Control of Cooperative Multi-Robotic
Systems with Limited Sensor Information
Xiaoming Hu
Optimization and Systems Theory
Royal Institute of Technology
Stockholm, Sweden
ICAI’06, Beijing
Main References
• Iain D. Couzin, Jens Krause, Nigel R. Franks & Simon A. Levin, Effective leadership
•
•
•
•
•
•
and decision-making in animal groups on the move,, Nature, vol. 433, 2005.
Larissa Conradt and Timothy J. Roper, Consensus decision making in animals,
TRENDS in Ecology and Evolution, Vol.20 No.8 August 2005.
Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbor Rules,
Ali Jadbabaie, Jie Lin, and A. Stephen Morse, IEEE TRANSACTIONS ON
AUTOMATIC CONTROL, VOL. 48, NO. 6, JUNE 2003.
Reza Olfati-Saber and Richard M. Murray, Consensus Problems in Networks of Agents
With Switching Topology and Time-Delays, IEEE TRANSACTIONS ON
AUTOMATIC CONTROL, VOL. 49, NO. 9, SEPTEMBER 2004.
Egersted, M. and X. Hu, Formation constrained multi-agent control. IEEE Trans. On
Robotics and Automation 17(6), 2001.
Das, A. K., R. Fierro, V. Kumar, J. P. Ostrowski, J. Spletzer and C. J. Taylor, A vision
based formation control framework. IEEE Transactions on Robotics and
Automation, 2002.
Tove Gustavi, Xiaoming Hu and Maja Karasalo, Multi-Robot Formation Control and
Terrain Servoing with Limited Sensor Information, Proc. of IFAC World Congress,
Prague, 2005.
Outline
•
•
•
•
•
Cooperative control in animal
Consensus problem
Multi-agent Control systems
Formation control
Closing remarks
Cooperative Control in Animals
Individual animals routinely face decisions that are crucial
to them. In social species, however, many of these
decisions need to be made jointly with other group
members because the group will be broken apart unless a
consensus is reached.
Natural Flocks, Herds, and Schools
• Bird in flock must have behavior that allows it to coordinate
movement with flock mates
• Two balanced, opposing behaviors
– Desire to stay close to flock
– Desire to avoid collisions within flock
• Individuals don’t pay much attention to each and every bird in flock
– Bird’s perception of the rest of flock is localized and
filtered
• Itself
• Two or 3 nearest neighbors
• Rest of flock
Different Types of Consensus
Table by L. Conradt and T. Roper,
Trends in Ecology and Evolution, 2005
Consensus Problem
An Example of Consensus Reaching
We begin by considering a flock of N birds.
Each bird flies with the same speed but with
possibly different directions. Namely
vi = (v cosµi ; v sin µi ) T ;
µi
where is the heading of bird i.
Now suppose for each bird, it changes its
heading
by the following model:
_
µi = ui ;
)
µi (t + 1) = µi (t) + ui (t):
An interesting question is how each bird should update its
heading so eventually we have
µ1(t) = ¢¢¢= µN (t);
provided only local information is available.
It turns out to be
1 X
ui (t) =
(µj (t) ¡ µi (t)):
Ni
)
j6
=i
1 XN i
µi (t + 1) =
µj (t)
Ni
j=1
Consensus Problem
• Now we consider a system of N agents:
x_i = ui ; i = 1; ¢¢¢; N
xi
where can be viewed as heading, position or other
quantities.
• We
follows:
F inddefine
u ( t ) the
suchconsensus
t hat as t !problem
1 we has
ave
i
x 1(t) = x 2(t) = ¢¢¢= x N (t);
here we assume t hat agent i can only det ect
relat ive errors x j ¡ x i of it s neighbors, namely j 2 N i .
Similar t o t he ° ocking problem, we consider
a cont roller of t he following type:
X
ui (t) =
ai j (x j ¡ x i );
j 2 Ni
where ai j = aj i are posit ive weight s. If we let
x = (x 1; ¢¢¢; x N ) T , t hen
x_ = ¡ L x;
where
X
L = D ¡ A = di ag(
X
a1j ; ¢¢¢;
j6
=1
aN j ) ¡ [ai j ]:
j6
=N
Now de¯ne
XN X
1
Á(x) = x T L x =
2
ai j (x j ¡ x i ) 2:
i = 1 j 2 Ni
T he consensus problem is solved, namely as
t ! 1 , x 1(t) = ¢¢¢= x N (t), if and only if
Á(x) = 0 ( ) x 1 = x 2 = ¢¢¢= x N :
In fact , in t his case
1 XN
lim x i (t) =
x i (0):
N
t! 1
i= 1
Connection to Graph
• We take graph as a collection of nodes (vertices), edges
that connect the nodes, and weights on the edges,
denoted by G=(V,E,A).
Node i
aik
aij
Node k
Node j
• We say a graph is connected if any two nodes are
connected by edges.
• The consensus problem is solved if and only if the
associated graph is connected.
Effective Leadership in Group
Behavior
• For a large group of animals, suppose a proportion of
the individuals are given information about a desired
direction xd.
• Then those individualsPwould modify their decision by
ui = (1 ¡ wi )
(x j ¡ x i ) + wi (x d ¡ x i )
Zoologists and biologists have found that
• For a given group size the accuracy of group motion (in
•
a preferred direction) increased asymptotically as the
proportion of informed individuals increased.
As the group size became larger this relationship became
increasingly nonlinear , meaning that the larger the
group, the smaller the proportion of informed individuals
needed to guide the group with a given accuracy.
• In a separate study it is found that in swarming honey
•
bees Apis mellifera only about five percent bees (scouts)
are involved in decision making.
Now an interesting question is: as the population tends
to infinity, what happens to the minimum percentage of
the informed individuals for a given accuracy?
Multi-agent Control Systems
• Winning by numbers (networked and robust)
• Distributed sensing and motoring
• Emergence, studied in Computer Science and Biology
Sensor, Actuator, Communication
and Environment Constraints
• Actuator, communication and environment:
– Limited accuracy and response time
– Very limited communication bandwidth
– At least partially unknown environment
• Sensor:
– Enteroception-Inner State
– Proprioception- Position of body and parts
– Extereoception-State of the environment
• Examples: encoders, gyros, force sensors, ultrasonic,
range sensors, and cameras
Vision:
Focal point
Optical axis
We suppose t he point 's posit ion is (x 1; x 2; x 3) in
3D. T hen in t he image plane we have (y1; y2) = (f x 1 ; f x 2 ).
x3
x3
T he relat ive mot ion of t he point (f = 1):
0 _1
x1
@x A
2
x
µ 3¶
y1
y2
0
=
=
1 0
1
0
1
0
¡ !3 !2
x1
v1
@!
0
¡ ! 1A @x 2A + @v2A
3
¡ !
!1
0
x3
v3
µ ¶2
x1
x3
x3 6
= 0;
x
2
x3
For nonlinear systems, the separation principle
does not hold in general.
We have to treat sensing and control in an
integrated fashion.
Integration of Sensing and Control
Consider several robots equipped with directional and range sensors
tracking a moving target:
Formation Control of Mobile Robots
with Limited Sensor Information
• Potential applications:
– Area coverage
– Infrastructure security (line of sight)
– Transportation
– Target tracking
– etc
Formation Control
• Mobile multi-agent systems with limited information from
range sensors.
• Some control algorithms that do not require global
information, and are easy to implement.
• Building blocks for complex formations.
System Model
We consider a mobile syst em t hat consist s of n
robot s. Each of t he robot s is modeled by:
x_i
=
vi cosÁi
y_i =
Á_i =
vi sin Ái
! i ; i = 1; ¢¢¢; n
where vi , Ái and ! i denot e t he speed, rot at ion and
angular velocity of robot i wit h respect t o some ¯xed
coordinat e syst em.
Formation Specification
Serial Formation
A serial formation in essence is to follow
Choose t he reference point (x 0; y0) t o be on t he
robot 's axis of orient at ion at a dist ance d0 from t he
cent er of t he robot :
x0 =
x i + d0 cosÁi
y0 =
yi + d0 sin Ái :
P r op osi t ion. Let vL and µL be t he speed and
orient at ion of t he point (x L ; yL ) t o be t racked by t he
robot . T hen for each robot , as t ! 1 , (x 0(t); y0(t))
converges t o (x L (t); yL (t)) wit h t he following cont rol
vi
!i
=
=
¡ k(d0 ¡ d cos¢ Ái ) + vL cos(µL ¡ Ái )
kd sin ¢ Á + vT sin(µ ¡ Á )
i
i
L
d0
d0
where ¢ Ái (¯i ) is t he relat ive angle t o t he t arget measured by t he robot , d is t he dist ance t o t he t arget and
k is any posit ive const ant .
Parallel Formation
Parallel formation in essence is to lead (predict)
¼. axis of orientation to
Ideally, the angle from the leader's
2
the following robot should be
Since our eventual control objective is to align the middle
points of two robots, the linearization technique used in
the previous section can not be applied here.
We can rewrit e t he syst em as
d_i =
¡ vi cos¯i ¡ vi ¡ 1 cos®i
°_i =
¯_i =
!i¡ !
; i = 2; ¢¢¢; n
vi ¡ 1
¡ !i+
¡
sin ®i :
i
di
di
i¡ 1
vi
sin ¯
Denot e t hat t he act ual dist ance between R i and R i ¡ 1
is di while d0 is t he desired dist ance, ° i = Ái ¡ Ái ¡ 1. ®i
is t he act ual relat ive angle from R i t o t he orient at ion
of R i ¡ 1 and ¯i is t he act ual relat ive angle from t he
orient at ion of R i t o R i ¡ 1.
Problem Formulation
Given v1(t) and ! 1(t), ¯nd control vi (t) and ! i (t)
i = 2; ¢¢¢; n such that for i = 2; ¢¢¢; n
di ! d0; ° i ! 0; ¯i !
¼
as t ! 1 :
2
T here are cont rols in t he form
· ¸
·
¸
vi
vi ¡ 1
=
+ Ci ;
!i
! i¡ 1
for example, for some const ant s k0 > 0; k1 > 0; k2 >
0; k3 > 0
·
¸
k0(di ¡ d0) + k1 cos¯i
Ci =
;
2
k2(di ¡ d0) sin ° i + k3(di ¡ d0)
t hat will st abilize t he parallel format ion locally. Or,
in pract ice
· ¸
µ·
¸
¶
vi
vi ¡ 1 ¡ vi
= k
+ Ci :
!i
! i¡ 1 ¡ ! i
Onboard Sensor Information Based
Control
T heor em . If t he cont rol for Robot i is given by
t he following dynamical syst em,
z_i =
k p ¢ ¯i
vi
=
zi
!
=
¡ ai ° i ¡ bi ¢ ¯i + ci ¢ di + !
i
i¡ 1
;
t hen, for some kp > 0; ai > 0; bi > 0; ci > 0, t he
equilibrium (di = d0; ° i = 0; ¯ = ¯0 = ¼ ¡ ±; zi =
2
vi ¡ 1) is locally exponent ially st able for all su± cient ly
small ± ¸ 0, when vi ¡ 1 is set t o a posit ive const ant
and ! i ¡ 1 is set t o zero.
Closing Remarks
• Cooperative behavior exists in animals
• Decision making with limited information is one of the
•
key issues with multi-agent mobile systems
Robust control with respect to environment uncertainties
and system uncertainties remains to be a research issue.