y, an z position of the vehicle, respectively ф, θ, ψ = roll, pitch, an yaw angle of the vehicle, respectively I xx, I yy, I zz = mass moment of inertia about x, y, an z axis, respectively g = acceleration of the gravity g = gravitational force in the earth frame gb = gravitational force in the vehicle frame C (x,y,z) = rag coefficients in the x, y, an z irection, respectively (x,y,z) = rag forces in the x, y, an z irection, respectively T(x,y,z) = thrust forces in the x, y, an z irection, respectively R (,,,) = the orientation angles of each propeller ( x, y ) = current position of the vehicle ( x, ~ y = comman position of the vehicle П = transformation matrix v (x,y,z) = velocity in the x, y, an z irection, respectively L, M, N = the moments prouce by the thrust force aroun x, y, an z axis, respectively Master stuent, Department of Mechanical an Inustrial Engineering, aa_melhem@yahoo.com. Associate Professor, Department of Mechanical an Inustrial Engineering, ymzhnag@encs.concoria.ca, Senior Member AIAA & IEEE, Corresponing author. Research Scientist, DRDC - Valcartier, 59 Pie-XI Blv. North, Camille-Alain.Rabbath@rc-rc.gc.ca. Copyright by the, Inc. All rights reserve.

2 U I. Introuction nmanne Aerial Vehicles (UAVs) are remotely operate or autonomous pilote vehicles. These vehicles have been use in applications such as security, management of natural risks, for intervention in hostile environments, management of groun installations, agriculture, an military -. Use of UAVs makes it possible to gather information in angerous environments without risk to flight crews. UAVs are classifie into two categories: fixe- an rotary-wing types. The rotary-wing type UAVs are more avantageous than the fixe-wing type UAVs in the sense of Vertical Take-Off an Laning (VTOL) capability, omni-irectional flying, an hovering performance, an can be ivie into qua-rotor an other helicopter types base on their shapes -5. The avantages of the qua-rotor versus comparably scale helicopters are that qua-rotor oes not require mechanical linkages to vary rotor angle of attack as they spin, the use of four rotors allows each iniviual rotor to have a smaller iameter than the equivalent helicopter rotor allowing them to store less kinetic energy uring flight an this reuces the amage cause by the rotors hitting any objects, an by enclosing the rotors within a frame, the rotors can be protecte uring collisions. Depening on the flying principal, we can classify the aerial vehicles in multiple categories epening on the flying principal as shown in ig. 5. In the motorize class, the bir like Micro Aerial Vehicle (MAV) can be consiere as the perfect example for the fast navigation. VTOL is classifie igure. Classification of aircraft base on the flying principal. within the same category. The most famous example of the qua-rotor UAV is the Draganflyer which is shown in ig.. This UAV is a raio-controlle four-rotor helicopter manufacture by Draganflyer 6. Although it can fly simply, stabilization of the hovering is a challenge. Control of the qua-rotor has been wiely stuie in a number of stuies 7-. The qua-rotor helicopter of the Stanfor Testbe of Autonomous Rotorcraft for Multi-Agent Control II (STARMAC II) has been cause by moments that affect altitue an control, an thrust variation that affects altitue control. The results of this work have proven the insufficiency for accurate trajectory tracking of the vehicle at spee an in uncontrolle environments but the control esign coul be improve by careful consieration of these isturbances. Omni-irectional Stationary lying Outstretche Robot OS is a qua-rotor project evelope towars fully autonomous operation. The main objective of this flying robot is the evelopment an implementation of an active control system for this vehicle 5,. Although many control algorithms have obtaine goo results in the previous works, those works focuse primarily on the control of the vehicle by taking into account only the spee of the propeller. igure. Draganflyer. The well-known example of the qua-rotor UAV. The objective of this work is to present the ynamic moel of the qua-rotor an propose a nonlinear Proportional- Integral-Derivative (PID) controller base on the ajustment of the orientation of the propeller in aition to control the spee of the rotor. The main contribution of our work is to esign PID controller for qua-rotor vehicle base on the orientation an the spee of each propeller. In this paper, we analyze the high-fielity moel of the qua-rotor

3 an obtain the equations of motion in Section II. In Section III, nonlinear PID controller is use to control the UAV moel. In this section there are two subsections: the first one shows the control of the orientation of each propeller an the secon one shows the control of the spee of the propellers. In Section VI, simulation results are presente an supporte by figures. inally, conclusions are presente. II. Motion of the Qua-Rotor UAV Qua-rotor is an aircraft without pilot whose lift is generate by its four rotors. It contains four propellers an each two of them forms a pair of propellers, the first pair is locate on the x-axis an rotate clockwise an the secon pair is locate on the y-axis an rotate counterclockwise as shown in ig.. Generally, its motion is controlle by varying the lift force prouce by the rotors. Each rotor prouces both a thrust an a torque about its center of rotation. The gyroscopic effects an the aeroynamic torques ten to cancel an these four rotors o not have a swash plate 5. The equations are presente about the boy frame (X, Y, Z). or any point of the airframe expresse in the earth-fixe frame (X E, Y E, Z E ), one can write transformation matrix as: Y θ Clockwise Counterclockwise Z ZE YE XE ψ Clockwise ф Counterclockwise igure. Qua-rotor configuration. The earth-fixe frame (X E, Y E, Z E ) is use to specify the location of the vehicle. The vehicle boy frame (X, Y, Z) an roll (ф), pith (θ), an yaw (ψ) angles are use to specify the vehicle orientation. Y Z X X Π cθcψ = cθsψ sθ cφsψ + sφsθsψ cφcψ + sφsθsψ sφcθ sφsψ + cφsθcψ sφcψ + cιsθsψ cφcθ () where s (ф, θ, ψ) = sin (ф, θ, ψ), c (ф, θ, ψ) = cos (ф, θ, ψ). The equation of motion of the qua-rotor at inertial coorinate system is given as : && x x m && y =Π y && z z () x, y,, z are the net forces in an inertial system represente in the boy coorinate system an (x, y, z) represent the location of the vehicle in the earth-fixe frame an П is efine as Eq. (). It can be seen that the accelerations in the earth-fixe frame are obtaine by multiplying the forces by the transformation matrix. The main forces acting on the boy are the propeller thrusts, rag forces an gravitational forces. The irection of the propeller thrust is perpenicular to the surface of rotating an its value is proportional with the spee of rotation. The propeller is riven by a DC motor mounte on the stepper motor locate at the en of a crossing boy frame as shown in ig.. The frame (x, y, z) of the DC motor has been assume. The DC motor frame (x, y, z) is parallel to the vehicle boy frame (X, Y, Z) when there is no orientation of the propeller, as emonstrate in the igure. Propeller configuration. The DC motor frame (x, y, z) is use to mention the orientation of the propeller with respect to the vehicle boy.

4 classical qua-rotor. The orientation of the propeller will take effect an the y-z plane of the propeller will orient at an angle as the propeller is tilte as a result of the rotation of the stepper motor. So the thrust forces will no longer be perpenicular on the X-Y plane of the vehicle. To fin the projection of the forces on each axis, we nee to multiply the thrust force by the sine or cosine of the angle that the propeller is tilte. igure 5 shows the orientation of propeller an the components of the thrust force. The rag force is proportional with the square of the velocity on each axis by coefficient of rag as shown in following equation. Thrust orce x c x v x () y = c y v y Π z c z v z Since the velocity of each axis is presente in earth frame, we nee to ivie it by the transformation matrix () to get its component on the boy frame. The main force acting on the qua-rotor is the gravity force equaling to the total mass of the qua-rotor multiplie by the acceleration of the gravity as following: igure 5. Propeller with certain orientation. This figure shows the propeller at an angle (R). The thrust force shoul be multiplie by cos(r) to get the component of the force on the Z-axis of the vehicle boy. This view is taken towars the positive X-axis. g = M g Its irection is towar the groun (negative z-irection of the earth frame). By iviing the gravitational force by the transformation matrix, we can calculate the effect on each axis on the boy frame. sin θ g gb (5) gb = cos θ sin φ g = gb cos θ cos φ g gb () The equations of the forces affecte to the qua-rotor are: = + + x y x y Tx = + + z z Ty = + + By the erivatives of the acceleration, we can get the velocity an then the location of the vehicle: t t x ξ& = && ξ τ ξ = ξ& τ where ξ = y ξ& ξ,, z ξ & an ξ are the initial conitions of the velocity an the position, respectively. Tz gb gb gb (6) (7) (8) The roll equations of motion are: I xx && φ L (9) I = yy && θ M I && N zz ϕ

5 The main moments acting on the qua-rotor vehicle are prouce by propellers, which is equal to the thrust force multiplie by the momentum arm. Note that the rag force was neglecte in computing the moment. This force was foun to cause a negligible isturbance on the total moment over the flight regime of interest. t t η& = && η τ, η = η & φ τ, where η = θ η& η ψ η& an η are the initial conitions of the angular velocity an the Euler angles rotational components, respectively. III. Controller Design Nonlinear (PID) controller is propose to esign trajectory tracking controller for the qua-rotor UAV an it is base on two funamentals: control of orientation of each propeller an control of spee of each motor. The parameters of the PID controller are tuning by trial an error. a) Control the orientation of each propeller We use the orientation primarily to change the yaw angle, which is the angle between a qua-rotor s heaing an a reference heaing, to get the commane angle. This is the yaw controller. The criterion of employing yaw control epens on the istance, which is enote as, between the current position ( x, y ) an commane position ( ~ x, ~ y ), where the istance is expresse as the following equation: where x e an y e are efine as: e e = x + y () x e = x ~ x y e = y ~ y Then, following yaw control will be activate when this istance is larger than a specific value γ, >γ, where γ is a positive constant. Assume γ =. x% = { x%, x%, x%, K, x% n } y% = { y%, y%, y%, K, y% n } where n is the number of the comman points. Then assuming the istance between the comman points is ~ ~ λ ~ x ~ x ) + ( y y () = ( n n n n ) The importance of the above part of the controller is coming from the istance λ. In other wors, we can neglect this part if the istance λ is less than γ. Also, we can ivie the yaw controller into two sub yaw controllers: one of them represents the controller for the pair at the x-axis an the other is for the pair at the y-axis. The pair on the x-axis will orient epenently on the commans coming from the yaw controller, which is the Proportional-Differential (PD) igure 6. The irection of orientation of the propellers on the x-axis. This view is taken towars positive y-axis. 5

6 controller output, an they orient in the opposite irection as shown in ig. 6. In other wors, when the istance () is larger than the specific value (γ), the yaw angle will be upate by a new angle (ψ ) which is equal to the angle of the trajectory; otherwise the yaw angle oes not be moifie. After that, a PD controller is propose before sening the comman to the actuators. The proceure to upate yaw angle is shown as: If > λ { ψ = ψ ; Otherwise ψ = ψ ; } where ψ = tan Δ y, Δ y = Δ x ~ y y, Δ x = ~ x x The pair on the y-axis will be ajuste by a stepper motor that takes comman from yaw controller as a constant value (υ) which is the step of the stepper motor, otherwise the stepper motor is still with zero rotation. The irection of each stepper motor in this pair is rotating in the same irection of each other to allow the qua-rotor to be capable of rotating. The equations of the orientation controller are: R = R = ± ( k p ( ) + k ( &)) () ψ ψ ψ ψ ψ = R v R = () where k pψ an k ψ are the parameters of the PD controller an v is the step of the stepper motor. Now, we can get the forces an moments components as escribe in the following equations: Tx = sin R + sin R () = sin R sin R (5) Ty (6) Tz = i cos R i i = M T φ = l cos R l cos R (7) M T θ = l cos R l cos R (8) (9) M Tψ = l i i sin R i i = l i is the length between the propeller an the Center Of Gravity (COG) an i=,,, an for each propeller. Observe that the forces an moments are taking effect of the orientation of each propeller by multiplying the forces by sin or cosine of the orientation angle. b) Control the spee of each motor We control the spee of motor mainly to reach the esire altitue an to get the right irection. This part of controller has three sub-controllers as shown below: i. Z-Controller. ii. Roll Controller. iii. Pitch Controller. In the following, we will propose a PID controller for each sub-controllers 6. i. Z-controller: 6

7 The purpose of this controller is to achieve the esire altitue by applying a PID controller on the altitue. The output of this controller shoul be ivie by four to generate the equivalent force on each propeller, hence stabilize the qua-rotor. The equation of this controller is shown as: C = k ( z z ) + k ( z ) + k ( z ) pz z & iz z where z is the esire value, an k pz, k z, an k iz are the parameters of PID controller. () ii. Roll Controller: To move into y-irection, it requires the qua-rotor to roll aroun the x-axis as shown in ig. 7. To obtain the roll rotation, we have to generate imbalance forces in the pair on the y-axis. This imbalance force has to rise above the inertial forces opposing the rotation. The roll angle (φ) is approximate as the first-orer system of (y y). In other wors the roll angle is output of PD controller of y as following relation: φ = k ( y y ) + k ( y& ) () where y an ф are the esire values, k py an k y are constant values. The roll controller will stabilize the roll angle to create the y-motion at its esire value if the gains are well-chosen 7. iii. Pitch Controller: The pitch controller is the same as the roll controller, except that the controller will be applie on the pitch angle (θ) as shown in ig. 8 which means the rotation about y-axis, to create the motion along the x-axis. This relation is: where x an θ are the esire values, k px an k x are constant values. The rotation of the boy yaw axis is applie to obtain the components of roll an pitch by using this transformation matrix: py y igure 7. Roll rotation. This rotation allows the vehicle to move in the y-irection. x ) + k ( & ) θ = k ( x x () px x R ψ = cos( ψ ) sin( ψ ) sin( ψ ) cos( ψ ) As mentione before, if the criterion of using orientation is satisfie, the yawing controller is propose to get the hovering motion of the quarotor. If the criterion is not satisfie, roll an pitch controllers are propose to keep the hovering motion of the qua-rotor. igure 8. Pitch rotation. This rotation allows the vehicle to move in the x- irection. Two PD controllers will be applie on roll an pitch angles accoring to the following relations: 7

8 C = k φ ( φ φ ) + φ ( φ& p k ) () C k ( θ θ ) + k ( &) () = pθ θ θ where C an C are the outputs of the controllers, an k pф, k ф, k pθ, an k θ are the parameters of the controllers. The output of each controller shoul be ivie by two, with opposite signs to istribute the input controller on each propeller in each pair, to guarantee the balance forces an these opposite signs mean that each propeller in the pair rotates in the opposite irection of the other. The equations of the istribution to generate the forces are: = C C = C C = C + C = C + C (5) (6) (7) (8) These forces are sent to the propeller which represents the spee of the propeller that is exactly the spee of the DC motor. In this paper, we focus on the controller to prouce the neee forces. Intereste reaers can refer to Ref.,, for the information about the DC motor configurations. The equations of the controller are mainly presente through the Eqs.,,,, an. IV. Simulation Results an Analysis We have applie several trajectories for the vehicle to follow. We can see that the result in ig. 9 emonstrates the effectiveness of the esigne controllers for the motion in the x-y plane. As we can see, the vehicle starts from position (-.5, ) an en at the same point (-.5, ) through following rectangular path with the constant altitue. In this simulation test, the qua-rotor is commane to fly from (-.5,,), (-.5,-.5,), (.5,-.5,), (.5,5.5,), an (-.5,5.5,) an then back to (-.5,,). These results show that the vehicle can track the commane path an it oes not rift away from the path more than.m. We have approve other paths as shown in ig. which y- axis position 6 5 Qua-Rotor Trajectory Tracking The esire path The Qua-Rotor response x- axis position igure 9. Trajectory tracking for the vehicle in x-y plane. 6 y-axis position x-axis postion (a) 8 (b) igure. Trajectory of American the vehicle. Institute The of projection Aeronautics of the an trajectory Astronautics in x-y plane is shown in (a). The projection in x-z plane is expose in (b). The vehicle is changing the altitue from to 6m.

9 shows the x-y projection of the path in part (a), part (b) shows the x-z projection of the path an the three imension view is expose in ig.. These figures show the tracking of the vehicle to the esire path in three imensions. The vehicle follows the path an passes through the commane points. The use of the orientation controller epens on the istance between the commane points. In this z-axis position test, the vehicle is commane to follow the esire path in the hexagonal form. The vehicle is commane to fly from (,-6,) an then back to (,-6,) through the esire path. These results show that the vehicle oes not rift away from the path more than.5m at any point uring the flight. The trajectory of the vehicle is consistent with the esire path most of the time as shown in ig. 9. These results also show the effectiveness of the esigne controller. y-axis position x-axis postion The esire path The vehicle response igure. Trajectory tracking for the vehicle in three imensional view. V. Conclusions We have erive the equations of motion for qua-rotor UAV starting from basic Newton s secon law an propose a metho to control the qua-rotor UAV base on a nonlinear PID controller. This work focuse mainly on the control of the vehicle from the sie of the orientation of the propeller. We have esigne the controller as two parts: the first part controls the orientation of the propeller base on the yaw angle an the secon part controls the spee of each propeller. This type of esigne controller increases the performance of the qua-rotor in tracking the esire trajectory. As shown in the simulation results, the vehicle was skille to follow the esire path while the operating points were something like far from each other. This way increases the control actuators which support the reliability of the system. The ifficulties of this system are coming from the implementation of the stepper motors with DC motors together from the harware sie an the big size of the coe from the software sie. The criterion of using the first part epens mainly on the specific value ( λ ) which gives the inication about the variation between the commane points. The propose controller has been implemente in a nonlinear six-egree of freeom simulation moel of a qua-rotor UAV an goo tracking performances have been obtaine through ifferent shapes of commane trajectories. References Castillo, P., Lozano, R., an Dzul, A., Moelling an Control of Mini-lying Machines, Lonon: Springer, 5, pp. -5. McKerrow, P., Moelling the Draganflyer our-rotor Helicopter, in Proceeings of the IEEE International Conference on Robotics an Automation, pp ,. Canttea, C., Chin, J., Mehrabian, S., Montejo, L., an Thompson, H., Qua-rotor Unmanne Aerial Vehicle, inal Report, Columbia University., New York, 7. Park, S., Won, D.H., Kang, M.S., Kim, T.J., Lee, H.G., an Kwon, S.J., RIC (Robust Internal-loop Compensator) Base light Control of a Qua-Rotor Type UAV, in Proceeings of the IEEE/RSJ International Conference, 5, pp Bouaballah, S., Murrieri, P., an Siegwart, R., Design an Control of an Inoor Micro Qua-rotor, in Proceeings of the IEEE Int. Conf. on Robotics an Automation,, pp Draganlyer 7 RCtoys, 8 Bouaballah, S., Design an Control of Quarotors with Application to Autonomous lying, Master s thesis, Aboubekr Belkai University, Tlemcen, Algeria, 7. 9

10 9 Soumeliis, A., Gaspar, P., Bauer, P., Lantos, B., an Prohaszka, Z., Design of an Embee Microcomputer Base Mini Quarotor UAV, in Proceeings of the European Control Conference, 7, Kos, Greece. Castillo, P., Albertos, P., Gracia, P., an Lozano, R., Simple Real-time Attitue Stabilization of a Qua-rotor Aircraft with Boune Signals, in Proceeings of the 5th IEEE Conference on Decision & Control, 6 San Diego, CA, USA. Camlica,., Demonstration of a Stabilize Hovering Platform for Unergrauate Laboratory, Master s thesis, Istanbul Technical University, Ankara, Turkey,. Hoffmann, G.M., Haung, H., Waslaner, S. L., Tomlin, C. J., Quarotor Helicopter light Dynamics an Control: Theory an Experiment, AIAA Guiance, Navigation, an Control Conference, Hilton Hea, South Carolina, August 7. Bouaballah, S., Noth, A., an Siegwart, R., PID vs LQ Control Techniques Applie to an Inoor Micro Quarotor, in Proceeings of the IEEE/RSJ International Conference on Intelligent Robots an Systems,, pp Das, A., Subbarao, K., an Lewis,., Dynamic Inversion with Zero-ynamics Stabilization for Quarotor Control, IET Control Theory an Appl., 9, Vol., No., pp Castillo P., Lozano R., an Dzul A., Stabilization of a Mini Rotorcraft Having our Rotors, IEEE Control Systems Magazine, 5, Vol. 5, No. 6, December 5, pp Kenoul,., Lara, D., antoni-coichot, I., an Lozano, R., Real-Time Nonlinear Embee Control for an Autonomous Quarotor Helicopter, Journal of Guiance, Control, an Dynamics, Vol., No., July-August 7, pp antoni, I., Zavala, A., an Lozano, R., Global Stabilization of a PVTOL Aircraft with Boune Thrust, in Proceeings of the st IEEE conference on Decision an Control, Las Vegas, Nevaa USA, December.