IEEE Robotics & Automation Magazine - September 2010 - 63

where / is a mapping of the higher-dimensional state space of
the team of robots q 2 Q to the lower-dimensional abstract
state x 2 M .
The abstract description of the team of robots, x, is given by
the position and orientation of the team, g, and the shape s. Therefore, / defines a transformation from the space of Q 2 R3N
to the lower and fixed dimensional space x 2 M , where M 2 R9
is defined by the six-dimensional pose and 3-D shape of
the formation.
We define the mean of the group by
l¼

N
1X
qi ,
N i¼1

(11)

and the orientation to be such that the coordinates of the robots
in the local frame, pi ¼ ½xi , yi , zi ŠT , satisfy
N
X

xi yi ¼

i¼1

N
X

xi zi ¼

N
X

i¼1

yi zi ¼ 0:

i¼1

Here, the position of each robot is pi ¼ RT (qi À l) with
respect to the frame fixed to the group of robots. The shape
space, s ¼ (s1 , s2 , s3 ), is defined by
s1 ¼ jI 11 ,
where j > 0 and
I¼

N
X
i¼1

s2 ¼ jI 22 ,
2

I 11
pi pTi ¼ 4 0
0

s3 ¼ jI 33 ,

(12)

3
0
0 5:
I 33

(13)

0
I 22
0

Choosing j ¼ (1=N À 1) gives the shape variables a geometric interpretation as the semimajor and semiminor axes for a
concentration ellipse for a group of robots whose coordinates
are chosen to satisfy a normal distribution.
Using the natural kinetic energy metric on Q, it is possible to
derive the optimal velocity (tangent vector) at any point q 2 Q
for a desired x_ at the corresponding point x ¼ /(q) 2 M . It
was shown in [19] that this input, u? , for the system can be
found by considering the time derivative of the transformation
described by (10),
d/_q ¼ x:
_

(14)

Thus, the minimum-energy solution satisfying (14) is obtained
using the Moore-Penrose inverse
u? ¼ d/T (d/d/T )À1 x:
_

(15)

In [19], the individual control law for each agent is found
to be
s2 À s3 2
s3 À s1 1
T (qi À l)x1 þ
T (qi À l)x2
u?i ¼ q_ i ¼ l_ þ
s2 þ s3 3
s1 þ s3 3
3
X
s1 À s2 1
s_k
þ
T2 (qi À l)x3 þ
Hk (qi À l),
(16)
s1 þ s2
2s
k¼1 k
SEPTEMBER 2010

The quadrotor simulator provides
state estimates and emulates the
performance and time delays that
appear in the actual system.
with (x1 , x2 , x3 ) as the angular velocity change of the ensemble shape and
Hji ¼ Rei eTj RT ,

j

Tji ¼ Hji þ Hi ,

where ½e1 e2 e3 Š ¼ I3 and i, j ¼ f1, 2, 3g.
Note that the controller for each agent is only dependent
on its state qi and the abstract state x. A global observer that is
able to acquire knowledge of the state and provide the values
of the abstract state is sufficient to control the entire system.
We will use the Vicon system in the experiments as the
global observer.
Introducing Interrobot Collision Avoidance
and Aerodynamic Interaction Effects

From the "Aerodynamic Interactions" section, we see that the
interrobot aerodynamic effects are considerable as the robots
get closer together, but it may be avoided by ensuring that the
robots maintain at least a distance of separation of 0:5 m in the
x and y directions and 1:5 m in the z direction. Therefore, we
approximate these regions as cylinders with a radius of 0:5 m
and height of 1:5 m centered about qi . Assuming that the
robots start outside the cylindrical region of other robots, we
ensure that robots avoid collisions and reduce aerodynamic
interactions by requiring that
(qi À qj ) Á (_qi À q_ j ) ! 0,

(17)

for all robots, qj , that lie on the boundary of the cylinder surrounding qi .
Monotonic Convergence with Interactions

In the absence of collisions and aerodynamic interactions, the
easiest way to guarantee convergence to an abstract state xdes is
to require the error, ~x, to converge exponentially to zero:
x_ ¼ K ~x,
where K is any positive-definite matrix.
We relax the requirement of exponential convergence to
an abstract state and instead of insisting on the minimumenergy solution (16), find the solution closest to the minimumenergy solution satisfying (17). Additionally, we require that the
error in the abstract state decrease monotonically
x~T K x_ ! 0:

(18)

It is shown in [19] that a sufficient condition to satisfy this
monotonic convergence condition is to require that each robot
select inputs that satisfy
IEEE Robotics & Automation Magazine

63



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

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