Nonlinear dynamics of unicycles in

leader-follower formation

Siming Zhao, Abhishek Halder, Tamas Kalmar-Nagy∗

February 13, 2009

Abstract

In this paper, dynamical analysis is presented for a group of unicycles

in leader-follower formation. The equilibrium formations were character-

ized along with the local stability analysis. It was demonstrated that

with the variation in control gain, the collective dynamics might undergo

Andronov-Hopf and Fold-Hopf bifurcations. An increase in the number of

unicycles increase the vigor of quasi-periodicity in the regime of Andronov-

Hopf Bifurcation and heteroclinic bursts between quasi-periodic and chaotic

behavior in the regime of Fold-Hopf bifurcation. Numerical simulations

also suggest the occurrence of global bifurcation involving the destruction

of heteroclinic orbit.

1 Introduction

Equilibrium formations for nonholonomic systems have been an active area ofresearch in recent times among many disciplines like biological sciences [1, 2,3], computer graphics [4] and systems engineering [5, 6, 7, 8]. One particularproblem studied in this context has been the consensus seeking [9] or the state

agreement problem [10] which deals with designing feedback controllers to makemultiple agents converge to a common configuration in the global coordinates. Aspecial case to this is the rendezvous problem [11, 12] where the agents convergeat a single location.

In addition to the stability and control aspects, considerable efforts havealso been put in effective modeling of the nonholonomic systems to make theanalysis tractable. Starting from the n-bug problem in mathematics [13], theself-propelled planar particles were later [14, 15] replaced by wheeled mobileagents with single nonholonomic constraint i.e. unicycles. Lie group formulation[16] and oscillator models [17] have been attempted for dynamic modeling ofsuch agents. In particular, Klein and Morgansen [18] extended the oscillatormodel to account for the intermediate centroid velocity of the unicycles to maketrajectory tracking possible.

Several researchers ([14], [15], [19], [20], [21]) proposed laws for designingcontrol strategies of such nonholonomic vehicles. One possible approach to de-sign the control law is to use a centralized cooperative control scheme for the

∗S. M. Zhao, A. Halder and T. Kalmar-Nagy are with the Department of Aerospace Engi-neering, Texas A&M University, College Station, Texas, 77843, USA. {s0z0239, a0h7710,kalmarnagy}@aeromail.tamu.edu

1

entire agent collective. However, such a control law is susceptible to band-width limitation as well as external disturbances and hence not scalable for ateam having large number of mobile agents. As a result, distributed controllaws have been investigated by the researchers for this problem, where the feed-back is constructed through local interactions of the vehicles leading to a globalformation convergence. In particular, Yang et al. ([22], [23]) proposed a de-centralized framework where a distributed controller accounts for local controldecision based on the interaction of each agent with its neighbors. Moreover,their algorithm was also capable of estimating the global statistics of the swarm(for example, overall swarm shape), thereby enabling simultaneous estimationand control. A special research topic has been to design the distributed con-troller with asynchronous communication constraints. For a detailed accounton this topic, the reader may refer [24], [25], [26] and [27].

The present paper is part of a research endeavor which aims to addressthe nonholonomic multi-agent dynamics and distributed control problem. Theauthors earlier studied [28] the cyclic pursuit of 2-unicycle problem with a con-troller similar to [14] in modified form. These preliminary results showed thatthe system may exhibit very different dynamics depending on the choice of con-troller gains and such regimes were calculated. As a next step, in this paper,the authors present nonlinear dynamics of multiple nonholonomic unicycles inleader-follower configuration to characterize similar regimes and system param-eter dependence which, the authors believe, throws light in many non-trivialareas of the complex dynamics of the agents leading to greater understandingof the overall system. In this paper, the local stability analysis has been per-formed and numerical results are presented to illustrate the dynamics of theagent collective.

As outlined in the brief literature review, the differences in the recent re-search directions in multi-agent systems has generally varied with the varietyof control strategies and the types of consensus demanded. To the best of theauthors knowledge, very few attempts (like [29]) have been made to character-ize the local stability of the system from the standpoint of nonlinear dynamics.While this is probably owing to the highly complex dynamics of the system, theauthors must underline the fact that a successful analysis to even slightly sim-pler systems like leader-follower configuration, can guide us in better designingof controllers.

As mentioned above, choice of leader-follower configuration was partly dueto its slightly simpler dynamics and partly due to the fact that many biologicalsystems (like birds) also exhibit this configuration. This choice, in the biologicalworld was long believed to be for energy efficiency [30]. Some recent results [31]tell that leader-follower configuration may also enhance communication andorientation of the flock. It is a topic of research whether this form may have anysuperiority in inter-agent communication and performance for the bio-mimeticcollectives.

The rest of this paper is organized as follows. Section II describes the math-ematical model considered in this paper and transforms the equations of motionfrom global coordinates to relative coordinates. Section III provides the deriva-tion of fixed points followed by corresponding equilibrium formations. SectionIV presents the stability boundary based on local stability analysis and associ-ated Hurwitz stability criteria. Section V presents the existence of Andronov-Hopf bifurcation depending on the value of scaled control gain followed by nu-

2

merical simulation results presented in section VI. Section VII concludes thepaper.

2 Mathematical model

The focus of this paper is to investigate the dynamics of a n-unicycle systemwhere the trajectory of the leader is characterized by constant linear and angularvelocities (V and ω)

υ0 = V,

ω0 = ω. (1)

The case ω = 0 represents straight line motion, while ω 6= 0 corresponds tocircular motion. The position and orientation of the jth vehicle (j = 0 for the

leader and j > 1 for the follower) are denoted by (xj , yj)T ∈ R

2 and θj ∈ [−π, π),respectively. The kinematic equations for the follower are

xj(t)yj(t)

θj(t)

=

cos θj(t) 0sin θj(t) 0

0 1

(

υj

ωj

)

, (2)

where j ∈ Z, j = 0, 1, . . . , n−1, (υj, ωj)T ∈ R

2 are control inputs (linear velocityand angular velocity).

i

i+1

ria

i

bi

Figure 1: Relative coordinates with vehicle i + 1 pursuing vehicle i

The configuration of n-unicycle system is shown in Fig. 1, where ri is therelative distance between the two vehicles, αi is the angle between the currentorientation of the ith unicycle and the line of sight, and βi is the angle betweenthe current orientation of i + 1th unicycle and the line of sight. Both angles arepositive in the sense of counterclockwise rotation to the line of sight. Following

3

[14], the kinematic equations are written in relative coordinates:

r0 = −υ0 cosα0 − υ1 cosβ0,

α0 = 1r0

(υ0 sin α0 + υ1 sin β0) − ω0,

β0 = 1r0

(υ0 sin α0 + υ1 sin β0) − ω1,

ri = −υi cosαi − υi+1 cosβi,

αi = 1ri

(υi sin αi + υi+1 sin βi) − ωi,

βi = 1ri

(υi sin αi + υi+1 sin βi) − ωi+1,

(3)

where i ∈ Z, i = 1, . . . , n − 2, ri ∈ R+ and (αi, βi) ∈ S1 × S1. The pursuit

control law for the ith follower is chosen as

υi = ri−1,

ωi = k sin βi−1, (4)

where the gain k is positive. The choice of this control law is inspired by thegoal to align the follower’s instantaneous velocity vector with its line of sight.

Substituting the control laws (1) and (4) into the relative dynamics (3) yieldsn − 1 sets of 3-D ODEs:

r0 = −V cosα0 − r0 cosβ0,

α0 = 1r0

(V sinα0 + r0 sin β0) − ω,

β0 = 1r0

(V sin α0 + r0 sin β0) − k sinβ0,

ri = −ri−1 cosαi − ri cosβi,

αi = 1ri

(ri−1 sin αi + ri sinβi) − k sin βi−1,

βi = 1ri

(ri−1 sin αi + ri sin βi) − k sin βi,

(5)

The parameters of this system are V , ω and k and are restricted to be positive.

3 Characterization of equilibria

3.1 Derivation of the fixed points

Setting the right hand side of (5) to zero results 3n−3 transcendental equationsfor the fixed points of the system

V

r∗0cosα∗

0 = − cosβ∗0 , (6)

V

r∗0sinα∗

0 = − sinβ∗0 + ω, (7)

V

r∗0sin α∗

0 = (k − 1) sinβ∗0 , (8)

r∗i−1

r∗icosα∗

i = − cosβ∗i , (9)

r∗i−1

r∗isin α∗

i = − sin β∗i + k sin β∗

i−1, (10)

r∗i−1

r∗isin α∗

i = (k − 1) sinβ∗i . (11)

4

Subtracting (7) from (8) and (10) from (11) yields

sin β∗0 = sin β∗

1 = · · · = sin β∗n−2 =

ω

k. (12)

Fixed point(s) exist when | sin β∗| 6 1, i.e. k > ω. When k = ω, fixed pointscoalesce in a saddle-node bifurcation.

Squaring and adding (6) and (7), (9) and (10) yields

(V

r∗0)2 = (

r∗i−1

r∗i)2 = 1 + ω2 − 2ω2

k,

which results the equilibrium relative distance as

r∗0 =V

√

1 + ω2 − 2ω2

k

,

r∗i =V

(1 + ω2 − 2ω2

k)

i+1

2

, 1 + ω2 − 2ω2

k> 0, i = 1, 2, . . . , n − 2. (13)

Further, substituting (12) and (13) into (7) and (10) yields

sinα∗0 = sinα∗

1 = · · · = sin α∗n−2 =

ω − ωk

√

1 + ω2 − 2ω2

k

. (14)

From (6) and (9), it can be noted that cosα∗i and cosβ∗

i must have differentsigns.

When k > ω, every α∗i can assume two distinct values in [0, 2π] and thus there

are 2n−1 possible fixed points of system (5). However 2n−1 − 2 of these fixedpoints are spurious, as the geometric constraint of the equilibrium formationdemands all unicycles to perform unidirectional translation in case of straightline formation and unidirectional rotation in case of cyclic formation. Thisconstraint makes only the following two fixed points (A and B) possible:

A

r∗0 = Vq

1+ω2− 2ω2

k

, r∗i = V

(1+ω2− 2ω2

k)

i+12

,

α∗0 = α∗

1 = · · · = α∗n−2 = π − arcsinω k−1

k

q

1+ω2− 2ω2

k

,

β∗0 = β∗

1 = · · · = β∗n−2 = arcsin ω

k,

B

r∗0 = Vq

1+ω2− 2ω2

k

, r∗i = V

(1+ω2− 2ω2

k)

i+12

,

α∗0 = α∗

1 = · · · = α∗n−2 = arcsinω k−1

k

q

1+ω2− 2ω2

k

,

β∗0 = β∗

1 = · · · = β∗n−2 = π − arcsin ω

k,

When k = ω, the two fixed points A and B coalesce in a saddle-node bifurcation.

3.2 Equilibrium formations

Fixed points A and B correspond to equilibrium formations in global coordinates(x, y, θ). The goal of this section is to characterize these formations, as these

5

−6 −4 −2 0 2 4

−10

−8

−6

−4

−2

0

2

4

6

8

x

y

Leader

Follower1

Follower2

Follower3

0

1

2

3

(a) Equilibrium formation for fixed point A

−6 −4 −2 0 2 4

−10

−8

−6

−4

−2

0

2

4

6

8

x

y

Leader

Follower1

Follower2

Follower3

0

1

2

3

(b) Equilibrium formation for fixed point B

Figure 2: Corresponding straight-line motion in global coordinates for fixedpoints A and B

correspond to the physical behavior of the leader-follower system. When ω = 0,the trajectory of the leader can be expressed explicitly as

x0(t) =(V cos θ0)t + x0(0),

y0(t) =(V sin θ0)t + y0(0),

θ0(t) = θ0(0), (15)

where (x0(0), y0(0), θ0(0)) are its initial positions and orientation. It is straight-forward to observe that fixed points A = (r∗i , α∗

i , β∗i ) = (V, π, 0) and B =

(r∗i , α∗i , β

∗i ) = (V, 0, π) correspond to rectilinear motion of the followers. Fig. 2

shows the corresponding “pursuit graph” (parametric plots of {xi (t) , yi (t)}) offixed A and fixed point B for this rectilinear motion. It can be noted that forfixed point A, the leader “leads the pack” and for fixed point B, it “trails thepack”.

When ω 6= 0, the trajectory of the leader becomes

x0(t) =V

ωsin(ωt + θ) + xc,

y0(t) = − V

ωcos(ωt + θ) + yc, (16)

θ0(t) = ωt + θ0(0), (17)

where xc = x0(0) − Vω

sin θ0(0) and yc = y0(0) + Vω

cos θ0(0) are the center of

the circle of radius R0 = Vω

traversed by the leader. Without loss of generalitywe choose xc = yc = 0.

Both fixed points A and B yield the following two equations for the locus ofthe ith follower

r∗2i =V 2

(1 + ω2 − 2ω2

k)i

= x2i + y2

i + x2i+1 + y2

i+1 − 2(xixi+1 + yiyi+1), (18)

6

sin α∗i =

ω − ωk

√

1 + ω2 − 2ω2

k

= sin arctan

(

yi+1 − yi

xi+1 − xi

− θi

)

=x2

i + y2i − xixi+1 − yiyi+1

r∗i

√

x2i+1 + y2

i+1

. (19)

Since x20 +y2

0 = V 2

ω2 corresponds to the leader’s (j = 0) trajectory, combining(18) and (19) results

x21 + y2

1 =V 2

ω2

1

1 + ω2 − 2ω2

k

= R21. (20)

In general, using method of induction

x2j + y2

j =V 2

ω2

1

(1 + ω2 − 2ω2

k)j

= R2j . (21)

This means that in the equilibrium formation, the jth follower is circling theorigin with radius Rj . Fig. 3 shows the corresponding “pursuit graph” for fixedA and fixed point B for circular motion (ω > 0).

It can be noted from (21) that depending on the value of k, the concentriccircles traced out by the followers can be inside (k > 2), on (k = 2) or outside(k < 2) the leader’s circle. Also, analogous to Fig. 2, for k > 2 case, correspond-ing to fixed point A, the leader “leads the pack” i.e. the followers have positivephase difference with respect to the leader (Fig. 3a). Similarly, for fixed pointB, the leader “trails the pack” i.e. the followers have negative phase differencewith respect to the leader (Fig. 3b).

As discussed in the previous section, when k = ω > 1, the two fixed points A

and B coalesce to give rise to the single fixed point (r∗i , α∗i , β

∗i ) = ( V

(ω−1)i+1 , π2 , π

2 )

(Fig. 4a). When k = ω < 1, the single fixed point becomes (r∗i , α∗i , β

∗i ) =

( V(1−ω)i+1 ,−π

2 , π2 ) (Fig. 4b).

4 Local stability analysis

4.1 Linearization about the fixed points

The local stability of the fixed points is determined by the eigenstructure of theJacobian evaluated at the fixed point. The Jacobian of (5) is given by:

Jp =

A0 0 · · · 0

B1 A1 · · · 0...

. . .. . .

...0 · · · Bn−2 An−2

, (22)

where Ai, Bi and 0 are all 3 × 3 matrices.

A0 =

− cosβ∗0 V sin α∗

0 r∗0 sin β∗0

− Vr∗20

sin α∗0

Vr∗

0

cosα∗0 cosβ∗

0

− Vr∗20

sin α∗0

Vr∗

0

cosα∗0 (1 − k) cosβ∗

0

,

7

−4 −2 0 2 4 6

−4

−2

0

2

4

6

x

y

Leader

Follower1

Follower2

Follower30

12

3

(a) Equilibrium formation for fixed point A

−4 −2 0 2 4 6

−4

−2

0

2

4

6

xy

Leader

Follower1

Follower2

Follower30

1

2

3

(b) Equilibrium formation for fixed point B

Figure 3: Corresponding circular motion in global coordinates for fixed pointsA and B

−4 −2 0 2 4 6

−4

−2

0

2

4

6

x

y

Leader

Follower1

Follower2

Follower30

1

23

(a) Equilibrium formation when k = ω > 1

−4 −2 0 2 4 6

−4

−2

0

2

4

6

x

y

Leader

Follower1

Follower2

Follower30

2

3

1

(b) Equilibrium formation when k = ω < 1

Figure 4: Corresponding circular motion in global coordinates for the coalescedfixed point

8

Ai =

− cosβ∗i r∗i−1 sin α∗

i r∗i sin β∗i

− r∗

i−1

r∗2i

sin α∗i

r∗

i−1

r∗

i

cosα∗i cosβ∗

i

− r∗

i−1

r∗2i

sin α∗i

r∗

i−1

r∗

i

cosα∗i (1 − k) cosβ∗

i

, i = 1, 2, . . . , n − 2,

Bi =

− cosα∗i 0 0

sin α∗

i

r∗

i

0 −k cosβ∗i−1

sin α∗

i

r∗

i

0 0

, i = 1, 2, . . . , n − 2,

The eigenvalues of (22) are also the eigenvalues of all Ai’s since the Jacobian(22) is lower triangular block matrix. The characteristic polynomial for any ofthe Ai’s evaluated at the fixed points have the same form and are given by (+and − corresponds to fixed point A and B, resp.):

λ3 ± p2λ2 + p1λ ± p0 = 0, (23)

where p2 = (1+ 1k)√

k2 − ω2, p1 = ω2+2k− 3ω2

kand p0 = (1+ω2− 2ω2

k)√

k2 − ω2.The characteristic equation corresponding to fixed point B can be obtained fromthat of fixed point A by the transformation λ → −λ, so the spectrum of B isthe reflection of that of A about the imaginary axis.

4.2 Linear stability boundary

A fixed point is stable when the corresponding characteristic polynomial is Hur-witz. Necessary and sufficient condition on Hurwitz stability of a third orderpolynomial is given on page 132 of [32], requiring p0, p1, p2 > 0 and p1p2 > p0

for (23), which results

2k3 + k2 − 3ω2 > 0, (24)

1 + ω2 − 2ω2

k> 0. (25)

From (12), the existence of the fixed points requires

k > ω. (26)

Inequalities (24), (25) and (26) determine regions in the k −ω parameter spacewhere fixed points exists, as well as their stability. These regions are charac-

terized by the three curves ω1(k) = k, ω2(k) =√

2k3+k2

3 and ω3(k) =√

k2−k

(k < 2). Notice that when k > 2, inequality (25) is alway satisfied, the stabilityregion is determined only by the remaining two curves. It can be easily verifiedthat ω1(k) 6 ω3(k) when 0 < k < 2. Fig. 5 depicts the stability boundaries ofthis system. It can be observed that when ω = 0 (straight line motion), fixedpoint A is always a stable node while B is always an unstable one.

5 Andronov-Hopf bifurcation

When 0 < k < 1, the characteristic polynomial on the curve ω2(k) can bewritten as

(

λ + (1 +1

k)√

k2 − ω2)

(λ2 + ω2 + 2k − 3ω2

k) = 0.

9

1

k

w

A B

A B

A B

A B

d

e

a

b

c

w ( )2 k

w ( )1 k

f

Figure 5: Linear stability boundary for system (5) with spectra of A and B

This implies that for the fixed point A, the Jacobian has one negative realeigenvalue and a complex conjugate pair on the imaginary axis. Below thecurve ω2(k), the Jacobian has one negative real eigenvalue and a pair of com-plex conjugates on the left half plane, i.e. the fixed point is a stable node-focus.Between the curve ω2(k) and ω1(k), the pair of complex conjugate eigenvalueshave positive real part, which suggests the occurrence of Andronov-Hopf bifur-cation by increasing k through ω2(k). On this curve, the critical bifurcation

value of ω is ωc =√

2k3+k2

3 , and the root crossing velocity can be calculated as

Redλ

dω|(k,ωc) =

3√

2

7k + 2

√

1 + 2k

1 − k> 0. (27)

Transversal root crossing is a necessary condition for the Andronov-Hopf bi-furcation. It can be noted that for the case of n unicycles, we have n − 1identical triplets of such eigenvalues. This implies that the system undergoesAndronov-Hopf bifurcation when n = 2, double-Hopf bifurcation when n = 3and in general, a bifurcation with (n − 1) pairs of pure imaginary eigenvalues.The rest of this section analyzes the two unicycle case in detail.

To show that the fixed point of the dynamical system (5) is weakly at-tracting/repelling on the stability boundary, one needs to compute the so-calledPoincare-Lyapunov constant [33]. To find this constant, the original equation(5) is expanded up to third order around fixed point A [34]

w = Ψ(w) = Jpw +1

2f (2)(w) +

1

6f (3)(w) + O(w4), (28)

10

where w = (r − r∗A, α − α∗A, β − β∗

A)T defines new coordinates which shift thefixed point A to the origin. In these new coordinates, f (2)(w) and f (3)(w) aremultilinear vector functions given by

f(2)i =

n∑

j,k=1

∂2Ψi(ξ)

∂ξj∂ξk

|ξ=0wjwk i = 1, 2, 3,

and

f(3)i =

n∑

j,k,l=1

∂3Ψi(ξ)

∂ξj∂ξk∂ξl

|ξ=0wjwkwl i = 1, 2, 3.

In order to obtain the real Jordan canonical form, a linear transformation T

needs to be constructed using the eigenvectors of the Jacobian evaluated at ωc.At the critical point, the pair of complex conjugate eigenvalues have the formλ2,3 = ±iω0,

ω0 =

√

k(1 − 2

3k)(1 − k) > 0.

Let q2 ∈ C3 be the complex eigenvector corresponding to the eigenvalue λ2.Then,

Jpq2 = iω0q2, Jpq2 = −iω0q2

Also, let q1 ∈ R3 be the real eigenvector corresponding to the eigenvalue

λ1 = −(1 + k)√

23 (1 − k), i.e. Jpq1 = λ1q1. The transformation matrix T

is composed by 1‖q1‖ (Req2, − Imq2, ‖q1‖q1) where q1 and q2 are given by

q2 =

2√

6V k

9γ√

1−k+ i

V k(1− 23k)

3ω0γ

1

1 − 23k + i

√6ω0

3√

1−k

,q1 =

−√

6V (1+2k)

3γ√

1−k

1k + 1

γ =1

3

√

(3 − 2k)(1 + 2k)(1 − k2).

Introducing the transformation y = T−1w

y = Jy +1

2g(2)(y) +

1

6g(3)(y) + O(y4), (29)

where the Jordan canonical form J is given by

J = T−1JpT =

0 −ω0 0ω0 0 00 0 λ1

.

In (29), the nonlinear vector functions in transformed coordinates are given by

g(2)(y) = T−1f (2)(w)|w=Ty,

g(3)(y) = T−1f (3)(w)|w=Ty.

Assuming that the center manifold has the quadratic form y3 = 12 (h1y

21 +

2h2y1y2 +h3y22), one can reduce (29) into a two-dimensional system up to third

11

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−100

−80

−60

−40

−20

0

20

40

60

k

∆

V=1V=9V=13V=16V=20

Figure 6: Variation of Poincare-Lyapunov constant (∆) with control gain (k)

order

y1 = −ω0y2 + a20y21 + a11y1y2 + a02y

22

+ a30y31 + a21y

21y2 + a12y1y

22 + a03y

32 ,

y2 = ω0y1 + b20y21 + b11y1y2 + b02y

22

+ b30y31 + b21y

21y2 + b12y1y

22 + b03y

32 . (30)

Using the 10 out of these 14 coefficients ajk, bjk, the so called Poincare-Lyapunovconstant △ can be calculated as [33]

△ =1

8ω((a20 + a02)(a11 − b20 + b02)

+ (b20 + b02)(a02 − a20 − b11))

+1

8(3a30 + a12 + b21 + 3b03). (31)

Fig. 6 illustrates the variation of ∆ with respect to k. Note that based onthe value of parameter V , the Andronov-Hopf bifurcation can be supercritical(∆ > 0) or subcritical (∆ < 0).

6 Numerical Results

6.1 Andronov-Hopf bifurcation

Fig. 7 shows the phase portrait corresponding to point a (k = 0.500, ω = 0.301)on the stability chart (Fig. 5) and the associated pursuit graph. Fixed pointA is exponentially attracting here. Fig. 8 depicts the phase portrait associatedwith point b (k = 0.500, ω = 0.408) showing a weakly attracting fixed point

12

A. There is a stable limit cycle born (supercritical Andronov-Hopf bifurcation)around the fixed point A (point c) when ω is increased through its criticalvalue ωc (phase portrait and pursuit graph are shown in Fig. 9). The pursuittrajectory in global coordinates has two harmonic components.

Fig. 10a shows the “pursuit graph” of five unicycles (initial conditions arechosen slightly away from the equilibrium formation) when ω lies slightly abovethe curve ω2(k) (point c). It shows that the fourth follower exhibits most quasi-periodic behavior. The vigor of such quasi-periodicity decreases with the prox-imity to the leader. Fig. 10b corroborates this fact by showing that further thefollower is, the wider its frequency spectrum becomes.

6.2 Fold-Hopf bifurcation

When k > 1, the characteristic polynomial on the stability curve ω1(k) can bewritten as

λ3 + (ω2 − ω)λ = 0

implying that there is zero eigenvalue together with a pair of pure imaginaryones. This is a Fold-Hopf (a codimension-two) bifurcation [33]. Fig. 11.a andFig. 11.c show the phase portrait of point f (k = 1.20, ω = 1.20) on the stabilitycurve k = ω and the corresponding pursuit graph, while Fig. 11.b shows thephase portrait of point e situated slightly below the point f .

For the case of five unicycles (initial conditions are chosen slightly away fromthe equilibrium formation), the time series and the corresponding FFT for thefollowers are plotted in Fig. 12. It shows that the third and fourth follower ex-hibit complex behavior with possibly heteroclinic bursts between quasi-periodicand chaotic behavior.

6.3 Global Bifurcations

In addition to the Andronov-Hopf and Fold-Hopf bifurcations, preliminary sim-ulations indicate global bifurcations involving the destruction of heteroclinic or-bits. Fig. 13 shows the phase portrait corresponding to point d (k = 1.20, ω =0.60), with a heteroclinic orbit (trajectory 1) connecting fixed points A andB. When k is decreased below 1, it was observed that the heteroclinic orbitdisappear and the region of attraction for fixed point A shrinks significantly.

The dynamics is very interesting when k is around 1 (the intersection of thesaddle-node and Andronov-Hopf bifurcation curves). When (k, ω) = (1.01, 1),one can observe a heteroclinic orbit or periodic motions containing higher har-monics as shown in Fig. 14.

7 Conclusions

In this paper, the leader-follower pursuit of unicycles is studied. Local stabilityanalysis around the equilibrium formation has been performed. Analysis andnumerical simulations have shown the existence of Andronov-Hopf and Fold-Hopf bifurcations on the stability boundary. In addition to the results providedhere, the authors have also studied the effect of constant communication delaybetween the unicycles in leader-follower configuration. Both analytical and nu-merical results (not provided here) show that for a suitable distributed control

13

3.6

3.65

3.7

3.75

0.8

0.85

0.9

0.9512.5

13

13.5

14

14.5

15

αβ

r

Fixed point A

(a) Phase portrait at point a

−40 −30 −20 −10 0 10 20 30 40−20

−10

0

10

20

30

40

50

60

70

X position

Y p

ositi

on

leader

follower

Starting Point

Equilibrium Formation

(b) Pursuit graph for point a in global coordinates

Figure 7: Phase portrait at point a when fixed point A is attracting and thecorresponding pursuit graph in global coordinates

14

3.66 3.68 3.7 3.72 3.74 3.76 3.780.88

0.9

0.92

0.94

0.96

0.9813.5

14

14.5

15

15.5

α

β

r

Fixed point A

Figure 8: Phase portrait at point b where the fixed point A is a weakly attractingone

law, where one may ignore the transients of the agent response, the commu-nication has no qualitative effect on the final consensus dynamics of the agentcollective. As briefly stated in Section I, the future research direction includesthe more generalized nonlinear dynamic analysis of two unicycles, not necessarilyin leader-follower configuration and extending the result for multiple unicycleswith communication delay.

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1.54 1.545 1.55 1.555 1.56 1.565 1.57

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18

0 10 20 30 40 50−2

−1.5

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1

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10000

Frequency

PS

D

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0

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1

time

x3

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500

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Frequency

PS

D

(f) FFT of follower 3

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−3

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0

1

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time

x4

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500

1000

1500

2000

2500

3000

3500

4000

Frequency

PS

D

(h) FFT of follower 4

Figure 12: Time series and corresponding FFT for five unicycle case at point f

19

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