Control of Quadrotor Unmanned Aerial Vehicles Using Exact Linearization Technique with the Static State Feedback

Control of Quadrotor Unmanned Aerial Vehicles Using Exact Linearization Technique with the Static State Feedback Yasuhiko Mutoh and Shusuke Kuribara Department of Engineering and Applied Sciences, Sophia University, Tokyo, Japan Email: y_mutou@sophia.ac.jp, s-kuribara@sophia.jp Abstract—It is known that the exact linearization technique by the static state feedback can not be applied to the quadrotor unmanned aerial vehicle (q-UAV), and also that, to apply this control technique, the total dynamic equations of q-UAV should be augmented by introducing two integrators. In this paper, we present a method to apply the exact linearization technique to the q-UAV without augmenting its dynamic model. For this purpose, using the vertical dynamic equation, the vertical position control input is designed separately so that the vertical acceleration is piecewise constant. By using this controller, the vertical dynamic equation and the rest of the total dynamic equations can be decoupled. It will be shown that the exact linearization technique by the static state feedback can be applied to the rest of the total equations. The simulation result will be presented to show the validity of this controller. Index Terms—quadrotor aerial vehicle, nonlinear control system, exact linearization, decoupling control I. INTRODUCTION A quadrotor unmanned aerial vehicle (q-UAV) becomes very popular because it can be used for many purposes. Basically, PID controller works well for q- UAV. And, since its position can be changed through pitch and roll angles, some angle controllers are also used advanced controllers have been applied in many , the back stepping control and the sliding mode control [6], the adaptive control [7], the robust control [8], et al. In this paper, the exact linearization controller is considered. The q-UAV is Affine nonlinear system. For this type of system, it is well known that if the total relative degree from the input signals to the output signals is equal the system degree, the exact linearization controller by the static state feedback can be designed. But, unfortunately, this design strategy can not be applied to q-UAV, because the nonlinear decoupling matrix is singular. But, it is still possible to apply the exact linearization technique to q-UAV by augmenting the system using two integrators [4]. This is called exact linearization by dynamic state feedback. Since q-UAV is  Manuscript received June 1, 2015; revised October 12, 2015. the system of degree 12, the controller should be designed for the augmented system of degree 14. In this paper, we assume that, using the vertical dynamic equation, the vertical position control is designed separately so that the vertical acceleration is piecewise constant. By using such a control input, the system of the vertical motion and the rest of the total system are almost decoupled. The latter system is also an Affine nonlinear system with degree 10. And, it will be shown that this latter dynamic system is exact linearizable by the static state feedback. This implies that the q-UAV system is exact linearizable by the static state feedback under the assumption of the piecewise constant vertical acceleration control. In the following, the motion equation of the q-UAV is summarized in Section 2, and the exact linearization controller by the static state feedback for the q-UAV is obtained in Section 3. In Section 4, simulation results will be shown. II. MOTION EQUATION OF Q-UAV Consider the q-UAV depicted in Fig. 1. Figure 1. The model of the q - UAV The inertial coordinate system is represented by x y z  for the position of the center of the gravity of the body, and the Euler angles  ,  and  are yaw, pitch and roll angles respectively. The relation of angular velocity with respect to the body reference frame p , q and r and  ,  and  is expressed as follows. 340©2016 Journal of Automation and Control Engineering Journal of Automation and Control Engineering Vol. 4, No. 5, October 2016 doi: 10.18178/joace.4.5.340-346 [1] [2]. Furthermore, for higher performance, various literatures. For example, the exact linearization [4][5] 1 1 0 sin cos cos cos 0 cos sin 1 sin tan cos tan p q r                                            (1) It is well known that the q-UAV has the following motion equations [4], [5]. 1 2 1 1 2 1 2 1 2 1 1 1 2 2 3 2 2 4 2 3 3 (cos sin cos sin sin ) (cos sin sin sin cos ) cos cos sin tan cos tan cos sin 1 1 sin cos cos cos x x u m y y u y m z z u z g m p q r p a b u q r q a b u q r a b x r u q p pq r r                                                (2) where 1x x , 2x x , 1y y , 2y y , 1z z , and 2z z . The input 1u is the total thrust of four rotors, and 2u , 3u , and 4u are torque in the coordinate  , and  respectively. Let 1 , 2 , 3 and 4 be the rotation velocities of four rotors shown in Fig. 1. Then, i and iu (i=1, 2, 3, 4) satisfy the following relation using the nonsingular matrix. 2 1 1 2 2 2 2 3 3 2 4 4 1 1 1 1 0 1 0 1 1 0 1 0 1 1 1 1 u u K u u                                        (3) K is a conversion factor. In equation (2), m and g are the mass of the q-UAV and the gravity acceleration, respectively. The parameters ia and ib ( 1, 2,3i  ) are constants defined by 1 2 3 1 2 3 1 y z x yz x x y z x y z I I I II I a a a I I I l l b b b I I I               (4) where xI , yI and zI are the moment of inertia in pitch, roll and yaw respectively about the center of gravity of the q-UAV, and l is the distance between the center of gravity and the rotor. It should be noted that, in this equations, the gyroscopic effects and aerodynamic forces and moments are omitted for simplicity of design procedure as is the case in many literatures. These effects are small for the q- UAV and can be controlled by some additional controller like the sliding mode control, the adaptive control, the robust control, et al. If the Euler angles Now, we define the state vector 12 R  by 1 2 1 2 1 2 [ , , , , , , , , , , , ] [ , , , , , , , , , , , ] T T x x y y z z p q r x x y y z z p q r          (5) The output signals of this system are the three positions ( 1 1 1 , ,x y z ) and the yaw angle  . This implies that the q-UAV system is 4-input 4-output Affine nonlinear system. Using the state vector  , the input vector 1 2 3 4 [ , , , ] T u u u u u and the output vector 1 1 1 [ , , , ] T x y z  , the equation (2) can be written compactly as follows. ( ) ( ) ( ) f G u h         (6) where 2 2 2 1 1 1 1 2 3 0 0 ( ) , ( )sin tan cos tan cos sin 1 1 sin cos cos cos x y z x g y f hp q r z a q r a q r a qr pr pq                                                                (7) Using Lie derivatives of ( )h  , it is readily checked that the nonlinear decoupling matrix is singular, which implies that this system is not exact linearizable by the static state feedback. In the reference [4], the design method of exact linearization controller using the dynamic state feedback for this type of the q-UAV is presented. It was shown there that the system equation should be augmented using two integrators, and the controller is designed for the nonlinear system of degree 14. 341©2016 Journal of Automation and Control Engineering Journal of Automation and Control Engineering Vol. 4, No. 5, October 2016 1 2 3 ( ) 0 0 0 0 (cos sin cos sin sin ) / 0 0 0 0 0 0 0 (cos sin sin sin cos ) / 0 0 0 0 0 0 0 (cos cos ) / 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G m m m b b b                                                    (8) III. EXACT LINEARIZATION BY THE STATIC STATE FEEDBACK In this section, the exact linearization technique by the static state feedback is applied to the q-UAV. This implies the decoupling controller from u to  . For this purpose, the input 1u is first designed only for the vertical position control using the following vertical motion equations. 1 2 1 2 cos cos z z u z g m       (9) It is supposed that the desired vertical position *1z is given by the following equation with the desired initial condition *1 (0)z . * * 1 2 * * 2 1 z z z g u     (10) here, *1u is a desired vertical acceleration which is supposed to be a piecewise constant. If *1u g , this implies the hovering control at the initial vertical position. Then, from (9), 1u is determined by the following equation. * 1 1 ( ) cos cos m u v u     (11) here, v is the following stabilizing PD feedback input 0 1 1 2 v e e   (12) where * 1 1 1 * 2 2 2 . e z z e z z     (13) From (9) - (13), 1e and 2e satisfy 1 2 2 0 1 1 2. e e e e e      (14) The parameter 0 and 1 are chosen so that the error equation (14) is asymptotically stable. This implies that v converges to 0 exponentially. From the above, the motion equation (2) is modified as follows. First, the vertical dynamics is as follows. 2 1 2 * 1 z z z g u v      (15) And, the rest of all dynamic equations are 1 2 * 2 1 1 2 2 * 1 1 1 2 2 2 3 3 3 4 1 (tan cos tan sin )( ) cos 1 (tan sin tan cos )( ) cos sin tan cos tan cos sin 1 1 sin c os cos cos x x x u v y y y u v p q r p a b u q r q a b u qr pr q r a b ur pq                                            (16) This implies that the dynamics of the q-UAV is represented by the vertical dynamics (15) and the rest of all dynamic model (16). Define the reduced state vector 10R  by 1 2 1 2[ , , , , , , , ] , , Tx x y y p q r    (17) Then, the equation (16) can be written as follows. ( ) ( ) ( ) f G u h         (18) where 2 1 3 1 4 , ( ) u x u u h y u                       Since v converges to 0 exponentially, v is omitted in (18) and (19) for simplicity of the controller design procedure. The system (18), (19) is Affine nonlinear system of degree 10 with 3-input 3-output. The output function ( )h  can be written as follows. 342©2016 Journal of Automation and Control Engineering Journal of Automation and Control Engineering Vol. 4, No. 5, October 2016 2* 1 2 * 1 1 2 3 1 2 1 (tan cos tan sin ) cos 1 (tan sin tan cos ) cos sin tan cos tan( ) cos sin 1 1 sin cos cos cos 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( ) 0 0 0 0 0 0 0 0 qr p x u y u p q rf a q r a q r a G b r pq b                                                                3 0 0 0 0 b                                (19) 1 2 3 ( ) ( ) ( ) ( ) h h h h                 (20) here, 1 1( )h x  , 2 1( )h y  and 3( )h   . Then, using Lie derivative, we have 2 1 1 1 2 2 2 2 3 3 ( ) ( ) ( ) 0 ( ) ( ) ( ) 0 ( ( ) ) 0 G G Gf f G G Gf f G G f L h L L h L L h L h L L h L L h L h L L h                 (21) And the nonlinear decoupling matrix ( ) becomes 3 1 1 11 2 12 3 13 3 2 1 21 2 22 3 2 2 3 3 3 sin cos cos c ( ) ( ) ( ) 0 s ( ) o G f G f G f L L h b b d b d b d L L h b d b d b d L L h b                               (22) where (23) And *1 11 12 1 1 *1 1 1 1 12 12 1 *1 1 13 1 1 1 1 *1 21 12 1 1 *1 1 1 1 22 1 1 1 2 1 2 2 3 1 1 2 2 sin cos cos cos tan sin sin cos tan cos ( tan sin ) cos cos cos cos sin tan cos sin cos (tan co u u u u u                                                                     *1 1 1 1 1 tan sin s ) cos u      (24) In (24), 1 and 2 are defined by 1 2 1 tan cos tan sin cos 1 tan sin tan cos cos                 (25) Which appeared in the second and the fourth elements of ( )f  . From the above, the determinant of the nonlinear decoupling matrix ( ) of the system (18) becomes * 2 1 2 3 142 1 det ( ) ( ) cos cos b b b u    (26) Which implies that ( ) is nonsingular if 1 / 2   and 1 / 2   . It is also required * 1 0u  for the non- singularity of ( ) . This means that rotors should rotate to control the q-UAV. Furthermore, from (22), the vector relative degree of the system (18) and (19) from u to  is (4, 4, 2), i.e., the total relative degree of this system is the same as the system degree, 10. This implies that the system (18) and (19) can be exact linearized and, at the same time, can be decoupled by the static state feedback. By differentiating 1x , 1y and  successively, the following equations are obtained using the differential operator s . 343©2016 Journal of Automation and Control Engineering Journal of Automation and Control Engineering Vol. 4, No. 5, October 2016 (27) Define arbitrarily stable characteristic polynomials of s as follows. 4 3 2 1 13 12 11 10 4 3 2 2 23 22 21 20 2 3 31 30 ( ) ( ) ( ) s s ss s s s s s s s s s                           (28) From the above, using (27) and (28), we have the following input-output equation. 1 1 2 1 3 ( ) ( ) ( ) ( ) ( ) p x p y F u p                          (29) where 2 3 4 10 1 11 1 12 1 13 1 1 2 3 4 20 2 21 2 22 2 23 2 2 2 30 3 31 3 3 ( ) f f f f f f f f f f h L h L h L h L h F h L h L h L h L h h L h L h                                (30) Then, by the state feedback with new input signal 1 2 3[ , , ]    , 1( )( ( ) )u F      (31) The following exact linearized system is obtained. 1 1 1 2 1 2 3 3 ( ) ( ) ( ) s x s y s                                (32) Now, the total system of the q-UAV is represented by (12)(15) and (32) which are decoupled linear systems. Note that these decoupled equations are obtained by only static state feedback. Using these equations, we can design the control input, *1u (piecewise constant) and PD feedback v , and then, 1 , 2 and 3 separately for the output signal 1z , 1x , 1y and  respectively. Let the desired output of 1x , 1y and  be * 1x , * 1y and * , then, one example of the control input 1 2 3[ , , ]    in (31) can be simply determined as follow. * 1 1 1 * 2 2 1 * 3 3 ( ) ( ) ( ) s x s y s           (33) From this, 1z , 1x , 1y and  follow the desired trajectory *1z , * 1x , * 1y and * , and the controller can be designed for each output signal separately by the static state feedback. IV. SIMULATION RESULT In this section, the simulation result is presented to show the validity of the controller stated in the previous section. The values of parameters used for the simulation are shown in Table I. TABLE I. VALUES OF PARAMETERS PARAMETER VALUE UNIT g 9.8 2/m s m 0.5 kg l 0.25 m xI 4.9E-3 2 kg m yI 4.9E-3 2 kg m zI 8.8E-3 2 kg m K 2.9E-5 The desired trajectories of 1x x , 1y y , 1z z ,  are as follows. 2sin 0.5 2cos0.5 1 0.1 0.35sin 0.5 d d d d x t y t z t t         (30) The initial conditions for x , y , z ,  are (0) 0, (0) 0, (0) 1, (0) / 6x y z      (31) From the previous Sections, the controller design procedure is as follows. [STEP 1] Design 1u as (11) first, based on the desired piecewise constant acceleration *1u for the desired vertical position, * dz z and the stabilizing feedback (12). [STEP 2] Design the static state feedback control input u as (31) for the exact linearization, based on the equation (18) (19). [STEP 3] Based on the desired positions, dx , dy and the desired yaw angle, d , calculate the control input, 1 , 2 and 3 , separately as (33). In this example, we use the stable characteristic polynomial, as follows. 344©2016 Journal of Automation and Control Engineering Journal of Automation and Control Engineering Vol. 4, No. 5, October 2016 4 3 2 1 4 3 2 2 2 3 ( ) 15 18 10 2 ( ) 15 18 10 2 ( ) 2 1 s s s s s s s s s s s s s                 (32) Fig. 2 shows the response of the position of the center of gravity of the q-UAV and its desired trajectory in the x y z  space. The response of x , y , z and the yaw angle  are shown in Fig. 3, 4, 5, and 6 respectively with their desired trajectories. Fig. 7 shows the roll angle  and the pitch angle  . The simulation result shows that the exact linearization controller by the static state feedback works very well. Figure 2. Response of the center of gravity of the q-UAV. Figure 3. Response of x and desired x. Figure 4. Response of y and desired y. Figure 5. Response of z and desired z. Figure 6. Response of yaw and desired yaw angles. Figure 7. Response of the roll and pitch angles V. CONCLUSIONS It is known that the q-UAV system is 4-input 4-output Affine nonlinear system of degree 12 and that it can not be exact linearizable by the static state feedback. So, to apply this control technique, it is necessary to augment the total system of the q-UAV by introducing two integrators. In this paper, we suppose that the vertical position is controlled so that the vertical acceleration is piecewise constant. By using this control input, the total system is presented by the vertical motion dynamics and the rest of the total dynamic system. The latter system is 3-input 3-output Affine nonlinear system of degree 10. It was shown in this paper that this system is exact linearizable by the static state feedback. The simulation result shows the high performance of this controller. REFERENCES [1] L. R. G. Garrillo, A. E. D. Lopez, R. Lozano, and C. Pegard, Quad Rotercraft Control, Springer, 2013. [2] A. Tayebi and S. MacGilcary, “Attitude stabilization of a VTOL quadrotor aircraft,” IEEE Trans. Control Systems Technology, vol. 14 [3] F. Kendoul, D. Lala, I. Choichot, and R. Lozano, “Real-time nonlinear embedded control for an autonomous quadrotor helicopter,” J. of Guidance, Control and Dynamics, vol. 30, no. 4, pp. 1049-1061, 2007. [4] V. Mistler, A. Benallegue, and N. K. M'Sirdi, “Exact linearization and noninteracting control of a 4 rotors helicopter via dynamic feedback,” in Proc. 10th IEEE International Workshop on Robot and Human Interactive Communication, 2001, pp. 586-593. [5] A. Mokhtari, N. K. M'Sirdi, and K. Meghriche, “Feedback linearization and linear observer for a quadrotor unmanned aerial vehicle,” Advanced Robotics, vol. 20, no. 1, pp. 71-91, 2006. [6] L. Besnarda, Y. B. Shtessel, and B. Landrum, “Quadrotor vehicle control via sliding mode controller driven by sliding mode disturbance observer,” J. of Franklin Institute, vol. 349, no. 2, pp. 658-684, 2012. [7] T. Madani and A. Benallegue, “Adaptive control via backstepping technique and neural networks of a quadrotor helicopter,” in Proc. the 17th IFAC World Congress, 2008, pp. 6513-6518. 345©2016 Journal of Automation and Control Engineering Journal of Automation and Control Engineering Vol. 4, No. 5, October 2016 , no. 3, pp. 562-572, 1993. [8] G. V. Raffo, M. G. Ortega, and F. R. Rubio, “MPC with nonlinear H-Infinity control for path tracking of a quad-rotor helicopter,” in Proc. the 17th IFAC World Congress, 2008, pp. 8564-8569. Yasuhiko Mutoh was born in Tokyo, Japan. He received the B.S., M.S. and Ph.D. degrees in mechanical engineering from Sophia University, Tokyo, Japan, in 1975, 1977 and 1981 respectively. In 1981, he joined the Department of Mechanical Engineering, Sophia University, where he is currently Professor. His research interests include multivariable systems, adaptive control systems, multivariable nonlinear systems, time-varying control system. Shusuke Kuribara received the B.E. Degree in engineering and applied science from Sophia University, Tokyo Japan, in 2014. He is now a master course student of Sophia University. His current research interests include navigation and control with applications to unmanned aerial vehicles. 346©2016 Journal of Automation and Control Engineering Journal of Automation and Control Engineering Vol. 4, No. 5, October 2016