Automatica 43 (2007) 1954 – 1960
www.elsevier.com/locate/automatica
Brief paper
Generalization of nonlinear cyclic pursuit夡
A. Sinha ∗ , D. Ghose
GCDSL, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India
Received 16 July 2006; received in revised form 10 December 2006; accepted 16 March 2007
Available online 23 August 2007
Abstract
In this paper, some generalizations of the problem of formation of a group of autonomous mobile agents under nonlinear cyclic pursuit is
studied. Cyclic pursuit is a simple distributed control law, in which the agent i pursues agent i + 1 modulo n. Each agent is subjected to a
nonholonomic constraint. A necessary condition for equilibrium formation to occur among a group of agents with different speeds and different
controller gains is obtained. These results generalize equal speed and equal controller gain results available in the literature.
䉷 2007 Elsevier Ltd. All rights reserved.
Keywords: Autonomous system; Multi-vehicle formations; Cooperative control; Pursuit problems; Decentralized control; Nonlinear dynamical systems
1. Introduction
This paper deals with the problem of cyclic pursuit in a
multi-vehicle system. Multi-vehicle systems are groups of autonomous mobile agents used for search and surveillance tasks,
rescue missions, space and oceanic explorations, and other automated collaborative operations. Cyclic pursuit uses simple local interaction between these vehicles to obtain desired global
behavior.
The pursuit strategies are designed to mimic the behavior
of biological organisms like dogs, birds, ants or beetles. They
are commonly referred to as the ‘bugs’ problem. Bruckstein,
Cohen, and Efrat (1991) modeled the behavior of ants, crickets
and frogs with continuous and discrete pursuit laws and examined the possible evolution of global behavior such as the convergence to a point, collision, limit points or periodic motion.
Bruckstein (1993) describes the stability and convergence of a
group of ants in linear and cyclic pursuit. Convergence to a point
in linear pursuit is the starting point to the analysis of achievable
global formation among a group of autonomous mobile agents
as discussed in Lin, Broucke, and Francis (2004), which also
夡 This paper was not presented at any IFAC meeting. This paper was
recommended for publication in revised form by Associate Editor Henk
Nijmeijer under the direction of Editor Hassan Khalil.
∗ Corresponding author. Tel.: +91 80 2293 3023; fax: +91 80 2360 0134.
E-mail addresses: [email protected] (A. Sinha),
[email protected] (D. Ghose).
0005-1098/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.automatica.2007.03.024
deals with the evolution of the formation of these agents with
respect to the possibility of collision. Kinematics of agents with
single holonomic constraint is discussed in Marshall, Broucke,
and Francis (2004, 2006). The equilibrium and stability of identical agents with this motion constraint is dealt within these
papers. The mathematics governing cyclic pursuit are studied in Richardson (2001a,b), Behroozim and Gagnon (1979),
Klamkin and Newman (1971), and Bernhard (1959). A generalization of the linear cyclic pursuit problem has been presented
in Sinha and Ghose (2006).
This work has been inspired by the problem addressed in
Marshall et al. (2004), which considers n identical autonomous
mobile agents in cyclic pursuit with nonholonomic constraint.
The agents are assumed to have same speed and controller
gains. In our paper, a more general case is discussed where the
speeds and controller gains for different agents may vary, thus
giving rise to a heterogenous system of agents. In such a case,
determination of conditions under which the system converges
to an equilibrium becomes more complicated. In this paper, we
provide a necessary condition for the existence of equilibrium
for the general case. Some preliminary results on this were
earlier presented in Sinha and Ghose (2005).
We would like to point out that, in this paper, we do not
address the issue of stability of the equilibrium points. Stability issues in the case of equal gain and equal speed of all the
agents are discussed in Marshall et al. (2004). The system is
linearized about an equilibrium point and the local stability is
A. Sinha, D. Ghose / Automatica 43 (2007) 1954 – 1960
determined from the eigenvalues of the linear system. Even
for the equal gain and equal speed case as in Marshall et al.
(2004), the stability analysis is rather involved. It is expected
that stability analysis for a general system with heterogenous
gains and speeds will be even more complicated and is beyond
the scope of this short paper.
2. Equations of cyclic pursuit
1955
3. Analysis for possible formations
For a system of n agents, with kinematics (3)–(5), equilibrium is said to have been achieved if the relative position of
the agents do not change with time. Therefore, at equilibrium
ṙi = 0
and
˙ i = ˙ i+1 ∀i.
(6)
In addition, we assume that at equilibrium
˙ i = 0
Nonlinear cyclic pursuit is an extension of the classical n
bugs problem or linear pursuit. Each bug is modeled as a point
mass autonomous mobile agent. The agents are ordered from
1 to n, and the agent i follows agent i + 1 modulo n.
Each agent has a constant velocity Vi , and orientation i
(variable), with respect to a fixed reference (Fig. 1). The distance between the ith and (i + 1)th agent is ri and the angle
from the reference to the line of sight (LOS) from agents i to
i + 1 is given by i . The control input to the ith agent is the
lateral acceleration ai which is given as
which is similar to the equilibrium condition considered in
Marshall et al. (2004). For a given system, there can be more
than one equilibrium point depending on the initial conditions.
In this paper, we present and analyze a necessary condition for
equilibrium in the heterogenous gain and speed case. Assuming
that the n agents eventually converge to a equilibrium formation,
we have the following theorem.
ai = ki i ,
Theorem 1. At equilibrium, a system of n agents, with kinematics (3)–(5), move in concentric circles with equal angular
velocities.
(1)
where (with reference to Fig. 2)
0
if 0 i i ,
i + i − i =
2 otherwise.
(2)
Here, ki is the controller gain. The lateral acceleration, so
defined, guarantees that all the agents move in the counterclockwise direction. Thus, the kinematics of the ith agent is
given as follows:
r˙i = Vi+1 cos(i+1 − i ) − Vi cos(i − i ),
(3)
ri ˙i = Vi+1 sin(i+1 − i ) − Vi sin(i − i ),
(4)
˙i = ai /Vi = ki i /Vi .
(5)
Vi+1
αi+1
Ref
Pi+1
θi
Vi
˙ = 0 ⇒ = constant ⇒ ai = constant. An agent
Proof. i
i
i, having constant speed Vi and constant lateral acceleration
ai , will move in a circular trajectory. Considering Fig. 3, let
the position of the ith agent be Pi and the center of the circle traversed by it be O. Let, Ri = d where d is a constant.
Again, i = constant implies OP i A = (90 − i ) is a constant.
Therefore, the position of (i + 1)th agent should be on the line
Pi A or its extension. The point at which (i + 1)th agent lies on
Pi A is determined from ṙi = 0 which implies that ri =constant.
Therefore, $OP i Pi+1 forms a rigid triangle. Thus, agent i + 1
also has the center of its circular trajectory at O. Again, since
the configuration of the $OP i Pi+1 remains rigid at equilibrium, we have i = i+1 , where i is the rate of rotation of
OP i . Hence, all agents move in concentric circles with equal
angular velocity. As pointed out before, there could be zero, one or several
equilibrium points for a given system. The necessary and sufficient condition for equilibrium point to exist or for stability of
ri
Pi
(7)
τ
Ref
αi
Pi
φi
Fig. 1. Basic formation geometry.
Vi
Ri
π/2–φi
O
ri
Pi+1
Pi+1
Ri+1
Vi
αi
Pi
φi
θi
Ref
θi
Pi
φi
αi
Vi
π/2–βi+1
Pi+1
βi+1
Ref
Fig. 2. Representation of the angles with respect to a fixed reference.
φi+1
Vi+1
A
Fig. 3. Multi-vehicle formation with circular trajectory.
1956
A. Sinha, D. Ghose / Automatica 43 (2007) 1954 – 1960
βi
Vi
φi
Pi
Vi+1
Pi+2
Ref
m=0
LO
S
ri–1
ri
Pi+1
Ref
φi–1
cos–1(γi+2)
Vi–1
Pi–1
cos–1(γi+1)
S
LO
equilibrium points are bound to be complicated for the general
case. In the following, we will derive only a necessary condition
for the existence of equilibrium base upon the geometry of the
final configuration of equilibrium.
Let the radius of the circle traversed by the first agent at
equilibrium be R1 = . Since the angular velocity is the same
for all the agents, we have
i = i+1 ⇒ Vi /Ri = Vi+1 /Ri+1 ⇒ Ri = Vi /V1 .
(8)
Now,
ai = Vi2 /Ri = ki i ⇒ i = (Vi V1 )/(ki ).
(9)
Again, from OP i Pi+1 ,
(10)
i+1 = cos−1 [(Vi /Vi+1 ) cos{(V1 Vi )/(ki )}].
(11)
i=1
ki Fig. 5. Representation of the range of in polar coordinate.
For a given q, Eq. (13) is a function of only. Therefore,
the values of that satisfies Eq. (13) gives the radius of the
circle of the first agent. The radius for the other agents can be
obtained from Eq. (8). Let us denote the left-hand side (LHS)
of Eq. (13) by f ().
Theorem 2 (Necessary condition for equilibrium). Consider
an n agent system with kinematics given by (3)–(5). A necessary
condition for equilibrium is
+ cos−1
1
Vi−1 V1
cos
i
ki−1 i∈{j :Vj >Vj +1 }
ai min
i∈{j :Vj >Vj +1 }
, (12)
where the subscript indices are modulo n.
Now, consider any n-sided polygon (not necessary regular),
a part of which is shown in Fig. 4. Each node i represents the
position of the agent i and the vector from that node represents
the velocity of the ith agent. Thus the angle (i + i ) is measured counter-clockwise from the extension of the line Pi−1 Pi
to the line Pi Pi+1 , as shown in Fig. 4. Therefore,
considering all
possible polygonal topologies for a given n, (i + i ) = 2q,
where, q = 1, 2, . . . , (n − 1). The q considered here is similar
to the variable d used in Definition 2 in Marshall et al. (2004).
Thus,
n Vi V 1
Vi−1 V1
−1 1
cos
+ cos
= 2q.
(13)
ki i
ki−1 i=1
Hence, if a system of n vehicles with arbitrary speeds and
controller gains ki attains equilibrium, Eq. (13) must be satisfied
for some and for some q = 1, 2, . . . , n − 1. We will refine
this condition further by analyzing Eq. (13).
bi ,
(14)
where
ai = [m + cos−1 (i+1 )]
Let the velocity ratio Vi+1 /Vi = i+1 . Therefore,
n V i V1
Vi
Pi
m=0
max
Ri+1 /Ri = sin(/2 − i )/ sin(/2 − i+1 ),
i=1
cos–1(γi+2)
π
–
2
cos–1(γi+1)
(i + i ) =
m=1
m=1
Vi+1
Fig. 4. Angle calculation for a general polygon of n sides.
n
Pi+1
Ref
φi+1
βi+1
π
–
2
ki
,
Vi V 1
bi = [(m + 1) − cos−1 (i+1 )]
m = 0, 1,
ki
,
Vi V1
m = 0, 1.
(15)
(16)
Proof. Eq. (13) has a solution if and only if the argument of
the cos−1 term in (13) is in [−1, 1], i.e.,
| cos{(Vi V1 )/(ki )}| i+1
∀i.
(17)
Let X1 = {i : Vi > Vi+1 } and X2 = {i : Vi Vi+1 }. Note that
both X1 and X2 are nonempty sets if not all the Vi ’s are same.
For all i ∈ X2 , (17) is always satisfied irrespective of the values
of . For a given i ∈ X1 , the range of values that can take is
given by Ri ,
Ri = { : ai 1/ bi },
(18)
where ai and bi are given by (15) and (16) and m =
0, ±1, ±2, . . . . Since, (Vi V1 )/(ki i ) = i , from Fig. 5, it is
evident that the possible values of m are 0 or 1. Note that
Ri has a center at (ki /Vi V1 )((2m + 1/2)) and a spread of
( − 2(ki /Vi V1 )cos−1 (i+1 )). Since both the center and spread
of Ri are functions of Vi ,Vi+1 and ki , they are different for
different i’s. Similarly, the distance between the centers also
vary with i. The range of i is shown in Fig. 6.
A. Sinha, D. Ghose / Automatica 43 (2007) 1954 – 1960
Now, there will exist some value of that will belong to Ri
given in Eq. (18) for all i ∈ X1 if
Ri = ∅.
(19)
i∈X1
1957
we do not discuss the stability of these equilibrium points in
this paper.
Special case: We may consider a special case in which Vi =V
and ki = k, for all i, which is the assumption made in Marshall
et al. (2004). Then (13) reduces to
This implies that for equilibrium to exist, (14) should be satisfied. Also, we can find a that will satisfy
(2V 2 n)/(k) = 2q ⇒ = (V 2 n)/(qk).
max ai (1/) min bi
Thus, all the agents will move in a fixed circle of radius
(V 2 n)/(qk) at equilibrium. This is the same result as that
obtained in Marshall et al. (2004). The difference in the exact
expression is due to the choice of the controller gain which,
in Marshall et al. (2004), is defined as kV. Again, from (9),
= (q)/n, which is the same as in Marshall et al. (2004).
Note that in the equal speed case, the bounds on 1/, given in
Theorem 2, do not exist.
∀i ∈ X1 .
(20)
It is interesting to speculate what happens when max ai >
min bi . From Fig. 5, at equilibrium each velocity vector Vi
should be within the shaded cone shown in the figure. Now,
we can write i = (k1 Vi /ki V1 )1 . Hence, all the cones can be
translated to the point P1 where each cone represents the bound
within which V1 should lie for ṙi =0, ∀i. When max ai > min bi ,
the intersection of the cones is empty and thus, there does not
exist a feasible direction for the velocity V1 at equilibrium to
satisfy ṙi = 0, ∀i. In other words, for some i, there will be no i
that can meet the requirement of the LOS speed components
to match. Hence, there will be no equilibrium.
The set of all values of obtained from (20) need not satisfy
(13). Only that which satisfies (13), for a given value of q,
gives the radius of the circle at equilibrium.
The equilibrium may or may not exist for a given q. Also,
equilibrium may occur for more than one value of q. Hence, the
system can attain any one of the equilibrium states. However,
^
a
^
^
a
bi
i
^
i
bi
φ
^
a
j
π
–
2
^
^
a
bj
aj
bj
j
ai
3π
––
2
^
bj
bi
ρ
kj
–
––
–– π
vjv1 2
k1 π
––
–– –
viv1 2
Fig. 6. A representation of the ranges of and for all agents.
(21)
4. Simulation results
Case 1: Consider a system of n = 5 agents. The velocities of
the agents are V = [25 20 15 10 5] while the gains are taken
to be the same for all, i.e., ki = 5, ∀i. So, X1 = {1 2 3 4} and
X2 = {5}.
The range of Ri are shown in Table 1. For both m = 0
and 1, i∈X1 Ri = ∅. The simulation result (Fig. 7) also shows
that the system does not have an equilibrium.
Case 2: Consider the previous example with all parameters
same except for V5 = 6. Here, X1 and X2 are same as in Case
1. But now, for m = 0, R1 , . . . , R4 have nonempty intersection. The ranges of for which f () exist is 50.0 53.9
as shown in Table 1. We plot f () for this range of in
Fig. 8. There is no value of and q that satisfies (13). The
simulation result given in Fig. 9 shows that this case indeed
does not have an equilibrium. This shows that the existence
of that satisfies (20) is not sufficient for (13) to have a
solution.
Case 3: Again, consider the previous example, but now use
V5 = 7, so that X1 and X2 are the same as before. The overlapping region of for m = 0 is 50.0 62.9. The range of
and f () are shown in Table 1 and Fig. 8, respectively. At
= 61.7, (13) is satisfied. This gives the radius of the other
agents (from (8)) as = [61.7 49.3 37.0 27.7 17.3]. The simulation, shown in Fig. 10, confirms the radius of the circles
evaluated analytically.
Table 1
The range of Ri
Case 1
Case 2
Case 3
Case 4
Case 5
[50.0
[41.3
[32.6
[23.9
[50.0
[41.3
[32.6
[22.6
[50.0
[41.3
[32.6
[21.3
[50.0 194.2]
[41.3 138.4]
[32.6 89.2]
–
[29.7
[27.0
[24.3
[21.5
[22.2 33.0]
[17.9 25.9]
[13.8 18.8]
–
[13.7 22.3]
[12.4 19.9]
[11.1 17.5]
[9.8 15.2]
m=0
R1
R2
R3
R4
194.2]
138.4]
89.2]
47.7]
194.2]
138.4]
89.2]
53.9]
194.2]
138.4]
89.2]
62.9]
177.4]
151.3]
126.6]
103.5]
m=1
R1
R2
R3
R4
[22.2 33.0]
[17.9 25.9]
[13.8 18.8]
[9.5 11.9]
[22.2 33.0]
[17.9 25.9]
[13.8 18.8]
[9.3 12.3]
[22.2 33.0]
[17.9 25.9]
[13.8 18.8]
[9.1 12.7]
1958
A. Sinha, D. Ghose / Automatica 43 (2007) 1954 – 1960
80
80
60
60
40
40
20
20
0
0
–20
–20
–40
–40
–60
–40
–20
0
20
40
60
80
–60
100
Fig. 7. Trajectories of n = 5 agents for Case 1 (•—initial position, —final
position of the UAVs).
25
–40
–20
0
20
40
60
80
100
Fig. 10. Trajectories of n = 5 agents for Case 3 (•—initial position, —final
position of the UAVs).
100
m=0
20
15
6π
f(ρ)
Case 2
Case 4
Case 3
10
4π
2π
5
80
60
40
20
0
–20
–40
0
40
50
60
70
ρ
80
90
100
–60
–100 –80 –60 –40 –20
0
20
40
60
80 100
Fig. 8. The roots of Eq. (13) for Cases 2–4.
Fig. 11. Trajectories of n=5 agents for Case 4 (stable equilibrium) (•—initial
position, —final position of the UAVs).
80
60
40
20
0
–20
–40
–60
–40
–20
0
20
40
60
80
100
Fig. 9. Trajectories of n = 5 agents for Case 2 (•—initial position, —final
position of the UAVs).
Case 4: When the necessary condition (14) is satisfied, (13)
can have more than one solution depending on the value of
q. For example, again consider the previous case (Case 1) but
now V5 = 22, so that X1 = {1 2 3} and X2 = {4, 5}. The range
of values of is shown in Table 1 and f () is plotted in
Fig. 8. From the figure it can be seen that more than one solution
( = 50.9 and 74.0) exists for this particular problem. However,
=50.9 is unstable. Simulation shows that even if we start from
a equilibrium point corresponding to =50.9, due to numerical
inaccuracies, it slowly migrates to the other equilibrium point
at = 74. This is shown in Figs. 11 and 12.
Case 5: However, if more than one equilibrium point is stable, the system converges to one of the equilibrium formations
depending on the initial configuration. For example, consider
V = [20 18 16 14 12] and k = [5 5 5 5 5]. For this case the
range of and f () are shown in Table 1 and Fig. 13, respectively. The final formation for two different initial conditions
is shown in Fig. 14. The dependence of the final configuration on initial configuration has also been observed in Marshall
et al. (2004). We conjecture that at equilibrium, all agents
move
in a counter-clockwise direction, therefore, the condition
i∈X1 Ri = ∅, from Eq. (19), occurs only at m = 0. This simplifies the effort required to find the overlapping region.
A. Sinha, D. Ghose / Automatica 43 (2007) 1954 – 1960
t=0
150
150
t = 240 sec
150
100
100
100
50
50
50
0
0
0
–100
0
100
–100
0
100
1959
t = 475 sec
–100
0
100
Fig. 12. Trajectories of n = 5 agents showing unstable equilibrium: (a) initial equilibrium configuration, (b) intermediate configuration, (c) final stable
configuration corresponding to q = 2.
25
20
6π
15
f(ρ)
4π
10
2π
5
0
0
20
40
60
80
100
120
ρ
Fig. 13. Roots of Eq. (13) for Case 5.
80
200
60
150
40
100
20
50
0
0
–20
–200
–150
–100
–50
0
50
100
–40
–20
0
20
40
60
80
Fig. 14. Trajectories of n = 5 agents for Case 5 for different initial conditions (•—initial position, —final position of the UAVs).
5. Conclusions
Nonlinear cyclic pursuit strategies have recently been of
much interest among researchers. In this paper, the formation
of a group of heterogenous mobile agents (with different speeds
and controller gains) is studied. A necessary condition and its
refinements are presented that generalizes several of the results
given in Marshall et al. (2004) and gives analytical expressions
for computing the radius of the circles and relative angles between agents at equilibrium.
1960
A. Sinha, D. Ghose / Automatica 43 (2007) 1954 – 1960
References
Behroozim, F., & Gagnon, R. (1979). Cyclic pursuit in a plane. Journal of
Mathematical Physics, 20, 2212–2216.
Bernhard, A. (1959). Polygons of pursuit. Scripta Mathematica, 24, 23–50.
Bruckstein, A. M. (1993). Why the ant trails look so straight and nice. The
Mathematical Intelligencer, 15(2), 59–62.
Bruckstein, A. M., Cohen, N., & Efrat, A. (1991). Ants, crickets and frogs
in cyclic pursuit. Center for Intelligence Systems, Technical Report 9105,
Technion-Israel Institute of Technology, Haifa, Israel.
Klamkin, M. S., & Newman, D. J. (1971). Cyclic pursuit or “the three bugs
problem”. The American Mathematical Monthly, 78(6), 631–639.
Lin, Z., Broucke, M., & Francis, B. (2004). Local control strategies for groups
of mobile autonomous agents. IEEE Transactions on Automatic Control,
49(4), 622–629.
Marshall, J. A., Broucke, M. E., & Francis, B. A. (2004). Formations of
vehicles in cyclic pursuit. IEEE Transactions on Automatic Control, 49(11),
1963–1974.
Marshall, J. A., Broucke, M. E., & Francis, B. A. (2006). Pursuit formations
of unicycles. Automatica, 42(1), 3–12.
Richardson, T. J. (2001a). Stable polygons of cyclic pursuit. Annals of
Mathematical and Artificial Intelligence, 31, 147–172.
Richardson, T. J. (2001b). Non-mutual captures in cyclic pursuits. Annals of
Mathematical and Artificial Intelligence, 31, 127–146.
Sinha, A., & Ghose, D. (2005). Generalization of cyclic pursuit problem. In
Proceedings of the American control conference (pp. 4997–5002). Portland,
USA.
Sinha, A., & Ghose, D. (2006). Generalization of linear cyclic pursuit
with application to rendezvous of multiple autonomous agents. IEEE
Transactions on Automatic Control, 51(11), 1819–1824.
A. Sinha received her Bachelor’s Degree in
Electrical Engineering from Jadavpur University, Kolkata, India and M.Tech. from Indian Institute of Technology, Kanpur, India. At present
she is a graduate student at the Department of
Aerospace Engineering, Indian Institute of Science, Bangalore, India. Her research interests include cooperative control of autonomous agents,
team theory, and game theory.
D. Ghose is a Professor in the Department of
Aerospace Engineering at the Indian Institute of
Science, Bangalore, India. He obtained a B.Sc.
(Engg.) degree from the National Institute of
Technology (formerly the Regional Engineering
College), Rourkela, India, in 1982, and an M.E.
and a Ph.D. degree, from Indian Institute of Science, Bangalore, in 1984 and 1990, respectively.
His research interests are in guidance and control of aerospace vehicles, collective robotics,
multiple agent decision-making, distributed
decision-making systems, and scheduling
problems in distributed computing systems. He is an author of the book
Scheduling Divisible Loads in Parallel and Distributed Systems published
by the IEEE Computer Society Press (presently John Wiley). He is in the
editorial board of the IEEE Transactions on Systems, Man, and Cybernetics,
Part A: Systems and Humans, and the IEEE Transactions on Automation
Science and Engineering. He has held visiting positions at the University of
California at Los Angeles and several other universities. He is an elected
fellow of the Indian National Academy of Engineering.