IEEE Robotics & Automation Magazine - September 2012 - 25

z (Height Above Ground)

pﬃﬃ
Assuming r $ d, we get - $ (1= r ). Thus, Mach
scaling predicts
1

1
a$ ,
d

0.8

1
,
d2

(14)

0.6

while Froude scaling leads to the conclusion

0.4
0.2

a $ 1,

0
1
0.5
0
-0.5
y

-1 -1

-0.5

0.5

(15)

Figure 4. Frame from the Simulink animation of quadrotor
dynamics.

performance control of quadrotor vehicles has been
demonstrated using the simple static thrust model [23],
[24]. The detailed modeling of the blade flapping and
induced drag is provided due to its importance in understanding the state estimation algorithms introduced
later the tutorial.
Size, Weight, and Power (SWAP)
Constraints and Scaling Laws
Reducing the scale of the quadrotor has an interesting
effect on the inertia, payload, and ultimately the maximum
achievable angular and linear acceleration. To gain insight
into scaling, it is useful to develop a simple physics model
to analyze a quadrotor's ability to produce linear and angular accelerations from a hover state.
If the characteristic length is d, the rotor radius r scales
linearly with d. The mass scales as d3 and the moments of
inertia as d 5 . On the other hand, from (3) and (4), it is
clear that the lift or thrust, T, and drag, Q, from the rotors
scale with the square of the rotor speed, -2 . In other
words, T $ -2 d4 and Q $ -2 d 4 , the linear acceleration
a ¼ v_ , which depends on the thrust and mass, and
_ which depends on
the angular acceleration a ¼ X,
thrust, drag, the moment arm, and the moments of inertia, scale as
-2 d 4
a $ 3 ¼ -2 d,
d

1
a$ :
d

-2 d 5
a $ 5 ¼ -2 :
d

To explore the scaling of rotor speed with length, it is
useful to adopt the two commonly accepted approaches
to study scaling in aerial vehicles [9]. Mach scaling is
used for compressible flows and essentially assumes that
the blade tip speed, vb , is a constant leading to
- $ (1=r). Froude scaling is used for incompressible
flows and assumes that, for similar aircraft configurations, the Froude number, (vb2 =dg) ¼ (-2 r 2 =dg),
is constant. Here, g is the acceleration due to gravity.

Of course, Froude or Mach number similitudes take
neither motor characteristics nor battery properties into
account. While motor torque increases with length, the
operating speed for the rotors is determined by matching
the torque-speed characteristics of the motor to the drag
versus speed characteristics of the rotors. Further, the
motor torque depends on the ability of the battery to
source the required current. All these variables are tightly
coupled for smaller designs since there are fewer choices
available at smaller length scales. Finally, as discussed in
the previous subsection, the assumption that rotor blades
are rigid may be wrong. Further, the aerodynamics of the
blades may be different for blade designs optimized for
smaller helicopters and the quadratic scaling of the lift with
speed may not be accurate.
In spite of the simplifications in the above similitude
analysis, the key insight from both Froude and Mach number similitudes is that smaller quadrotors can produce
faster angular accelerations while the linear acceleration is
at worst unaffected by scaling. Thus, smaller quadrotors
are more agile, a fact that is easily validated from experiments conducted with the Ascending Technologies Pelican
quadrotor [10] (approximately 2 kg gross weight when
equipped with sensors, 0.75 m diameter, and 5,400 r/min
nominal rotor speed at hover), the Ascending Technologies Hummingbird quadrotor [11] (approximately 500 g
gross weight, 0.5 m diameter, and 5,000 r/min nominal
rotor speed at hover), and laboratory experimental prototypes developed at GRASP laboratory at the University of
Pennsylvania (approx. 75 g gross weight, 0.21 m diameter,
and approximately 9,000 r/min nominal rotor speed).
Estimating the Vehicle State
The key state estimates required for the control of a quadrotor are its height, attitude, angular velocity, and linear
velocity. Of these states, the attitude and angular velocity
are the most important as they are the primary variables
used in attitude control of the vehicle. The most basic
instrumentation carried by any quadrotor is an inertial
measurement unit (IMU) often augmented by some form
of height measurement, either acoustic, infrared, barometric, or laser based. Many robotics applications require
more sophisticated sensor suites such as VICON systems,
global positioning system (GPS), camera, Kinect, or scanning laser rangefinder.
SEPTEMBER 2012

IEEE ROBOTICS & AUTOMATION MAGAZINE

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - September 2012