System Design of Quadrotor

Transcription

1 Syste Design of Quadrotor Yukai Gong, Weina Mao, Bu Fan, Yi Yang Mar. 29, 2016 A final project of MECHENG 561. Supervised by Prof. Vasudevan. 1

2 Abstract In this report, an autonoous quadrotor is designed. Based on two control techniques drawn fro a classical PID schee in classical control theory and a backstepping control schee in nonlinear control theory, We proposed a siplified atheatical odel and conduct a siulation in Siulink. Different perturbation has been exerted to the syste, the perforance in stability and robustness has been analyzed and copared between two different controller schees. For siplicity of the project, we do not involve an experiental validation even if the we can consider to construct a real quadrotor controlled be an ebedded Arduino Board for the next step. 2

3 Contents 1 Introduction Background Aerodynaic Modelling of Quadrotor Aircraft Syste Design Based on PID Method Design the PID Controller Siulation Results and Analysis Suary Syste Design Based on Backstepping Method Backstepping controller design Siulation of Backstepping control with siplified inputs Siulation of Backstepping control with real inputs Siulation of Backstepping control in discrete tie Backstepping control in discrete tie under disturbance Siulation of Adaptive Backstepping control in discrete tie under disturbances Siulation of Adaptive Backstepping control with noise Coparison 10 A Backstepping control 11 A.1 Backstepping attitude control A.1.1 Backstepping position control A.2 Adaptive Backstepping control in discrete tie under disturbance

4 1 Introduction 1.1 Background Flying objects have always exerted a great fascination on an encouraging all kinds of research and developent. [1]The iportant recent technological progress in sensors, actuators, processors and power storage devices represents a real jup ahead, which enable the eergence of new applications like the indoor icro aerial robots. Copared with the other flying principles, quadrotor systes have specific characteristics which allow the execution of applications that would be difficult or ipossible otherwise, such as building surveillance and intervention in hostile environents. 1.2 Aerodynaic Modelling of Quadrotor Aircraft The odel will have 12 states: displaceent in three directions(x,y,z), velocity in three directions(ẋ, ẏ, ż), roll angle(φ), pitch angle(θ), yaw angle(ψ) and three angular velocities( φ, θ, ψ) with respect to these three angles.[2] Let [φ, φ, θ, θ, ψ, ψ, z, ż, x, ẋ, y, ẏ] = [x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9, x 10, x 11, x 12 ]. The original nonlinear dynaic odel for this syste is provided as: 1 = x 2 2 = (I y I z ) I x x 4 x 6 + U 2l I x 3 = x 4 4 = (I z I x ) I y x 2 x 4 + U 3l I y 5 = x 6 6 = (I x I y ) I z x 2 x 4 + U 4 I z 7 = x 8 8 = cos(x 1 )cos(x 3 ) U 1 9 = x 10 g x 10 = (sin(x 3 )cos(x 1 )cos(x 5 ) + sin(x 1 )sin(x 5 )) U 1 x 11 = x 12 x 12 = (sin(x 3 )cos(x 1 )sin(x 5 ) sin(x 1 )cos(x5)) U 1 where U 1 = b(ω Ω2 2 + Ω2 2 + Ω2 4 ) U 2 = b(ω 2 4 Ω2 2 ) U 3 = b(ω Ω2 1 ) U 4 = d( Ω Ω2 2 Ω2 2 + Ω2 4 ) We choose the syste odel paraeters as follows: g L I x I y I z d b e e e 2 7.5e e 5 1

5 2 Syste Design Based on PID Method 2.1 Design the PID Controller Fro the aerodynaic equations of quadrotors we know the attitude angles and angular velocities are independent of translation velocity. However, the translation otions depend on angular otion. Therefore, we can divide the whole syste into two subsystes: linear otion subsyste and angular otion subsyste, they have sei-coupling relations as illustrated by the following figure: Figure 1: Syste with controller According to this sei-coupled relation, we can first design the inner loop(attitude control) and then outer loop(position control), as shown in Figure2b. The inner loop is to regulate the attitude angles and ensure the stability when the flight is steadily flying or staying in a fixed position. The outer loop is to control the flight to fly in our target trajectory and finally ake sure the quadrotor attain our goal destination.[3] (x d, y d, z d ) is the reference target position, (φ d, θ d, ψ d ) is the expected value of our inner loop subsyste. (a) Exaple of PID control (b) A siplified chart of how the control syste work Figure 2: PID control syste First we siplify the dynaic equation where x 10 = Ux U 1 x 12 = Uy U 1 U x = sin(x 3 )cos(x 1 )sin(x 5 ) sin(x 1 )cos(x5) U y = sin(x 3 )cos(x 1 )cos(x 5 ) + sin(x 1 )sin(x 5 ) 2

6 The XYZ controller use the inforation fro feedback and desired x, y, z to decide the desired U xd and U yd, then U xd and U yd is used to copute the desired x 1d (φ) and x 3d (θ), where x 1d = arcsin(u x sin(x 5 ) u y cos(x 5 )) x 3d = arcsin[ u x sin(x 1 )sin(x 5 ) ] cos(x 1 )cos(x 5 ) Use desired x1d and x3d as reference for the Angle Controller to copute the desired U1d U2d U3d and U4d. Finally copute the needed voltage, which is the real input. The PID paraeter we have chosen is shown in Table1. These paraeters are used in the continuous case and in the 0.05s sapling case. However in the 0.1s sapling case the paraeters of Angle PID Controller are odified to keep stable. K p K i K d φ θ ψ x y z Table 1: Paraeters for PID Controller Use the linearized syste we plot the open loop Bode plot for the phi,see Figure 3 The gain argin is 24.1 db, the phase argin is 46.4deg, which is satisfactory. Figure 3: Bode plot of the linearized syste 2.2 Siulation Results and Analysis We first test the controller in continuous syste. The initial condition is set to be φ = 0.4, θ = 0.3, ψ = 0.2, x = 0, y = 0, z = 0. The destination is x = 1, y = 1.5, z = 2. The result is shown in the Figure 4. In fact a ore optial choice of the PID paraeters can be chosen for the continuous syste, however for consistency, we chose the optial paraeters that can ake the 0.05s discrete syste stable. Figure 5 shows the result of sapling tie 0.05s. It can be observed that the Angles oscillate ore and has higher aplitude. As for positions, Oscillation is introduced during rise. The advantage of this discrete controller is that the rise tie and the overshoot of x y is reduced. Figure 6 show the result of sapling tie

7 The stable range of paraeters shrank as the sapling tie increase, so we chose a new set of paraeter for this case, the perforance is liited copared with previous cases because any aggressive choice of paraeters will cause unstable. Figure 4: PID control in continuous tie Figure 5: PID control in sapling tie=0.05s Figure 6: PID control in sapling tie=0.1s We perfor ore tests in the case of sapling tie 0.05s. First we add disturbances to the syste. A 2/s upwind occur at 15s and end at 20s, an extra weight of 0.5kg is added to the quadrotor at 15s 4

8 and is not reoved until the end. The result is shown in Figure 7. It can be seen that the PID controller successfully handle these two case. However the controller fail when the wind speed reach 18/s. There is no liitation for the extra weight in our siulation. However the overshoot will be very large if the extra weight is too high. It is also known that in real life the extra weight is liited by the otors capacity. Figure 7: PID control in discrete tie under disturbance Then we add noise to our feedbacks, the gaussian noise is ajorly liited in 0.03 and 0.03rad. Figure 8 shows the result. Fro the plot we can observe that z is not significantly affected by the noise, because the noise influence in four rotors is neutralized. The converge process of x and y basically reain the sae except that they preserve the noise after settle. In the case of phi and theta, the noise is aplified in the steady-state response. Figure 8: PID control with noise 2.3 Suary In conclusion, the PID controller perfor well in both continuous and discrete case. It also show robustness under disturbances and noises. 5

9 3 Syste Design Based on Backstepping Method Linear feedback is the ost coon controller design ethod for MIMO syste. It abstract linear ingredient fro the nonlinear odel by linearization. To control the syste with high precision, it requires the syste with highly linear characteristic. On the contrary, Backstepping control does not have to eliinate the nonlinear portion, which akes it efficient in controlling nonlinear syste. In this chapter, we will show a ethod naed backstepping and induce an expression of the control variables. Then, trying to control the quadrotor syste by Backstepping ethod[4]. 3.1 Backstepping controller design Siulation of Backstepping control with siplified inputs According to Appendix A.1 and A.1.1, we could built a Backstepping controller in Matlab Siulink as it is shown in Figure 9. In the following siulation, we test the Backstepping, a nonlinear control ethod. We siulate the syste in continuous tie, tying to control the quadrotor fly fro point (0,0,0) to point (1,1.5,2). Total siulation tie is 30 seconds with the initial condition X = [φ φ θ θ ψ ψ z ż x ẋ y ẏ] T = [0.4, 0, 0.3, 0, 0.2, 0, 0, 0, 0, 0, 0, 0] T. Moreover, paraeters c 1 c 12 also have influence on the siulation results. As c 1 c 12 getting large, control ability gets stronger, as well as the control error. On the other hand, if c 1 c 12 becoe saller, oscillation and overshoot happens ore frequently. In the results shown in Figure 10, we choose c 1 = 10; c 2 = 3; c 3 = 5; c 4 = 8; c 5 = 3; c 6 = 2; c 7 = 2; c 8 = 2; c 9 = 2; c 10 = 2; c 11 = 2; c 12 = 2; (Readers could run run_backstepping. file first, then run backstepping.slx file) (a) Syste with controller (b) Attitude controller (c) Position controller Figure 9: Backstepping control syste Siulation result in Figure 10 sees quite satisfying. The whole syste becoe steady in about 3 seconds without over shoot along x,y,z axes Siulation of Backstepping control with real inputs In practical cases, the only inputs we can directly reach are the voltages of four rotors. In the next siulation, we solved the desired inputs of rotor voltage with a "U trans V" sub-block and considered the 800 transfer function of electric rotor as 0.01s+1. The rotating speed of rotors grow 800r/s as the input voltage grow 1V. Initial condition and paraeters reain unchanged. (Readers could run run_backstepping. file first, then run backstepping_real.slx file) 6

10 Figure 10: Backstepping control in continuous tie Figure 11: Backstepping control with inputs V1, V2, V3, V4 in continuous tie Fro Figure 11, we noticed a slight difference fro the real situation to the siplified condition. It took about 2 seconds for the quadrotor stay steady after tie 0. Fortunately, though the rotor conducted a inor delay of the rotating speed, the response tie of the syste were still kept at about 3 seconds Siulation of Backstepping control in discrete tie As we know, in practical researches, controllers are working under discrete tie. Figure 12 and Figure 13 tested the controlling results in different sapling tie, 0.05s and 0.1s. (Readers could run run_backstepping_discrete. file first and change the paraeter saplingtie, then run backstepping_real_discrete.slx file) 7

11 Figure 12: Backstepping control in sapling tie=0.05s Figure 13: Backstepping control in sapling tie=0.1s As shown in those pictures, vibration occurs when sapling tie is changed. The aplitude of vibration is about 0.01rad 0.573deg axiu, which is acceptable under after all. To keep the sae pace of PID control, we choose sapling tie=0.05s in the following sections Backstepping control in discrete tie under disturbance To test the robustness of backstepping control, we applied several disturbances on the quadrotor. A 2/s horizontal wind on x axis starts at t=15s and ends at t=20s. An additional weight =0.5kg will be add on the quadrotor at t=15 and aintained until t=30s. Siulation results are shown in Figure 14. (Readers could run run_backstepping_discrete_disturb. file first and then run backstepping_real_disc rete_disturb.slx file) The siulation tie in Figure 14 is 30 seconds. We noticed, fro plot of z, that the syste will be stable after disturbance had happened for about 3 seconds and stay at the new stable point. Due to the plot of position x, the quadrotor will not flying back to point (1,1.5,2) as long as the disturbance exists. To conduct 8

12 Figure 14: Backstepping control in discrete tie under disturbance Figure 15: Adaptive Backstepping control in discrete tie under disturbance the syste adaptive to the disturbances, an Adaptive Backstepping approach will be deduced in the next subsection Siulation of Adaptive Backstepping control in discrete tie under disturbances We are capable to solve the unknown disturbance proble after building an Adaptive Backstepping Controller as written in Appendix A.2. Figure 15 shows the siulation result of Adaptive Backstepping control when disturbances are applied,i.e. a 2/s horizontal wind on x axis starts at t=15s and ends at t=20s, an unknown additional weight =0.5kg added on the quadrotor at t=15 and aintained until t=30s. Fro the plot of attitude, we can conclude that adaptive backstepping control converges faster than the nonadaptive one. Moreover, adaptive control successfully estiate the unknown disturbance and fly back to the desired position. (Readers could run run_backstepping_discrete_disturb. file first and then run backstepping_real_disc rete_disturb_adp.slx file) 9

13 Figure 16: Adaptive Backstepping control with sensor noise Siulation of Adaptive Backstepping control with noise In the next siulation, disturbances and other paraeters are kept sae as before. Furtherore, noise of aplitude 0.6 are added in ẍ, ÿ, z, φ, θ, ψ. Note that, in this case, noises will also be added back to the syste itself, which could be regarded as a cobination of otion error and sensor error. We could discover fro the siulation result(figure 16) that vibration will occur due to noises. The aplitude of vibration is about along x and y axis and 0.01rad 0.05deg on rolling angleφ and pitch angle θ. (Readers could run run_backstepping_discrete_disturb. file first and then run backstepping_real_disc rete_disturb_adp_noise.slx file) 4 Coparison Response Properties PID Backstepping tie to the expected altitude 5s 4s tie to the target horizontal position 15s 4s tie to recover fro disturbance 10s 5s Table 2: Perforance of Different Methods Fro the coparison we can find that the Backstepping has a ore keen response than PID. What s ore, the Backstepping has less overshoot, which eans that in reality it has less chance hit obstacles which is not in the expected trajectory. The angle oscillation frequency and aplitude in Backstepping are also less than those in PID. 10

14 A Backstepping control A.1 Backstepping attitude control As shown in chapter two, the attitude controller will not influence the position controller. Hence, we could design three Backstepping controllers to control the three angles φ, θ, ψ respectively. Take the rolling angle φ as an exaple, we could deduce the expression of U 2 by Backstepping ethod: { x1 = φ = x 2 x 2 = a 1 θ ψ + b 1 U 2 = a 1 x 4 x 6 + b 1 U 2 where a 1 = (I y I z )/I x, b 1 = L/I x. Step 1 Denote the desired rolling angle as φ d = x 1d, and the error variable z 1 = x 1 x 1d. Then take the derivative of z 1, we have: z 1 = 1 x 1d = x 2 x 1d Then, we define a virtual control variable α 1, such that z 2 = x 2 α 1, and choose the Lyapunov function V 1 = 1 2 z2 1. Thus: V 1 = z 1 = z 1 (x 2 x 1d ) = z 1 (z 2 + α 1 x 1d ) According to Lyapunov stability criterion, we need to choose α 1 = x 1d c 1 z 1 such that V 1 < 0, where c 1 > 0. Then we could have: V 1 = c 1 z z 1z 2 z 2 = x 2 x 1d + c 1 z 1 We could see that if z 2 = 0, then z 1 is asyptotically stable. Since, z 2 is not always zero, we need another control variable for z 1 z 2. Step 2 Define Lyapunov function V 2 = 1 2 z2 2 + V 1, since z 2 = 2 x 1d + c 1 z 1, we have: Then, the control input should be: V 2 = z 2 z 2 + V 1 = z 2 ( 2 x 1d + c 1 z 1 ) c 1 z z 1z 2 (1) = z 2 (b 1 U 2 + a 1 x 4 x 6 x 1d + c 1 z 1 ) c 1 z z 1z 2 = z 2 (b 1 U 2 + a 1 x 4 x 6 x 1d + c 1 (z 2 c 1 z 1 )) c 1 z z 1z 2 where c 2 > 0, U 2 = 1 ( x b 1d z 1 a 1 x 4 x 6 c 1 z 2 + c 2 1 z 1 c 1 z 2 ) (2) 1 = 1 ( x b 1d + (c 2 1 1)z 1 a 1 x 4 x 6 (c 1 + c 2 )z 2 ) 1 x 1d = 0, z 1 = x 1 x 1d, z 2 = x 2 x 1d + c 1 z 1. Put (2) into (1), we have: V 2 = c 1 z 2 1 c 2z 2 2 < 0 According to Lyapunov stability criterion, U 2 can ensure that z 1 and z 2 are asyptotically stable. Sae reasoning, we could deduce the control variable for pitch angle θ and yaw angle ψ as: { U3 = b 1 2 [ x 3d + (c 2 3 1)z 3 a 2 x 2 x 6 (c 3 + c 4 )z 4 ] U 4 = b 1 3 [ x 5d + (c 2 5 1)z 5 a 3 x 2 x 4 (c 5 + c 6 )z 6 ] 11

15 where: A.1.1 Backstepping position control z 3 = θ θ d = x 3 x 3d z 4 = x 4 x 3d + c 3 z 3 z 5 = ψ ψ d = x 5 x 5d z 6 = x 6 x 5d + c 5 z 5 Position control is a cobination of altitude control and horizontal control. The altitude z is controlled by input U 1, and horizontal position x,y are not directly controlled by inputs. Since x, y are directly correlated with φ and θ, we could define soe virtual input U x, U y, expressed by φ and θ, parallel to U 1. Take the altitude controller as an exaple, we could design a Backstepping controller. The dynaic equations of z and ż are: { x7 = x 8 8 = g + (cos(x 1 )cos(x 3 )) U (3) 1 Step 1 For a given desired altitude z d = x 7d, define the error variable z 7 = x 7 x 7d and calculate its derivation: z 7 = 7 x 7d = x 8 x 7d Then, define α 7, where z 8 = x 8 α 7, and choose the Lyapunov function V 7 = 1 2 z2 7, then: V 7 = z 7 z 7 = z 7 (x 8 x 7d ) + z 7 z 8 To ensure z 7 asyptotically stable at the origin, according to Lyapunov stability criterion, we should have V 7 < 0, i.e. α 7 = x 7d c 7 z 7 (c 7 > 0). Thus: V 7 = c 7 z z 7z 8 z 8 = x 8 x 7d + c 7 z 7 Sae as we did in attitude control, we should define another Lyapunov function V 8 such that z 8 0, in order to express our control variables. Step 2 Define Lyapunov function V 8 = 1 2 z2 8 + V 7, since z 8 = 8 x 7d + c 7 z 7, then: To guarantee V 8 < 0, choose: V 8 = z 8 z 8 + V 1 = z 8 ( 8 x 7d + c 7 z 7 ) c 7 z z 7z 8 (4) = z 8 [ 8 x 7d + c 7 (z 8 c 7 z 7 )] c 7 z z 7z 8 = c 7 z z 8[z 7 x 7d + c 7 (z 8 c 7 z 7 ) + 8 ] z 7 x 7d + c 7 (z 8 c 7 z 7 ) + 8 = c 8 z 8 (c 8 > 0) (5) Put (5) into (4), we have: V 8 = c 7 z z 8[z 7 x 7d + c 7 z ] = c 7 z 2 7 c 8z 2 8 < 0 Moreover, (5) can be written as: 8 = x 7d + (c 2 7 1)z 7 (c 7 c 8 )z 8 (c 7 > 0, c 8 > 0) (6) 12

16 Put (6) into (3): U 1 = cos(x 1 )cos(x 3 ) [ x 7d + (c 2 7 1)z 7 + g (c 7 + c 8 )z 8 ] Now, we have U 1 such that z 7 and z 8 are asyptotically stable. Sae reasoning, we could deduce the expression for virtual control variables: { ux = U1 [ x 9d + (c 2 9 1)z 9 (c 9 + c 10 )z 10 ] u y = U 1 [ x 11d + (c )z 11 (c 11 + c 12 )z 12 ] (7) where z 9 = x x d = x 9 x 9d z 10 = x 10 x 9d + c 9 z 9 z 11 = y y d = x 11 x 11d z 12 = x 12 x 11d + c 11 z 11 Siilar to what we did in PID control, we need the outputs of rolling angle and pitch angle of the position controller and take the as the desired inputs of the attitude controller. Thus, according to the syste odel, we acquire the nonlinear relationship between φ d, θ d and u x, u y : φ d = arcsin(u x sin(ψ) u y cos(ψ)) = arcsin(u x sin(x 5 ) u y cos(x 5 )) (8) θ d = arcsin[ u x sin(φ)sin(ψ) cos(φ)cos(ψ) ] = arcsin[ u x sin(x 1 )sin(x 5 ) ] (9) cos(x 1 )cos(x 5 ) Put (8) and (9) into (7), we could deduce the final expression of inputs: U 1 = cos(x 1 )cos(x 3 ) [ x 7d + (c 2 7 1)z 7 + g (c 7 + c 8 )z 8 ] U 2 = b 1 [ x 1d + (c 2 1 1)z 1 a 1 x 4 x 6 (c 1 + c 2 )z 2 ] U 3 = b 1 2 [ x 3d + (c 2 3 1)z 3 a 2 x 2 x 6 (c 3 + c 4 )z 4 ] = b 1 3 [ x 5d + (c 2 5 1)z 5 a 3 x 2 x 4 (c 5 + c 6 )z 6 ] U 4 A.2 Adaptive Backstepping control in discrete tie under disturbance In the real environent, a quadrotor will always be influenced by unknown paraeters and unexpected disturbances denoted as D. In order to online estiate disturbance D and calculate adaptive feedback by Backstepping ethod, we could express the dynaic equation of position subsyste as following: ẍ = u x U 1 + D x ÿ = u y U 1 + D y z = cos(φ)cos(θ)u 1 g + D z where u x = sin(θ)cos(φ)cos(ψ) + sin(φ)sin(ψ), u y = sin(θ)cos(φ)sin(ψ) sin(φ)cos(ψ) and D i (i = x, y, z) is denoted as disturbance on x,y,z axes. Take the altitude state z as an exaple, it s state space representation is: { x7 = x 8 8 = 1 (cos(x 3)cos(x 1 )U 1 g + D z ) where x 7 = z, x 8 = ż, x 1 = φ, x 3 = θ and D z is the disturbance added on z axis. Since D z is unknown, we need to online estiate it and define the estiated D z as Dˆ z, the estiate error D z = D z Dˆ z. 13

17 For a given desired altitude z d = x 7d, define the altitude error variable as z 7 = x 7 x 7d, speed error variable z 8 = x 8 α 7, where α 7 is a virtual control variable. Step 1: design for z 7 subsyste. z 7 = 7 x 7d = x 8 x 7d Take x 8 as the control variable, if x 8 = x 7d c 7 z 7, then z 7 subsyste is asyptotically stable. Since x 8 is a transient control variable, we can only express x 8 by α 7. Thus, design: α 7 = x 7d c 7 z 7 (c 7 > 0) Choose the lyapunov function of the first subsyste as V 7 = 1 2 z2 7, then: V 7 = z 7 z 7 = z 7 (x 8 x 7d ) = z 7 (z 8 + α 7 x 7d ) = z 7 z 8 c 7 z 2 7 If V 7 < 0, z 7 z 8 will be canceled in the next step and the first subsyste would be stable. Step 2: calculate for the actual control law and the adaptive control law. z 8 = 8 x 7d + c 7 z 7 Define Lyapunov function as V 8 = 1 2 z2 8 + V 7, then: V 8 = z 8 z 8 + V 7 = z 8 ( 8 x 7d + c 7 z 7 ) c 7 z z 7z 8 Now, the syste input: = z 8 [ 8 x 7d + c 7 (z 8 c 7 z 7 )] c 7 z z 7z 8 = z 8 [ 1 (cos(x 3)cos(x 1 )U 1 g + D z ) x 7d + c 7 (z 8 c 7 z 7 )] c 7 z z 7z 8 U 1 = cos(x 3 )cos(x 1 ) (x 7d z 7 c 8 z 8 c 7 z 8 + c 2 7 z 7 + g) D z Since D z is unknown, we could substitute D z by ˆ D z, then: U 1 = cos(x 3 )cos(x 1 ) (x 7d z 7 c 8 z 8 c 7 z 8 + c 2 7 z D 7 + g) ˆ z To declare the adaptive law of Dˆ z, set the Lyapunov function asv 8 = V z2 8 + we could have its derivative as: V 8 = z 8 z 8 + V 7 1 D λ z D ˆ z = c 7 z 2 7 c 8z z 8(D z Dˆ z ) 1 D λ z D z = c 7 z 2 7 c 8z D ˆ z (z 8 1 D ˆ z ) λ 2 D z 2λ λ, (λ > 0), then where D ˆ z = λ 1 z 8, (λ > 0) Thus, we could keep Lyapunov function V 8 = c 7 z 2 7 c 8z 2 8 < 0,(c 7, c 8 > 0) and guarantee that z 7, z 8, are asyptotically stable. D z 14

18 Sae reasoning, we could apply adaptive backstepping ethod on D x and D y : u x = u y = where U1 [ x 9d + (c 2 9 1)z D 9 (c 9 + c 10 )z 10 ˆ x ] U 1 [ x 11d + (c )z D 11 (c 11 + c 12 )z 12 ˆ y ] (10) { D ˆ x = λ 2 z 10 (λ 2 > 0) D ˆ z = λ 3 z 12 (λ 3 > 0) (11) z 9 = x x d = x 9 x 9d z 11 = y y d = x 11 x 11d (12) z 10 = x 10 x 9d + c 9 z 9 z 12 = x 12 x 11d + c 11 z 11 Then, put (10),(11),(12)into (8) and (9), we could solve for the desired rolling angle and pitch angle adaptive to unknown disturbances. Reference [1] Y. C. Choi and H. S. Ahn. Nonlinear control of quadrotor for point tracking: Actual ipleentation and experiental tests. IEEE/ASME Transactions on Mechatronics, 20(3): , June [2] Sair Bouabdallah. Backstepping and sliding-ode techniques applied to an indoor icro quadrotor. In In Proceedings of IEEE Int. Conf. on Robotics and Autoation, pages , [3] A. L. Salih, M. Moghavvei, H. A. F. Mohaed, and K. S. Gaeid. Modelling and pid controller design for a quadrotor unanned air vehicle. In Autoation Quality and Testing Robotics (AQTR), 2010 IEEE International Conference on, volue 1, pages 1 5, May [4] F. Yacef, O. Bouhali, and M. Haerlain. Adaptive fuzzy backstepping control for trajectory tracking of unanned aerial quadrotor. In Unanned Aircraft Systes (ICUAS), 2014 International Conference on, pages , May