Homoclinic solutions and motion chaotization in attitude dynamics of a multi-spin spacecraft

Commun Nonlinear Sci Numer Simulat 19 (2014) 2528–2552

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Commun Nonlinear Sci Numer Simulat

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Homoclinic solutions and motion chaotization in attitudedynamics of a multi-spin spacecraft

1007-5704/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cnsns.2013.09.043

⇑ Tel.: +7 9276563036.E-mail addresses: [email protected], [email protected]

Anton V. Doroshin ⇑Space Engineering Department (Division of Flight Dynamics and Control Systems), Samara State Aerospace University (National Research University), SSAU, 34,Moskovskoe Shosse Str., Samara 443086, Russia

a r t i c l e i n f o

Article history:Received 21 February 2013Received in revised form 19 July 2013Accepted 2 September 2013Available online 18 November 2013

Keywords:Multi-rotor systemMulti-spin spacecraftDual-spin spacecraftGyrostatExact solutionsHomoclinic chaosPolyharmonic perturbationsMelnikov functionPoincaré mapRoll-walking robot

a b s t r a c t

The attitude dynamics of the multi-spin spacecraft (MSSC) and the torque-free angularmotion of the multi-rotor system are considered. Some types of homoclinic and generalsolutions are obtained in hyperbolic and elliptic functions. The local homoclinic chaos inthe MSSC angular motion is investigated under the influence of polyharmonicperturbations. Some possible applications of the multi-rotor system are indicated, includ-ing gyrostat-satellites, dual-spin spacecraft, roll-walking robots, and also the inertialessmethod of the spacecraft attitude (angular) reorientation/control.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

The fundamental problem of the rigid body angular motion and corresponding applied tasks of the space flightmechanics and especially a spacecraft attitude dynamics remain very important and attract the attention of many scien-tists. The classical results in the framework of the indicated problem are described in [1–7] and in other well-knowntreatises. The modern research directions include many aspects [8–45]: the analytical/numerical modeling, the analy-sis/synthesis of the regular/chaotic motion of multi-body systems/spacecraft/vehicles with the constant/variable structureunder the influence of perturbations. Here as parts of the problem we can indicate the comprehensive investigation ofthe attitude motion of a dual-spin spacecraft and gyrostats [8–17], the analysis/synthesis of the multi-rotor systems/spacecraft/vehicles dynamics [18–21], the general/homoclinic/heteroclinic solutions obtaining [22–31], the local/globalchaos exploration [35–45].

First of all, we should indicate the dual-spin spacecraft (DSSC) and gyrostat-satellites (GS) as the important space-flightmechanics’ application of the multi-body systems. The two-body construction of the DSSC allows fulfill the gyroscopic atti-tude stabilization with the help of the rotation of only one of the DSSC’s bodies (the ‘‘rotor’’-body) at the ‘‘quiescence’’ of thesecond body (the ‘‘platform’’-body). The GS also contains the rotating rotors (usually one or three) for the creation of the

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A.V. Doroshin / Commun Nonlinear Sci Numer Simulat 19 (2014) 2528–2552 2529

gyrostatic momentum, and, moreover, the relative angular velocities of its rotors (with respect to the main GS’ body) areconstant. The investigation of various aspects of the DSSC/GS attitude dynamics is the urgent task which is considered indifferent formulations, e.g., [8–17].

As the further generalization of the GS construction we can present a multi-rotor (multi-spin) spacecraft. So, the multi-spin spacecraft (MSSC) is constructed as the multi-rotor system with conjugated pairs of rotors placed on the all inertiaprinciple axes of the main body (Fig. 1). General properties of the MSSC motion are connected with the internal multi-rotorsystem (the multi-rotor kernel). This ‘‘spider’’-type multi-rotor system was described in [18–21] where the attitude dynam-ics, spatial (attitude) reorientations of the MSSC and also roll-walking motions of multi-rotor robots are considered. The mul-ti-rotor kernel allows to perform the attitude gyroscopic stabilization of the MSSC with the help of a compound spinup of therotors. Also we can use the MSSC in the traditional DSSC/GS-regimes. One of the important features of the MSSC is numerousindependent internal degrees of freedom corresponding to the rotors’ rotations. It is the powerful instrument for the space-craft’s attitude control and/or the angular reorientation with the help of an internal redistribution of the system angularmomentum between the rotors and the main body.

For the purposes of the synthesis of necessary (in the framework of space missions) regimes of the DSSC/GS/MSSC atti-tude dynamics we need, first of all, to obtain the essential solutions and to analyze the properties of the torque-free angularmotion of the corresponding mechanical systems – coaxial bodies, gyrostats and multi-rotor systems. These importantsolutions for DSSC/GS motion were obtained in different formulations in the papers [22–31]. The mentioned articles includethe general and particular (homo/heteroclinic) solutions which can be used for the study of the SC weakly perturbed motionunder the influence of external/internal disturbances (the gravitation/magnetic influence, the resistant environment, theconstruction asymmetry, etc.).

Here we have to note the importance of the indicated homo/heteroclinic solutions for the investigation of the local cha-otic motion with the help of Melnikov’s [35], Wiggins’ [36] and Holmes’–Marsden’s [37] approaches. The research results forthe chaotic DSSC/GS motion analysis were presented, for example, in the papers [29–31,38–41]. Also the study of the non-regular motion modes or, on the contrary, the study of the regularization/synchronization properties can be performed withthe help of the special methods [42–45].

So, in this paper the MSSC attitude dynamics is considered, some homoclinic and general exact solutions are obtained,and cases of chaotic and regular motion under the influence of polyharmonic disturbances are investigated.

2. Mechanical and mathematical models of the system

Let us investigate the torque-free attitude dynamics of the MSSC and the angular motion of the multi-rotor mechanicalsystem (Fig. 1) about its mass center (the ‘‘fixed’’ point O).

First of all, we need to consider the multi-rotor system with six identical rotors symmetrically placed on the general axesof the main body (Fig. 1a). Then the angular momentum of the system can be written in projections onto the frame Oxyz axesconnected with the central main body

Fig. 1. The multi-rotor systems (the multi-spin spacecraft).

2530 A.V. Doroshin / Commun Nonlinear Sci Numer Simulat 19 (2014) 2528–2552

K ¼ Km þ Kr ð1:1Þ

Km ¼~Ap~Bq~Cr

264

375þ ð4J þ 2IÞ

p

q

r

264375; Kr ¼ I

r1 þ r2

r3 þ r4

r5 þ r6

264

375 ð1:2Þ

where Km is the angular momentum of the main body with the fixed (‘‘frozen’’) rotors; Kr is the relative angular momentumof the rotors; x = [p,q, r]T – the vector of the absolute angular velocity of the main body; ri – the relative angular velocity ofthe i-th rotor; ~A; ~B; ~C are the general moments of inertia of the main body; I is the longitudinal inertia moment of the rotor; Jis the equatorial inertia moment of the rotor calculated relatively the point O.

The angular motion equations of the system can be written with the help of the law of the angular momentum’s variationin the moving coordinates frame Oxyz

dKdtþx� K ¼Me ð1:3Þ

where Me is the external torque. The vector Eq. (1.3) can be rewritten

A _pþ I _r12 þ ðC � BÞqr þ Iðqr56 � rr34Þ ¼ Mex

B _qþ I _r34 þ ðA� CÞpr þ Iðrr12 � pr56Þ ¼ Mey

C _r þ I _r56 þ ðB� AÞpqþ Iðpr34 � qr12Þ ¼ Mez

8><>: ð1:4Þ

where

rij ¼ ri þ rj; A ¼ ~Aþ 4J þ 2I

B ¼ ~Bþ 4J þ 2I; C ¼ ~C þ 4J þ 2Ið1:5Þ

The rotors’ relative motion equations are written in the form

Ið _pþ _r1Þ ¼ Mi1 þMe

1x; Ið _pþ _r2Þ ¼ Mi2 þMe

2x

Ið _qþ _r3Þ ¼ Mi3 þMe

3y; Ið _qþ _r4Þ ¼ Mi4 þMe

4y

Ið_r þ _r5Þ ¼ Mi5 þMe

5z; Ið_r þ _r6Þ ¼ Mi6 þMe

6z

8>><>>: ð1:6Þ

where Mij is the torque of the internal forces acting between the main body and the j-th rotor; Me

jx; Mejy; Me

jz are the torquesfrom the external forces acting only on the j-th rotor.

The Eqs. (1.4) and (1.6) form the complete dynamical system.We can generalize the dynamical equations for the multi-rotor system with 6N rotors (Fig. 1b). This system contains N

layers with rotors on the six general directions coinciding with the principle axes of the main body. We also assume thateach layer contains the equal rotors, but the rotors in the different layers are different. For the angular momentum of thegeneralized 6N-rotors-system we have

Km ¼ApBq

Cr

264

375; Kr ¼

XN

l¼1

Il

r1l þ r2l

r3l þ r4l

r5l þ r6l

264

375 ð1:7Þ

A ¼ ~Aþ 4�J þ 2�I; B ¼ ~Bþ 4�J þ 2�I; C ¼ ~C þ 4�J þ 2�I

�J ¼XN

l¼1

Jl;�I ¼

XN

l¼1

Il

Here rkl is the relative angular velocity of the kl-th rotor (relatively the main body); Il and Jl are the longitudinal and theequatorial inertia moments of the l-layer-rotor relatively the point O.

Then the Eq. (1.3) can be written in the following scalar form

A _pþXN

l¼1

Ilð _r1l þ _r2lÞ þ ðC � BÞqr þ qXN

l¼1

Ilðr5l þ r6lÞ � rXN

l¼1

Ilðr3l þ r4lÞ" #

¼ Mex

B _qþXN

l¼1

Ilð _r3l þ _r4lÞ þ ðA� CÞpr þ rXN

l¼1

Ilðr1l þ r2lÞ � pXN

l¼1

Ilðr5l þ r6lÞ" #

¼ Mey

C _r þXN

l¼1

Ilð _r5l þ _r6lÞ þ ðA� CÞqpþ pXN

l¼1

Ilðr3l þ r4lÞ � qXN

l¼1

Ilðr1l þ r2lÞ" #

¼ Mez

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

ð1:8Þ


The relative motion equations of the rotors are (l = 1, . . . ,N)

Ilð _pþ _r1lÞ ¼ Mi1l þMe

1lx; Ilð _pþ _r2lÞ ¼ Mi2l þMe

2lx

Ilð _qþ _r3lÞ ¼ Mi3l þMe

3ly; Ilð _qþ _r4lÞ ¼ Mi4l þMe

4ly

Ilð_r þ _r5lÞ ¼ Mi5l þMe

5lz; Ilð_r þ _r6lÞ ¼ Mi6l þMe

6lz

8>><>>: ð1:9Þ

The equation system (1.8) with N systems (1.9) completely describe the angular motion of the multi-rotor system with6N rotors and the attitude dynamics of the multi-spin spacecraft (Fig. 1b).

3. The Hamiltonian form of the motion equations

Now we construct the mathematical model of the N-layers-multi-rotor system’s motion in the Hamiltonian form.The kinetic energy expressions for the six rotors in the j-layer are

2T1j ¼ Jjðq2 þ r2Þ þ Ijðpþ r1jÞ2; 2T2j ¼ Jjðq2 þ r2Þ þ Ijðpþ r2jÞ2

2T3j ¼ Jjðp2 þ r2Þ þ Ijðqþ r3jÞ2; 2T4j ¼ Jjðp2 þ r2Þ þ Ijðqþ r4jÞ2

2T5j ¼ Jjðp2 þ q2Þ þ Ijðr þ r5jÞ2; 2T6j ¼ Jjðp2 þ q2Þ þ Ijðr þ r6jÞ2ð2:1Þ

Then the full system’s kinetic energy is equal to

T ¼ T0 þXN

j¼1

Tj; 2T0 ¼ ~Ap2 þ ~Bq2 þ ~Cr2; Tj ¼X6

i¼1

Tij

2Tj ¼ ð2Ij þ 4JjÞ p2 þ q2 þ r2� �

þ 2Ij p½r1j þ r2j� þ q½r3j þ r4j� þ r½r5j þ r6j��

þ Ij

X6

i¼1

r2ij

ð2:2Þ

In purposes of the Hamiltonian formalism application we introduce the well-known Andoyer–Deprit canonical variables[6,7]. With the help of these variables the angular motion of the system’s main body is described by the angles ðu3; u2; lÞ ofthe rotations about the axis OZ, about the angular momentum direction, and about the axis Oz (Fig. 2). The canonical Ando-yer–Deprit momentums are defined by the following expressions

L ¼ @T

@_l¼ K � k; G ¼ @T

@ _u2¼ K � s ¼ K; H ¼ @T

@ _u3¼ K � k0; Dij ¼

@T

@ _dij

¼ @T@rij

ð2:3Þ

where dij is the relative rotation angle of the ij-rotor ð _dij ¼ rijÞ.The components of the system angular momentum can be expressed as the functions of the Andoyer–Deprit variables:

Kx ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG2 � L2

psin l ¼ Apþ

XN

j¼1

Ijðr1j þ r2jÞ

Ky ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG2 � L2

pcos l ¼ Bqþ

XN

j¼1

Ijðr3j þ r4jÞ

Kz ¼ L ¼ Cr þXN

j¼1

Ijðr5j þ r6jÞ; ðL 6 GÞ

ð2:4Þ

The canonical momentums for the relative motion of the j-layer-rotors are

D1j ¼ Ijðpþ r1jÞ; D2j ¼ Ijðpþ r2jÞD3j ¼ Ijðqþ r3jÞ; D4j ¼ Ijðqþ r4jÞD5j ¼ Ijðr þ r5jÞ; D6j ¼ Ijðr þ r6jÞ

ð2:5Þ

From (2.5) the expressions follow

XN

j¼1

Ijðr1j þ r2jÞ ¼XN

j¼1

ðD1j þ D2jÞ � 2pXN

j¼1

Ij ð2:6Þ

XN

j¼1


j¼1

ðD3j þ D4jÞ � 2qXN

j¼1

Ij ð2:7Þ

Fig. 2. The Andoyer–Deprit variables and the system’s main body frame.


XN

j¼1


j¼1

ðD5j þ D6jÞ � 2rXN

j¼1

Ij ð2:8Þ

Using the expressions (2.5)–(2.8) the components of the main body angular velocity can be written

p ¼ 1

A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG2 � L2

psin l�

XN

j¼1

ðD1j þ D2jÞ" #

ð2:9Þ

q ¼ 1

B


pcos l�

XN

j¼1

ðD3j þ D4jÞ" #

ð2:10Þ

r ¼ 1

CL�

XN

j¼1

ðD5j þ D6jÞ" #

ð2:11Þ

where A ¼ A� 2PN

j¼1Ij; B ¼ B� 2PN

j¼1Ij; C ¼ C � 2PN

j¼1Ij.

Also we can rewrite the system’s kinetic energy and the angular momentum’s components:

T ¼ T0 þXN

j¼1

Tj; 2T0 ¼ ~Ap2 þ ~Bq2 þ ~Cr2; Tj ¼X6

i¼1

Tij

2T1j ¼ Jjðq2 þ r2Þ þ D21j

Ij; 2T2j ¼ Jjðq2 þ r2Þ þ D2

2j

Ij

2T3j ¼ Jjðp2 þ r2Þ þ D23j

Ij; 2T4j ¼ Jjðp2 þ r2Þ þ D2

4j

Ij

2T5j ¼ Jjðp2 þ q2Þ þ D25j

Ij; 2T6j ¼ Jjðp2 þ q2Þ þ D2

6j

Ij

2Tj ¼ 4Jjðp2 þ q2 þ r2Þ þ 1Ij

X6

i¼1

D2ij

2XN

j¼1

Tj ¼ ðp2 þ q2 þ r2ÞXN

j¼1

ð4JjÞ þXN

j¼1

X6

i¼1

D2ij

Ij

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

ð2:12Þ

2T ¼ Ap2 þ Bq2 þ Cr2 þ 2TR; TR ¼12

XN

j¼1

X6

i¼1

D2ij

Ijð2:13Þ


Kx ¼ ApþXN

j¼1

ðD1j þ D2jÞ ¼ Apþ D12

Ky ¼ BqþXN

j¼1

ðD3j þ D4jÞ ¼ Bqþ D34

Kz ¼ Cr þXN

j¼1

ðD5j þ D6jÞ ¼ Cr þ D56

ð2:14Þ

K2 ¼ ðApþ D12Þ2þ ðBqþ D34Þ

2þ ðCr þ D56Þ

2ð2:15Þ

where Dnm are the following axial summarized angular momentums of the rotors:

D12 ¼XN

j¼1

½D1j þ D2j�; D34 ¼XN

j¼1


j¼1

½D5j þ D6j� ð2:16Þ

Taking into account the expressions (2.16), (2.6)–(2.8), the main dynamical equations (1.8) are rewritten in the unbal-anced-gyrostat-form

A _pþ _D12 þ ðC � BÞqr þ ½qD56 � rD34� ¼ Mex

B _qþ _D34 þ ðA� CÞrpþ ½rD12 � pD56� ¼ Mey

C _r þ _D56 þ ðB� AÞpqþ ½pD34 � qD12� ¼ Mez

8>><>>: ð2:17Þ

Based on (1.9) we obtain the equations for the rotors’ summarized angular momentums (2.16):

_D12 ¼ Mi12 þMe

12; _D34 ¼ Mi34 þMe

34; _D56 ¼ Mi56 þMe

56 ð2:18Þ

where

Mi12 ¼

XN

l¼1

ðMi1l þMi

2lÞ; Me12 ¼

XN

l¼1

ðMe1lx þMe

2lxÞ

Mi34 ¼

XN

l¼1

ðMi3l þMi

4lÞ; Me34 ¼

XN

l¼1

ðMe3ly þMe

4lyÞ

Mi56 ¼

XN

l¼1

ðMi5l þMi

6lÞ; Me56 ¼

XN

l¼1

ðMe5lz þMe

6lzÞ

With the help of (2.9)–(2.11) we express the system kinetic energy (2.2) as the function of the Andoyer–Deprit variables:

2T ¼ ðG2 � L2Þ sin2 l

Aþ cos2 l

B

" #þ 1

CL�

XN

j¼1

½D5j þ D6j� !2

� 2

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG2 � L2

p sin l

A�XN

j¼1

½D1j þ D2j� þcos l

B�XN

j¼1

½D3j þ D4j�( )

þ 1

A

XN

j¼1

½D1j þ D2j� !2

þ 1

B

XN

j¼1

½D3j þ D4j� !2

þXN

j¼1

X6

i¼1

D2ij

Ijð2:19Þ

The Hamiltonian and canonical equations have the form (P – the system potential energy)

H ¼T þ P ¼ G2 � L2

2sin2 l

Aþ cos2 l

B

" #þ 1

2CðL� D56Þ2 �


p D12 sin l

Aþ D34 cos l

B

� �þ D2

12

2Aþ D2

34

2Bþ 1

2

XN

j¼1

X6

i¼1

D2ij

Ij

þ Pðl;u2;u3; dijÞD12 ¼XN

j¼1


j¼1


j¼1

½D5j þ D6j�; ð2:20Þ

_x ¼ @H@X

; _X ¼ � @H@x

x ¼ fl;u2;u3; dijg; X ¼ fL;G;H;Dijgð2:21Þ

Fig. 3. The angular momentum ellipsoid with the plane projections A = 0.5, B = 0.5, C = 0.7 (kg m2); D34 = 0.1, K⁄ = 1 (kg m2/s).


In the case of the absence of internal/external forces and torques (P = 0) the system’s kinetic energy and the angularmomentum are constant

T ¼ E ¼ const; K ¼ G ¼ const ð2:22Þ

Here we can to note, that the expression (2.20) is the important generalization of the Hamiltonian’s form corresponded tothe task of the one-rotor-gyrostat’s angular motion [14,24–28].

4. The system phase portraits and the homoclinic orbits

Let’s investigate the system torque-free motion ðMex ¼ Me

y ¼ Mez ¼ Me

ij ¼ 0Þ and obtain the exact solutions for thehomoclinic phase trajectories in the three-dimensional space corresponding to the angular velocities components (or inthe topologically equivalent space of the angular momentum components). In this research we assume

A 6 B < C; D12 ¼ D56 ¼ 0; D34 ¼ const > 0: ð2:23Þ

Here we note that the assumption (2.23) is valid for the realization of the symmetrical spinup of the rotors on the rays 1–2 and 5–6 (Fig. 1). Therefore the total values of the summarized angular momentums of these rotors are equal to zero(D12 = D56 = 0). The equatorial axis Oy (the intermediate inertia axis) contains the rotors with the nonzero total angularmomentum D34.


4.1. The case of the dynamical symmetry of the main body

First of all let’s consider the angular motion of the system with the dynamical symmetry of the main body ðA ¼ BÞ.In this case we have the following structure of the polhodes on the ellipsoid of the angular momentum (Fig. 3)1 (e.g.,

[2,24,27,28,31]). This structure has one homoclinic trajectory (the homoclinic polhode) with the saddle pointSðp ¼ 0; q ¼ q� > 0; r ¼ 0Þ and with two loops (blue line). Also we need to note that in the considering case the followinginequality takes place along the homoclinic trajectory:

1 Theon the

8t 2 ð�1;þ1Þ : qðtÞ 6 q� ð2:24Þ

The constants (2.22) for the homoclinic trajectory (at the arbitrary value q⁄) are equal to

2E ¼ 2E� ¼ Aq2� þ 2TR; K2 ¼ K2

� ¼ ðAq� þ D34Þ2

ð2:25Þ

Now we can write the dependences for the homoclinic polhode with the help of the combination of expressions (2.15),(2.13) ((2.13) with the multiplier A) and (2.25) [28,31]

CðC � AÞr2 þ 2AD34q ¼ P

P ¼ K2� � 2AE� þ 2ATR � D2

34 ¼ 2Aq�D34

ð2:26Þ

From (2.26) the parabola’s dependence follows (for the corresponding curves on the Oqr-projection)

q� � q ¼ CðC � AÞ2AD34

r2

r ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2AD34ðq� � qÞ

CðC � AÞ

vuut ð2:27Þ

Using expressions (2.15), (2.13) ((2.13) with the multiplier C) and the perfect square separating we can obtain:

AðC � AÞp2 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAðC � AÞ

qq� D34

ffiffiffiA

pffiffiffiffiffiffiffiffiffiffiffiffiC � A

p" #2

¼ R

R ¼ CAq2� � ðAq� þ D34Þ

2þ D2

34 þA

C � AD2

34 ¼ A q�ffiffiffiffiffiffiffiffiffiffiffiffiC � A

p� D34ffiffiffiffiffiffiffiffiffiffiffiffi

C � Ap

!2ð2:28Þ

From (2.28) the ellipse’s equation follows (for the corresponding curves on the Opq-projection)

p ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq� �

D34

C � A

� �2

� q� D34

C � A

� �2s

¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq� þ q� 2D34

C � A

ðq� � qÞ

sð2:29Þ

In the considered case the second equation (2.17) can be rewritten

A _q ¼ ðC � AÞpr ð2:30Þ

Here we can note that the following coordinates for the point S and the vertices V1,2 (Fig. 3) take place

pðSÞ ¼ 0; qðSÞ ¼ q�

pðV1;2Þ ¼ 0; qðV1;2Þ ¼ qV ¼2D34

C � A� q�

Using (2.27) and (2.29) we rewrite the last Eq. (2.30) in the form

A _q ¼ �ðC � AÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2AD34

CðC � AÞ

vuut ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqþ q� �

2D34

C � A

sðq� � qÞ ð2:31Þ

The integration of (2.31) gives us

Z q

q0

dqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqþ q� � 2D34

C�A

� �rðq� � qÞ

¼ �Z t

0Mdt; M ¼ ðC � AÞ

A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2AD34

CðC � AÞ

vuut ð2:32Þ

numerical modeling (and the numerical results plotting) in the angular momentum (or in the angular velocity) components’ space is performed basednumerical integration of the Eqs. (2.17) and (2.18).

Fig. 4. The homoclinic dependences: the analytical results (points) and the numerical integrations (lines) A = 0.5, B = 0.5, C = 0.7 (kg m2); D34 = 0.1, K⁄ = 1(kg m2/s).


where we can take the vertex V1,2 as the start of the homoclinic trajectory with the corresponding initial q-value:qjt¼0 ¼ q0 ¼ qV ¼ 2D34

C�A� q�; then the saddle point S is the end of the homoclinic trajectory at t ? ±1. The reduced form of

the integral (2.32) can be written as follows

Z q

qV

dqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðq� q� þ bÞ

pðq� q�Þ

¼ �Z t

0Mdt

M ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðC � AÞD34

AC

s; b ¼ 2q� �

2D34

C � A> 0

ð2:33Þ

The last integral can be evaluated with the help of the standard quadrature [33]

Zdx

xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaxþ bÞ

p ¼ 1ffiffiffibp ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiaxþ bp

�ffiffiffibp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiaxþ bp

þffiffiffibp

As the result we obtain

�Mt ¼ 1ffiffiffibp ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq� q� þ b

p�

ffiffiffibp


pþ

ffiffiffibp

q

qV

¼ 1ffiffiffibp ln


p�

ffiffiffibp


pþ

ffiffiffibp

ð2:34Þ

Using (2.24) we can solve the problem with modules in (2.34):

�Mt ¼ 1ffiffiffibp ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2q� � 2D34

C�A

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqþ q� � 2D34

C�A

q� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqþ q� � 2D34

C�A

qþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2q� � 2D34

C�A

q� � ð2:35Þ

After some transformations the analytical dependences for the homoclinic polhode follow from (2.35):

qðtÞ ¼ 2q� �2D34

C � A

1� expð�ffiffiffibp

MtÞ� �2

expð�ffiffiffibp

MtÞ þ 1� �2 � q� þ

2D34

C � A

pðtÞ ¼ pðqðtÞÞ; rðtÞ ¼ rðqðtÞÞ

ð2:36Þ

where the explicit homoclinic dependences p(t) and r(t) are obtained with the help of the substitution of the solution (2.36)into expressions (2.29) and (2.27). The verification (Fig. 4) demonstrates the full coincidence of the analytical results (2.36)and the numerical calculations.

Fig. 5. The correspondence between the angular momentum’s ellipsoid and the Andoyer–Deprit phase space A = 0.5, B = 0.5, C = 0.7 (kg m2); D34 = 0.1,K⁄ = 1 (kg m2/s).


As we can see (Fig. 5)2 the Andoyer–Deprit phase space has the virtual heteroclinic phase trajectories (red lines) with thevirtual saddles-points (SV) connected with the following initial conditions

2 The(2.21).

fLð0Þ ¼ �G ¼ �K�g () rð0Þ ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2� � D2

34

qC

; pð0Þ ¼ qð0Þ ¼ 0

8<:

9=;

These virtual saddles points and the corresponding virtual heteroclinic trajectories vanish at the topological transforma-tion of the Andoyer–Deprit plane to the angular momentum’s spherical structure: in the angular momentum’s space thesephase trajectories correspond to the ordinary polhodes with the maximal absolute values of Kz-components (|Kz| = K⁄).

4.2. The case of the three-axial inertia tensor of the main body

Let’s consider the motion of the system with the general inertia tensor of the main body ðA < B < CÞ. Corresponding struc-tures of the angular momentum ellipsoid (e.g., [2,24,27,28,31]) and the Andoyer–Deprit phase space are presented at Figs. 6and 7. We have two homoclinic two-loop orbits with the saddle points S1, S2 and the loop-vertex points V1,2 and W1,2.

In this case we have the following values for constants (2.22) at the defined value q⁄

2E ¼ 2E� ¼ Bq2� þ 2TR; K2 ¼ K2

� ¼ ðBq� þ D34Þ2

ð2:37Þ

The coordinates of the saddle points are

S1ðp ¼ 0; q ¼ q�; r ¼ 0Þ

S2 p ¼ 0; q ¼ �K� þ D34

B¼ �q� �

2D34

B; r ¼ 0

ð2:38Þ

Similarly to the previous case we can write the dependences for the homoclinic polhode. Using expressions (2.15), (2.13),(2.37) ((2.13) with the multiplier A) and the perfect square separating we can obtain the ellipse’s dependence:

CðC � AÞr2 þ BffiffiffiffiffiffiffiffiffiffiffiffiB� A

pqþ D34ffiffiffiffiffiffiffiffiffiffiffiffi

B� Ap

!2

¼ P

P ¼ K2� � 2AE� þ 2ATR � D2

34 þBD2

34

B� A¼ B

ffiffiffiffiffiffiffiffiffiffiffiffiB� A

pq� þ

D34ffiffiffiffiffiffiffiffiffiffiffiffiB� A

p" #2

ð2:39Þ

From (2.39) the expression follows

numerical modeling (and the numerical results plotting) in the Andoyer–Deprit phase space is performed based on the numerical integration of the Eq.

Fig. 6. The angular momentum ellipsoid with the plane projections A = 0.5, B = 0.7, C = 0.9 (kg m2); D34 = 0.1, K⁄ = 1 (kg m2/s).


r ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB

CðC � AÞ

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB� A

pq� þ

D34ffiffiffiffiffiffiffiffiffiffiffiffiB� A

p !2

�ffiffiffiffiffiffiffiffiffiffiffiffiB� A

pqþ D34ffiffiffiffiffiffiffiffiffiffiffiffi

B� Ap

!2vuut

and in the rewritten form

r ¼ �


CðC � AÞ

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðB� AÞ½q� þ q� þ 2D34

� �½q� � q�

rð2:40Þ

Using expressions (2.15), (2.13), (2.37) ((2.13) with the multiplier C) and the perfect square separating we obtain onceagain the ellipse’s dependence:

AðC � AÞp2 þ BffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðC � BÞ

qq� D34ffiffiffiffiffiffiffiffiffiffiffiffi

C � Bp

" #2

¼ R

R ¼ CBq2� � Bq� þ D34

� �2þ D2

34 þB

C � BD2

34 ¼ BffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðC � BÞ

qq� �

D34ffiffiffiffiffiffiffiffiffiffiffiffiC � B

p !2

ð2:41Þ

From (2.41) the expression follows

p ¼ �


AðC � AÞ

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðC � BÞ

qq� �

D34ffiffiffiffiffiffiffiffiffiffiffiffiC � B

p !2

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðC � BÞ

qq� D34ffiffiffiffiffiffiffiffiffiffiffiffi

C � Bp

" #2vuut ;

Fig. 7. The correspondences between the angular momentum ellipsoid and the Andoyer–Deprit phase space A = 0.5, B = 0.7, C = 0.9 (kg m2); K⁄ = 1 (kg m2/s); (a) D34 = 0.10, (b) D34 = 0.12, (c) D34 = 0.15 (kg m2/s).



or in the rewritten form

p ¼ �


AðC � AÞ

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiððC � BÞ½q� þ q� � 2D34Þ½q� � q�

qð2:42Þ

The coordinates of the loop-vertex points V1,2 and W1,2 are

qðV1;2Þ ¼ qV ¼ �q� þ2D34

ðC � BÞ; pðV1;2Þ ¼ 0

qðW1;2Þ ¼ qW ¼ �q� �2D34

ðB� AÞ; rðW1;2Þ ¼ 0

ð2:43Þ

Now we can rewrite the second equation (2.17)

_q ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðB� AÞðC � BÞ

qffiffiffiffiffiffiAC

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½q� þ q� þ 2D34

ðB� AÞ

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½q� þ q� � 2D34

ðC � BÞ

s½q� � q�

and in the rewritten form

_q ¼ �Mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½q� � q� � a

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½q� � q� � b

p½q� � q�

a ¼ 2q� þ 2D34

ðB�AÞ > 0; b ¼ 2q� � 2D34ðC�BÞ > 0; M ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðB�AÞðC�BÞp ffiffiffiffi

ACp

The following quadrature takes place after the integration

�Z t

0Mdt ¼

Z q

q0

d½q� q��ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½q� q�� þ a

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½q� q�� þ b

p½q� q��

ð2:44Þ

It is the well-known standard integral [33]
Zdx
xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia0 þ a1xþ a2x2

p ¼ � 1ffiffiffiffiffia0p ln

2a0 þ a1xþ 2ffiffiffiffiffia0p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a0 þ a1xþ a2x2p

xð2:45Þ

Using (2.45) the quadrature (2.44) can be rewritten

�Z t

0Mdt ¼

Z q

q0

d½q� � q�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½q� � q� � a

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½q� � q� � b

p½q� � q�Z

dx

xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia0 þ a1xþ a2x2

p ¼ � 1ffiffiffiffiffia0p FðxÞ

FðxÞ ¼ ln2a0 þ a1xþ 2

ffiffiffiffiffia0p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a0 þ a1xþ a2x2p

xa2 ¼ 1; a1 ¼ aþ b; a0 ¼ ab

�Mt ¼ � 1ffiffiffiffiffiffiabp Fðq� q�Þ

q

q0

¼ � 1ffiffiffiffiffiffiabp ½Fðq� q�Þ � Fðq� q�Þ�

ð2:46Þ

Here we note the following circumstances:

(1) At the solutions finding for the homoclinic loops with the saddle S1 we have to take the coordinates of the vertices V1,2

as the initial q-values:

q0 ¼ qV ¼ �bþ q�; Fðq0 � q�Þ ¼ FðqV � q�Þ ¼ ln EV ; EV ¼ b� a: ð2:47Þ

(2) At the solutions finding for the homoclinic loops with the saddle S2 we have to take the coordinates of the vertices W1,2

as the initial q-values:

q0 ¼ qW ¼ �aþ q�; Fðq0 � q�Þ ¼ FðqW � q�Þ ¼ ln EW ; EW ¼ a� b: ð2:48Þ

So, the following solution takes place:

�ffiffiffiffiffiffiabp

Mt ¼ ln2abþ ðaþ bÞ½q� q�� þ 2

ffiffiffiffiffiffiabp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð½q� q�� þ aÞð½q� q�� þ bÞp

½q� q��Const; ð2:49Þ

where Const ¼ fEV or EWg.


After some transformations the homoclinic dependence q(t) follows from (2.49); and the reverse reductions give thehomoclinic dependencies pðtÞ; rðtÞ:

Fig. 8.(kg m2/

qðtÞ ¼ q� þ4ab exp½�

ffiffiffiffiffiffiabp

Mt�Const

exp½�ffiffiffiffiffiffiabp

Mt�Const� a� b� �2

� 4ab

pðtÞ ¼ pðqðtÞÞ; rðtÞ ¼ rðqðtÞÞ

ð2:50Þ

At the Fig. 8 the full coincidence of the analytical results (2.50) with the numerical calculations is demonstrated.

4.3. The general exact explicit solutions

Let’s obtain the general form of exact solutions in the Jacobi elliptic functions. Similarly to the previous cases, usingexpressions (2.13), (2.15) ((2.13) with the multiplier A) we can write the ellipse’s dependence in the general form for thearbitrary values of the constants (2.22):

CðC � AÞr2 þ B B� A� �

qþ D34

B� A

2

¼ P

P ¼ K2 � 2AEþ 2ATR � D234 þ

BD234

B� A> 0

ð2:51Þ

Also with the help of (2.13) and (2.15) ((2.13) with the multiplier C) we can write the following expression:

AðC � AÞp2 þ BðC � BÞ q� D34

C � B

� �2

¼ R

R ¼ 2EC � 2TRC � K2 þ D234 þ

B

C � BD2

34 > 0

ð2:52Þ

With the help of (2.51) and (2.52) it is possible to express the p- and r-components of the angular velocity as the functions ofthe q-component, then the substitution of these expressions fp ¼ pðqÞ; r ¼ rðqÞg into the second equation (2.17) gives us

dqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� c2ðqþ aÞ2

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� d2ðq� bÞ2

q ¼ � cd

BffiffiffiffiffiffiAC

p dt

a ¼ D34=ðB� AÞ > 0; b ¼ D34=ðC � BÞ > 0

c2 ¼ BðB� AÞ=P > 0; d2 ¼ BðC � BÞ=R > 0

ð2:53Þ

The homoclinic dependences: The analytical results (points) and the numerical integrations (lines) A = 0.5, B = 0.7, C = 0.9 (kg m2); K⁄ = 1, D34 = 0.15s).


Now we can use the change of the variables

z ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaþ 1=cþ b� 1=dÞðaþ 1=cþ bþ 1=dÞ

ðq� b� 1=dÞðq� bþ 1=dÞ

sð2:54Þ

Then after integration of (2.53) the quadrature follows

Fðz; kÞ ¼ �N½t � t0� þ I0 ð2:55Þ

where the elliptic integral of the first kind F(z,k) takes place [34]

Fðz; kÞ ¼R z

0dzffiffiffiffiffiffiffiffi

1�z2p ffiffiffiffiffiffiffiffiffiffiffi

1�k2z2p ; k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaþbÞ2�ð1=dþ1=cÞ2

ðaþbÞ2�ð1=d�1=cÞ2

r< 1

I0 ¼ Fðz0; kÞ ¼ const; N ¼ c2d2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaþbÞ2�ð1=d�1=cÞ2p

2BffiffiffiffiACp

After the inversion of the elliptic integral (2.55) we obtain the exact explicit solution in terms of elliptic functions (‘‘ampli-tude’’ and ‘‘elliptic sine’’):

zðtÞ ¼ snð�N½t � t0� þ I0; kÞ ð2:56Þ

where

snðu; kÞ ¼ sinðuðuÞÞ; uðuÞ ¼ amðu; kÞ ¼ U�1ðuÞu ¼ UðuÞ ¼

Ru0

d#ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�k2 sin2 #

p ¼ �N½t � t0� þ I0

The reverse reductions give us the exact explicit solutions for the angular velocity’s components fp; q; rg:

qðtÞ ¼ ðaþ 1=cþ bþ 1=dÞðb� 1=dÞsn2 �N½t � t0� þ I0; kð Þ � aþ 1=cþ b� 1=dð Þðbþ 1=dÞaþ 1=cþ bþ 1=dð Þsn2 �N½t � t0� þ I0; kð Þ � aþ 1=cþ b� 1=dð Þ

p ¼ pðqðtÞÞ; r ¼ rðqðtÞÞð2:57Þ

Also similarly to the articles [1–3,28] if the vector of the angular momentum K is directed precisely along the axis OZ (italways can be realized by the OXYZ-frame rotation), we obtain the exact solutions for the Euler angles (w – the precessionangle, h – the nutation angle, u – the intrinsic rotation angle):

cos hðtÞ ¼ CrðtÞ=K; tguðtÞ ¼ ApðtÞBqðtÞ þ D34

wðtÞ � w0 ¼Z t

t0

KKxpþ Kyq

K2x þ K2

y

dt ¼Z t

t0

KAp2ðtÞ þ BqðtÞ þ D34

h iqðtÞ

A2p2ðtÞ þ BqðtÞ þ D34

h i2 dtð2:58Þ

In this case the following correspondences between Andoyer’s–Deprit’s and Euler’s variables take place

cos h ¼ L=K; l ¼ u

The obtained solutions for the MSSC angular velocity (2.57) and for the Euler angles (2.58) are connected with resultsgeneralizing the classical tasks of the rigid body dynamics and corresponding applications, especially, with the angular mo-tion of gyrostats, coaxial bodies and the dual-spin spacecraft – here we can indicate the important results [1–4,22–29].

At the end of this section it is worth to underscore that as opposed to the previous MSSC-results [19] in this article thehomoclinic and general solutions were found with the help of the polhodes’ geometry analysis [2,28,31]. Also we addition-ally note that the MSSC homo/heteroclinic solutions in the article [19] were obtained by analogy with the DSSC solutions[30], which were found by V.S. Aslanov using the classical method of the differential equations integration [32] with the tra-ditional analysis of the quantity/disposition/multiplicity of the polynomials roots at the elliptic (and in limits hyperbolic/trigonometric) quadratures evaluating – this approach is quite useful in the classical and modern tasks of rigid bodiesdynamics [1–5,24–27].

5. Chaos in the MSSC attitude dynamics

5.1. The perturbed system equations

Using the Hamiltonian (2.20) we can write the following equations for the positional part of the Andoyer–Deprit variables(for the subsystem {l,L}):


_l ¼ L1

C� sin2 l

A� cos2 l

B

" #þ Lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

G2 � L2p D12 sin l

Aþ D34 cos l

B

� �� D56

C

_L ¼ ðG2 � L2Þ 1

B� 1

A

sin l cos lþ


p D12 cos l

A� D34 sin l

B

� � ð3:1Þ

Let us investigate the local chaos in the MSSC dynamics close to the homoclinic orbits under the influence of small har-monic perturbations in the internal engines of the rotors (1.9):

_Dij ¼ �edij sinðmijt þ wijÞ ð3:2Þ

where dij; mij; wij are the constants, corresponding to the ij-rotor (some of them can be equal to zero), and e is the smallparameter (e 1).

These harmonic perturbations are possible, for example, at the presence of harmonic errors in the signals of control/reg-ulating/stabilizing systems – it can be due to the delays in the rate gyro units.

From (3.2) the solution directly follows

Dij ¼ Dijð0Þ þ edij

mij½cosðmijt þ wijÞ � cos wij� ð3:3Þ

As it was realized in the previous sections, we take the same assumption (2.23). Then we have

D12 ¼XN

j¼1

½D1jð0Þ þ D2jð0Þ� ¼ 0; D56 ¼XN

j¼1

½D5jð0Þ þ D6jð0Þ� ¼ 0

D34 ¼XN

j¼1

½D3jð0Þ þ D4jð0Þ�–0

ð3:4Þ

Now it is possible to rewrite the canonical equations with the separation of the unperturbed (fl,L) and e-perturbed (gl,L)parts

_l ¼ fl þ egl;_L ¼ fL þ egL ð3:5Þ

where

fl ¼ L1

C� sin2 l

A� cos2 l

B

" #þ D34L cos l

BffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG2 � L2

pfL ¼ ðG2 � L2Þ 1

B� 1

A

sin l cos l� D34

B


psin l

gl ¼Lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

G2 � L2p Q 12 sin l

Aþ Q 34 cos l

B

� �� Q56

C

gL ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG2 � L2

p Q 12 cos l

A� Q 34 sin l

B

� �ð3:6Þ

Q 12 ¼XN

j¼1

d1j

m1jcosðm1jt þ w1jÞ � cos w1j

� �þ d2j


� ��

Q 34 ¼XN

j¼1

d3j


� �þ d4j


� ��

Q 56 ¼XN

j¼1

d5j


� �þ d6j


� �� ð3:7Þ

5.2. The Melnikov function construction

In the case with polyharmonic perturbations the Melnikov function can be constructed as the ‘‘quasiperiodic Melnikovfunction’’ [36].

As indicated in [36] (Definition 4.1):.Function f : R! R is said to be quasiperiodic if there exists a Cr function f : Rl ! R where F is 2p-periodic in each var-

iable, i.e.,

Fðn1; . . . ; ni; . . . ; nIÞ ¼ Fðn1; . . . ; niþ2p; . . . ; nlÞ;8n 2 Rl; 8i ¼ 1; . . . ; l

and


f ðtÞ ¼ Fðx1t; . . . ;xltÞ; t 2 R ð3:8Þ

The real numbers x1; . . . ;xl are called the basic frequencies of f(t). A vector-valued function is said to be quasiperiodic ifeach component is quasiperiodic in the above sense. N

So, in compliance with the above mentioned definition, we have the quasiperiodic vector-valued function Q with 6N basicfrequencies mij ði ¼ 1; . . . ;6; | ¼ 1; . . . ;NÞ:

QðtÞ ¼ Fðm11t; . . . ; m6NtÞ ¼Q 12ðm1jt; . . . ; m2jtÞQ 34ðm3jt; . . . ; m4jtÞQ 56ðm5jt; . . . ; m6jtÞ

264

375 ð3:9Þ

Here we note that the values dij; wij in the functions are the parameters connected only with the mechanical properties ofthe system (this set of parameters is essential for the system and cannot vary during the mathematical transformations).

Then, following [36], for the one-degree-of-freedom Hamiltonian system with l-quasiperiodic perturbations we have thecorresponding equations

_x ¼ JDHðxÞ þ egðx;x1t; . . . ;xlt;l; eÞ

J ¼0 1�1 0

� �; D ¼ @

@x1;@

@x2

� �T ð3:10Þ

This system also can be written in the autonomous form

_x ¼ JDHðxÞ þ egðx; h1; . . . ; hl;l; eÞ_h1 ¼ x1

. . ._hl ¼ xl

8>>><>>>:

ð3:11Þ

where ðx; h1; . . . ; hl;l; eÞ 2 R2 � Tl � Rp � R, l – is the parameters vector.For the homoclinic orbit xhðtÞ of the unperturbed (e = 0) system (3.11) and for the Poincaré section by the angle hi (the

repetition of the initial value hið0Þ ¼ �hi0: �hi0 ! �hi0 þ 2p) the following quasiperiodic Melnikov function takes place [36]

Mðt0;h10; . . . ;�hi0; . . . ;hl0;lÞ¼Z þ1

�1DHðxhðtÞÞ; gðxhðtÞ;x1ðtþ t0Þþh10; . . . ;xiðtþ t0Þþ�hi0; . . . ;xlðtþ t0Þþhl0;l;0Þ� �

dt ð3:12Þ

where h; i denotes the scalar product.It is well-known fact, if at some point ð~t0; ~h10; . . . ; �hl0; . . . ; ~hl0;lÞ the Melnikov function has the simple zero-root, then the

intersection of stable and unstable manifolds of the homoclinic orbit takes place. Usually to the detection of the zero-roots ofthe Melnikov function only parameter t0 may be used at the fixed ‘‘angle’’-parameters ð~h10; . . . ; �hl0; . . . ; ~hl0Þ [36].

So, in our case we have the perturbed one-degree-of-freedom Hamiltonian system (3.5) with the quasiperiodic perturba-tion, where x ¼ ðl; LÞ – is the main coordinates and hij ¼ mijt – are the angle-variables. Using the expressions (2.9)–(2.11) and(3.4) it is possible to rewrite the functions (3.6) in the following form

flðl; LÞ ¼ flðp; q; rÞ ¼ Cr1

C� BAp2 þ ðBqþ D34Þ

2þ D34ðBqþ D34Þ

BðG2 � C2r2Þ

" #

fLðl; LÞ ¼ fLðp; q; rÞ ¼1

B� 1

A

ApðBqþ D34Þ

glðl; L; hijÞ ¼ glðp; q; r; hijÞ ¼Cr

ðG2 � C2r2ÞQ12ðh1j; . . . ; h2jÞAp

Aþ Q 34ðh3j; . . . ; h4jÞðBqþ D34Þ

B

( )� Q56ðh5j; . . . ; h6jÞ

C

gLðl; L; hijÞ ¼ gLðp; q; r; hijÞ ¼Q 12ðh1j; . . . ; h2jÞðBqþ D34Þ

A� Q 34ðh3j; . . . ; h4jÞAp

B

( )ð3:13Þ

The mechanical characteristics of the rotors’ real angular motion are connected with the parameters dij; wij, therefore inthe framework of the mathematical part of the task we can assign zero starting conditions for the associated angle-variableshij ðhijð0Þ ¼ 0; 8i; jÞ. Also we can define the Poincaré map as the section by the h31-angle (certainly, we could take any otherangle-variable):

Ph310 : �h310 ! �h310 þ 2p

With the help of expressions (3.13) the quasiperiodic Melnikov function is written in the form

Mðt0;h110; . . . ;�h310; . . . ;h6N0Þ¼Z þ1

�1fLðpðtÞ;qðtÞ;rðtÞÞglðpðtÞ;qðtÞ;rðtÞ;hijðtþ t0ÞÞ� flðpðtÞ;qðtÞ;rðtÞÞgLðpðtÞ;qðtÞ;rðtÞ;hijðtþ t0ÞÞ� �

dt;

ð3:14Þ


where hijðt þ t0Þ ¼ mijðt þ t0Þ þ hij0 ð8i; jÞ; the homoclinic dependences fpðtÞ; qðtÞ; rðtÞg correspond to the exact homoclinicsolutions (2.50) and/or (2.36).

In this paper we focus only at the numerical analysis and simulations of the system’s chaotic behavior. The analyticalinvestigation of the Melnikov function (3.14) and the antichaotization conditions obtaining are the prospective tasks for

Fig. 9. The results of the numerical simulations.


independent research. So, in the next section we will present the main simulations of the perturbed system behavior, includ-ing the case of the system regular response under the influence of the polyharmonic disturbances.

5.3. Numerical simulations of the system chaotic regimes

5.3.1. The system chaotic motion under the influence of the single-harmonic perturbationFirst of all, we can simulate the system motion with only one disturbance corresponding to the small single-harmonic

perturbation at the rotor #31:

Fig. 10.the homimagesarticle.)

Q34 ¼d31

m31½cosðm31t þ w31Þ � cos w31� ð3:15Þ

In this case the numerical investigation gives us the following results: the time dependence (the time-history) (Fig. 9a),the perturbed S1-homoclinic trajectory (Fig. 9b), the Poincaré section of the S1-homoclinic trajectory in the space of the angu-lar velocity components (Fig. 9c), the Poincaré section of the S2-homoclinic trajectory in the space of the angular velocitycomponents (Fig. 9d), the Poincaré section of the system’s phase flow in the space of the angular velocity components(Fig. 9e), the Poincaré section of the system’s phase flow in the Andoyer–Deprit space (Fig. 9f).

As we can see, the chaotic features of the MSSC motion appear ex facte. It also could be confirmed with the help of theMelnikov function evaluation (Fig. 10a). We should note that at the Melnikov function evaluation (Fig. 10a) only t0 parameterwas varied [36] with all fixed hij0 ¼ 0. The Melnikov function has the harmonic form with the infinite set of the simple roots,and, therefore, the intersections of stable and unstable manifolds of the homoclinic trajectory take place – this effect

The Melnikov function (a) and the homoclinic nets: (a) – the Melnikov function; (b) – five forward (red) and back (blue) Poincaré-map-images ofoclinic separatrix; (c) – ten forward Poincaré-map-images of the homoclinic separatrix; (d) – ten forward (red) and back (blue) Poincaré-map-

of the homoclinic separatrix. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this

Fig. 11. The results of the numerical simulation (the chaotic motion): (a) – the time-history of the angular velocity components (p(t)-black, q(t)-red, r(t)-blue); (b) – the perturbed homoclinic trajectory; (c) – the Poincaré-map in the space of the angular velocity components; (d) – the Poincaré-map in theAndoyer–Deprit space; (e) – the Melnikov function; (f) – the homoclinic net as ten forward and back images of the homoclinic separatrix. (For interpretationof the references to color in this figure legend, the reader is referred to the web version of this article.)


Fig. 12. The results of the numerical simulation (the regular motion).


generate so-called homoclinic nets. The homoclinic nets (Fig. 10b–d) were plotted as the sets of the Poincaré-map-images ofthe unperturbed homoclinic trajectory [31]: we separately plot the forward (in the forward direction of the time t : 0! þ1)Poincaré-map-iterations and the backward (in the back direction of the time t : 0! �1) Poincaré-map-iterations.

The numerical simulations (Figs. 9 and 10) were performed at the following parameters:

Fig. 13. The Fourier transform spectrum of the p(t)-component: blue – the chaotic case (4.3.1); red – the chaotic case (4.3.2); green – the regular case(4.3.3). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


A ¼ 0:5; B ¼ 0:7; C ¼ 0:9 ðkg m2Þ; D34 ¼ 0:5; K ¼ 2:45 ðkg m2=sÞd31 ¼ 1 ðkg m2=s2Þ; m31 ¼ 2 ð1=sÞ; w31 ¼ 0; e ¼ 0:06

5.3.2. The system chaotic motion under the influence of the three-harmonic perturbationNow we present the simulation of the system motion under the influence of the small perturbations at the rotors ## 11,

31, 51:

Q 12 ¼ d11m11½cosðm11t þ w11Þ � cos w11�



8>><>>: ð3:16Þ

with the following parameters:

A ¼ 0:5; B ¼ 0:7; C ¼ 0:9 ½kg m2�; D34 ¼ 0:5; K ¼ 3:16 ½kg m2=s�; e ¼ 0:01d11 ¼ 1 ½kg m2=s2�; m11 ¼ 3:5 ½1=s�; w31 ¼ 1d31 ¼ 1 ½kg m2=s2�; m31 ¼ 2:5 ½1=s�; w31 ¼ 0d51 ¼ 1 ½kg m2=s2�; m51 ¼ 1:5 ½1=s�; w51 ¼ 2

As the result we see the typical chaotic features of the motion (Fig. 11): the phase portraits have the well-defined chaoticlayers; the Melnikov function has the polyharmonic form with the simple roots; the homoclinic net is very complicated.

5.3.3. The system regular motion under the influence of the three-harmonic perturbationIn this section we present the results (Fig. 12) of the system motion modeling with the same disturbances (3.16), but at

the different values of the parameters:

A ¼ 0:5; B ¼ 0:7; C ¼ 0:9 ½kg m2�; D34 ¼ 0:5; K ¼ 2:45 ½kg m2=s�; e ¼ 0:02d11 ¼ 1 ½kg m2=s2�; m11 ¼ 8 ½1=s�; w31 ¼ 1d31 ¼ 1 ½kg m2=s2�; m31 ¼ 10 ½1=s�; w31 ¼ 0d51 ¼ 1 ½kg m2=s2�; m51 ¼ 6 ½1=s�; w51 ¼ 2

As the result we obtain the regular dynamical process (Fig. 12): the time-history of the angular velocity components isperiodic (Fig. 12a); the perturbed homoclinic polhode is the regular twisted closed curve (Fig. 12b); the Melnikov functionis negative and has not any roots (Fig. 12c); the Poincaré sections include only regular invariant curves without any chaoticregions (Fig. 12d and f). Also we’d like to emphasize the simple spectrum (with separated frequencies) of the Fourier trans-form for the p(t)-signal (Fig. 12e). All of the listed features are the typical attributes of the regular dynamical process.

Fig. 14. The possible form of the roll-walking robot.


The mentioned aspects of the regularization of the local homoclinic dynamics under the influence of the polyharmonicperturbations are connected, probably, with the following phenomena [42–45]:

– Regular synchronization of complicated nonlinear quasiperiodic oscillations (autooscillations) in dynamical systems withcorresponding bifurcations and phase portraits’ reconfigurations;

– Chaos synchronization as a phenomenon of periodic regimes’ occurrences in chaotic oscillations (autooscillations) underexternal influences (periodic/chaotic/stochastic); and also coupled (drive-respond) chaotic systems synchronizations;

– Resonance initiation phenomenon (resonance tuning) in chaotic motions.

The advanced study of the listed phenomena in the MSSC’s dynamics is the important independent research task, whichcan be considered in further separate publications. Here we have to repeat the statement that the Melnikov function is one ofthe main mathematical instruments for the local homoclinic chaotization/regularization investigation: the analytical struc-ture of this function allows to obtain explicit conditions of the realization of local chaotic/regular modes.

Also the harmonic analysis and the Fourier transformation are the very useful tool: it is well-known that the chaoticmodes have broad ‘‘continuous’’ spectrums, and, on the contrary, the regular regimes have simple discrete spectrums (withdiscrete frequencies). For the illustration of the last assertion we present (Fig. 13) the Fourier transform spectrum of the‘‘p(t)-signal’’ for the considered cases 4.3.1–4.3.3.

6. Additional applications

In this section the possible applications of the multi-rotor system are shortly described. Also some future-technologyvehicles are offered.

6.1. Gyroscopic stabilization, gyrostats and gyrostat-satellites

In the previous sections we considered the attitude dynamics of the multi-spin spacecraft. The main features of the MSSCmotion are connected with the multi-rotor system (the multi-rotor kernel). This kernel allows to perform the attitude gyro-scopic stabilization by the rotors’ spinup. This method is usually applied to the dual-spin spacecraft and gyrostat-satellitesattitude stabilization. So, we can use the multi-rotor MSSC in the DSSC-mode and also in the gyrostat-mode.

We would like to note that the considered multi-rotor system has various independent internal degrees of freedom cor-responding to the rotors’ rotations. It is the powerful instrument for the support of the spacecraft’s attitude stabilization withthe help of the internal redistribution of the gyrostatic angular momentum between the rotors and the main body.

Also this multi-rotor system can be used for the investigation of the non-regular motion (with strange attractors) of gyro-stats in resistant environments and gyrostat-satellites with complex feedbacks [41].


6.2. The attitude reorientation of the spacecraft

The considered in this paper multi-rotor systems allow fulfill series of the rotors’ spinups and captures to perform theangular reorientation of the SC [18]. For the realization of the reorientation maneuver we have to perform the conjugatedspinup (in the opposite directions) of the pair of conjugated rotors, and then stop one of them. After this capture the rotor’sangular momentum is transmitted to the main rigid body of the SC, which performs the rotation around the correspondingaxis. The capture of the second conjugated rotor compensates the angular momentum and stops the rotation of the SC’s mainbody.

This approach of the attitude reorientation is characterized by the minimal motion inertia (or, moreover, by the inertiaabsence): it means the immediate redistribution of the angular momentum between the rotors and the SC’s main body. Itis possible if we fulfill the immediate rotors’ captures with the help of the large friction generation or the gear meshing(or other types of slip-free engagements). This inertialess feature of the SC’s angular motion is unique as compared withthe classical attitude control realization by the reaction-wheels: as it was presented in [18], we can synthesize the piecewiseconstant angular velocity of the SC’s main body without any noticeable transient regimes. Also it is possible to perform thecompound spatial reorientation (the spatial rotation series) with the help of the corresponding conjugated spinup-capture-series for the rotors placed in several independent layers (Fig. 1b).

The inertialess reorientation method is applicable for many types of the SC because the multi-rotor kernels can be con-structed as quite small devices, which can be placed even in the nanosatellites.

6.3. Roll-walking multi-rotor robots and vehicles

One of the interesting applications of the considered multi-rotor system is roll-walking (somersaulting) multi-rotor ro-bots and vehicles [20,21].

So, it is possible to use the internal multi-rotor system as the drive kernel of the roll-walking robots and vehicles (Fig. 14).The multi-rotor drive kernel allows to perform the roll-walking somersaulting motion: to make one step we should turn overthe robot body with the help of the conjugated spinup-capture.

For the realization of the roll-walking somersaulting step (the ‘‘somersault’’ around the edge) with the one-axis conju-gated spinup/capture we can use two ground-fixed points (G1 and G2). And only one ground-fixed point (G1) is needed toturn over the robot’s body at the double-axis conjugated twofold spinup/capture with the gyroscopic torque initiation.

6.4. Suppressions of the SC chaotic regimes with the help of the internal multi-rotor kernels

As it was described in the previous Section 5.3.3 the regularization of the local chaotic motion takes place under theinfluence of the special defined polyharmonic torques in the rotors’ engines. This feature we can use for the initiation ofcorresponding small polyharmonic torques in the rotors’ engines to avoid the chaotic regimes. This important ‘‘antichaoti-zation’’ task with constructing of ‘‘the internal antichaotization multi-rotor kernel’’ of the spacecraft can be considered in theindependent separate research.

In this framework we can use many dynamical tools such as control techniques with the feedbacks, the resonance tuning,the initiation of dominating oscillations with suppressions of the ‘‘chaotic’’ frequencies.

6.5. The approach to modeling of the angular motion dynamics of an elastic body at small elastic vibrations

The mechanical and mathematical multi-rotors models developed in the previous sections can be applied to the modelingof the angular motion of an elastic body. We can consider the layers of the rotors as rotationally oscillating masses attachedto the rigid body (the core body) by elastic connections (angular springs). Choosing the angular spring’s stiffnesses cij

corresponding to the vibration eigenfrequencies of the elastic body) and the mass-inertia parameters of the rotors’ (corre-sponding to the eigenmodes) we can consider the angular motion of the elastic body with the help of the simple dynamicalsystem in the Hamiltonian form:

H ¼ T þP; x ¼ fl;u2;u3; dijg; X ¼ fL;G;H;Dijg

T ¼ G2�L2

2sin2 l

Aþ cos2 l

B

h iþ 1

2CðL� D56Þ2 �


pD12 sin l

Aþ D34 cos l

B

n oþ D2

12

2Aþ D2

342Bþ TR

D12 ¼XN

j¼1


j¼1


j¼1

½D5j þ D6j�; TR ¼ 12

XN

j¼1

X6

i¼1

D2ji

Ij

P ¼ PðdijÞ ¼ 12

Xi

Xj

cijd2ij; R ¼ RðrijÞ ¼ 1

2

Xi

Xj

bijr2ij

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

Certainly, the selection of the springs’ stiffness and the rotors’ mass-inertia parameters should be carried out with the help ofthe special methods of the definition of the distributed oscillations’ parameters of elastic bodies, or experimentally. For theadditional modeling of the dissipation properties of the elastic body the standard Rayleigh’s dissipation function RðrijÞ isquite useful.


7. Conclusion

So, in this paper the MSSC attitude dynamics was considered, the homoclinic and general exact solutions were obtained,the cases of the chaotic and regular motion under the influence of the polyharmonic disturbances were investigated.

Acknowledgment

This work was supported by the Russian Foundation for Basic Research (RFBR ## 11-08-00794-a, 12-08-09202-mob-z).

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Homoclinic solutions and motion chaotization in attitude dynamics of a multi-spin spacecraft

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