**4. Control problem formulation**

*<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

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

> 2222 1234 2 2 2 4 2 2 3 1

( )

wwww

 ww

æ ö æ ö -+- ç ÷ ç ÷ - ç ÷

*kl kl*

 ww

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

ç ÷ - è ø è ø

( )

 fq y fy

+ -

constitute the total thrust *T* along *z* axis. Therefore, the distributed forces and moments from

*l l f*

*c cc c f*

where *c* is the drag coefficient. Since the quadrotor UAV is under-actuated system, the design

*l l f*

1 111

*T f*

0 0 <sup>=</sup> 0 0

é ù é ùé ù ê ú ê úê ú - ê ú ê úê ú ê ú ê ú - ê ú ê ú ê úê ú ê ú ë û ë û - - ë û

 fy

f q

== ( )

*k*

y

t

 t

t

q y fq y fy

**3. Quadrotor forces and moments**

qy

q

, pointed upward, where *i* = 1,2,3,4, and *ki*

four actuator motors for the quadrotor are given by:

y

t

t

t

q

f

of control inputs for this kind of systems is a challenging topic.

*a*

q

f

a rotation matrix R from B to I, where *R* ∈*SO*3 is expressed as follows:

 fqy

The four rotors of quadrotor, rotating at angular velocities *omegai*

=

*R RRR*

f q

yqf

æ ö - + ç ÷

*CC SSC CS CSC SS CS SSS CC CSS SS S SC CC*


 f y fqy

t

where *ω<sup>i</sup>* being the speed of the rotor *i, k* is the thrust factor and

(2)

(3)

(4)

, produce the four forces

are positive constants. These four forces

where *Ir* denotes the moment of inertia. The generalized moments on the

four rotors given as *T* =*k*∑<sup>1</sup>

140 Recent Progress in Some Aircraft Technologies

*Ir*(Ω*e*3)(−1)

Θ variable are as follows.

*i*+1 *ωi*

*Ga* <sup>=</sup>∑<sup>1</sup> 4

*f <sup>i</sup>* =*kω<sup>i</sup>* 2 4 *ωi* 2

In this work, we aim to design control laws for the total thrust *T* and torques *τa* allowing the quadrotor UAV to track a desired trajectory *η<sup>d</sup>* = *xd* ,*yd* ,*zd* and desired heading *ψd*. The set of trajectory and their derivatives are smooth enough so that they are uniformly continuous and bounded. The linear velocity is assumed to be not available for feedback and only the position and acceleration are therefore accessible by the translational subsystem. However, we want to design a global feedback control law in the absence of measurements of linear velocity that guarantee the position, heading and velocity tracking errors that are bounded and converge asymptotically to zero.

$$e\_{\nu} = \overline{\nu} - \overline{\nu}\_d, e\_{\eta} = \overline{\eta} - \eta\_d, \ \dot{e}\_{\eta} = \overline{\nu} - \nu\_d \tag{5}$$

The main goal of this paper is to solve the position control problem; meanwhile, guaranteeing all closed-loop signals are bounded and the corresponding constraints of the attitude system are violated. In the following, we will introduce the methodology of control design to achieve our objectives.
