**1. Introduction**

Unmanned aerial vehicle (UAV) research has attracted tremendous attention during the last decade. This interest is mainly given due to the low cost of this type of vehicles and its large application range in diverse areas such as surveillance, delivery, maintenance, inspection, transportation, work assistant, and aerial photography. For instance, UAVs could be provided with cameras so as to observe nature and wildlife. In addition, arms, grippers, or tethers might be installed to UAVs, for which UAVs can assist in construction, transportation, and carrying payloads. Different types of UAVs are considered as complex systems since their dynamic models are nonlinear, dynamically coupled, and the difficulty to establish a very accurate mathematical model. The design of UAV control systems has attracted many researchers worldwide, and many control techniques have been proposed for the aim of accomplishing a 6-DoFs dynamic and trajectory tracking

control of UAVs. This chapter focuses on advanced nonlinear control approaches in order to enhance the dynamic performance of both dynamic and trajectory tracking control of UAVs. Nonlinear control theories have been developed among other control strategies due to their capacity to deal with the nonlinearity and the coupling components of the UAV state variables.

and without adaptive control is investigated. An adaptive controller based on backstepping technique is employed for the trajectory tracking of quadrotor incorporating a fuzzy monitoring strategy to compensate the undesired dynamic error caused by lumped disturbances and total thrust input saturation [12]. Reference [13] introduces adaptive sliding mode controller for distributed control systems with mismatched uncertainty that exists in communication channels. A linear sliding surface is adopted to guarantee asymptotic stability of each subsystem, and an adaptive scheme that can update the unknown upper bound of uncertainty is applied. The distributed controller is constructed based on the information from the adaptive scheme and neighboring subsystems, such that each subsystem can keep

*Advanced UAVs Nonlinear Control Systems and Applications*

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

On the other hand, a tracking control system for the quadrotor UAV based on Takagi-Sugeno (T-S) fuzzy control has been presented in [14]. At first, T-S fuzzy error model has been presented as three independent subsystems for altitude, attitude, and position. Then, T-S fuzzy feedback controller design procedure is applied for altitude, attitude, and position subsystems of the quadcopter. LMI algorithm has been utilized in order to calculate the controller's gains. In [15], a sliding mode underactuated control (SMUC) is designed for the quadrotor UAV model with small uncertainty. In order to enhance the tracking response of the quadrotor UAV, recurrent-neural-network-based sliding-mode underactuated control (RNN-SMUC) with online recurrent neural network modeling and compensation of dynamical uncertainty is designed, and the RNN performs as an approximator. Finally, the combination of SMUC and RNN-SMUC with a transition as so-called hybrid neural-network-based sliding-mode underactuated control (HNN-SMUC) is developed. This development has the advantages of SMUC and RNN-SMUC; e.g., a better transient response of SMUC and an improved tracking performance of RNN-SMUC are accomplished. Furthermore, researchers of [16] compare between LPV controllers and LTI *H*<sup>∞</sup> controllers with S/KS loop shaping to test the performance of a quadrotor while tracking fast trajectories and aggressive maneuvers. Reference [17] combines nonlinear model predictive control (NMPC) and PID controller for better stabilizing of quadrotor UAV under different noises and disturbance conditions; the proposed controller has been applied for the altitude and attitude control loops, whereas switching model predictive controllers for attitude, altitude, and translational motion are derived based on piecewise affine linearized dynamic model in [18], where the effects induced by wind gusts disturbances are considered as affine outputs. The experimental platform utilizes inertial measurement unit IMU, sonar and an optic-flow sensor to produce feedback to the system for indoor applications. Various flight cases including position hold and altitude set-point, trajectory tracking, hovering, and aggressive attitude control have been performed

in order to justify the efficiency of the proposed control system.

**81**

Nonlinear control methods cover the majority of the applied approaches in the literature. For instance, [19] proposes nonlinear hybrid controller that utilizes the time response characteristics of the PID and the stability characteristics of the LQR; differential-flatness-based feedforward control is incorporated with the LQR to enhance the performance of the position system, whereas PID controllers are designed to control the attitude of the quadcopter. Authors of [20] utilize LQR, PID, and feedback linearization in order to design position-tracking model. The LQR controller is added to the feedback linearization model to optimize the control algorithm by determining a suitable cost function; the attitude of dynamic control was modeled so as to maintain desired quadcopter's position despite the presence of disturbances. The performance of tracking position is optimized by adding PID loop control for pitch, roll, and yaw movement, and a comparison between the performance of the two nonlinear control techniques, including backstepping and

stable and have good performance.

Quadcopters are one of the very common UAV platforms; in fact, the literature related to control design of quadcopters is extensive, and this type of UAVs is underactuated, nonlinear, and strongly coupled, which is hard to cope properly with conventional control methods. On the other hand, they have many advantages over conventional helicopters, which may be concluded as follows: capability of vertical take-off and landing (VTOL), hovering and maneuverability, and low power consumption, since it has four small-scale propellers for thrust and orientation.

In the area of quadcopter literature, there is a variety of applications as aerial manipulation [1, 2], quadcopter pendulum [3], navigation and localization [4, 5], obstacle avoidance [6], altitude control [7], and cooperative and formation control [8, 9]. Moreover, several control schemes have been proposed including adaptive control [10–13], fuzzy control [14], neural network control [15], linear parameter varying (LPV) control [16], predictive control [17, 18], nonlinear control methods [19–23], and sliding mode control [24, 25]. In [4], researchers propose localization, navigation, and mapping methods based on the characteristic map; feature map is selected to localize and navigate the UAV under investigation, while drawing up navigation strategy and avoidance strategy. In [5], PID controllers for the attitude, altitude, and position of a quadrotor are designed, and an outdoor experiment is conducted based on GPS to verify the performance, and desired trajectory's waypoints are determined using Mission Planar software. The application of ultrasonic sensor is used to detect barriers during the flight, so that the position of the quadrotor is adjusted depending on the signal of the ultrasonic sensor in order to avoid collision [6]. Cascaded PID controller with the usage of laser range finder combined with accelerometer in order to determine the height of the vehicle has been presented in [7]; the proposed system is compared with the performance of the system using GPS combined with pressure gauge. However, the results of the proposed system exhibit better performance especially in the range of low altitude. Centralized formation flight control of a leader/follower structure of three quadcopters is proposed in [8] using LQR-PI, the trajectory of the leader defines the desired trajectory for the followers, and a pole placement controller is used for the leader and LQR-PI controllers for the followers. In case of communication loss between leader and any of the followers, the other follower quadcopter provides the leader's states to the affected follower quadcopter in order to keep the formation intact. Whereas a multiagent consensus control incorporated with collision avoidance using model predictive control is presented in [9], the term of achieving formation and the term of repulsive potential are set in the index function to realize the formation control considering collision avoidance. The experiment is carried out using three quadrotor UAVs.

By looking to the quadcopter control systems, dynamic inversion and linear neural-network-based adaptive attitude control of a quadrotor UAV is introduced in [10]. Based on the time-scale separation principle, an attitude dynamic inverse controller and a trajectory dynamic inverse controller are deduced, respectively; the inverse error dynamics is regulated using PD controller, and a sigma-pi neural network is introduced to eliminate the inverse error adaptively to improve the robustness of the controller. Authors of [11] propose a compound adaptive backstepping and sliding mode control subject to unknown external disturbances and parametric uncertainties. A comparison study for the proposed method with

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

control of UAVs. This chapter focuses on advanced nonlinear control approaches in order to enhance the dynamic performance of both dynamic and trajectory tracking control of UAVs. Nonlinear control theories have been developed among other control strategies due to their capacity to deal with the nonlinearity and the cou-

Quadcopters are one of the very common UAV platforms; in fact, the literature

In the area of quadcopter literature, there is a variety of applications as aerial manipulation [1, 2], quadcopter pendulum [3], navigation and localization [4, 5], obstacle avoidance [6], altitude control [7], and cooperative and formation control [8, 9]. Moreover, several control schemes have been proposed including adaptive control [10–13], fuzzy control [14], neural network control [15], linear parameter varying (LPV) control [16], predictive control [17, 18], nonlinear control methods [19–23], and sliding mode control [24, 25]. In [4], researchers propose localization, navigation, and mapping methods based on the characteristic map; feature map is selected to localize and navigate the UAV under investigation, while drawing up navigation strategy and avoidance strategy. In [5], PID controllers for the attitude, altitude, and position of a quadrotor are designed, and an outdoor experiment is conducted based on GPS to verify the performance, and desired trajectory's waypoints are determined using Mission Planar software. The application of ultrasonic sensor is used to detect barriers during the flight, so that the position of the quadrotor is adjusted depending on the signal of the ultrasonic sensor in order to avoid collision [6]. Cascaded PID controller with the usage of laser range finder combined with accelerometer in order to determine the height of the vehicle has been presented in [7]; the proposed system is compared with the performance of the system using GPS combined with pressure gauge. However, the results of the proposed system exhibit better performance especially in the range of low altitude.

Centralized formation flight control of a leader/follower structure of three

quadcopters is proposed in [8] using LQR-PI, the trajectory of the leader defines the desired trajectory for the followers, and a pole placement controller is used for the leader and LQR-PI controllers for the followers. In case of communication loss between leader and any of the followers, the other follower quadcopter provides the leader's states to the affected follower quadcopter in order to keep the formation intact. Whereas a multiagent consensus control incorporated with collision avoidance using model predictive control is presented in [9], the term of achieving formation and the term of repulsive potential are set in the index function to realize the formation control considering collision avoidance. The experiment is carried out

By looking to the quadcopter control systems, dynamic inversion and linear neural-network-based adaptive attitude control of a quadrotor UAV is introduced in [10]. Based on the time-scale separation principle, an attitude dynamic inverse controller and a trajectory dynamic inverse controller are deduced, respectively; the inverse error dynamics is regulated using PD controller, and a sigma-pi neural network is introduced to eliminate the inverse error adaptively to improve the robustness of the controller. Authors of [11] propose a compound adaptive backstepping and sliding mode control subject to unknown external disturbances and parametric uncertainties. A comparison study for the proposed method with

related to control design of quadcopters is extensive, and this type of UAVs is underactuated, nonlinear, and strongly coupled, which is hard to cope properly with conventional control methods. On the other hand, they have many advantages over conventional helicopters, which may be concluded as follows: capability of vertical take-off and landing (VTOL), hovering and maneuverability, and low power consumption, since it has four small-scale propellers for thrust and

pling components of the UAV state variables.

*Unmanned Robotic Systems and Applications*

orientation.

using three quadrotor UAVs.

**80**

and without adaptive control is investigated. An adaptive controller based on backstepping technique is employed for the trajectory tracking of quadrotor incorporating a fuzzy monitoring strategy to compensate the undesired dynamic error caused by lumped disturbances and total thrust input saturation [12]. Reference [13] introduces adaptive sliding mode controller for distributed control systems with mismatched uncertainty that exists in communication channels. A linear sliding surface is adopted to guarantee asymptotic stability of each subsystem, and an adaptive scheme that can update the unknown upper bound of uncertainty is applied. The distributed controller is constructed based on the information from the adaptive scheme and neighboring subsystems, such that each subsystem can keep stable and have good performance.

On the other hand, a tracking control system for the quadrotor UAV based on Takagi-Sugeno (T-S) fuzzy control has been presented in [14]. At first, T-S fuzzy error model has been presented as three independent subsystems for altitude, attitude, and position. Then, T-S fuzzy feedback controller design procedure is applied for altitude, attitude, and position subsystems of the quadcopter. LMI algorithm has been utilized in order to calculate the controller's gains. In [15], a sliding mode underactuated control (SMUC) is designed for the quadrotor UAV model with small uncertainty. In order to enhance the tracking response of the quadrotor UAV, recurrent-neural-network-based sliding-mode underactuated control (RNN-SMUC) with online recurrent neural network modeling and compensation of dynamical uncertainty is designed, and the RNN performs as an approximator. Finally, the combination of SMUC and RNN-SMUC with a transition as so-called hybrid neural-network-based sliding-mode underactuated control (HNN-SMUC) is developed. This development has the advantages of SMUC and RNN-SMUC; e.g., a better transient response of SMUC and an improved tracking performance of RNN-SMUC are accomplished. Furthermore, researchers of [16] compare between LPV controllers and LTI *H*<sup>∞</sup> controllers with S/KS loop shaping to test the performance of a quadrotor while tracking fast trajectories and aggressive maneuvers. Reference [17] combines nonlinear model predictive control (NMPC) and PID controller for better stabilizing of quadrotor UAV under different noises and disturbance conditions; the proposed controller has been applied for the altitude and attitude control loops, whereas switching model predictive controllers for attitude, altitude, and translational motion are derived based on piecewise affine linearized dynamic model in [18], where the effects induced by wind gusts disturbances are considered as affine outputs. The experimental platform utilizes inertial measurement unit IMU, sonar and an optic-flow sensor to produce feedback to the system for indoor applications. Various flight cases including position hold and altitude set-point, trajectory tracking, hovering, and aggressive attitude control have been performed in order to justify the efficiency of the proposed control system.

Nonlinear control methods cover the majority of the applied approaches in the literature. For instance, [19] proposes nonlinear hybrid controller that utilizes the time response characteristics of the PID and the stability characteristics of the LQR; differential-flatness-based feedforward control is incorporated with the LQR to enhance the performance of the position system, whereas PID controllers are designed to control the attitude of the quadcopter. Authors of [20] utilize LQR, PID, and feedback linearization in order to design position-tracking model. The LQR controller is added to the feedback linearization model to optimize the control algorithm by determining a suitable cost function; the attitude of dynamic control was modeled so as to maintain desired quadcopter's position despite the presence of disturbances. The performance of tracking position is optimized by adding PID loop control for pitch, roll, and yaw movement, and a comparison between the performance of the two nonlinear control techniques, including backstepping and

feedback linearization with LQR, has been performed in [21]. The control laws have been derived depending on the nonlinear model with no linearization, and experiments for the attitude have been performed. Whereas in [22], the performance of sliding mode techniques has been verified and *sat* function has been used in order to obtain a continuous control law instead of *sign* function [23]. This shows nonlinear control laws applied for optimal trajectory tracking depending on minimum snap theory, and differential flatness method is utilized to derive control laws that link between the system outputs and its inputs. Reference [24] focuses on sliding mode control of the quadcopter; the proposed approach consisted of a sliding mode observer with finite-time process, a hybridization of a PID conventional controller, and a continuous sliding-mode one. The main aim is to estimate the system's state vector based on the measured system's output states and to identify a certain type of the inherited system's disturbances simultaneously. It is also to track a desired timevarying trajectory in spite of the influence of external disturbances and uncertainties. Finally, fractional order sliding mode control is used to derive the attitude control laws of a quadcopter, where PD tracking controllers are used to control the position of the quadcopter in [25].

Let us define the following error function as

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

*Advanced UAVs Nonlinear Control Systems and Applications*

*<sup>V</sup>*\_ ¼ �*ke*<sup>2</sup>

*<sup>V</sup>* <sup>¼</sup> <sup>1</sup> 2 *e* 2

In order to obtain the derivative of the proposed Lyapunov function with a

where *k* is a positive constant, so that the control law will be as follows:

Note: it is remarkable to mention that the parameters of a system would appear in the derived control law when using backstepping, so that an integral action is added to each virtual control during the procedure of deriving the control law, which is termed integral backstepping, and more details about backstepping

Feedback linearization is also one of the major nonlinear design tools. It is used to cancel the nonlinear terms in a system's model; this cancellation resulting in a linear system allows designing and incorporating linear controllers for a nonlinear system with the feedback linearization laws. To introduce the procedure of this strategy, we first introduce the notions of *full-state linearization*, where the state equation is completely linearized, and *input-output linearization*, where the input-output map is linearized, while the state equation may be only partially

In this chapter, we will pay attention to input-output linearization method. To obtain the input-output feedback linearization law, we simply repeat the calculation of the derivative of the system output along the state variables. Let us consider the

The input-output linearization law would become:

*<sup>u</sup>* <sup>¼</sup> <sup>1</sup>

Sliding mode control is considered one of the control tools of the variable structure systems (VSS), since it produces a discontinuous controller. It has the advantage of stabilizing and achieving robustness criteria against model uncertainty and disturbances. Sliding mode control theory depends on a sliding surface *s*, where the sliding mode controller constrains a system to it. The motion toward the sliding surface consists of a *reaching phase* during which trajectories starting off

*<sup>u</sup>* <sup>¼</sup> <sup>1</sup>

and Lyapunov function as

method are described in [27].

**2.2 Feedback linearization**

linearized [26].

system in (1) as,

**2.3 Sliding mode control**

**83**

negative definite,

*e*<sup>1</sup> ¼ *x*1*<sup>d</sup>* � *x*<sup>1</sup> (2)

<sup>1</sup> þ *e*1ð Þ *x*1*<sup>d</sup>* þ *k*1*e*<sup>1</sup> � *f x*ð Þ� *g x*ð Þ*u* (4)

*g x*ð Þð Þ *<sup>x</sup>*1*<sup>d</sup>* <sup>þ</sup> *ke*<sup>1</sup> � *f x*ð Þ (5)

*y* ¼ *x*2*, y*\_ ¼ *x*\_ <sup>2</sup> (6)

*g x*ð Þð Þ �*f x*ð Þþ *<sup>υ</sup>* (7)

<sup>1</sup> (3)

## **2. Nonlinear control approaches**

Nonlinear control theory is the area of control theory that deals with nonlinear systems, time variant systems, or both. Different engineering applications motivate researchers to develop powerful nonlinear control methods, since a majority of these systems are considered to be nonlinear. The key reason behind the use of nonlinear control techniques is their capability to deal with the nonlinear characteristics of nonlinear systems such as underactuations, models uncertainty, and dynamic coupling. This chapter focuses on the following nonlinear control approaches:


#### **2.1 Backstepping**

It is a widely used nonlinear control technique, due to its significant inherited characteristics including: being a recursive controller approach, which depends on a proposed Lyapunov function for deriving the system control law; higher flexibility, to some extent, in avoiding key nonlinearity cancellation; and verifying the desired objective of stabilization and tracking [26, 27]. The procedure of deriving control laws depending on backstepping technique is concluded in, at first, determining the error function between the desired input and the system actual output, then outlining a Lyapunov function and determining virtual controls to make the derivative of the proposed Lyapunov function with a negative definite. Finally, these steps are repeated until obtaining the control law.

Consider the following system:

$$\begin{aligned} \dot{\mathbf{x}}\_1 &= \mathbf{x}\_2 \\ \dot{\mathbf{x}}\_2 &= f(\mathbf{x}) + \mathbf{g}(\mathbf{x})u \\ \mathbf{y} &= \mathbf{x}\_2 \end{aligned} \tag{1}$$

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

Let us define the following error function as

$$
\varkappa\_1 = \varkappa\_{1d} - \varkappa\_1 \tag{2}
$$

and Lyapunov function as

feedback linearization with LQR, has been performed in [21]. The control laws have been derived depending on the nonlinear model with no linearization, and experiments for the attitude have been performed. Whereas in [22], the performance of sliding mode techniques has been verified and *sat* function has been used in order to obtain a continuous control law instead of *sign* function [23]. This shows nonlinear control laws applied for optimal trajectory tracking depending on minimum snap theory, and differential flatness method is utilized to derive control laws that link between the system outputs and its inputs. Reference [24] focuses on sliding mode control of the quadcopter; the proposed approach consisted of a sliding mode observer with finite-time process, a hybridization of a PID conventional controller, and a continuous sliding-mode one. The main aim is to estimate the system's state vector based on the measured system's output states and to identify a certain type of the inherited system's disturbances simultaneously. It is also to track a desired timevarying trajectory in spite of the influence of external disturbances and uncertainties. Finally, fractional order sliding mode control is used to derive the attitude control laws of a quadcopter, where PD tracking controllers are used to control the

Nonlinear control theory is the area of control theory that deals with nonlinear systems, time variant systems, or both. Different engineering applications motivate researchers to develop powerful nonlinear control methods, since a majority of these systems are considered to be nonlinear. The key reason behind the use of nonlinear control techniques is their capability to deal with the nonlinear characteristics of nonlinear systems such as underactuations, models uncertainty, and dynamic coupling. This chapter focuses on the following nonlinear control

It is a widely used nonlinear control technique, due to its significant inherited characteristics including: being a recursive controller approach, which depends on a proposed Lyapunov function for deriving the system control law; higher flexibility, to some extent, in avoiding key nonlinearity cancellation; and verifying the desired objective of stabilization and tracking [26, 27]. The procedure of deriving control laws depending on backstepping technique is concluded in, at first, determining the error function between the desired input and the system actual output, then outlining a Lyapunov function and determining virtual controls to make the derivative of the proposed Lyapunov function with a negative definite. Finally, these

*x*\_ <sup>1</sup> ¼ *x*<sup>2</sup>

*y* ¼ *x*<sup>2</sup>

*x*\_ <sup>2</sup> ¼ *f x*ð Þþ *g x*ð Þ*u*

(1)

position of the quadcopter in [25].

*Unmanned Robotic Systems and Applications*

**2. Nonlinear control approaches**

approaches:

1. Backstepping

**2.1 Backstepping**

**82**

2. Sliding mode control

3. Feedback linearization

steps are repeated until obtaining the control law.

Consider the following system:

$$V = \frac{1}{2}e\_1^2 \tag{3}$$

In order to obtain the derivative of the proposed Lyapunov function with a negative definite,

$$\dot{V} = -k e\_1^2 + e\_1(\varkappa\_{1d} + k\_1 e\_1 - f(\varkappa) - \mathbf{g}(\varkappa)\mathbf{u}) \tag{4}$$

where *k* is a positive constant, so that the control law will be as follows:

$$u = \frac{1}{g(\mathbf{x})} (\mathbf{x}\_{1d} + k e\_1 - f(\mathbf{x})) \tag{5}$$

Note: it is remarkable to mention that the parameters of a system would appear in the derived control law when using backstepping, so that an integral action is added to each virtual control during the procedure of deriving the control law, which is termed integral backstepping, and more details about backstepping method are described in [27].

#### **2.2 Feedback linearization**

Feedback linearization is also one of the major nonlinear design tools. It is used to cancel the nonlinear terms in a system's model; this cancellation resulting in a linear system allows designing and incorporating linear controllers for a nonlinear system with the feedback linearization laws. To introduce the procedure of this strategy, we first introduce the notions of *full-state linearization*, where the state equation is completely linearized, and *input-output linearization*, where the input-output map is linearized, while the state equation may be only partially linearized [26].

In this chapter, we will pay attention to input-output linearization method. To obtain the input-output feedback linearization law, we simply repeat the calculation of the derivative of the system output along the state variables. Let us consider the system in (1) as,

$$
\mathbf{y} = \mathbf{x}\_2 \,\, \dot{\mathbf{y}} = \dot{\mathbf{x}}\_2 \tag{6}
$$

The input-output linearization law would become:

$$
\mu = \frac{1}{g(\varkappa)} (-f(\varkappa) + \nu) \tag{7}
$$

#### **2.3 Sliding mode control**

Sliding mode control is considered one of the control tools of the variable structure systems (VSS), since it produces a discontinuous controller. It has the advantage of stabilizing and achieving robustness criteria against model uncertainty and disturbances. Sliding mode control theory depends on a sliding surface *s*, where the sliding mode controller constrains a system to it. The motion toward the sliding surface consists of a *reaching phase* during which trajectories starting off

velocity, *η* ¼ ½ � *φ; θ; ψ*

where *fi* <sup>¼</sup> *<sup>b</sup>ω*<sup>2</sup>

drag coefficient *d* in N<sup>2</sup>

*Ti* <sup>¼</sup> *<sup>d</sup>ω*<sup>2</sup>

*b* in N�s 2

**85**

where *<sup>B</sup>*

quadcopter, and *<sup>η</sup>*\_ <sup>¼</sup> *<sup>φ</sup>*\_ *;* \_

in the body-fixed frame *A*.

*BRA* <sup>¼</sup>

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

*RA* is the transformation matrix

*Advanced UAVs Nonlinear Control Systems and Applications*

The equations of motion can be written as follows (10):

*x*€ ¼ ð Þ *cφsθcψ* þ *sφsψ*

*y*€ ¼ ð Þ *cφsθsψ* � *sφsψ*

*<sup>m</sup>* � *<sup>g</sup>*

\_ *θψ*\_ <sup>þ</sup> *Jr Ixx* Ω*<sup>r</sup>* \_ *θ* þ 1 *Ixx U*<sup>2</sup>

*φ*\_ \_ *θ* þ 1 *Izz U*<sup>4</sup>

*U*<sup>1</sup> ¼ *f* <sup>1</sup> þ *f* <sup>2</sup> þ *f* <sup>3</sup> þ *f* <sup>4</sup>

*U*<sup>4</sup> ¼ *T*<sup>1</sup> � *T*<sup>2</sup> þ *T*<sup>3</sup> � *T*<sup>4</sup>

center of propeller in m, *I* is the inertia matrix, and *Ixx*, *Iyy*, and *Izz* are moments of

motors' angular speed. Based on the above derivation and discussion,

where *Jr* is the moment of inertia of the propeller and Ω<sup>r</sup> is the sum of the four

where *m* is the mass of the quadcopter given in kilograms. With,

*U*<sup>2</sup> ¼ *l f* <sup>4</sup> � *f* <sup>2</sup> � �

*U*<sup>3</sup> ¼ *l f* <sup>3</sup> � *f* <sup>1</sup> � �

*<sup>φ</sup>*\_*ψ*\_ � *Jr Iyy*

*<sup>z</sup>*€ <sup>¼</sup> ð Þ *<sup>c</sup>φc<sup>θ</sup> <sup>U</sup>*<sup>1</sup>

*<sup>φ</sup>*€ <sup>¼</sup> *Izz* � *Iyy Ixx*

€ <sup>¼</sup> *Izz* � *Ixx Iyy*

*<sup>ψ</sup>*€ <sup>¼</sup> *Ixx* � *Iyy Izz*

/m and *ω<sup>i</sup>* is the angular speed of motor *i*.

inertia about *x, y,* and *<sup>z</sup>* axes, respectively, in kg�m2

*U*<sup>1</sup> *U*<sup>2</sup> *U*<sup>3</sup> *U*<sup>4</sup>

*U* ¼

*θ*

*<sup>T</sup>* are the roll-pitch-yaw angles describing the attitude of the

*cψcθ sφ:sθ:cψ* � *cφsψ cφ:sθ:cψ* þ *sφ:sψ*

*sψcθ sφ:sθ:sψ* þ *cφcψ cφ:sθ:sψ* � *sφ:cψ*

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

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

Ω*rφ*\_ þ

*<sup>i</sup>* is the thrust force produced by propeller *i* with thrust coefficient

s, *l* is the distance between center of the quadcopter and

.

*ω*2 1 *ω*2 2 *ω*2 3 *ω*2 4

*<sup>i</sup>* is the drag torque produced by propeller *i* in N�m with corresponding

*b b bb* 0 �*bl* 0 *bl* �*bl* 0 *bl* 0 *d* �*d d* �*d*

1 *Iyy U*<sup>3</sup>

�*sθ sφcθ cφcθ*

*θ; ψ*\_ � �*<sup>T</sup>* are the Euler angle rates of the quadcopter described

(9)

(10)

(11)

(12)

**Figure 1.** *Illustration of sign function.*

the surface *s* = 0 move toward it and reach it in finite time, followed by a *sliding phase* during which the motion is confined to the surface *s* [26, 28].

Equivalent control law is one of the sliding mode control strategies; it consists of two terms, the first is produced by equaling the derivative of sliding surface *s* to 0. The other term is called reaching law that has some common formulas such as [28]:

Constant rate reaching law, i.e., *s*\_ ¼ �*K* sgn ð Þ*s ,*

and constant plus proportional rate reaching law, i.e., *s*\_ ¼ �*Qs* � *K* sgn ð Þ*s* .

where *Q* and *K* are positive constants and sign(s) is illustrated in **Figure 1**. With *<sup>V</sup>* <sup>¼</sup> <sup>1</sup> 2 *s* <sup>2</sup> as a Lyapunov function candidate, hence the condition of the stability is *V*\_ to be negative definite. In order to ensure that error *e* converges to zero, the sliding surface might be supposed as a function of the error as follows [26]:

$$\mathfrak{s} = \mathfrak{c}\_0 \mathfrak{e} + \mathfrak{c}\_1 \dot{\mathfrak{e}} + \dots + \mathfrak{c}\_{d-1} \mathfrak{e}^{\rho - 1(.)} + \mathfrak{e}^{\rho(.)} \tag{8}$$

where *ρ* is the relative degree.

The procedure for designing a sliding mode controller can be summarized by the following steps:


## **3. Quadcopter modeling**

The dynamic model of the quadcopter is delivered in this section; the details of the model can be seen in the literature [29–31]. The state variables of the quadcopter are defined as, *<sup>X</sup>* <sup>¼</sup> *<sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>x</sup>*\_*; <sup>y</sup>*\_*; <sup>z</sup>*\_*; <sup>φ</sup>; <sup>θ</sup>; <sup>ψ</sup>; <sup>φ</sup>*\_ *;* \_ *<sup>θ</sup>; <sup>ψ</sup>*\_ *<sup>T</sup>* where *<sup>ζ</sup>* <sup>¼</sup> ½ � *<sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>T</sup>* is the position described in the inertial coordinate frame *<sup>B</sup>*, *<sup>V</sup>* <sup>¼</sup> ½ � *<sup>x</sup>*\_*; <sup>y</sup>*\_*; <sup>z</sup>*\_ *<sup>T</sup>* is the translational

velocity, *η* ¼ ½ � *φ; θ; ψ <sup>T</sup>* are the roll-pitch-yaw angles describing the attitude of the quadcopter, and *<sup>η</sup>*\_ <sup>¼</sup> *<sup>φ</sup>*\_ *;* \_ *θ; ψ*\_ � �*<sup>T</sup>* are the Euler angle rates of the quadcopter described in the body-fixed frame *A*.

where *<sup>B</sup> RA* is the transformation matrix

$${}^{B}R\_{A} = \begin{bmatrix} c\wp c\theta & s\wp s\theta.c\wp - c\wp s\wp & c\wp s\theta.c\wp + s\wp.s\wp\\ s\wp c\theta & s\wp s\theta.s\wp + c\wp c\wp & c\wp s\theta.s\wp - s\wp.c\wp\\ & -s\theta & s\wp c\theta & c\wp c\theta \end{bmatrix} \tag{9}$$

The equations of motion can be written as follows (10):

$$\begin{aligned} \ddot{\mathbf{x}} &= \left(c\rho s\theta c\mathbf{y} + s\rho s\mathbf{y}\right) \frac{U\_1}{m} \\\\ \ddot{\mathbf{y}} &= \left(c\rho s\theta s\mathbf{y} - s\rho s\mathbf{y}\right) \frac{U\_1}{m} \\\\ \ddot{\mathbf{z}} &= \left(c\rho c\theta\right) \frac{U\_1}{m} - \mathbf{g} \\\\ \ddot{\boldsymbol{\rho}} &= \frac{I\_{xx} - I\_{\gamma\gamma}}{I\_{xx}} \dot{\boldsymbol{\theta}} \dot{\mathbf{y}} + \frac{J\_r}{I\_{xx}} \boldsymbol{\Omega}\_r \dot{\boldsymbol{\theta}} + \frac{1}{I\_{xx}} U\_2 \\\\ \ddot{\boldsymbol{\theta}} &= \frac{I\_{xx} - I\_{xx}}{I\_{\gamma\gamma}} \dot{\boldsymbol{\rho}} \dot{\mathbf{y}} \dot{\boldsymbol{\theta}} - \frac{J\_r}{I\_{\gamma\gamma}} \boldsymbol{\Omega}\_r \dot{\boldsymbol{\phi}} + \frac{1}{I\_{\gamma\gamma}} U\_3 \\\\ \dddot{\mathbf{y}} &= \frac{I\_{xx} - I\_{\gamma\gamma}}{I\_{xx}} \dot{\boldsymbol{\rho}} \dot{\boldsymbol{\theta}} \dot{\boldsymbol{\theta}} + \frac{1}{I\_{xx}} U\_4 \end{aligned} \tag{10}$$

where *m* is the mass of the quadcopter given in kilograms. With,

$$\begin{aligned} U\_1 &= f\_1 + f\_2 + f\_3 + f\_4\\ U\_2 &= l \left( f\_4 - f\_2 \right) \\ U\_3 &= l \left( f\_3 - f\_1 \right) \\ U\_4 &= T\_1 - T\_2 + T\_3 - T\_4 \end{aligned} \tag{11}$$

where *fi* <sup>¼</sup> *<sup>b</sup>ω*<sup>2</sup> *<sup>i</sup>* is the thrust force produced by propeller *i* with thrust coefficient *b* in N�s 2 /m and *ω<sup>i</sup>* is the angular speed of motor *i*.

*Ti* <sup>¼</sup> *<sup>d</sup>ω*<sup>2</sup> *<sup>i</sup>* is the drag torque produced by propeller *i* in N�m with corresponding drag coefficient *d* in N<sup>2</sup> s, *l* is the distance between center of the quadcopter and center of propeller in m, *I* is the inertia matrix, and *Ixx*, *Iyy*, and *Izz* are moments of inertia about *x, y,* and *<sup>z</sup>* axes, respectively, in kg�m2 .

where *Jr* is the moment of inertia of the propeller and Ω<sup>r</sup> is the sum of the four motors' angular speed. Based on the above derivation and discussion,

$$U = \begin{bmatrix} U\_1 \\ U\_2 \\ U\_3 \\ U\_4 \end{bmatrix} = \begin{bmatrix} b & b & b & b \\ 0 & -bl & 0 & bl \\ -bl & 0 & bl & 0 \\ d & -d & d & -d \end{bmatrix} \begin{bmatrix} a\_1^2 \\ a\_2^2 \\ a\_3^2 \\ a\_4^2 \end{bmatrix} \tag{12}$$

the surface *s* = 0 move toward it and reach it in finite time, followed by a *sliding*

and constant plus proportional rate reaching law, i.e., *s*\_ ¼ �*Qs* � *K* sgn ð Þ*s* .

where *Q* and *K* are positive constants and sign(s) is illustrated in **Figure 1**.

stability is *V*\_ to be negative definite. In order to ensure that error *e* converges to zero, the sliding surface might be supposed as a function of the error as follows [26]:

The procedure for designing a sliding mode controller can be summarized by the

3. Equaling the derivative of sliding surface with the appropriate reaching law

The dynamic model of the quadcopter is delivered in this section; the details of the model can be seen in the literature [29–31]. The state variables of the quadcopter

*<sup>θ</sup>; <sup>ψ</sup>*\_ *<sup>T</sup>* where *<sup>ζ</sup>* <sup>¼</sup> ½ � *<sup>x</sup>; <sup>y</sup>; <sup>z</sup>*

described in the inertial coordinate frame *<sup>B</sup>*, *<sup>V</sup>* <sup>¼</sup> ½ � *<sup>x</sup>*\_*; <sup>y</sup>*\_*; <sup>z</sup>*\_ *<sup>T</sup>* is the translational

*s* ¼ *c*0*e* þ *c*1*e*\_ þ …… þ *cd*�<sup>1</sup>*e*

<sup>2</sup> as a Lyapunov function candidate, hence the condition of the

*<sup>ρ</sup>*�1ð Þ*:* <sup>þ</sup> *<sup>e</sup>*

*<sup>ρ</sup>*ð Þ*:* (8)

*<sup>T</sup>* is the position

Equivalent control law is one of the sliding mode control strategies; it consists of two terms, the first is produced by equaling the derivative of sliding surface *s* to 0. The other term is called reaching law that has some common formulas such as [28]:

*phase* during which the motion is confined to the surface *s* [26, 28].

Constant rate reaching law, i.e., *s*\_ ¼ �*K* sgn ð Þ*s ,*

With *<sup>V</sup>* <sup>¼</sup> <sup>1</sup>

**Figure 1.**

*Illustration of sign function.*

*Unmanned Robotic Systems and Applications*

following steps:

2 *s*

where *ρ* is the relative degree.

1. Designing the sliding surface *s*

**3. Quadcopter modeling**

**84**

2. Determining the derivative of the sliding surface *s*\_

4.Deriving the control law from the previous step

are defined as, *<sup>X</sup>* <sup>¼</sup> *<sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>x</sup>*\_*; <sup>y</sup>*\_*; <sup>z</sup>*\_*; <sup>φ</sup>; <sup>θ</sup>; <sup>ψ</sup>; <sup>φ</sup>*\_ *;* \_
