Robust Chattering Free Backstepping Sliding Mode Control Strategy for Autonomous Quadrotor Helicopter

Transcription

1 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:4 No: 6 Robust Chattering Free Backstepping Sliding Mode Control Strategy for Autonomous Quadrotor Helicopter Mohd Ariffanan Mohd Basri, Abdul Rashid Husain, Kumeresan A. Danapalasingam Department of Control & Mechatronics Engineering, Faculty of Electrical Engineering, Universiti Teknologi Malaysia, 8 Skudai, Johor, Malaysia. ariffanan@fke.utm.my, rashid@fke.utm.my, kumeresan@fke.utm.my Abstract In this paper a robust chattering free backstepping sliding mode controller is developed for the attitude stabilization and trajectory tracking control of quadrotor helicopter with external disturbances. The control scheme is developed based on backstepping technique and a sliding surface is introduced in the final stage of the algorithm. To attenuate the chattering problem caused by a discontinuous switching function, a simple fuzzy system is used. The asymptotical stability of the system can be guaranteed since the control law is derived based on Lyapunov theorem. Simulation results of the developed controller which applied for a highly nonlinear quadrotor helicopter are presented. From the simulation results, it is shown that the developed control system not only achieves satisfactory control performance, but also eliminates the chattering phenomena in the control effort. Index Term Quadrotor helicopter, backstepping control, sliding mode control, chattering free. I. INTRODUCTION Nowadays, autonomous helicopters are used in different military and civilian applications such as border patrolling, geological surveying, traffic monitoring, area mapping, search and rescue and reconnaissance missions. Quadrotors are a special class among different types of helicopters. Quadrotor helicopters have gained a lot of research interest in the past few years, due to the clear advantages posed by their vertical take-off and landing VTOL), hovering capability, and slow precise movements. Belonging to the helicopter rotorcraft class, quadrotors are highly nonlinear systems that are difficult to stabilize. The main challenges of the quadrotor helicopters are mainly due to its unstable nature and complexity of the dynamic model. Furthermore, this type of helicopter is a highly nonlinear system and very susceptible to external disturbances. Many approaches and techniques have been proposed to control a quadrotor helicopter, such as linear quadratic regulator LQR) control [], proportional-integralderivative PID) control [], fuzzy logic FL) control [], sliding mode control [4], and backstepping control [-8]. The backstepping control scheme is a nonlinear control method based on the Lyapunov theorem. The backstepping control design techniques have received great attention because of its systematic and recursive design methodology for nonlinear feedback control. The advantage of backstepping compared with other control methods lies in its design flexibility, due to its recursive use of Lyapunov functions. Sliding mode control SMC) is a type of nonlinear control systems. Such controller has proven very robust to model uncertainties and external disturbances [9-]. Successfully implemented in varieties forms and numerous real-world applications such as satellite [], microelectromechanical system MEMS) gyroscope [], aircraft [], chemical reactor [4], flexible spacecraft [], robot manipulator [6], electrohydraulic actuator [7], and wheeled inverted pendulum systems [8], SMC has proven reliable and much more robust. Thus, owing to the merits of backstepping control and sliding mode control, in this paper, both control schemes are combined for attitude stabilization and trajectory tracking control of a quadrotor helicopter. In the design procedure, the sliding manifold is introduced in the final step of backstepping method. A discontinuous sign function of sliding manifold will excite undesired phenomena called chatter in the control input. The chatter can deteriorate system performance and also cause undesired wear and tear in mechanical devices. Therefore, it is essential to develop a strategy to eliminate the chatter in the control input. To solve the issue, a simple fuzzy system which proposed in [9] is utilized to replace the sign function. Through this job, the chatter is eliminated effectively, because the control input is smooth with respect to time. To demonstrate the effectiveness and feasibility of the developed control strategy, a simulation of quadrotor helicopter with external disturbances is examined. Compared with the method presented in [], the developed control scheme has the advantage of free from the chattering phenomena. The main contribution of this paper is a successful development of a robust chattering free backstepping sliding mode controller for quadrotor helicopter perturbed by external disturbances. II. QUADROTOR DYNAMIC MODEL Let consider earth fixed frame * + and body fixed frame * +, as seen in Fig.. Let be the generalized coordinates for the quadrotor, where denote the absolute position of the rotorcraft and are the three Euler angles roll, pitch and yaw) that describe the orientation of the aerial vehicle. Therefore, the model could be separated in two coordinate IJMME-IJENS June 4 IJENS

2 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:4 No: 7 subsystems: translational and rotational. They are defined respectively by: ) where is the rotor inertia, represent the distance from the rotors to the centre of mass and is the drag factor. Then, by recalling ) and 6), the dynamic model of the quadrotor in terms of position and rotation is written as: ) Fig.. Quadrotor UAV configuration. ) ) ) The dynamic model of quadrotor is derived from Newton- Euler approach. The translational dynamic equations of quadrotor can be written as follows: where denotes the quadrotor mass, the gravity acceleration, the unit vector expressed in the frame E and the total thrust produced by the four rotors. where. Consequently, quadrotor is an underactuated system with six outputs and four control inputs. Finally, the quadrotor dynamic model can be written in the following form: where and denote respectively, the thrust force and speed of the rotor and is the thrust factor. The orientation matrix R is given by: where s and c are abbreviations for sin and cos, respectively. The rotational dynamic equations of quadrotor can be written as follows: where is the inertia matrix, and are the gyroscopic effect due to rigid body rotation and propeller orientation change respectively, while is the control torque obtained by varying the rotor speeds. and are defined as: with a renaming of the control inputs as: IJMME-IJENS June 4 IJENS

3 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:4 No: 8 III. BACKSTEPPING CONTROL SYSTEM The dynamic model ) with the consideration of external disturbance can be represented into nonlinear dynamic equation described as follows: where, and are respectively the input, state and external disturbance vector given as follows:, -, -, -, - The bound of the external disturbance is assumed to be given, that is, where is a given positive constant. From ) and ), the nonlinear dynamic function and nonlinear control function matrices can be written accordingly as: ) can be viewed as a virtual control. The desired value of virtual control known as a stabilizing function can be defined as follows: where is a positive constant. By substituting the virtual control by its desired value, Eq. ) then becomes: Step : The deviation of the virtual control from its desired value can be defined as: The derivative of is expressed as: ) The second Lyapunov function is chosen as: with the abbreviations,,,,,,,,,, The control objective is to design a suitable control law so that the state trajectory of the quadrotor system can track a desired reference trajectory, - despite the presence of external disturbance. The design of ideal backstepping control IBC) is described step-by-step as follows: Step : Define the tracking error: Then the derivative of tracking error can be represented as: The first Lyapunov function is chosen as: Finding derivative of 6) yields: ) ) ) ) Step : Assuming the external disturbance is well known, an IBC can be obtained as: ) where is a positive constant. The term is added to stabilize the tracking error. Substituting 8) into ), the following equation can be obtained: Since, is negative semi-definite. The derivative of is: IJMME-IJENS June 4 IJENS

4 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:4 No: 9 Therefore, the IBC in 8) will asymptotically stabilize the system. ) IV. BACKSTEPPING SLIDING MODE CONTROL SYSTEM However, if unpredictable perturbations from the unknown external disturbance occur, the optimized IBC 8) effort cannot ensure the favorable control performance. Thus, auxiliary control effort should be designed to eliminate the effect of the unpredictable perturbations. The auxiliary control effort is referred as hitting control effort represented by. The hitting control effort is designed such that the system state trajectories are forced toward the sliding surface and stay on it. This effort is known as the sliding mode. Procedures to design of the backstepping sliding mode control BSMC) can be described by the following steps: Step : Similar as step in the design of ideal backstepping control IBC). Step : Define a sliding surface in terms of the error such as: Thus, the Lyapunov function 6) can be written as: V. CHATTERING FREE BACKSTEPPING SLIDING MODE CONTROL SYSTEM The utilization of discontinuous sign function will excite undesired phenomenon called chatter. In order to eliminate the chattering phenomena, a simple fuzzy system is utilized to mimic the hitting control effort. Let the sliding surface be the input linguistic variable of the fuzzy logic system, and the hitting control effort be the output linguistic variable. To reduce the calculation, the hitting control effort is designed by three simple fuzzy rules as given in the following form [9]: Rule : IF is P THEN is PE Rule : IF is Z THEN is ZE Rule : IF is N THEN is NE where the triangular-typed functions and singletons are used to define the membership functions of IF-part and THEN-part, which are depicted in Figs. a) and b), respectively. Differentiating ) with respect to time leads to: ) ) Step : Since the external disturbance is unknown, a backstepping control can be obtained as: Fig.. Membership functions. a) Input fuzzy sets. b) fuzzy sets. The defuzzification of the output is accomplished by the method of center-of-gravity as follows: ) Step 4: Define the hitting control effort such as: where is a constant and is a sign function: { where, and are the firing strengths of rules, and, respectively;, and are the center of the membership functions PE, ZE and NE, respectively. The relation is valid according to the special case of triangular membership function-based fuzzy system. Hence, the following four conditions can be deduced. For any value of input, only one of four conditions will occur according to Fig. a) [9]. Totally, the robust control law known as backstepping sliding mode control law for nonlinear systems with the present of external disturbance, which guarantees the stability and convergence can be represented as: Condition. Only rule is triggered. ) IJMME-IJENS June 4 IJENS

5 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:4 No: 4 Condition. Rules and are triggered simultaneously. Condition. Rules and are triggered simultaneously. Condition 4. Only rule is triggered. ) According to four possible conditions, the 4)-44) can be rewritten as: Totally, the control law for a quadrotor nonlinear system with the present of external disturbance can be represented as: ) Theorem: For the nonlinear dynamic equation of quadrotor with external disturbance as represented by ), if the control law in 46) is applied, the system will asymptotically stable. Proof: Consider the Lyapunov function ) and its derivative ). Substituting the control law from 46), then ) becomes: ) Hence, the design parameter should be chosen in such a way that is always satisfied. As aforementioned, it is assumed that is bounded with. If the following inequality holds: then can be guaranteed. The configuration of the developed control system is depicted in Fig.. Fig.. Block diagram of the chattering free backstepping sliding mode control system. VI. SIMULATION RESULTS In this section, the performance of the proposed approach is evaluated. The corresponding algorithm is implemented in MATLAB/SIMULINK simulation environment. The model parameter values of the quadrotor system are adopted from [] and listed in Table. To explore the effectiveness of the developed controller, two simulation experiments have been performed on the quadrotor. In the first experiment, the simulation results of the developed controller in attitude stabilizing problem are given. In the second, the performance of the scheme is investigated in trajectory tracking problem. TABLE I. PARAMETERS OF THE QUADROTOR. Parameter Description Value Units g Gravity 9.8 m/s m Mass. kg l Distance. m I xx Roll inertia kg m I yy Pitch inertia kg m I zz Yaw inertia kg m b Thrust factor.9-6 d Drag factor. -7 A. Simulation experiment : attitude stabilizing problem In this simulation experiment, the control objective is to regulate a quadrotor at a certain desired attitude angles. Attitude control is crucial importance for a quadrotor to implement a hover mission. The desired attitude is given by, -, -. In the simulation, first, the IBC system is employed. The response of attitude angles and the associated control signals for stabilizing a quadrotor are respectively depicted in Figs. 4 and. As it can be seen from Fig. 4, the IBC system is unable to stabilize the quadrotor effectively. It is noted that the regulation errors of the roll and pitch angles are achieved with amplitudes of. Under the same simulation condition, the backstepping sliding mode control system using sign function and also using simple fuzzy system are simulated accordingly. The simulation results of both controllers for stabilizing a quadrotor are depicted in Figs. 6 and 8, respectively. The respective control signals are shown in Figs. 7 and 9. From the simulation results, the robust IJMME-IJENS June 4 IJENS

6 Yaw [rad] U4 [N] Pitch [rad] U [N] Roll [rad] U [N] U4 [N] Yaw [rad] U [N] Pitch [rad] U [N] Roll [rad] Yaw [rad] Pitch [rad] Roll [rad] International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:4 No: 4 control performance of both control systems in the quadrotor stabilization can be observed. It can be clearly seen that the attitude of the quadrotor can be maintained at the desired value with almost zero regulation errors even in the influences of the external disturbances. However, the control effort of the backstepping sliding mode controller with sign function produces a serious chattering phenomena as can be seen in Fig. 7. On the contrary, the chattering phenomena of the backstepping sliding mode controller with a simple fuzzy system are suppressed.. Roll angle Pitch angle Yaw angle.. Roll angle Fig. 6. Attitude angles response using BSMC with sign function Pitch angle Yaw angle Fig. 4. Attitude angles response using IBC Fig. 7. Control inputs U, U & U4) of BSMC with sign function Roll angle Fig.. Control inputs U, U & U4) of IBC Pitch angle Yaw angle Fig. 8. Attitude angles response using BSMC with fuzzy system IJMME-IJENS June 4 IJENS

7 zm) U [N] U [N] U [N] U4 [N] U [N] zm) U [N] International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:4 No: End position Initial position - - ym) xm) Fig.. Helical trajectory tracking response using IBC Fig. 9. Control inputs U, U & U4) of BSMC with fuzzy system. A. Simulation experiment : trajectory tracking problem In this simulation experiment, the control objective is to ensure the quadrotor can effectively track the desired reference trajectory. The helical trajectory is adopted to test the trajectory tracking capability of the quadrotor by the IBC and BSMC system. The desired trajectory is generated using the following command: { The initial state of the quadrotor is set to be, -, -m. The simulation results of helical trajectory tracking for IBC approach under the occurrence of external disturbances are shown in Fig.. The associated control inputs are shown in Fig.. As it can be seen from Fig., the IBC system is still able to track the desired reference trajectory, but severe deviations are notable during maneuvering. In contrast, the BSMC system is able to deal effectively with external disturbances to achieve satisfactory tracking performance as depicted in Figs. and 4. From the simulation results, although the satisfactory tracking performance can be achieved, however, there still exists the undesirable chattering problem in the control inputs. This problem which arises from the utilization of sign function can be clearly seen in Fig.. Nevertheless, the time response curve of associated control input as shown in Fig. is smooth by using the proposed fuzzy system, that is to say, there is no chatter in the control inputs. It is evident from the simulation results that the proposed fuzzy system not only can provide robustness to BSMC system, but also can solve a chattering problem in the control inputs Fig.. Control inputs U, U & U) of IBC. ym) - - Initial position Fig.. Helical trajectory tracking response using BSMC with sign function. End position xm) IJMME-IJENS June 4 IJENS

8 U [N] U [N] U [N] zm) U [N] U [N] U [N] International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:4 No: Fig.. Control inputs U, U & U) of BSMC with sign function. - - Fig. 4. Helical trajectory tracking response using BSMC with fuzzy system. ym) Initial position Fig.. Control inputs U, U & U) of BSMC with fuzzy system. End position xm) VII. CONCLUSIONS In this paper, a robust chattering free backstepping sliding mode controller is successfully developed for attitude stabilization and trajectory tracking of a quadrotor helicopter perturbed by external disturbances. First, the mathematical model of the quadrotor is introduced. Then, the proposed robust control system comprises a backstepping and a switching function is developed. The backstepping control design is derived based on Lyapunov function, so that the stability of the system can be guaranteed, while switching function is used to attenuate the effects caused by external disturbances. In order to eliminate the chattering phenomena, the sign function is replaced by the fuzzy system. Finally, the developed control scheme is applied to autonomous quadrotor helicopter. Simulation results show that a satisfactory control performance can be achieved by using the developed control system. Furthermore, it can be seen that a fuzzy system can be utilized to eliminate a chattering phenomena in the control inputs of the BSMC. REFERENCES [] P. Castillo, R. Lozano, A. Dzul, Stabilization of a mini rotorcraft with four rotors, In Proc. of The IEEE Control Systems Magazine, vol., no. 6, pp. 4,. [] A.L. Salih, M. Moghavvemi, H.A.F. Mohamed, K.S. Gaeid, Modelling and PID controller design for a quadrotor unmanned air vehicle, In Proc. of The IEEE International Conference on Automation Quality and Testing Robotics AQTR), pp.,. [] M. Santos, V. López, F. Morata, Intelligent fuzzy controller of a quadrotor, In Proc. of The IEEE International Conference on Intelligent Systems and Knowledge Engineering ISKE), pp. 4-46,. [4] R. Xu, U. Ozguner, Sliding mode control of a quadrotor helicopter, In Proc. of The IEEE Conference on Decision and Control, pp , 6. [] T. Madani, A. Benallegue, Backstepping Control for a Quadrotor Helicopter, In Proc. of The IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. -6, 6. [6] H. Bouadi, M. Bouchoucha, M. Tadjine, Modelling and Stabilizing Control Laws Design Based on Backstepping for an UAV Type- Quadrotor, In Proc. of the IFAC Symposium on IAV, Toulouse, France, 7. [7] G.V. Raffo, M.G. Ortega, F.R. Rubio, Backstepping/nonlinear H control for path tracking of a quadrotor unmanned aerial vehicle, In Proc. of The American Control Conference, pp. 6-6, 8. [8] G. Regula, B. Lantos, Backstepping-based control design with state estimation and path tracking to indoor quadrotor helicopter, Period. Polytech. Electr. Eng, pp. -,. [9] Y. Shtessel, et al., "Tailless aircraft flight control using multiple time scale reconfigurable sliding modes," IEEE Transactions on Control Systems Technology, vol., pp ,. [] I. Shkolnikov, et al., "Robust missile autopilot design via high-order sliding mode control," in Proceedings of AIAA guidance, navigation, and control conference,. [] T. E. Massey and Y. B. Shtessel, "Continuous traditional and high-order sliding modes for satellite formation control," Journal of Guidance, Control, and Dynamics, vol. 8, pp. 86-8,. [] J. Fei and C. Batur, "Adaptive sliding mode control with sliding mode observer for a microelectromechanical vibratory gyroscope," Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol., pp , 8. [] Y. Huang, et al., "Robust vertical takeoff and landing aircraft control via integral sliding mode," IEE Proceedings-Control Theory and Applications, vol., pp. 8-88, IJMME-IJENS June 4 IJENS

9 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:4 No: 44 [4] R. Aguilar-López and J. Alvarez-Ramírez, "Sliding-mode control scheme for a class of continuous chemical reactors," IEE Proceedings- Control Theory and Applications, vol. 49, pp. 6-68,. [] Q.-L. Hu, et al., "Sliding mode and shaped input vibration control of flexible systems," Aerospace and Electronic Systems, IEEE Transactions on, vol. 44, pp. 9, 8. [6] S. Huh and Z. Bien, "Robust sliding mode control of a robot manipulator based on variable structure-model reference adaptive control approach," IET Control Theory & Applications, vol., pp. -6, 7. [7] A. G. Loukianov, et al., "Robust trajectory tracking for an electrohydraulic actuator," IEEE Transactions on Industrial Electronics, vol. 6, pp. -, 9. [8] J. Huang, et al., "Sliding-mode velocity control of mobile-wheeled inverted-pendulum systems," IEEE Transactions on Robotics, vol. 6, pp. 7-78,. [9] R.-J. Wai, "Fuzzy sliding-mode control using adaptive tuning technique," IEEE Transactions on Industrial Electronics, vol. 4, pp. 8694, 7. [] H. Bouadi, et al., "Sliding Mode Control based on Backstepping Approach for an UAV Type-Quadrotor," International Journal of Applied Mathematics & Computer Sciences, vol. 4, 8. [] H. Voos, "Nonlinear control of a quadrotor micro-uav using feedbacklinearization," in IEEE International Conference on Mechatronics ICM), pp. -6, IJMME-IJENS June 4 IJENS