**5. Simulations**

In order to determine the efficiency of the proposed controller, a MATLAB quadrotor simulator is used to test it numerically. The design parameters of the quadrotor used in the simulator are listed in **Table 1**. Two paths were presented in the simulation to show the performance of using the proposed controller with four different circumstances for quadrotors team formation. The first desired path to be tracked by the leader was.

$$\begin{cases} x\_{Ld} = 2\cos\left(t\pi/80\right); & y\_{Ld} = 2\sin\left(t\pi/80\right) \\ x\_{Ld} = 1 + 0.1t; & y\_{Ld} = \pi/6 \end{cases} . \tag{41}$$

The IBS controllers were tested in simulation to track a desired path by the leader and maintain the desired distance, desired incidence angle and desired bearing angle between them for the follower. The parameters chosen for both paths were *bL* ¼ *diag*ð Þ 180, 0*:*34, 0*:*34 , *cL* ¼ *diag*ð Þ 0*:*7, 0*:*02, 0*:*02 , *kL* ¼ *diag*ð Þ 0*:*0516, 0*:*0081, 0*:*0081 , *bF* ¼ *diag*ð Þ 12, 0*:*7, 0*:*7 , *cF* ¼ *diag*ð Þ 1*:*4, 0*:*02, 0*:*02 and *kF* ¼ *diag*ð Þ 0*:*01, 0*:*001, 0*:*001 .

The leader initial positions were *xL*, *yL*, *zL <sup>T</sup>* <sup>¼</sup> ½ � 2, 0, 0 *<sup>T</sup>* metres and the initial angles were *<sup>φ</sup>L*, *<sup>θ</sup>L*, *<sup>ψ</sup><sup>L</sup>* ½ �*<sup>T</sup>* <sup>¼</sup> ½ � 0, 0, 0 *<sup>T</sup>* radian. Then the follower followed the leader and maintained the desired distance between them *d* ¼ 2 metres, the desired incidence and bearing angles *ρ* ¼ �*π=*6, *σ* ¼ *π=*6 radian, respectively. The follower initial positions were *xF*, *yF*, *zF <sup>T</sup>* <sup>¼</sup> ½ � <sup>0</sup>*:*5, 0, 0 *<sup>T</sup>* metres and the initial angles were *<sup>φ</sup>F*, *<sup>θ</sup>F*, *<sup>ψ</sup><sup>F</sup>* ½ �*<sup>T</sup>* <sup>¼</sup> ½ � 0, 0, 0 *<sup>T</sup>* radian.

The second desired path to be tracked by the leader was

$$\begin{cases} \varkappa\_{Ld} = 4 \cos \left( t \pi / 4 \mathbf{0} \right); & \varkappa\_{Ld} = 4 \sin \left( t \pi / 4 \mathbf{0} \right) \\ z\_{Ld} = \mathbf{1} + \mathbf{0}.\mathbf{1}t; & \varkappa\_{Ld} = \pi / \mathbf{6} \end{cases} . \tag{42}$$

The leader initial positions were *xL*, *yL*, *zL <sup>T</sup>* <sup>¼</sup> ½ � 4, 0, 0 *<sup>T</sup>* metres and the initial angles were *<sup>φ</sup>L*, *<sup>θ</sup>L*, *<sup>ψ</sup><sup>L</sup>* ½ �*<sup>T</sup>* <sup>¼</sup> ½ � 0, 0, 0 *<sup>T</sup>* radian. Then the follower followed the leader and maintained the desired distance between them *d* ¼ 3 metres, the desired incidence and bearing angles *ρ* ¼ 0, *σ* ¼ *π=*6 radian, respectively. The follower initial positions were *xF*, *yF*, *zF <sup>T</sup>* <sup>¼</sup> ½ � <sup>1</sup>*:*4, �1*:*5, 0 *<sup>T</sup>* metres and the initial angles were *<sup>φ</sup>F*, *<sup>θ</sup>F*, *<sup>ψ</sup><sup>F</sup>* ½ �*<sup>T</sup>* <sup>¼</sup> ½ � 0, 0, 0 *<sup>T</sup>* radian.

The four circumstances included: (17) no disturbance, (32) force disturbance *dvix* ¼ �2 Nm during 10 ≤*t*≥10*:*25 seconds, *dviz* ¼ 2 Nm during


**Table 1.** *Quadrotor parameters.* 20≤*t*≥ 20*:*25 seconds, *dviy* ¼ 2 Nm during 30≤ *t*≥30*:*25 seconds in the first path, *dvix* ¼ �0*:*5 Nm during 20≤ *t*≥20*:*25 seconds, *dviz* ¼ 0*:*5 Nm during 60≤*t* ≥60*:*25 seconds, *dviy* ¼ 0*:*5 Nm during 100≤ *t*≥100*:*25 seconds in the second path, and the attitude part for the leader and the follower is disturbed using (43), applied at the same time for both the leader and the follower, (33) þ30% model parameter uncertainty, and (44) �30% model parameter uncertainty.

$$\mathbf{d} = \mathbf{0}.01 + \mathbf{0}.01\sin\left(0.024\pi t\right) + \mathbf{0}.05\sin\left(1.32\pi t\right) \tag{43}$$

**Figures 5** and **6** indicate the response of the IBS controller while the leader was tracking the first and second desired path, respectively. **Figure 7** shows the distance

**Figure 5.** *Leader-follower formation in first path.*

**Figure 6.** *Leader-follower formation in second path.*

**Figure 7.** *The distance between the leader and the follower in (a) the first path, (b) the second path.*

between the leader and the follower via the two paths, and **Figures 8**–**11** illustrate the yaw angles' behaviour for the leader and the follower via the two paths respectively.

It can be noticed from these figures that not only the overshoot but also the error in distance between the leader and the follower was low. It was also rejecting the disturbances in the two paths.

**Table 2** demonstrates the RMSE values of the two paths positions and yaw angle. It is clear that the RMSE values of the IBS controller were almost the same

**Figure 8.** *Leader yaw angle in first path.*

when using the IBS controller in normal conditions and with 30% model parameter uncertainty in both paths, while they significantly increased with the disturbance. It can be seen that the IBS controller was able to track the desired trajectories with small position tracking errors in less than 3 s and it could reject the disturbances and cover the change in model parameter uncertainties.

**Figure 9.** *Follower yaw angle in first path.*

**Figure 10.** *Leader yaw angle in second path.*

**Figure 11.** *Follower yaw angle in second path.*


#### **Table 2.**

*Position and ψ RMSE values for the two paths.*

In conclusion, it is obvious that the proposed IBS controller maintained the distance between the leader and the follower and keep them in the desired formation.
