IEEE Robotics & Automation Magazine - September 2010 - 62

One quadrotor can influence the
dynamic behavior of neighboring
quadrotors because of the
downwash from its rotors.
can easily switch between the quadrotor simulator and the
actual robot. The command messages to which the simulator
subscribes are identical to the messages sent to the onboard
controller. This feature allows for quick transitions from simulations to experiments.
The quadrotor simulator provides state estimates and emulates the performance and time delays that appear in the actual
system. The core of the quadrotor simulator is the numerical
integration of the 16-state quadrotor model detailed in the
"Modeling" section. A limitation is that we do not simulate aerodynamic effects between robots discussed in the "Aerodynamic
Interactions" section, and so we characterize these effects experimentally and assume conservative approximations in our algorithms based on these experiments.

Group Behaviors
Aerodynamic Interactions
To further cooperation among robots, it is essential to have
the robots operate in close proximity and maintain constraints
on separation. Thus, formation flight is a key component technology. However, for 3-D flight, the interaction between MAVs
goes beyond constraints on relative positions because of specifications on the task and collision avoidance. One quadrotor
can influence the dynamic behavior of neighboring quadrotors
because of the downwash from its rotors. This effect has consequences in many applications including formation control,
because additional requirements must be imposed on the relative poses of quadrotors. To design algorithms for near-hover

3.5

z (m)

2.5

−0.1

2
−0.15
1.5
−0.2

1
0

0.5

1 1.5
x (m)
(a)

z (m)

−0.05

4
3.5
3
2.5
2
1.5
1
−2
x (m)

controllers that consider these effects, we need to first quantify
the extent of the influence. We designed two experiments to
measure this disturbance. The purpose of the first experiment
is to characterize the aerodynamic effect of a quadrotor hovering
above another quadrotor. In the experimental results shown in
Figure 5(a), the first quadrotor, A, uses a hover controller to maintain the position (À0:03, À0:03, 40) m. A second quadrotor,
B, uses a hover control to execute a sequence of waypoints on
a 20 3 20 grid on the xW À zW plane, dwelling 5 s at each
waypoint. At each waypoint, the position data are recorded
for a 5-s interval after B's speed falls below the threshold of
0:005 m=s. In Figure 5(a), we show the z displacement error
in gray scale with the steady-state xz displacement error vector. It is apparent that A's influence is mostly concentrated to a
cylindrical region with a radius of approximately 0:5 m extending to a height of 1:5 m below the quadrotor. This cylindrical
region bounds the volume where the z displacement error is
greater than 5 cm. In this cylindrical region, the x displacement error for B ranges from À0:12 to À0:37 m and z errors
from À0:05 to À0:22 m. Although the severe effects are contained in this region, it is clear from the data that there is a
larger region in which B is affected to a lesser extent.
The second experiment characterizes the errors in trajectories of B induced by A. With the same position for A as
before, B executes the trajectory controller with a desired
speed of 1:0 m=s along a trajectory parallel to the x axis from
x ¼ À1:5 to x ¼ 1:5 m, with the y and z coordinates being
chosen from a 5 3 5 grid on the yW À zW plane, as shown in
Figure 5(b). Once again, we see a trend similar to the one in
Figure 5(a). Trajectories, that are closer to A, show the largest
deviations from the straight-line trajectories. We observe that
the trajectories close to A are pushed down and pulled into the
downwash of A.
Control of Aerial Robot Ensembles
In this section, we present the experimental results with a team
of MAVs controlling to a desired ensemble pose and shape.
We begin by briefly detailing the theory
behind the control law. We provide the
description of the experiment design and
review the results. The underlying theory
of this section is provided in [19].
Statistical Representation and
Control of Aerial Robot Formations

0
2

−2

−1.5

−1 −0.5
y (m)
(b)

0.5

Figure 5. (a) Displacement error in the xW À zW plane for a quadrotor hovering
below a quadrotor hovering at (À0.03, À0.03, 40) m. (b) The trajectories for
Quadrotor B moving parallel to the xW axis at a desired speed of 1.0 m/s in the
proximity of Quadrotor A hovering at (À0.03, À0.03, 4) m [location marked by the
black dot].
62

IEEE Robotics & Automation Magazine

We consider a team of N-point robots in
3-D space, with the position of the ith
robot denoted as qi 2 R3 . We wish to
control the ensemble pose and shape,
which is decomposed into a shape space,
S, and a lie group, G, which in our case
is SE(3). Define an abstract space, M ,
whose dimension is smaller and independent of the number of robots by a smooth,
differentiable map
/ : Q ! M,

/(q) ¼ x,

(10)

SEPTEMBER 2010

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - September 2010