**6. Numerical simulation results**

Consider the following positive definite Lyapunov function candidate:

Replacing the third equation of system (6, Σ2) in Eq. (33) yields

The time derivative of *Vbf* ,using Eqs. (29) and (34), is given by

= [ <sup>1</sup>

= [( ) <sup>1</sup>

cos( ) cos( )

*a ee*

f

q

=

*V ke*

&

t

and hence

a

In view of last equation, we propose the following torque input for the rotational dynamics

22 22

ç÷ ç÷ W- - ç÷ ç÷ - - èø èø

*ee G I I*

f f

2

f f

*k e*

*k e*

2

 q


*k I*

f f

cos( ) = 10

*<sup>e</sup> <sup>e</sup> <sup>k</sup> ke ke*

W W


f

ff

y

*r a*

2 1 2 2


y y

> q q

q


2 2

*T T ee e e e*

cos( )

*k e k e*

f

0 0

æö æö

f

0

0 1

 qq

and its time derivative is

148 Recent Progress in Some Aircraft Technologies

Substituting Eq. (30) into Eq. (32) yields

1 <sup>1</sup> <sup>=</sup> <sup>2</sup>

=[ ] <sup>1</sup> +W W-

a

t a] + W -W ´ W - + - <sup>W</sup> & & & *<sup>T</sup> VV IG I BF <sup>e</sup> a a* (35)

 ta] +W - W + ´ W- + - W W & & & *<sup>T</sup> V V BF e e a a IG I* (36)

q

(37)

(38)

+WW*<sup>T</sup> VV I BF e e* (32)

&& & = <sup>1</sup> +W W*<sup>T</sup> VV I BF e e* (33)

<sup>W</sup> && & & *<sup>T</sup> VV II BF <sup>e</sup>* (34)

To show the effectiveness of the proposed control algorithm, we have introduced the simula‐ tion results. In order to test the overall system using Matlab simulink, the quadrotor UAV modelled as a rigid body of mass *m*=1.169*Kg* with inertial matrix *I* =*dia*(*col*(0.024,0.024,0.032))*Kg*.*m*<sup>2</sup> , *k* =1.41×10−<sup>5</sup> , *c* =1.84×10−<sup>6</sup> , *Ir* =7.81×10−<sup>5</sup> *Kg*.*m*<sup>2</sup> and *l* =0.2125*m*. The numerical simulation results have been obtained using Runge-Kutta's method with variable time step 0.001*s*. The desired trajectory is given by *xd* (*t*)=0, *yd t* =0, *zd* (*t*)= −(1−*e t* <sup>10</sup> ), *<sup>ψ</sup><sup>d</sup>* (*t*)= *<sup>π</sup>* <sup>4</sup> *for*20≤*t* ≤30with initial conditions *x*(0)=0, *y*()=0, *z*(0)=0, *ψ*(0)=0. The controller gains are as follows: *A*<sup>=</sup> 40,40,40 *<sup>T</sup>* ,*<sup>B</sup>* <sup>=</sup> 30,30,30 *<sup>T</sup>* ,*kp* <sup>=</sup> 160,160,160 *<sup>T</sup>* ,*kd* <sup>=</sup> 20,20,20 *<sup>T</sup>* ,*k<sup>φ</sup>* =4,*k<sup>θ</sup>* =8,*kψ*4, *ke* <sup>=</sup> 7,7,7 *<sup>T</sup>* . The obtained results are shown in **Figures 3**, **4**, **5**, **6**, **7**, **8** and **9**.

**Figure 3.** Position tracking.

**Figure 4.** Position tracking error.

**Figure 5.** Estimated velocity tracking.

**Figure 6.** Estimated velocity tracking error.

**Figure 7.** Attitude tracking.

**Figure 8.** Attitude tracking error.

**Figure 5.** Estimated velocity tracking.

150 Recent Progress in Some Aircraft Technologies

**Figure 6.** Estimated velocity tracking error.

**Figure 7.** Attitude tracking.

**Figures 3**, **4**, **5** and **6** show the three components of the position and estimated velocity tracking errors. **Figures 7** and **8** illustrate the attitude tracking and its tracking errors. To show the quadrotor's position tracking, a 3D plot of the quadrotor's position with desired trajectory is given in **Figure 9**.

**Figure 9.** 3D trajectory of the quadrotor UAV (dashed blue) with the desired trajectory (red).

#### **7. Conclusion**

In this work, we addressed the trajectory tracking problem of quadrotor UAV without linear velocity measurements. The presented algorithm was based on a separate translational and rotational control design, and the overall closed loop system showed a global asymptotic stability. Exploited the nature structure of the model, the controller is designed in a hierarchical manner by considering the separation between the translational and orientation dynamics. In the first phase of our control design, the linear velocity measurement has been avoided with the introduced nonlinear filter without using any observer. During this phase, the position controller computes the total thrust, considered as input for translational dynamics, and the desired orientation. In the second phase of our control design, barrier Lyapunov function has been employed to design the control torque that ensuring the tracking of the desired attitude derived at first phase of the control design. The stability of the closed loop system has been investigated and the simulation results have been provided to show the good performance of the proposed algorithm. Our next work will be to accomplish real time experiments for the proposed algorithm.
