Constrained Motion Approach to the Synchronizationof the Multiple Coupled Slave Gyroscopes

Harshavardhan Mylapilli1

Abstract: A set of n gyroscopes are coupled to form a system of slave gyroscopes. A simple approach is developed for synchronizing themotion of these slave gyroscopes whose individual motion may be regular or chaotic, with the motion of an independent master gyroscopeirrespective of the chaotic or regularmotion exhibited by themaster. The problemof synchronization of thesemultiple gyroscopes is approachedfrom a constrained motion perspective through the application of the fundamental equation of mechanics. The approach yields explicit, closed-form expressions for the control torques that are required to be applied to each of the coupled slave gyroscopes to achieve exact synchronizationwith the master’s motion. The influence of different types of interactions between the slave gyroscopes is investigated with an incidence matrixthat describes the coupling between any two of them. The effect of the so-called sleeping condition on the synchronization of the gyroscopes isalso explored. To illustrate the efficacy of the methods presented in this paper, we consider numerical examples involving systems of multiplegyroscopes and synchronize them with the motion of a master gyroscope.DOI: 10.1061/(ASCE)AS.1943-5525.0000192. © 2013 AmericanSociety of Civil Engineers.

CE Database subject headings: Control systems; Coupling; Motion; Aerospace engineering.

Author keywords:Multiple gyroscopes; Fundamental equation; Synchronization; Constrained motion; Nonlinear control; Chain-coupled;General-coupled.

Introduction

Gyroscopes have long been used in navigation to measure orien-tation. From aerospace vehicles to consumer electronic devices,gyroscopes have played an important role in our everyday lives.From an engineering point of view, gyroscopes are highly nonlinearmechanical systems that exhibit a plethora of complicated motionssuch as periodic behavior, period-doubling behavior, quasi-periodicbehavior, and chaotic behavior. Studies (Tong and Mrad 2001; Geand Chen 1996; Ge et al. 1996; Van Dooren 2003) have shownthat when a symmetric gyroscope with linear plus cubic damping issubjected to a harmonic vertical base excitation, the motion thusobtained can range from regularmotion to chaoticmotion dependingon the parameters selected.

In analytical mechanics, the problem of synchronization ofmultiple chaotic systems has been of particular interest in recenttimes (Leipnik and Newton 1981; Pecora and Carroll 1990). Inparticular, the problem of synchronizing a set of slave gyroscopes,which may or may not be coupled to each other, to an independentmaster gyroscope is an important problem in nonlinear dynamicsthat has received considerable attention (Chen 2002; Lei et al. 2005;Udwadia 2008). This problem gains prominence in the field ofspacecraft dynamics where attitude control of spacecraft is almostalways performed using gyroscopes. When there are multiplegyroscopes onboard a spacecraft, it is often required that they besynchronized to indicate a specific, usable spacecraft pointing. And

given that these gyroscopes exhibit a wide range of dynamic be-havior, the problem of synchronization becomes even more chal-lenging. The synchronization problem also finds applications inareas of secure communication (e.g., transmission of encryptedmessages) and in areas of signal processing (Strogatz 2000). A largenumber of papers written on this subject only seem to consider thesynchronization problem of two identical gyroscopes. Chen (2002)considers two identical chaotic gyroswith different initial conditionsand uses numerous classical control laws to show that when thefeedback gain (obtained through experimentation) exceeds a par-ticular value, the slave gyro synchronizes with the master gyro. Leiet al. (2005) approach the same problemusing feedback linearizationwherein the difference in response between the master and slavegyro is taken as an error signal and a time-varying control is chosento drive the error signal to zero. Aghababa (2011), on the other hand,considers an adaptive robust finite-time controller to synchronizetwo chaotic gyros despite unknown uncertainties in the system.These approaches to the synchronization of master/slave gyrosappear to work when the number of slaves considered is small innumber. However, when a large number of slaves with nonidenticalphysical and geometric properties are considered, the need to de-velop efficient control methods arise. It is also important to note thatmany papers on this subject assume that the gyroscopes are in theso-called sleeping position, which in general is an oversimplifyingcondition, because most gyroscopes do not satisfy it.

Recently, Udwadia and Han (2008) approached the problemof synchronization of multiple chaotic gyroscopes from a newperspective—using the constrained motion approach. In their paper,Udwadia and Han consider a system of n gyroscopes, each in the so-called sleeping position, which are all uncoupled from one another,and have varying physical and geometrical properties, some or allof which exhibit chaotic behavior. They attempt to synchronizethese uncoupled slave gyros with a master gyro. The problem ofsynchronization, which is classically considered as a tracking con-trol problem in control theory, is recast as a problem of constrained

1Ph.D. Candidate, Dept. of Aerospace andMechanical Engineering, Univ.of Southern California, Los Angeles, CA 90089. E-mail: mylapill@usc.edu

Note. This manuscript was submitted on August 16, 2011; approved onDecember 14, 2011; published online on December 16, 2011. Discussionperiod open until March 1, 2014; separate discussions must be submittedfor individual papers. This paper is part of the Journal of AerospaceEngineering, Vol. 26, No. 4, October 1, 2013. ©ASCE, ISSN 0893-1321/2013/4-814–828/$25.00.

814 / JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

motion. The slaves are suitably constrained and the fundamentalequation (Udwadia and Kalaba 1996a) is used in arriving at thenonlinear control torques that force exact synchronization of eachslave gyro with the master gyro. However, the problem withsynchronizing a master gyro with n uncoupled slave gyroscopesis that it can effectively be simplified into n separate problems ofsynchronization of the master gyro with n individual slave gyros.

In this paper, we consider a set of n coupled gyros such thatthey form a single system of interacting slave gyroscopes. Next, weattempt to synchronize this nonlinear system of slave gyros with anindependentmaster gyroscopewhosemotion can be either regular orchaotic. We adopt the methods developed by Udwadia and Han(2008) and build on them to derive the constrained equations ofmotion for the case when the slave gyroscopes are all coupled. In theprocess, we also obtain explicit and closed-form expressions for thegeneralized nonlinear control torques that are required to be appliedto each of the slave gyros to obtain synchronization with the mastergyro.We consider both linear and highly nonlinear types of coupling[which include linear plus cubic coupling, sinusoidal coupling, andToda coupling (Toda 1981)] between the slave gyroscopes. Theeffect of the sleeping condition on the synchronization of the gyros isalso studied. Further, utilizing the concept of an incidencematrix, wealso examine the case when the slave gyroscopes are coupled inamore general fashion (instead of the usual chain coupling). Finally,to illustrate the efficacy of the methods presented in this paper, weprovide three numerical simulations. We consider a system of fivegyros with one master and four chain-coupled slaves where theslaves are coupled (1) using sinusoidal coupling and then (2) usingToda coupling. In the Toda coupling case, the effect of the no-sleeping condition on the synchronization of the gyros is studied.The third example consists of a system of six gyroscopes with onemaster and five slaves where the slaves are coupled to one anotherthrough an incidence matrix with different types of strongly non-linear couplings.

Equations of Motion

Consider the system of n1 1 symmetric gyroscopes as shown inFigs. 1 and 2. The master gyro whose parameters are denoted by thesubscript m is an independent system separate from the slave

gyroscopes. The other n gyroscopes (denoted by subscripts 12 n)are all coupled together with torsion springs to form the system ofslave gyros. In the current study, we use Euler angles to describe theorientation of each gyroscope—u (nutation), w (precession), andc (spin).

Master Gyroscope

The nonlinear equation of motion of a symmetric gyroscope whosepoint of support O is subjected to a vertical harmonic excitation offrequency vm and amplitude ~dm has been derived by Udwadia andHan (2008) and is reproduced as

Im€um þ�pwm

2 pcmcos um

��pcm

2 pwmcos um

�Im sin3um

2mmgrm sin um

2mmrm sin umd€mðtÞ ¼ Fd

(1)

where the angular momenta pcm5 I3mð _cm 1 _wm cos umÞ and pwm

5 Im _wm sin2um 1 pcmcos um are conserved quantities as cm, wm are

cyclic coordinates; mm 5 mass of the master gyroscope, Im5 I1m 1mmr2m, where I1m 5 I2m5 principal equatorial moment ofinertia of the master gyroscope and I3m 5 polar moment of inertia;um, cm, wm 5 Euler angles of rotation associated with the mastergyroscope; rm 5 distance along the polar axis of the center of massof the gyro from its point of support (see Fig. 1); anddmðtÞ5 ~dm sinðvmtÞ5 time-varying amplitude of the vertical supportmotion that has frequencyvm. The nonconservative damping forceacting on the master gyroscope Fd 5 2~cm _um 2~em _u

3m is assumed to

be of linear plus cubic type (Ge et al. 1996). A stationary symmetricgyro positioned with its axis along the vertical usually falls over.However, when it is given a sufficiently large spin about thevertical axis, it begins to rotate in a stable fashion, with its axisremaining very close to the vertical. The symmetric gyro showingthis type of sleeping-top motion is said to be in a sleeping positionand this usually happens when the angular momenta ( pc 5 pw 5 p)is constant for all time t. The equation of motion of the verticallyexcited master gyro [Eq. (1)] in the sleeping position is then givenby

€um þ a2mð12 cos umÞ2

sin3umþ cm _um þ em _u

3m 2bm sin um þ gm sin um sinðvmtÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

2Fgyrom

¼ 0 (2)

where

Pm ¼�am ¼ pm

Im, bm ¼ mmgrm

Im, cm ¼ ~cm

Im, em ¼ ~em

Im,

gm ¼ mmrm ~dmv2m

Im, vm, Im

�

is the parameter set that describes the physical characteristics of themaster gyroscope.

Slave Gyroscopes

Consider the system of n symmetric gyroscopes that are coupled inchain fashion as shown in Fig. 2. The subscript i (where i5 12n)denotes each individual slave gyroscope. Each of the slave gyros issubjected to a vertical harmonic base excitation of amplitude ~di andfrequency vi. The slaves are also subjected to a nonconservativedamping force that is of linear plus cubic type related only to theu-coordinate. The following types of torsion spring couplings arestudied in the present paper:1. Harmonic coupling

Vti ðuiþ12 uiÞ ¼ si½12 cosðuiþ1 2 uiÞ� (3)

JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013 / 815

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

2. Toda torsion coupling (Toda 1981)

Vti ðuiþ12 uiÞ ¼ ai

biebiðuiþ12uiÞ 2 aiðuiþ12 uiÞ2 ai

bi(4)

3. Linear plus cubic coupling

Vti ðuiþ12 uiÞ ¼ fi

2ðuiþ1 2 uiÞ2 þ gi

4ðuiþ12 uiÞ4 (5)

whereai, bi, fi, gi, and si 5 spring constants corresponding to the ithspring element and Vt

i 5 torsional potential energy of the ith springelement in the chain.

Consider the ith gyroscope in the slave system. Besides the forcesFgyroi [see Eq. (6)] that act on a symmetric gyro as a consequence of it

being subjected to nonlinear damping and vertical excitation, theonly additional forces that act on it are the coupling forces. Thus, theequation of motion of the ith slave gyroscope in the ui direction canbe computed using Newton’s laws as follows:

€ui þ�pwi

2 pcicos ui

��pci

2 pwicos ui

�I2i sin

3uiþ ~ci

Ii_ui þ ~ei

Ii_u3i 2

migriIi

sin ui 2miriIi

sin uid€iðtÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}2Fgyro

i

¼ 1Ii

∂Vtðuiþ1 2 uiÞ

∂ui2

∂Vtðui 2 ui2 1Þ∂ui

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

FSiþ1,i 2 FS

i,i2 1

(6)

Fig. 1. Symmetrical master gyro (independent) subjected to a vertical harmonic excitation

Fig. 2.A system of chain-coupled slave gyroscopes; each slave gyro in the chain is subjected to a vertical harmonic excitation which, when uncoupledfrom the system, results in regular or chaotic motion

816 / JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

where pci5 I3ið _ci 1 _wi cos uiÞ, pwi

5 Ii _wi sin2ui 1 pci

cos ui, diðtÞ5 ~di sinðvitÞ, and Fsine

i11,i 5si sinðui11 2 uiÞ for a harmonic cou-pling; FToda

i11,i 5 ai½ebiðui1 12uiÞ 2 1� for a Toda coupling; andFLCi11,i 5 fiðui11 2 uiÞ1 giðui1 1 2 uiÞ3 for a linear plus cubic cou-

pling. Because ci and wi are cyclic coordinates, Eq. (6) alone issufficient to describe the equation of motion of the ith slave gyro.Eq. (6) can be represented more conveniently as

€ui ¼ Fgyroi þ 1

Ii

�FSiþ1,i 2FS

i,i2 1

�(7)

whereFgyroi 5 gyroscopic forces acting on the ith gyro from its being

subjected to vertical excitation and nonlinear damping and FSi11,i is

the coupling force exerted by the ði1 1Þth gyro on the ith gyro. In thecurrent study,weapproach theproblemof synchronizationofmultiplegyroscopes from a constrained motion perspective, which includesthree vital steps: (1) derivation of the equations of motion of theunconstrained system; (2) formulation of the constraint equations; and(3) use of the fundamental equation (Udwadia and Kalaba 1996a, b)to obtain the constrained equations ofmotion of synchronizedmaster-slave system.

In the process, we also find a closed-form expression for theexplicit nonlinear control torques that are required to be applied toeach of the slave gyroscopes to synchronize them precisely with themotion of the master gyroscope.

Unconstrained Equations of Motion

In the present paper, the independent master gyroscope [Eq. (1)]along with the system of coupled slave gyroscopes [Eq. (7)] formsthe unconstrained system (Udwadia and Kalaba 1996a, b). Theunconstrained equation of motion of the system can be written inmatrix form [using Eqs. (2) and (6)] as follows:

Mðnþ1Þ � ðnþ1Þ

26666666666664

€um€u1«

€ui«

€un

37777777777775

|fflfflffl{zfflfflffl}€Xnþ1�1

¼

2666666664

Fgyrom

Fgyro1 þ FS

2, 1=I1«

Fgyroi þ

�FSiþ1, i2FS

i, i2 1

�.Ii

«

Fgyron þ

�2FS

n, n2 1

�.In

3777777775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Fnþ1�1

(8)

where M 5 identity matrix of size n1 1; and a5M21F5F5 unconstrained acceleration of the system. To this unconstrainedsystem, we impose a set of constraints such that the system ofcoupled slave gyros precisely follows the master gyroscope in itsmotion.

Constraint Equations

The unconstrained system fromEq. (8) is subjected to the followingset of constraints:

No Control Force on the Master GyroBecause the motion of the master gyroscope is considered to beindependent of the motion of the slave gyroscopes, the need toapply a control force on the master gyro does not arise in the

constrained system. Hence, the unconstrained motion of the mastergyroscope can itself be considered as a constraint equation. Thus,we have

€um ¼ Fgyrom (9)

Synchronization ConstraintIn the present paper, our goal is to synchronize the motion of each ofthe coupled slave gyros with that of the master gyro. Hence, thefollowing set of n constraints are imposed on the unconstrainedsystem and are given by

yi ¼ um 2 ui ¼ 0, i ¼ 1, 2, 3, . . . , n (10)

The initial conditions of these slave gyros are assumed to satisfy theconstraint equations. However, it may not always be possible toinitiate the slave gyros from positions in phase space that satisfy theconstraints. This problem is surmounted by choosing appropriatetrajectory stabilization parameters d, k (Udwadia 2003, 2008) suchthat

€yi þ d _yi þ k yi ¼ 00€um2 €ui ¼ 2d�_um 2 _ui

�2 kðum2 uiÞ

(11)

Thus, the unconstrained system in Eq. (8) is subjected to a total ofn1 1 cumulative constraints to achieve synchronization of eachindividual coupled slave gyro with the motion of the master gyro.When the constraints in Eqs. (9) and (11) are expressed in the con-straint matrix form, we obtain

"½1� ½ 0 ⋯ 0 �24 1«1

35 �

2In by n�#

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Anþ1 � nþ1

26666666666664

€um€u1«

€ui«

€un

37777777777775

|fflfflffl{zfflfflffl}€Xnþ1�1

¼

266666666664

Fgyrom

2d�_um 2 _u1

�2 kðum 2 u1Þ«

2d�_um 2 _ui

�2 kðum 2 uiÞ«

2d�_um 2 _un

�2 kðum 2 unÞ

377777777775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}bnþ1�1

(12)

Constrained Equations of Motion

With all the requisite matrices (M, F, A, b) at our disposal, theconstrained equations of motion can now be calculated using thefundamental equation (Udwadia and Kalaba 2002). BecauseM5 In11 and A 5 square matrix of size n1 1, the expression forthe control force reduces to

FC ¼ M1=2�AM2ð1=2Þ

�þðb2AaÞ ¼ Aþðb2AaÞ¼ A21ðb2AaÞ (13)

The constraint matrix A is structured in such a way that it possessesthe unique propertyA21 5A, and this reduces the explicit expressionof the nonlinear control torques to

JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013 / 817

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

FC ¼ A21ðb2AaÞ ¼ A21b2A21Aa ¼ Ab2 a

¼

26666664

½1� ½ 0 ⋯ 0 �266641

«

1

37775 ½2In� n�

37777775

266666666666666664

Fgyrom

2d�_um 2 _u1

�2 kðum 2 u1Þ

«

2d�_um2 _ui

�2 kðum 2 uiÞ

«

2d�_um 2 _un

�2 kðum 2 unÞ

377777777777777775

2

2666666666666664

Fgyrom

Fgyro1 þ FS

2,1=I1

«

Fgyroi þ

�FSiþ1,i 2FS

i,i21

�.Ii

«

Fgyron þ

�2FS

n,n21

�.In

3777777777777775

¼

8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:

0

d�_um 2 _u1

�þ kðum2 u1Þ

þ Fgyro

m 2

Fgyro1 þ FS

2,1

I1

!

«

d�_um2 _ui

�þ kðum 2 uiÞ

þ Fgyro

m 2

Fgyroi þ FS

iþ1,i 2FSi,i21

Ii

!

«

d�_um 2 _un

�þ kðum 2 unÞ

þ Fgyro

m 2

Fgyron þ2 FS

n,n21

In

!

9>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>;

ð14Þ

where d and k 5 trajectory stabilization parameters and Fgyro andFS 5 gyroscopic forces and coupling forces, respectively. Thus,Eq. (14) gives us the explicit, closed-formexpression for the nonlinearcontrol torques (devoid of any approximations or simplifications) thatneed to be applied to the unconstrained system to obtain the con-strained system. The constrained equations of motion of the system(Udwadia and Kalaba 1996a, b, 2002) can now be written as

M€X ¼ F þ FC (15)

whereF is given by the right-hand side of Eq. (8) and FC is given bythe Eq. (14). Eq. (15) represents the governing equations of motionof the coupled slave gyroscopes that follow a given independentmaster gyroscope, irrespective of its chaotic or regular motion.

Incidence Matrix

Until now,we have considered the casewherein the slave gyros are allcoupled in a chain fashion. But it is indeed possible to consider amoregeneral interaction of the slave gyros wherein a particular slave gyrocan be coupled to any number of other slave gyros without anyrestrictions. Now, in such a case, it becomes imperative to keep trackof the coupling information between any two slave gyros in the sys-tem. This can be succinctly done by the use of an incidence matrix.The incidence matrix has the following salient features:• The incidence matrix, denoted by j, is a symmetric matrix of size n

(wheren denotes thenumber of slave gyros). It gives us informationabout the couplingbetween any twoslavegyroscopes in the system.

• Each element of the incidence matrix j is restricted to havea value of either 0 or 1. A zero value of jij indicates that the ith

and jth slaves are uncoupled whereas a unitary value of jijindicates that the ith slave is coupled to the jth slave.

• Because a gyroscope cannot be coupled to itself, the diagonalelements of the incidence matrix are all zero (i.e., jii 5 0;i 5 1, 2, 3, . . . , n).Thus, given an incidence matrix that describes the various dif-

ferent interactions between a set of coupled slave gyros, the un-constrained equations of motion of the ith slave gyro in the systemcan be written as follows:

€ui ¼ Fgyroi þ 1

Ii

0@Pn

j¼1j�i

jijFCoupleij

1A, i ¼ 1, 2, 3, . . . , n (16)

where FCoupleij 5 2FCouple

ji . This set of n equations along with theunconstrained motion of the independent master gyro [Eq. (2)] formsthe unconstrained system. The reader can now utilize the methodsdescribed in this section to rederive the constrained equations ofmotionof the master-slave system and generate the necessary control torquesthat are required to be applied to the slave system of gyroscopes suchthat they synchronizepreciselywith themotionof themastergyroscope.

Results and Simulations

To better illustrate the efficacy of the methods presented in this paper,we present three numerical simulations. In the first example, weconsider a systemoffivegyros (1Master1 4 Slaves) with the sleepingcondition imposed on all five gyros. We examine the effect of

818 / JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

sinusoidally chain coupling the slave gyros on the synchronization ofthe gyros. In the second example, we consider a system of six gyros (1Master15Slaves) and use an incidencematrix to describe the variousdifferent interactions between the five coupled slave gyros. Thesleeping condition is imposed on all six gyros and different types oflinear andnonlinear couplings are used to couple the individual slaves.In the third example, we consider a five-gyro system (1 Master 1 4Slaves) and explore the effect of the sleeping condition on the syn-chronization of the gyros.

Throughout this paper, the integration of the equations of motion(constrained as well as unconstrained) is performed using the ODE45solver from the software MATLAB with a relative error tolerance of1029 and an absolute error tolerance of 10212. Further, the Lyapunovexponents for the master-slave system of gyros are computed usingthe methods described in Udwadia and von Bremen (2000, 2001,2002) over a time span of 1,000 s. To determine these exponents,integration has been performed, once again, using the MATLABODE45 solver with a relative error tolerance of 1029 and an absoluteerror tolerance of 10213.

Synchronization with the Use of a Sleeping Condition

Five Gyroscopes (1 Master 1 4 Coupled Slaves) Usinga Sine Coupling

Consider a system of five nonidentical gyroscopes (1Master1 4Slaves) where the set of four slave gyroscopes are all connected

in a chain fashion using a sinusoidal coupling [described byEq. (3)] with coefficients s12 5 2, s23 5 1:5, and s34 5 2:5.All gyros are assumed to be in the sleeping position and wehave

pwi¼ pci

¼ pi, i ¼ m, 1, 2, 3, 4

Each individual slave gyro (although coupled to other slaves) isrequired to precisely track the motion of the master gyro. Table 1gives the parameter sets

Pi ¼ fai,bi, ci, ei,gi,vi, Iig, i ¼ m, 1, 2, 3, 4

that describe the characteristics of each of the gyros of the master-slave system and their initial condition sets

ICi ¼hu0i , _u

0i , t

0i

iUsing the parameters in Table 1, the Lyapunov exponents for themaster gyroscope and the system of slave gyros are computed,respectively, to be

lm ¼ f20:179707, 20:500232, 0g

ls ¼�

0:121455, 0:066865, 0:004824, 20:039699, 20:253232, 0:000107,20:568669, 20:645423, 0:000110, 20:728986, 20:777867, 0

�

Because the values of lm are either negative or zero, the master gyrois said to exhibit regular periodic motion. The slave system, on theother hand, appears to be chaotic.

The unconstrained system consists of the master gyroscope andthe set of four coupled slave gyroscopes whose equations of motioncanbewritten usingEq. (8). Fig. 3 shows a phase plot (um, _um) of theunconstrained motion of the master gyroscope. Clearly, the motionof the master is periodic, as also indicated by its Lyapunov expo-nents. Fig. 4, on the other hand, shows a phase plot (ui, _ui),i5 1, 2, 3, 4 of the unconstrained motion of the individual slavegyroscopes. The phase plots (Figs. 3 and 4) are all plotted in the timerange 150# t# 200 s. A superimposed image of the time history ofthe nutation angle u of the five gyros is plotted in Fig. 5 showing thatprior to synchronization, the five gyros exhibit highly nonlinearbehavior and their trajectories vary widely from each other.

To this unconstrained system, we impose a set of five constraints[asdescribedbyEq. (12) with trajectory stabilization constants d5 1and k5 2] and compute the constrained equations of motion usingEq. (15). A superimposed phase plot of the constrainedmotion of themaster-slave system of gyros is plotted in Fig. 6 for 150# t# 200.As can be seen from the figure, the four slave gyros which are allsinusoidally (chain) coupled to each other synchronize preciselywith the periodic motion of the master gyroscope. Fig. 7 shows thetime history of nutation angle u post-synchronization; the controlforces take less than 10 s to exponentially reduce the error betweenthe motion of master and the individual slaves to zero. Fig. 8 showsa time history of synchronization errors for 60# t# 200. Note thatthe error converges exponentially as demanded by Eq. (11). Thesynchronization errors are approximately 10213, which is lower thanthe tolerance levels used in the MATLAB ODE45 solver showing

Table 1. Parameter and Initial Conditions Sets for the Five Gyroscopes

Name Parameter sets Initial conditions

Master gyroscope (periodic) Pm5 f10:5, 1, 0:5, 0:02, 38:7, 2:2, 1g ICm5 ½21, 0:5, 0�1st slave gyroscope P1 5 f10 1, 0:5, 0:05, 35:5, 2, 1g IC1 5 ½0:5, 1, 0�2nd slave gyroscope P2 5 f10:5, 1, 0:5, 0:04, 38:5, 2:1, 2g IC2 5 ½1, 20:5, 0�3rd slave gyroscope P3 5 f10, 1, 0:5, 0:03, 35:8, 2:05, 1:5g IC3 5 ½20:5, 1, 0�4th slave gyroscope P4 5 f10:5, 1, 0:45, 0:045, 36, 2:05, 1:7g IC4 5 ½0:5, 0:5, 0�

JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013 / 819

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Fig. 3. A phase plot of the unconstrained motion of the periodic master gyroscope for 150# t# 200

Fig. 4.A phase plot of the unconstrained motion of the four slave gyroscopes which are all coupled to one another in a chain fashion (150# t# 200)

820 / JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

the efficacy of the control forces in achieving synchronization. Thus,a system of four nonidentical slave gyros which are sinusoidallychain-coupled to each other have been shown to precisely track themotion of a master gyroscope exhibiting periodic motion.

SixGyroscopes (1Master1 5CoupledSlaves) Using theIncidence Matrix

In the present example, we consider a system of six nonidenticalgyroscopes (1 Master 1 5 Coupled Slaves) where the interactionbetween the five slaves is more general and is described by an in-cidence matrix. The five coupled gyros are required to preciselytrack the motion of the independent master gyro [described by theparameter set Pm (see Table 2)], which in this case exhibits chaoticmotion as seen from its Lyapunov exponents

lm ¼ f 0:206668, 20:896620, 0 g

The coupling between the slaves (with parameter sets Pi,i5 1, 2, 3, 4, 5 given in Table 2) is described by the 535 incidencematrix j, shown as

j ¼

2666664

0 1T 0 1L 1S1T 0 1S 0 1L0 1S 0 1LC 0

1L 0 1LC 0 0

1S 1L 0 0 0

3777775 (17)

The interaction between the ith and jth slaves is characterized by thevalue of the ði, jÞth element of the matrix j. The subscripts L, S, T ,andLC of the ði, jÞth element of the incidencematrix denotes Linear,Sinusoidal, Toda, andLinear1Cubic couplings, respectively. Fig. 9is a pictorial representation of the incidence matrix showing thevarious types of couplings applied between the individual slave

Fig. 5. Time history of u for each individual gyroscope for 0# t# 20 prior to synchronization

Fig. 6. A superimposed phase plot of the constrained motion of the five gyroscopes for 150# t# 200; the slave gyroscopes have precisely syn-chronized with the motion of the master gyro

JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013 / 821

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

gyroscopes. The unconstrained equation of motion of the master[Eq. (2)] and the five coupled slave gyroscopes [Eq. (16)], which areall assumed to be in the sleeping position, can now be written as

€um ¼ Fgyrom (18)

€u1 ¼ Fgyro1 þ 1

I1

�FToda12 þ FLinear

14 þ FSine15

�(19)

€u2 ¼ Fgyro2 þ 1

I1

�FToda21 þ FSine

23 þ FLinear25

�(20)

€u3 ¼ Fgyro3 þ 1

I1

�FSine32 þ FLinear þ Cubic

34

�(21)

Fig. 7. Time history of u for each individual gyroscope for 0# t# 20 in the constrained system; slave gyros take less than 10 s to synchronize with themotion of the master gyro

Fig. 8. Time history of synchronization errors for 60# t# 200; errors are smaller than the tolerance levels of the integration scheme

Table 2. Parameter and Initial Condition Sets for the Six Gyroscopes

Name Parameter sets Initial conditions

Master gyroscope (chaotic) Pm 5 f10, 1, 0:5, 0:03, 35:8, 2:05, 1g ICm 5 ½20:5, 1, 0�1st slave gyroscope P1 5 f10, 1, 0:5, 0:05, 35:5, 2, 2g IC1 5 ½0:5, 1, 0�2nd slave gyroscope P2 5 f10:5, 1, 0:5, 0:04, 38:5, 2:1, 1:3g IC2 5 ½1, 20:5, 0�3rd slave gyroscope P3 5 f10:5, 1, 0:5, 0:02, 38:7, 2:2, 0:9g IC3 5 ½21, 0:5, 0�4th slave gyroscope P4 5 f10:5, 1, 0:45, 0:045, 36, 2:05, 1:5g IC4 5 ½0:5, 0:5, 0�5th slave gyroscope P5 5 f10:5, 1, 0:5, 0:05, 38:5, 2, 1:7g IC5 5 ½0:5, 21, 0�

Fig. 9. General-coupled slave gyroscope system depicting varioustypes of coupling/interactions between the slaves

822 / JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

€u4 ¼ Fgyro4 þ 1

I1

�FLinear41 þ FLinearþCubic

43

�(22)

€u5 ¼ Fgyro5 þ 1

I1

�FSine51 þ FLinear

52

�(23)

where Fgyroi 5 definition given by Eq. (6);FToda

ij 5 aij½ebijðuj2uiÞ 2 1�;FSineij 5sij sinðuj 2 uiÞ; FLinear

ij 5 lij ðuj 2 uiÞ; FLinear1Cubicij

5 fijðuj 2 uiÞ1 gijðuj 2 uiÞ3; and FCoupleij 5 2FCouple

ij " i, j;

i, j5 1, 2, 3, 4, 5. The spring constants are given by the followingparameters:

�a12 ¼ 1:5, b12 ¼ 0:05, l14 ¼ 0:5, s15 ¼ 2,

s23 ¼ 1:2, l25 ¼ 0:3, f34 ¼ 0:8, g34 ¼ 0:04

�

From these parameters, the Lyapunov exponents for the system ofslave gyros are calculated to be

ls ¼�

0:082678, 0:051033, 0:015758, 20:007526, 20:075204, 0:000197, 20:424779,20:483015, 0:000402, 20:523197, 20:646143, 0:000002, 20:723966, 20:787581, 0

�

Based on these exponents, the slave system appears to be chaotic.Fig. 10 shows phase plots ðui, _uiÞ, i5 m, 1, 2, 3, 4, 5 of the

unconstrained system from Eqs. (18)–(23) for 50# t# 100. Themaster gyroscope exhibits chaotic motion as predicted by lm. FromFig. 10, we can infer that all six gyros exhibit highly nonlinearbehavior and are unsynchronized. The constrained (synchronized)equations of motion for this system of six gyros can once again becomputed using Eq. (15) with the application of six constraints [seeEq. (12)]. The phase plots of the constrained system (ui vs: _ui),i5m, 1, 2, 3, 4, 5 have been plotted in Fig. 11 for (a) 50# t# 100and (b) 150# t# 200, where we observe that the plot of each gyrohas been superimposed on top of another, indicating synchronizationof the five slave gyros with the chaotic motion of the master. Fig. 12

shows a time history of the nutation angle u for 0# t# 20. As can beinferred from the figure, it takes less than 10 s for the slave gyros totrack the motion of the chaotic master gyro. Fig. 13 shows a timehistory of the synchronization errors in u for each slave gyroscope inthe time range of 60# t# 200. The vertical scale of the figure showsthat the error is approximately 10213, signifying that the errors in thesimulation are within the tolerance levels of the MATLAB ODE45solver. A time history of the generalized control torques applied toeach individual slave gyroscope to achieve synchronization with thechaotic master gyro is shown in Fig. 14. Thus, given a set of fivenonidentical general-coupled slave gyroswhere the coupling betweenthe gyros is prescribed by an incidence matrix, we managed toprecisely synchronize the motion of the each of the slave gyros with

Fig. 10. Phase plots of the unconstrained motion for each individual gyroscope for 50# t# 100; the master gyroscope is independent whereas theother remaining five slave gyroscopes are all coupled using the incidence matrix

JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013 / 823

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

that of the chaotic master gyro, something quite significant for chaoticsystems because they show extremely sensitive dependence to initialconditions.

Synchronization without the Use of theSleeping Condition

In this section, we attempt to synchronize a set of nonidenticalgyroscopes without using the so-called sleeping condition and showthat synchronization can indeed be obtained even without the needto assume that the gyros are in sleeping position. If we remove thesimplification of a sleeping condition, then we no longer assumethat pwi

5 pci5 pi and hence pwi

, pcican be chosen to be two dif-

ferent constants. Angular momenta pwi, pci

are associated with cy-clic coordinates wi and ci, respectively, and are therefore constantand cannot change with time. The imposition of the no-sleeping

condition brings an additional two parameters pwi, pci

into thedynamics of each individual gyroscope, and they are appended atthe end of the set P to give the new parameter set

Pi ¼(ai ¼ 1

Ii, bi ¼ migri

Ii, ci ¼ ~ci

Ii, ei ¼ ~ei

Ii,

gi ¼miri~div2

i

Ii, vi, pwi

, pci

)

Five Gyroscopes (1 Master 1 4 Coupled Slaves)Using a Toda Coupling

Consider a system of five nonidentical gyroscopes (with parametersets given by Table 3) where the set of four slave gyros are coupledin a chain fashion using a Toda torsion spring coupling. Each

Fig. 11. Superimposed phase plots of the constrained motion of the five gyroscopes for (a) 50# t# 100 and (b) 150# t# 200; slave gyros haveprecisely tracked the motion of the master gyro

Fig. 12. Time history of u for each individual gyroscope for 0# t# 20 in the constrained system

824 / JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

individual coupled slave gyro is required to exactly track the motionof the master gyro

lm ¼ f20:223403, 20:226832, 0 g

which, in this case, exhibits regular periodic motion. The co-efficients of the Toda spring elements are given by a125 1:20;a23 5 1:50; a34 5 1:25; b12 5 0:11; b23 5 0:13; and b34 5 0:105.The system of slave gyros appears to exhibit chaotic behavior, asindicated by its Lyapunov exponent set

ls ¼�

0:107805, 20:015994, 0:000394, 20:121534, 20:634956, 0:000173,21:390733, 21:850994, 0:000002, 21:881298, 21:976043, 0

�

Because the simplification of the sleeping condition is removed, theunconstrained equations of motion of the master and the individualslave gyros are given by Eqs. (1) and (6), respectively. A super-imposed phase plot of the unconstrained motion of the periodic

master gyroscope (um, _um) along with the unconstrained motionof each individual coupled slave gyro (ui, u

:i), i5 1, 2, 3, 4 is shown

in Fig. 15. The phase plots are all plotted in the time range50# t# 100. The nonidentical gyros exhibit highly nonlinear

Fig. 13. Time history of synchronization errors for 60# t# 200; errors are smaller than the tolerance levels of the integration scheme

Fig. 14. Generalized control torques acting on each individual slave gyroscope for 0# t# 200

Table 3. Parameter and Initial Condition Sets for the Five Gyroscopes

Name Parameter sets Initial conditions

Master gyroscope (periodic) Pm 5 f2:5, 1:5, 0:45, 0:045, 35, 3, 1:5, 2:5g ICm 5 ½1, 20:5, 0�1st slave gyroscope P1 5 f1:35, 1:1, 0:44, 0:044, 36:1, 3, 1:9, 2:1g IC1 5 ½0:5, 1, 0�2nd slave gyroscope P2 5 f1, 1:0, 0:44, 0:044, 35, 2:5, 2, 2:2g IC2 5 ½1, 20:5, 0�3rd slave gyroscope P3 5 f1:04, 1:0, 0:45, 0:045, 36:0, 2:06, 1:5, 1:8g IC3 5 ½1, 21, 0�4th slave gyroscope P4 5 f1, 1:1, 0:45, 0:045, 36:0, 2:05, 2:05, 2:15g IC4 5 ½0:5, 0:5, 0�

JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013 / 825

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Fig. 15. Superimposed phase plot depicting the unconstrained motion of the periodic master gyro along with the four chain-coupled slave gyros for50# t# 100

Fig. 16. A superimposed phase plot of the constrained motion of the five gyroscopes for 50# t# 100; the slave gyroscopes have precisely syn-chronized with the motion of the master gyro

826 / JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

behavior as expected, with their trajectories varying widely fromeach other. The constrained equations of motion of the master-slavesystem of gyroscopes are computed using Eq. (15) with the ap-plication of five constraints as described by Eq. (12). A super-imposed phase plot of the constrained motion of the master-slavesystem of gyros is plotted in Fig. 16 for 50# t# 100. As can be seenfrom the figure, the four slave gyros (which are all chain-coupled)synchronize exactly with the periodic motion of the master gyro-scope. The phase plot of each gyro has superimposed on top of theothers, indicating precise synchronization with the master gyro.Fig. 17 shows the time history of nutation angle u for the five gyrospost-synchronization. The control torques take less than 10 s tosynchronize the motion of individual slaves with that of the periodicmaster gyroscope. Fig. 18, on the other hand, shows the time history ofsynchronization errors for 60# t# 100. The exponential conver-gence of the errors to zero is evident. Andfinally, Fig. 19 shows a timehistory of the nonlinear control torques acting on the individual slavegyroscopes to obtain synchronization with the master gyro. Thus, weconclude that the removal of the sleeping condition has had little effecton the synchronization of the master-slave system of gyros.

Conclusions

In the current study, we consider the problem of synchronization ofa system of n chain-coupled or general-coupled slave gyroscopeswith that of a master gyroscope. However, this classical problem oftracking is approached from a constrained motion perspective. Thetracking requirements are recast as constraints and the fundamentalequation is used to obtain the constrained (synchronized) equationsof motion of the master-slave system of gyroscopes. In the process,explicit closed form expressions for the nonlinear control torques arederived that are used to drive the system of coupled slave gyroscopesto synchronize exactly with the motion of the master gyroscope(irrespective of the chaotic or periodic behavior displayed by themaster or the slave system of gyroscopes).1. Previous investigators concern themselves with the synchro-

nization of a set of one or two uncoupled slave gyroscopeswiththe motion of a master gyroscope. But the theory developed inthis paper allows for a set of n slave gyroscopes that are allcoupled to one another to synchronize exactly with the motionof the master gyro. Each slave gyro is a highly nonlinear

Fig. 17. Time history of u for each individual gyroscope for 0# t# 20 in the constrained system

Fig. 18. Time history of synchronization errors for 60# t# 100; errors are smaller than the tolerance levels of the integration scheme

JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013 / 827

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

system. In addition, the slaves are nonlinearly coupled to oneanother with different types of couplings, and this leads toa highly nonlinear, complex dynamical system. However, thecontrol torques are found with relative ease, showing thepower and efficacy of the underlying control methods.

2. The control torques derived in this paper are continuous in timeand in theory lead to exact synchronization of the master-slavesystem of gyroscopes. No approximations have been made inthe derivation of these nonlinear control torques, or in ap-proximating the highly nonlinear dynamics of either themasteror any of the coupled slave gyroscopes.

3. The equations of motion of a master-slave system of gyros arederived without applying the simplification of the sleepingcondition. Although many authors on this subject assume thegyros to be in sleeping position, in general, most gyros do notsatisfy this condition. In this paper, it is shown that removal ofthe sleeping condition has little effect on the synchronizationof the gyroscopes as is evident from the numerical simulation.

4. To show the efficacy of the methods presented in this paper,numerical simulations involving multiple nonidentical gyro-scopes, with various types of couplings between the slavegyros, is presented. In all the cases, the control torques pro-vided by the fundamental equation leads to an exact synchro-nization of the master-slave system of gyros irrespective of thechaotic or regular motions exhibited by the individual gyro-scopes. It should be noted that the methods presented in thispaper are applicable to any number of slave gyroscopes givenany type of general, nonlinear coupling among them. The speedwith which synchronization is achieved can be controlled byappropriately modifying the trajectory stabilization parameters.

5. The constrained motion approach and the fundamental equa-tion are powerful tools to obtain the equations of motion ofconstrained nonlinear mechanical systems. Thus, the meth-odologies provided in this paper can be readily adopted andexpanded to synchronize other general nonlinear dynamicalsystems.

References

Aghababa, M. P. (2011). “A novel adaptive finite-time controller for syn-chronizing chaotic gyros with nonlinear inputs.” Chin. Phys. B, 20(9),090505.

Chen, H. K. (2002). “Chaos and chaos synchronization of a symmetric gyrowith linear-plus-cubic damping.” J. Sound Vibrat., 255(4), 719–740.

Ge, Z. M., and Chen, H. H. (1996). “Bifurcation and chaos in rate gyro withharmonic excitation.” J. Sound Vibrat., 194(1), 107–117.

Ge, Z. M., Chen, H. K., and Chen, H. H. (1996). “The regular and chaoticmotions of a symmetric heavy gyroscope with harmonic excitation.”J. Sound Vibrat., 198(2), 131–147.

Lei, Y., Xu, W., and Zheng, H. (2005). “Synchronization of two chaoticnonlinear gyros using active control.” Phys. Lett. A, 343(1–3), 153–158.

Leipnik, R. B., and Newton, T. A. (1981). “Double strange attractors in rigidbody motion with linear feedback control.” Phys. Lett., 86(A), 63–67.

Pecora, L. M., and Carroll, T. L. (1990). “Synchronization in chaoticsystems.” Phys. Rev. Lett., 64(8), 821–824.

Strogatz, S. (2000). Nonlinear dynamics and chaos, Westview, Cambridge,MA.

MATLAB 7.13 [Computer software]. Natick, MA, The MathWorks.Toda, M. (1981). Theory of nonlinear lattices, Springer-Verlag, Berlin.Tong, X., and Mrad, N. (2001). “Chaotic motion of a symmetric gyro

subjected to harmonic base excitation.” J. Appl. Mech., 68(4), 681–684.Udwadia, F. E. (2003). “A new perspective on the tracking control of

nonlinear structural and mechanical systems.” Proc. Royal Soc. A,459(2035), 1783–1800.

Udwadia, F. E. (2000). “Nonideal constraints and Lagrangian dynamics.”J. Aerosp. Eng., 13(1), 17–22.

Udwadia, F. E. (2008). “Optimal tracking control of nonlinear dynamicalsystems.” Proc. Royal Society A, 464(2097), 2341–2363.

Udwadia, F. E., and Han, B. (2008). “Synchronization of multiple chaoticgyroscopes using the fundamental equation of mechanics.” J. Appl.Mech. 75(2), 02011.

Udwadia, F. E., and Kalaba, R. E. (1996a). Analytical dynamics: A newapproach, Cambridge University Press, Cambridge, U.K.

Udwadia, F. E., and Kalaba, R. E. (1996b). “Equations of motion formechanical systems.” J. Aerosp. Eng., 9(3), 64–69.

Udwadia, F. E., and Kalaba, R. E. (2002). “What is the general form of theexplicit equations of motion for constrained mechanical systems?”J. Appl. Mech., 69(3), 335–339.

Udwadia, F. E., and von Bremen, H. (2001). “An efficient and stable ap-proach for computation of Lyapunov characteristic exponents of con-tinuous dynamical systems.” Appl. Math. Comput., 121(2–3), 219–259.

Udwadia, F. E., and von Bremen, H. (2002). “Computation of Lyapunovcharacteristic exponents for continuous dynamical systems.” Z. Angew.Math. Phys., 53(1), 123–146.

Udwadia, F. E., von Bremen, H., and Proskurowski, W. (2000). “A note onthe computation of the largest p-Lyapunov characteristic exponents fornonlinear dynamical systems.” Appl. Math. Comput., 114(2–3), 205–214.

Van Dooren, R. (2003). “Comments on chaos and chaos synchronizationof a symmetric gyro with linear-plus-cubic damping.” J. Sound Vibrat.,268(3), 632–634.

Fig. 19. Time history of the control torques acting on each individual slave gyroscope for 0# t# 100

828 / JOURNAL OF AEROSPACE ENGINEERING © ASCE / OCTOBER 2013

J. Aerosp. Eng. 2013.26:814-828.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/25/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.