IEEE Robotics & Automation Magazine - December 2018 - 38

Rigid Body Model
For the modeling of the body dynamics, the Newton-Euler
formalism was used. Its general form is the following:

αi

mI 3 0 B vo
B ~ # (m $ B v)
BF
E; E+ ;
E = ; E,
;
0 J B ~o
B ~ # (I $ B ~)
BM

(1)

with I 3 being the three-dimensional identity matrix and J
the moment of inertia of the system.
R

ey i
R
ez i

ex i

Motor Dynamics
The dynamics of the rotors' dc motors are modeled as a firstorder system,
no i = 1 ^n i,des - n ih,
xn

Figure 5. The rotor coordinate system.

shown in Figure 5. Their origins are at the center of the rotor
blades, and their x axes are aligned with the axes to the center
of mass of the body. These coordinate frames rotate around
their x axes with respect to the body frame.
Notation
Throughout this article, we use left-hand subscripts for the
coordinate system in which a symbol is referenced. For example, I r indicates a vector r represented in coordinate system
FI . The matrix R AB stands for the rotation matrix that rotates
a vector represented in coordinate system FB to the coordinate system FA , i.e., A r = R AB · B r. The estimate of a state x
will be denoted with a hat on top of the symbol: xt .
Assumptions
For the sake of model simplicity, the following assumptions
were made:
● The body structure is rigid and symmetric.
● The rotors are rigid and on the same height as the center of
gravity, and they rotate around an axis that passes through
the center of gravity.
● The thrust and drag torques are proportional to the square
of the rotor speed.
● The linear and angular velocities of the body are small.
● The dynamics of the tilting motors are independent of the
rotational speed of the rotors.
● The thrust and drag torques produced by one rotor are
independent of the thrust and drag torques of the
other rotors, i.e., the resulting air flows do not produce
interference.
While most of these assumptions will cause little or no
deviation of the flight characteristics of the real-world system from those of the derived system model, the final
assumption on this list, which ignores rotor interference,
is likely to have a significant impact. However, in our
experience and flight tests, we have found that, with a
well-tuned controller, this influence can be treated as an unmodeled disturbance.
38

IEEE ROBOTICS & AUTOMATION MAGAZINE

december 2018

(2)

with n i,des being the desired angular velocity and x n the
motor's time constant. Similarly, the closed-loop tilting
.
velocity ~ a i = a i dynamics are modeled as a first-order
system, with time constant x a . Moreover, the tilting velocity is saturated to model the maximum possible tilting
velocity as follows:
# ~ a,max,

~ ai

(3)

where ~ a,max is the maximum possible tilting velocity.
Aerodynamics
The aerodynamics modeling transfers the actual angular
velocities n and tilting angles a into the forces F and moments
M acting on the center of gravity of the body. Given the stated
assumptions of low body velocities, we can neglect the aerodynamic drag of the main body and assume that the thrust and
reaction torques are proportional to the square of the rotational velocity of the rotors. The force and torque produced by one
rotor are given by
F = nn 2,
2
x = ln ,

(4)
(5)

with n being the lift force coefficient and l the drag torque
coefficient. These forces and moments are produced in the z
direction of the rotor's coordinate frames. The forces have a
negative sign because the z axis points downward.
The forces and moments are then rotated into the body
frame according to
F = - nn 2i $ R i e z ,

(6)

2
i Ri z

(7)

Ri i

Ri x i

= - c i ln $ e ,

with c i = 1 for i ! {1, 3, 6} and - 1 otherwise. Furthermore,
B
B

/ B Fi = / R BR $ Ri Fi ,

(8)

/ B x i + / R M ind,i ,
i
i
= / R BR $ R x i + B ri # B Fi ,

(9)

IEEE Robotics & Automation Magazine - December 2018

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - December 2018