**2. Quadrotor dynamic model**

ity character of the UAV vehicles and focuses on the position and attitude controllers. One of scenarios that could be achieved by the quadrotor is autonomous navigation. Implementing such a mission requires the measurement of the components of vehicle state (position, linear velocity,orientationanglesandangularvelocity)fromGPSsensors.Mostofthe control schemes that have been introduced in the literature rely on the available full state of feedback using definitionofanobserverorrelyontheestimationofvelocityofthevehiclebymeansofderivation of the successive measurements from the used sensors such GPS, etc. [1–3]. However, an additional disturbance may occur in the control loop as a result of the use of the observer. It is also important to validate first convergency of the observer. In addition, the stability of closed loop of complete system controlled by observer has to be assured through checking the compatibility of observerfrequency andcontrollerfrequency [4].In this estimation method,the errors produced by GPS may approach several meters, and in reality, velocity measurement error is growing fast by the induced measurement noise along with numerical integration. Few researchers have solved the problem of linear velocity estimation without the use of GPS, and some of them considered combination and artificial vision and the inertial sensors [5, 6].

In [7], the authors proposed a global exponential observer with a complete order to ensure a continued trajectory of a helicopter (VTOL). To reduce the size of the estimated state vector, a reduced observer introduced in [8], the linear velocity of the quadrotor has been estimated using a complete or partial measure of the acceleration such that the asymptotic stability of the error is obtained. Some other forms of observers have been introduced such as adaptive observer [9], and this approach provided an estimation of the velocity of quadrotor UAV from acceleration measurements calculated by the inertial system, but this method has disadvan‐ tages of slow convergence of the estimated parameters. While the sliding mode observer used in [10] has shown not only the ability to estimate the velocity of quadrotor but also showed its robustness against the external disturbances. The other technique used in the literature to estimate the linear velocity is the filters, and one of these filters is extended Kalman filter that widely applied to nonlinear dynamical systems. In this case, the principle is to use the standard equations of Kalman filter for nonlinear model linearized by Taylor formula of first order and the states of systems can be estimated from the noise measurements. These kinds of filters have

In this paper, we propose an approach to solve the problem of position tracking of quadrotor UAV when the linear velocity measurements are unavailable. The proposed controller provides a hierarchical design of the system using assign inner loop for position control and outer loop for attitude control. Each loop has its own control algorithm. The resulting outer control loop, which is based on backstepping, has a simple structure. The inner control loop is based on nonlinear filter-based control design with the aim of estimating the linear velocity

This paper is organized as follows. In Section 2, the dynamic model of quadrotor UAV is introduced. Section 3 presents the design of position controller based on the nonlinear dynamic model using nonlinear filter-based approach. In Section 4, the attitude controller of rotational system is presented. In Section 5, the overall closed loop system stability is analysed, and

been used to estimate the states of quadrotor [11].

138 Recent Progress in Some Aircraft Technologies

simulation results are shown in Section 6.

of quadrotor.

The quadrotor UAV is modelled as a rigid body of mass *m*∈*R* and of matrix of inertia *I* ∈*R* <sup>3</sup>*x*<sup>3</sup> (*I* =*dia*(*I*1,*I*2,*I*3)). The equations of motion can be described by two reference frames: first one is the reference frame (I) combined with the unit vector basis (*e*1,*e*2,*e*3) and second one is the body frame (B) fixed on the quadrotor UAV and combined with vector basis (*e*<sup>1</sup> *b* ,*e*2 *b* ,*e*3 *b* ) (see **Figure 1**). The kinematic and dynamic equations of motion for the quadrotor expressed in the body-fixed reference frame can be described as [12]. The dynamic model is derived using Newton-Euler formalism in the body fixed frame B.

$$\begin{aligned} \dot{\eta} &= RV \\ \dot{V} &= -s(\Omega)V + \frac{1}{m}f + \frac{1}{m}R^\top b\_v \\ \dot{\Theta} &= k(\Theta)\Omega \\ I\_f \dot{\Omega} &= -\Omega \times I\_f \Omega - G\_a + \tau\_a \end{aligned} \tag{1}$$

**Figure 1.** Quadrotor airframe and reference frames.

where *η* = *x*,*y*,*z <sup>T</sup>* represents the position of the centre of mass of the quadrotor, these positions are expressed relatively with respect to an inertial frame and Θ= *ψ*,*θ*,*φ* represents the three Euler angles (yaw, pitch, roll) that define the orientation of the quadrotor in space. *V* = *vx*,*vy*,*vz <sup>T</sup>* ∈*R* <sup>3</sup> and Ω∈*R* <sup>3</sup> denote the linear and angular velocities expressed in the bodyfixed frame B. *bv* represents the external force disturbance expressed in the inertial frame I. *m* is the mass of the quadrotor, *If* <sup>∈</sup>*<sup>R</sup>* 3×3 denotes a positive definite inertia matrix relative to the inertial frame I. The vector *<sup>f</sup>* ,*τa*∈*<sup>R</sup>* <sup>3</sup> denotes the external force and torque expressed in B. The external force is aligned with the *<sup>z</sup>* <sup>−</sup>*axis* and can be written as *<sup>f</sup>* <sup>=</sup> <sup>−</sup>*<sup>G</sup>* <sup>+</sup> *<sup>T</sup> <sup>e</sup>*3, where *<sup>G</sup>* <sup>∈</sup>*<sup>R</sup>* <sup>3</sup> is the gravity vector defined as *<sup>G</sup>* <sup>=</sup>*mgR <sup>T</sup> <sup>e</sup>*3 where *<sup>g</sup>* <sup>∈</sup>*R* denotes the gravitational acceleration and *<sup>e</sup>*<sup>3</sup> <sup>=</sup> 0,0,1 *<sup>T</sup>* is the unit vector in the inertial frame. T denotes the total thrust produced by the four rotors given as *T* =*k*∑<sup>1</sup> 4 *ωi* 2 where *ω<sup>i</sup>* being the speed of the rotor *i, k* is the thrust factor and it is a positive proportional constant parameter depending on the density of air, the radius, the shape, the pitch angle of the blade and other factors. *Ga*∈*<sup>R</sup>* <sup>3</sup> contains gyroscopic couples, due to the rotational motion of the quadrotor and the four rotors which is given by *Ga* <sup>=</sup>∑<sup>1</sup> 4 *Ir*(Ω*e*3)(−1) *i*+1 *ωi* where *Ir* denotes the moment of inertia. The generalized moments on the Θ variable are as follows.

$$\begin{aligned} \boldsymbol{\tau}\_a = \begin{pmatrix} \boldsymbol{\tau}\_\nu \\ \boldsymbol{\tau}\_\theta \\ \boldsymbol{\tau}\_\phi \end{pmatrix} = \begin{pmatrix} k(\boldsymbol{\alpha}\_1^2 - \boldsymbol{\alpha}\_2^2 + \boldsymbol{\alpha}\_3^2 - \boldsymbol{\alpha}\_4^2) \\\ kl(\boldsymbol{\alpha}\_2^2 - \boldsymbol{\alpha}\_4^2) \\\ kl(\boldsymbol{\alpha}\_3^2 - \boldsymbol{\alpha}\_1^2) \end{pmatrix} \end{aligned} \tag{2}$$

where *l* denotes the distance from the rotor to the centre of mass of the quadrotor. Using Euler angles parameterization and "*XYZ*" convention, the airframe orientation in space is given by a rotation matrix R from B to I, where *R* ∈*SO*3 is expressed as follows:

$$R = R\_{\psi} R\_{\theta} R\_{\phi}$$

$$\begin{pmatrix} C\_{\theta} C\_{\psi} & S\_{\phi} S\_{\theta} C\_{\psi} - C\_{\phi} S\_{\psi} & C\_{\phi} S\_{\theta} C\_{\psi} + S\_{\phi} S\_{\psi} \\ C\_{\phi} S\_{\psi} & S\_{\phi} S\_{\theta} S\_{\psi} + C\_{\phi} C\_{\psi} & C\_{\phi} S\_{\theta} S\_{\psi} - S\_{\phi} S\_{\psi} \\ -S\_{\theta} & S\_{\phi} C\_{\theta} & C\_{\phi} C\_{\theta} \end{pmatrix} \tag{3}$$

#### **3. Quadrotor forces and moments**

The four rotors of quadrotor, rotating at angular velocities *omegai* , produce the four forces *f <sup>i</sup>* =*kω<sup>i</sup>* 2 , pointed upward, where *i* = 1,2,3,4, and *ki* are positive constants. These four forces constitute the total thrust *T* along *z* axis. Therefore, the distributed forces and moments from four actuator motors for the quadrotor are given by:

$$
\begin{bmatrix} T \\ \tau\_{\psi} \\ \tau\_{\theta} \\ \tau\_{\phi} \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & 1 \\ -l & 0 & l & 0 \\ 0 & l & 0 & -l \\ c & -c & c & -c \end{bmatrix} \begin{bmatrix} f\_1 \\ f\_2 \\ f\_3 \\ f\_4 \end{bmatrix} \tag{4}
$$

where *c* is the drag coefficient. Since the quadrotor UAV is under-actuated system, the design of control inputs for this kind of systems is a challenging topic.
