Advanced Mathematics and Numerical Modeling of IoT

Figure 2: Physical appearance/shape of a quadrotor.

Table 1: Quadrotor physical parameters.

Parameters Value

Inertia around푋-axis 퐼푥 7.5 × 10−3kg⋅m^2

Inertia around푌-axis 퐼푦 7.5 × 10−3kg⋅m^2

Inertia around푍-axis 퐼푧 1.5 × 10−3kg⋅m^2

Distance to the center of the

quadrotor

퐿 0.23 m

Mass of the quadrotor 푚 0.65 kg

Gravitational acceleration 푔 0.98 m⋅s−^2

(ii) Thrust in all directions is proportional to the square
of the rotor speed.
(iii) The quadrotor is a symmetrical rigid body.
(iv) The origin of the inertial coordinate system is in
the same position as the geometric center and the
centroid of the quadrotor.
Two main effects are taken into consideration: generation
ofthethrustandthedragforce.Thethrust,푇,producedby
each motor is a force calculated as

퐹푖=휌퐶푡퐴휔^2 푖푅^2 =푘푡휔^2 푖, (1)

where퐶푡is the thrust coefficient,휌is the air density,퐴is the
rotor disk area, and푅is the blade radius. Further, the drag
force is defined as

퐷푖=

1

2

휌퐶푑V^2 =푘푑V^2 , (2)

where퐷is the drag force,퐶푑is the drag force coefficient, and
Vis the speed of the quadrotor.
In the fixed coordinates of the body, the direct inputs are
revolutions per minute (RPM) commands for the motors. The
resultant outputs are푍direction thrusts in these coordinates.
However, the outputs under consideration are the attitude
and position. To eliminate this gap, four control variables are
defined as follows:

[[

[

[

푈 1

푈 2

푈 3

푈 4

]]

]

]

=

[

[

[

[[

[

[

[

[

퐹 1 +퐹 2 +퐹 3 +퐹 4

푙(퐹 4 −퐹 2 )

푙(퐹 3 −퐹 1 )

퐹 2 +퐹 4 −퐹 3 −퐹 1

]

]

]

]]

]

]

]

]

=

[

[

[

[

[[

[

[

[[

[

푘푖

4

∑

푖=1

휔^2 푖

푘푡(휔^24 −휔^22 )

푘푡(휔^23 −휔^21 )

푘푑(휔^21 −휔^22 +휔 32 −휔^24 )

]

]

]

]

]]

]

]

]]

]

,

(3)

z

y x

g

F 4

F 3

F 1

F 2

Figure 3: Definition of axis and rotor output.

z

y x

z

y x

O

O

{E}

{B}

Figure 4: Model of the structure of the quadrotor.

where휔 1 ,휔 2 ,휔 3 ,and휔 4 are the respective rotational speed of

each rotor,퐹 1 ,퐹 2 ,퐹 3 ,and퐹 4 are the lift forces of the motor’s

axis,푙is the length of the quadrotor,푈 1 is the total lift force,

푈 2 is the rolling moment,푈 3 is the pitching moment, and푈 4

is the pitching moment.

3.3. The Attitude Control Layers.Let us now derive a math-

ematical model for the quadrotor shown inFigure 2(see

Figure 4). But this layer is not our main interest. For that

reason, this layer is designed using existing excellent research.

Descriptions, expressions, sentences, and equations also are

quoted in those papers (especially reference [ 7 ]).

The origin of the inertial coordinate system E is the

initial position of the quadrotor. The positive direction of the

푂푋axis is the designated heading of the quadrotor and is

perpendicular to the horizontal plane.

Thiscoordinatesystemisusedtostudytherelativemove-

ment of ground and quadrotor. The푂푌axis is perpendicular

to the푂푋푍plane. The quadrotor’s spatial coordinates (푋,푌,

푍) can be obtained through the inertial coordinate system.

The origin of quadrotor coordinate system B (표푥푦푧)is

the center of the quadrotor, and표푥is parallel to the center

connection of the front and rear rotors and the positive

direction points to the front.표푧is parallel to the center

connection of the left and right rotors and the positive

direction points to the right. The표푦axis is perpendicular