# Quadrotor Dynamics with Multiple Disturbances

## Quadrotor Dynamic Model

The quadrotor dynamic system is a nonlinear, underactuated, and strong coupled system. The quadrotor UAV has six degrees of freedom and is actuated by four motors. Fig. 4.1 shows the quadrotor UAV flying under both cable- suspended-payload disturbance and wind disturbance.

The configuration of a quadrotor UAV is illustrated in Fig. 4.2. Two coordinate systems are defined in this chaptert, namely, world-fixed frame W = [е1,в2,ез] and body-fixed frame *В =* [61,62,63]. In addition, we suppose that the origin of the body-fixed frame is located at the center of the quadrotor’s mass. 7 = [ж, *у, г] ^{т}* denotes the position of the quadrotor in the world-fixed frame and its linear velocity is represented by

*и*= [ж,

*у,*i]

^{T}= [u,

*v,*u']

^{T}. 77 =

*[ф, в, ф}"*represents the Euler angular between the world frame and the body frame, and its corresponding angular velocity is denoted by

^{1}*Cl*= [ф,

*0.*U]

^{1}in the world-fixed frame, u; = [p, <7, r]

^{T}represents the angular velocity of the quadrotor UAV in the body frame.

FIGURE 4.1: The quadrotor UAV flying under both cable-suspended pay- load and wind disturbance: the weight of the payload is 500 g, 70% of UAV, and the wind speed is up to 5m/s.

FIGURE 4.2: The quadrotor and the cable-suspended payload in wind environment.

The rotation matrix from the body-fixed frame to the world-fixed frame is given by

Thrust / € R and moment r € R^{3}, as the control ouputs, can be presented as

where /,, *i =* 1,2,3,4 is the thrust force generated by the ttli propeller, r = *тф. ту.* 7>]^{T} represents the equivalent control moment of the four thrust forces on the quadrotor, *d,-,* is the half of roll motor-to-motor distance, *dy* is the half of pitch motor-to-motor distance, and *c _{T}j* is a fixed constant reflecting the relationship between the thrust force /,; and its corresponding torque.

As can be seen from Fig. 4.2, the quadrotor UAV’s translational and rotational motions are driven by adjusting the speed of four motors, while the four lifts (/i, /2, /3, /4) are proportional to the square of the corresponding rotor speeds. The pitching motion is achieved by decreasing the forces /2 and /4 and increasing the forces *f* and /3, leading to a forward movement. In a similar manner, the rolling motion is achieved by increasing the forces /3 and /4 and decreasing the forces *f* and /2, resulting in a lateral movement. Yaw motion is achieved by accelerating the two clockwise turning rotors (/2,/3) and decelerating the two counter-clockwise turning rotors (/ь/4). The mathematical model can be divided into two portions: translational dynamics and rotational dynamics, respectively.

Based on the Newton’s second law, the dynamic of the position loop can be formulated as

where *m* is the total mass of the quadrotor. *F = [F _{x}. F_{y}.* F,]

^{T}is the equivalent control force of the position loop,

*g*is the gravity acceleration, and

*d*/ stands for the disturbance force vector constituted by environmental disturbance and model uncertainty of the position loop, respectively.

Based on the Lagrange-Euler formalism [91], the dynamic of attitude loop is formulated as

where *M*(r*))* € R^{3x3} represents the diagonal moment of inertia tensor, and *C(r). f})* € IR^{:!} is the centrifugal and Coriolis matrix. *d _{T}* is the disturbance torque matrix constituted by environmental disturbance and model uncertainty of the attitude loop. The expansions of

*M(rj)*and

*C(r),i))*are given by

and

where

## The Analysis of Disturbances

As the quadrotor UAV with a cable-suspended payload operating in a wind field, there are multifarious disturbances acting on the quadrotor such as the payload, wind, rotor drag, and model uncertainty [37, 105]. According to the characteristics, the disturbances concerned can be divided into two types: *d _{m }*represented by an exogenous system (e.g., periodic payload disturbance) and

*d*described as a derivative bounded variable (e.g., wind disturbance and model uncertainty).

In most situations, the suspended payload can be regarded as a disturbance with partially known information. For example, when the payload swings or the quadrotor has a periodic trajectory, the disturbance can be seen as a kind of *d _{m}.* Thus, the dynamics can be described as [35],

*А _{х}* 0

_{2}х2 0

_{2}х2 г Q -

where *А =* 0_{2Х}2 *А _{у}* 0

_{2х2}

*,A*

_{t}= __{Q}*

*В =*

0_{2}х2 0_{2х}2 * А_{г}* _ L '

**h=x,y,z**' 1 0 0 0 0 0 '

- 0 0 1 0 0 0 , and
is the frequency of the periodic disturbance - 0 0 0 0 1 0

along г-axis.

Suppose that the lashing point between the quadrotor and the cable is coincident with the center of gravity of the quadrotor, then the periodic disturbance *d _{m}* caused by the payload will directly affect the position of the quadrotor UAV. Thus

*d*can be classified as part of

_{m}*d f*given in Eq. (4.3a), denoted as

*d*

_{m}j.On the other hand, without loss of generality, suppose that the wind disturbance suffered by the quadrotor UAV has a bounded variation and is regarded as *di* in this chapter. The upper bound of *dt* is denoted by a positive scalar

*dh* be., ||d_{(}|| < *d _{t}.*

In nature, it is difficult to accurately predict wind disturbance, especially its direction and speed. For a quadrotor UAV, if the wind disturbance invades laterally, it mainly affects the UAV’s position. Its attitude will be influenced greatly when the wind blows upwards or downwards and is not precisely aligned with the center of mass. In most cases, the effect of wind disturbance on the quadrotor UAV is a mixture of the preceding two scenarios and its variation can be considered bounded. Thus, corresponding to the force and torque disturbances in Eqs. (4.3a) and (4.4). *di* herein is divided into two types: force disturbance part *dt j* and torque disturbance part *d[ _{T}.* The following assumption about wind disturbance is presented.

Assumption 4.1. *The force and torque portions of wind disturbance are assumed to have bounded variation and there exist two constants dif and d( _{T }such that di f* || <

*dif and*||

_{T}.

Note that it requires that the changing rate cannot be too large, otherwise the disturbance is difficult to be observed.

To sum up, the translational dynamic Eq. (4.3a) and rotational dynamic Eq. (4.4) of a quadrotor UAV under both cable-suspended-payload disturbance and wind disturbance can be rewritten as

where *dj* = *d _{m}f* +

*dif*and

*d~ = di~.*

With the translational and rotational dynamics of a quadrotor UAV in the presence of the disturbances being modelled, a MDOBAC scheme is presented to achieve high-precision control performance next.