帶機械爪的無人機設(shè)計與控制【全套含CAD圖紙、說明書】,全套含CAD圖紙、說明書,機械,無人機,設(shè)計,控制,全套,CAD,圖紙,說明書 Modelling and PID Controller Design for a Quadrotor Unmanned Air Vehicle Atheer L. Salih, M. Moghavvemi, Haider A. F. Mohamed and Khalaf Sallom Gaeid Centre for Research in Applied Electronics (CRAE) University of Malaya 50603 Kuala Lumpur, Malaysia Abstract this paper presents the modelling of a four rotor vertical take-off and landing (VTOL) unmanned air vehicle known as the quadrotor aircraft. The paper presents a new model design method for the flight control of an autonomous quad rotor .The paper describes the controller architecture for the quadrotor as well. The dynamic model of the quad-rotor, which is an under actuated aircraft with fixed four pitch angle rotors, will be described. The Modeling of a quadrotor vehicle is not an easy task because of its complex structure. The aim is to develop a model of the vehicle as realistic as possible. The model is used to design a stable and accurate controller. This paper explains the developments of a PID (proportional-integral-derivative) control method to obtain stability in flying the Quad-rotor flying object. The model has four input forces which are basically the thrust provided by each propeller connected to each rotor with fixed angle. Forward (backward) motion is maintained by increasing (decreasing) speed of front (rear) rotor speed while decreasing (increasing) rear (front) rotor speed simultaneously which means changing the pitch angle. Left and right motion is accomplished by changing roll angle by the same way. The front and rear motors rotate counter-clockwise while other motors rotate clockwise so that the yaw command is derived by increasing (decreasing) counter-clockwise motors speed while decreasing (increasing) clockwise motor speeds. Keywords- Quadrotor, PID controller, VTOL, UAV, MATHLAB simulink. I. INTRODUCTION UAVs, or Unmanned Aerial Vehicles, are defined as aircrafts without the onboard presence of pilots 1. UAVs have been used to perform intelligence, surveillance, and reconnaissance missions. The technological promise of UAVs is to serve across the full range of missions. UAVs have several basic advantages over manned systems including increased maneuverability, reduced cost, reduced radar signatures, longer endurance, and less risk to crews. Vertical take- off and landing type UAVs exhibit even further maneuverability features. Such vehicles are to require little human intervention from take- off to landing. Unmanned aerial vehicles (UAVs) have potential for full-filling many civil and military applications including surveillance, intervention in hostile environments, air pollution monitoring, and area mapping 2. Unmanned aerial vehicles (UAV) have shown a growing interest thanks to recent technological projections, especially those related to instrumentation. They made possible the design of powerful systems (mini drones) endowed with real capacities of autonomous navigation at reasonable cost. In this paper, we are studying the behavior of the quadrotor. This flying robot presents the main advantage of having quite simple dynamic features. Indeed, the quadrotor is a small vehicle with four propellers placed around a main body. The main body includes power source and control hardware. The four rotors are used to controlling the vehicle. The rotational speeds of the four rotors are independent. Thanks to this independence, its possible to control the pitch, roll and yaw attitude of the vehicle. Then, its displacement is produced by the total thrust of the four rotors whose direction varies according to the attitude of the quadrotor. The vehicle motion can thus be controlled. There have been numerous projects involving quadrotors to date, with the first known hover occurring in October, 1922 3. Recent interest in the quadrotor concept has been sparked by commercial remote control versions, such as the DraganFlyer IV 4. Many groups 58 have seen significant success in developing autonomous quadrotor vehicles. Nowadays, the mini- drones invade several application domains 9: safety (monitoring of the airspace, urban and interurban traffic); natural risk management (monitoring of volcano activities); environmental protection (measurement of air pollution and forest monitoring); intervention in hostile sites (radioactive workspace and mine clearance), management of the large infrastructures (dams, high- tension lines and pipelines), agriculture and film production (aerial shooting). In contrast to terrestrial mobile robots, for which it is often possible to limit the model to kinematics, the control of aerial robots (quadrotor) requires dynamics in order to account for gravity effects and aerodynamic forces 10. In general, existing quadrotor dynamic models are developed on the hypothesis of a unique rigid body which is a restrictive hypothesis that does not account for the fact that the system is composed of five rigid bodies: four rotors and a crossing body frame. This makes the explanation of several aspects, like gyroscopic effects, very difficult. Additionally, simplification hypotheses are generally introduced early in the model development and leads in general to misleading interpretations. II. MATHEMATICAL MODELLING A quadrotor is an under actuated aircraft with fixed pitch angle four rotors as shown in Fig. 1. Modeling a vehicle such as a quadrotor is not an easy task because of its complex structure. The aim is to develop a model of the vehicle as realistically as possible. yzxTh2Th3Th1Th4w2w4w1w3mgu1l Fig. 1. The quadrotor schematic. In the quadrotor, there are four rotors with fixed angles which represent four input forces that are basically the thrust generated by each propeller as shown in Fig. 1. The collective input (u1) is the sum of the thrusts of each motor. Pitch movement is obtained by increasing (reducing) the speed of the rear motor while reducing (increasing) the speed of the front motor. The roll movement is obtained similarly by increasing (reducing) the speed of the right motor while reducing (increasing) the speed of the left motor. The yaw movement is obtained by increasing (decreasing) the speed of the front and rear motors together while decreasing (increasing) the speed of the lateral motors together. This should be done while keeping the total thrust constant. Each of the controller inputs affects certain side of the quadrotor model, u2 here affects the rotation in the roll angle while u3 affect the pitch angle and u4 control the yaw angle during the flying process and u1 affect the altitude (z-axis) for this model. Each rotor produces moments as well as vertical forces. These moments have been experimentally observed to be linearly dependent on the forces for low speeds There are four input forces and six output states (x, y, z, , , ) therefore the quadrotor is an under- actuated system. The rotation direction of two of the rotors are clockwise while the other two are counterclockwise, in order to balance the moments and produce yaw motions as needed. The compensation of this torque in the center of gravity is established thanks to the use of contra rotating rotors 1- 3 and 2- 4. Recall that rotors 2 and 4turn counterclockwise while rotors 1 and 3 turn clockwise. In order to move the quadrotor model from the earth to a fixed point in the space, the mathematical design should depend on the direction cosine matrix as follows: +=CCSCSSCCSSCCSSSSCSSCSCCSSSCCCzxyR (1) where - S = Sin() ,C= Cos(), etc. - R: is the matrix transformation. - ?: is the Roll angle. - : is the Pitch angle. - : is the Yaw angle. The dynamic model of the quadrotor helicopter can be obtained via a Lagrange approach and a simplified model is given as follow 11. The equations of motion can be written using the force and moment balance. =+=mzKgCosCosuzmyKSinCosCosSinSinuymxKSinSinCosSinCosux/)(/)(/)(312111? ? ? ? (2) Where: x: Forward position in earth axes y: Lateral position in earth axes z: Vertical position in earth axes Ki: The Drag Coefficients for the system. The Kis given above are the drag coefficients. In the following we assume the drag is zero, since drag is negligible at low speeds. The center of gravity is assumed to be at the middle of the connecting link. As the center of gravity moves up (or down) d units, then the angular acceleration becomes less sensitive to the forces, therefore stability is increased. Stability can also be increased by tilting the rotor forces towards the center. This will decrease the roll and pitch moments as well as the total vertical thrust. For convenience, we will define the inputs to be: +=+=+=+=34321424321314321243211I / ) Th Th Th Th ( C UI / ) Th - Th Th Th - ( l UI / ) Th Th Th - Th - ( l Um / ) Th Th Th Th ( U (3) Where: u1 : Vertical thrust generated by the four rotors u2 : Pitching moment u3 : Yawing moment u4 : Rolling moment Thi : The thrusts generated by four rotors Ii : The moments of inertia with respect to the axes Where This are thrusts generated by four rotors and can be considered as the real control inputs to the system, and C the force to moment scaling factor. And Iis are the moment of inertia with respect to the axes. Therefore the equations of Euler angles become: =364253142/IKu/IlKu/IlKu? ? ? ? (4) where (x, y, z) are three positions; (, ,) three Euler angles, representing pitch, roll and yaw respectively; g the acceleration of gravity; l the half length of the helicopter; m the total mass of the helicopter; Iis the moments of inertia with respect to the axes; Kis the drag coefficients. This quadrotor helicopter model has six outputs (x, y, z, , , ) while it only has four independent inputs, therefore the quadrotor is an under- actuated system. We are not able to control all of the states at the same time. A possible combination of controlled outputs can be x, y, z and in order to track the desired positions, move to an arbitrary heading and stabilize the other two angles, which introduces stable zero dynamics into the system 11, 5. A good controller should be able to reach a desired position and a desired yaw angle while keeping the pitch and roll angles constant. By applying Pythagoras theorem and implementing some assumptions and cancellations as follows: 1- The quadrotor structure is symmetrical and rigid. 2- The Inertia matrix (I) of the vehicle is very small and to be neglected. 3- The center of mass and o coincides. 4- The propellers are rigid. 5- Thrust and drag are proportional to the square of the propellers speed. These above equations have been established assuming that the structure is rigid and the gyroscopic effect resulting from the propellers rotation has been neglected. The Phi (d) and (d) can be extracted in the following expressions: )2y)(y2x)(xzz(1tanand)xxyy(1tanddddddd+= (5) Where: - d :is the desired roll angle. - d :is the desired yaw angle. xd,yd,zdxyz? xd-xz-zd Fig. 2. The Quadrotor Angles Movements By supplying the four motors with the required voltage, the system will be on, the thrust here is directly proportional with these voltages, whenever increasing the voltage, the thrust for the motor increase and vice versa. III. PID CONTROL DESIGN The proportional Integral derivative (PID) design are pointed out in many references, such as 13, that PID controllers can be used only for plants with relatively small time delay for high performance devices like the quadroator . This controller takes many structures but the most important one as in the following form: +=t0dtde(t)dTe()1e(t)PKu(t)iT (6) where u(t) is the input signal to the plant model, the error signal e(t) is defined as y(t) - r(t)= e(t) (7) and r(t) is the reference input signal. In this paper, the PID controller for the quadrotor is developed based on the fast response. Using this approach as a recursive algorithm for the control- laws synthesis, all the calculation stages concerning the tracking errors are simplified. One other aspect of the controller selection depends on the method of control of the UAV. It can be mode- based or non- mode based. For the mode based controller, independent controllers for each state are needed, and a higher level controller decides how these interact. On the other hand for a non- mode based controller, a single controller controls all of the states together. However the adopted control strategy is summarized in the control of two subsystems; the first relates to the position control while the second is that of the attitude control. The quadrotor model above can be divided into two subsystems: A fully- actuated subsystem S1 that provides the dynamics of the vertical position z and the yaw angle (z and ). +=3634/1IKmzKugCosCosuz? ? ? (8) An under actuated subsystem S2 representing the under-actuated subsystem which gives the dynamic relation of the horizontal positions (x, y) with the pitch and roll angles. +=myKmxKSinCosSinCosuSinuSinuCosuyx/211111? ? ? (9) And: +=251432/IlK/IlKuu? ? ? (10) Since drag is very small at low speeds, the drag terms in the above equations can be considered as small disturbances to the system. The PID control is applied to the equations above with inputs u1, u2, u3, u4 and outputs , , and Zd . Though these methods were rather successful in local analysis of nonlinear systems affine in control they usually fail to work for a global analysis and nonlinear systems that are non-affine in control 12. For the fully- actuated subsystem we can construct a rate bounded PID controllers to move states z and , , to their desired values. IV. RESULTS AND SIMULATION STUDY The nominal parameters and the initial conditions of the quadrotor for simulation are: I1=I2=1.25 Ns2/rad. I3=2.5 Ns2/rad. K1=K2=K3=0.010 Ns/m. K4=K5=K6=0.012 Ns/rad. m=2 kg. l=0.2 m. g=9.8 m/s2. the proposed control algorithm shown in Fig.3 ,which is composed of all controllers, inputs, speed reference and the inner relationships of the thrust , the quadrotor system is supplied by a step function for the altitude and (z- axis) which is subject to the three step inputs at (3,10 ,20) and the response yields as can be seen in Fig 4 which is contains some transient overshot and another for the Yaw angle () which is subjected to step input after 5 second as shown in Fig.6 and the roll angle () which is respond after 3 second as it can be seen in Fig 5,the pitch angle response is shown in Fig.7 which 5% overshot when subjected to step input .these transient perturbation are due to many reasons such as an certain of some mechanical parameters in the design and the simplification of controller design . Theta dCSubsystemu1u4u2u3XPhix x Yy Phi Phi Thetay Zz Theta Theta Psiz Psi Psi Speed referenceScope 4Scope 3Scope 2Scope 1Controller 4Theta dTheta u2Controller 2,3xyzPhiPsizdu3u4Controller 1zdzu1z zzPhiPhiThetaThetaPsi Fig. 3.The final simulation model with the PID controllers for the quadrotor 0123456x 104051015202530Iteration(5*104=50sec)z-axis Fig. 4 Plot drawing represent the z- axiz moving to the desired z- point 0123456x 104-1.4-1.2-1-0.8-0.6-0.4-0.200.2Iteration (5*104=50sec)Phi (rad/sec) Fig. 5 Plot drawing represent the Phi (Roll) angle after 3 seconds to start moving to the desired point 0123456x 104-0.02-0.0100.010.020.030.040.050.060.070.08Iteration(5*104=50sec)Psi(rad/sec) Fig. 6 Plot drawing represent the Psi (Yaw) angle after 5 seconds to start moving to the desired point 0123456x 10400.20.40.60.811.21.4Iteration(5*104=50sec)Theta(rad/sec) Fig. 7 Plot drawing represent the Theta (Pitch) angle start moving to the desired point The simulation results show that the PID controllers are able to robustly stabilize the quadrotor helicopter and move it to a desired position with a desired yaw angle while keeping the pitch and the roll angles zero. And here in this design, its easy and with a fast response time, can get the Theta (Pitch angle) to its desired value. The reason of using the PID controllers in this system is to control z, which is sensitive to the changes for the other parameters, Through using the proposed PID controller method strategy. The good performance can be shown from the speed of response of the quadrotor; although the overshoot in the altitude response was removed, the transient response of the system became faster. The same speed of response can be also seen in the yaw, pitch and roll angles control of Figures 4, 5, 6. V. CONCLUSION This paper presented the design of a PID controller algorithm to control the quadrotor system. The model of the vehicle was first modified to simplify the controller design; a different state space representation was described in the paper. The resulting system and controller mathematical models were converted to their respective Simulink models for ease of simulations and studies of the system. These resulting Simulink models are ready to be used now by other researchers as the literature does not clearly explain modeling of the quadrotor or supply a working model and controller. REFERENCES 1 UAVs. New world vistas: Air and space for the 21st centry. Human systems and biotechnology systems, 7.0:1718,1997. 2 P. Castillo, R. Lozano, and A. Dzul, Stabilization of a mini rotorcraft with four rotors, IEEE Control Systems Magazine, vol. 25, pp. 45- 50, Dec. 2005. 3 Lambermont, P., Helicopters and Autogyros of the World, 1958. 4 DraganFly- Innovations, “,” 2003. 5 Pounds, P., Mahony, R., Hynes, P., and Roberts, J., “Design of a Four-Rotor Aerial Robot,” Australian Conference on Robotics and Automation, Auckland, November 2002. 5 Altug, E., Ostrowski, J. P., and Taylor, C. J., “Quadrotor Control Using Dual Camera Visual Feedback,” ICRA, Taipei, September 2003. 7 Bouabdallah, S., Murrieri, P., and Siegwart, R., “Design and Control of an Indoor Micro Quadrotor,” ICRA, New Orleans, April 2004. 8 Castillo, P., Dzul, A., and Lozano, R., “Real- Time Stabilization and Tracking of a Four- Rotor Mini Rotorcraft,” IEEE Transactions on Control Systems Technology, Vol. 12, No. 4, 2004. 9 Hamel T. Mahoney r. Lozano r. Et Ostrowski j. “Dynamic modelling and configuration stabilization for an X4- flyer.” In the 15me IFAC world congress, Barcelona, Spain. 2002. 10 Guenard N. Hamel t. Moreau V, modlisation et laboration de commande de stabilisation de vitesse et de correction dassiette pour un drone CIFA, 2004. 11 E. Altug, J. P. Ostrowski and R. Mahony, ”Control of a Quadrotor Helicopter using Visual Feedback”, Proceedings of the 2002 IEEE International Conference on Robotics and Automation, vol. 1, pp. 72- 77, 2002. 12 R.Olfati- Saber, Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles. PHD thesis in Electrical Engineering and Computer Science, Massachusetts Institute Of Technology, Feb 2001. 13 Khalaf Salloum Gaeid, Haider A. F. Mohamed, Hew Wooi Ping, Lokman H. Hassan “NNPID Controller for Induction Motors with Faults” University of Malaya& University of Nottingham Malaysia Campus, The 2 nd International Conference on Control, Instrumentation & Mechatronic (CIM2009).