**4.4 Results**

From the equation of motion, the second derivative of the error becomes

*m*

*m*

*<sup>c</sup>φc<sup>θ</sup> <sup>z</sup>*€*<sup>d</sup>* <sup>þ</sup> *<sup>g</sup>* <sup>þ</sup> *<sup>c</sup>*<sup>1</sup> *<sup>z</sup>*\_ ð Þþ *<sup>d</sup>* � *<sup>z</sup>*\_ *<sup>K</sup>*1*s*<sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>1</sup> f g sgn ð Þ *<sup>s</sup>*<sup>1</sup> (45)

*θ*Ω*<sup>r</sup>* þ *c*<sup>2</sup> *φ*\_ ð Þþ *<sup>d</sup>* � *φ*\_ *K*2*s*<sup>2</sup> þ *Q*2*:*sgn ð Þ *s*<sup>2</sup> � � (46)

� � (47)

*θ* þ *c*<sup>4</sup> *ψ*\_ ð Þþ *<sup>d</sup>* � *ψ*\_ *K*4*s*<sup>4</sup> þ *Q*4*:*sgn ð Þ *s*<sup>4</sup>

*<sup>θ</sup>* � � <sup>þ</sup> *<sup>K</sup>*3*s*<sup>3</sup> <sup>þ</sup> *<sup>Q</sup>*3*:*sgn ð Þ *<sup>s</sup>*<sup>3</sup>

*U*<sup>1</sup> þ *K*5*s*<sup>5</sup> þ *Q*5*:*sgn ð Þ *s*<sup>5</sup>

*U*<sup>1</sup> þ *K*6*s*<sup>6</sup> þ *Q*6*:*sgn ð Þ *s*<sup>6</sup>

(49)

(50)

*U*<sup>1</sup> þ *g* (42)

*U*<sup>1</sup> þ *g* þ *c*1ð Þ *z*\_*<sup>d</sup>* � *z*\_ (44)

*U*<sup>1</sup> þ *g* þ *c*1ð Þ¼ *z*\_*<sup>d</sup>* � *z*\_ 0 (43)

*<sup>e</sup>*€<sup>1</sup> <sup>¼</sup> *<sup>z</sup>*€*<sup>d</sup>* � *<sup>z</sup>*€ <sup>¼</sup> *<sup>z</sup>*€*<sup>d</sup>* � *<sup>c</sup>φc<sup>θ</sup>*

By equaling Eq. (41) to zero, we obtain

*Unmanned Robotic Systems and Applications*

*<sup>U</sup>*<sup>1</sup> <sup>¼</sup> *<sup>m</sup>*

*<sup>φ</sup>*€*<sup>d</sup>* � *<sup>a</sup>*<sup>1</sup> \_

*4.3.2 Attitude control*

quadcopter, we obtain

*<sup>U</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> *b*1

*<sup>U</sup>*<sup>3</sup> <sup>¼</sup> <sup>1</sup> *b*2 *θ* €*<sup>d</sup>* � *<sup>a</sup>*<sup>2</sup> \_

*4.3.3 Position control*

with

**90**

*<sup>U</sup>*<sup>4</sup> <sup>¼</sup> <sup>1</sup> *b*3

*<sup>θ</sup><sup>d</sup>* <sup>¼</sup> arcsin *<sup>m</sup>*

*<sup>φ</sup><sup>d</sup>* ¼ �arcsin *<sup>m</sup>*

*<sup>s</sup>*\_<sup>1</sup> <sup>¼</sup> *<sup>z</sup>*€*<sup>d</sup>* � *<sup>c</sup>φc<sup>θ</sup>*

�*K*1*s*<sup>1</sup> � *<sup>Q</sup>*<sup>1</sup> sgn ð Þ¼ *<sup>s</sup>*<sup>1</sup> *<sup>z</sup>*€*<sup>d</sup>* � *<sup>c</sup>φc<sup>θ</sup>*

So that the control law of the altitude will become:

*θψ*\_ � *<sup>a</sup>*<sup>2</sup> \_

*θψ*\_ <sup>þ</sup> *<sup>a</sup>*<sup>4</sup> \_

*<sup>ψ</sup>*€*<sup>d</sup>* � *<sup>a</sup>*5*φ*\_ \_

*cφ:cθ:U*<sup>1</sup>

*cψ:U*<sup>1</sup>

*m*

By using the constant and proportional rate reaching law formula

By following sliding mode control steps of design for the attitude of the

*<sup>θ</sup>*Ω*<sup>r</sup>* <sup>þ</sup> *<sup>c</sup>*<sup>3</sup> \_

*<sup>θ</sup><sup>d</sup>* � \_

� � (48) with

*s*<sup>2</sup> ¼ *c*2*e*<sup>2</sup> þ *e*\_<sup>2</sup> *s*<sup>3</sup> ¼ *c*3*e*<sup>3</sup> þ *e*\_<sup>3</sup> *s*<sup>4</sup> ¼ *c*4*e*<sup>4</sup> þ *e*\_<sup>4</sup> *e*<sup>2</sup> ¼ *φ<sup>d</sup>* � *φ e*<sup>3</sup> ¼ *θ<sup>d</sup>* � *θ e*<sup>4</sup> ¼ *ψ<sup>d</sup>* � *ψ*

Same strategy will be followed to derive the control laws of the position as in integral backstepping and feedback linearization. The control laws of both *x*, *y* will command the attitude loop with the references to accomplish the desired trajectory

� � n o

� � � �

*s*<sup>5</sup> ¼ *c*5*e*<sup>5</sup> þ *e*\_<sup>5</sup> *s*<sup>6</sup> ¼ *c*6*e*<sup>6</sup> þ *e*\_<sup>6</sup> *e*<sup>5</sup> ¼ *xd* � *x e*<sup>6</sup> ¼ *yd* � *y*

*<sup>x</sup>*€*<sup>d</sup>* � *<sup>s</sup>φ:s<sup>ψ</sup> m*

*<sup>y</sup>*€*<sup>d</sup>* � *<sup>c</sup>φ:sθ:s<sup>ψ</sup> m*

The discussed nonlinear approaches have been tested in MATLAB/Simulink based on the nonlinear quadcopter model of Eq. (10), as well as experimental verification is also conducted. For modeling and simulation of the proposed approaches, the simulation sample time was Ts = 100 μs and the solver used was Runge-Kutta with a fixed integration. **Figures 4** and **5** show the system's trajectory tracking response. **Figure 4a** depicts the system response when implementing the proposed integral backstepping approach. **Figure 4b** shows the system response using feedback linearization with LQI approach. **Figure 4c** represents the system response using sliding mode control. **Figure 4a**–**c** demonstrates the system trajectory tracking to a desired trajectory command signal, with the existing external disturbances. These disturbances are being added with the command signals at different time instances. The initial position of the desired trajectory was (2, 0, 0), but the quadcopter was initiated with a different initial position as (0, 0, 0). As seen from **Figure 4a–c**, for the three investigated control approaches, the actual trajectory at the start was a bit diverged from the desired trajectory. However, the actual trajectory was then converged to the desired one fast. **Figure 5** exhibits the reference signals and the responses for x-, y-, and z axes of the quadcopter in the 3D space. These references on x and y axes were selected to be sinusoidal signals with 2 m of magnitude and 0.05 Hz of frequency. The command along z axis was a ramp signal with 0.2 m.s<sup>1</sup> velocity rate. **Figure 6** shows the tracking errors of the

**Figure 4.**

*Desired and actual trajectory, proposed integral backstepping response (4-a), feedback linearization with LQI response (4-b), and sliding mode control response (4-c).*

#### **Figure 5.**

*Desired and actual trajectory, proposed integral backstepping response (5-a), feedback linearization with LQI response (5-b), and sliding mode control response (5-c).*

**Figure 6.**

*Trajectory tracking errors, proposed integral backstepping response (6-a), feedback linearization with LQI response (6-b), and sliding mode control response (6-c).*

quadcopter motion on x, y and z. However, as seen, the tracking error of the motion on the three axes converged to zero. But, a little divergence was observed, which

The practical implementation, of the proposed control strategies of the attitude control of the quadcopter, has been validated using Arduino MEGA board with an inertial measurement unit (IMU). **Figure 7** exhibits the practical UAV control system. **Figure 8** shows the practical implementation results and response of pitch angle using integral backstepping controller. As noticed earlier, there is a static error with oscillating response. **Figure 9a** and **b** demonstrates the practical result and response of the roll angle when implementing integral backstepping and feedback

As noticed from **Figure 9a**, there was an oscillating response for pitch angle control during transient state, of almost undesired of 20° of overshoot and downshoot when implementing the proposed backstepping controller. But, high dynamic performance and fast tracking control were obtained for pitch angle control when implementing the proposed LQI controller with feedback linearization

This chapter has discussed different advanced control techniques for UAV control. Nonlinear control theories have been reviewed among other control strategies due to their capacity to deal with the nonlinearity and the coupling components of the UAV state variables. This includes backstepping, feedback linearization, and sliding mode control. UAV nonlinear model has been derived and modeled in MATLAB®, and the proposed control strategies have been implemented. Simulation

results obtained from the developed model with the control strategies were presented and discussed. Different path tracking and trajectories have been examined with successful and high dynamic performance. The developed control strategies have exhibited robustness against the UAV parameter mismatch and

were due to the existence of disturbance with the command signals.

*Pitch practical response: (a) using integral backstepping and (b) feedback linearization with LQI.*

linearization with LQI controllers, respectively.

*Advanced UAVs Nonlinear Control Systems and Applications*

*DOI: http://dx.doi.org/10.5772/intechopen.86353*

approach as seen in **Figure 9b**.

**5. Conclusion**

**Figure 9.**

dynamic uncertainties.

**93**

**Figure 7.** *Practical UAV control scheme.*

**Figure 8.** *Pitch practical response using integral backstepping.*

*Advanced UAVs Nonlinear Control Systems and Applications DOI: http://dx.doi.org/10.5772/intechopen.86353*

**Figure 9.** *Pitch practical response: (a) using integral backstepping and (b) feedback linearization with LQI.*

quadcopter motion on x, y and z. However, as seen, the tracking error of the motion on the three axes converged to zero. But, a little divergence was observed, which were due to the existence of disturbance with the command signals.

The practical implementation, of the proposed control strategies of the attitude control of the quadcopter, has been validated using Arduino MEGA board with an inertial measurement unit (IMU). **Figure 7** exhibits the practical UAV control system. **Figure 8** shows the practical implementation results and response of pitch angle using integral backstepping controller. As noticed earlier, there is a static error with oscillating response. **Figure 9a** and **b** demonstrates the practical result and response of the roll angle when implementing integral backstepping and feedback linearization with LQI controllers, respectively.

As noticed from **Figure 9a**, there was an oscillating response for pitch angle control during transient state, of almost undesired of 20° of overshoot and downshoot when implementing the proposed backstepping controller. But, high dynamic performance and fast tracking control were obtained for pitch angle control when implementing the proposed LQI controller with feedback linearization approach as seen in **Figure 9b**.
