Exact solutions for angular motion of coaxial bodies and attitude dynamics of gyrostat-satellites

International Journal of Non-Linear Mechanics 50 (2013) 68–74

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International Journal of Non-Linear Mechanics

0020-74

http://d

n Corr

Moscov

E-m

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Exact solutions for angular motion of coaxial bodies and attitude dynamicsof gyrostat-satellites

Anton V. Doroshin n

Flight Dynamics and Control Systems Department, Samara State Aerospace University (National Research University), Samara, Russia

a r t i c l e i n f o

Article history:

Received 17 April 2012

Received in revised form

28 September 2012

Accepted 27 October 2012Available online 6 November 2012

Keywords:

Dual-spin spacecraft

Angular motion

Polhode

Explicit solutions

Elliptic integrals

62/$ - see front matter & 2012 Elsevier Ltd. A

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espondence to: Samara State Aerospace Univ

skoe shosse 34, Samara 443086, Russia. Tel.:

ail address: [email protected]

a b s t r a c t

Dynamics of the torque-free angular motion of gyrostat-satellites and dual-spin spacecraft are

examined. New analytical solutions for the angular moment components are obtained in terms of

Jacobi elliptic functions. Also analytical solutions for Euler’s angles are found. These solutions can be

used for a dual-spin spacecraft and gyrostat-satellites attitude dynamics analysis and synthesis.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Study of the angular motion of rigid bodies and attitude dynamicsof gyrostat-satellites (GS) and dual-spin spacecraft (DSSC) stillremains one of the main research areas of modern mechanics andspaceflight dynamics. This research area connected with classicaltasks of angular rigid body motion, gyrostats and coaxial bodies’systems motion [1–6]. Regular and irregular (chaotic) motion modes,attitude control and reorientation of gyrostats and coaxial spacecraftare being examined by many scientists [7–40].

Classical investigation results of rigid body and gyrostats motionhave been presented in many works, for example, in Refs. [1–7].In Refs. [8,9] important aspects of torque-free motion dynamics ofgyrostats were studied.

Analysis of angular motion of coaxial bodies and dual-spinspacecraft was conducted, for example, in Refs. [10–20] includingperturbed cases of motion. In works [17–19] research results forDSSC motion at rotor-body spinup realization (a momentumtransfer maneuver) were collected. In Refs. [20–33] compoundand chaotic modes of motion of gyrostats and DSSC at theabsence/presence of control were considered.

In work [34] angular motion of variable mass dual-spin spacecraftwas investigated. Corresponded research results showed non-trivialchanges of DSSC angular motion at the variability of mass–inertia

ll rights reserved.

ersity, SSAU,

þ7 8462674411.

parameters. Also the qualitative method for phase space analysisbased on the evaluation of a phase trajectory curvature wasdeveloped—this method can be used for the synthesis of realizationconditions of special motion modes (for example, monotonedecrease/increase of nutation angle).

In paper [35] a chaotic motion of gyrostats in resistant environ-ment was considered with the help of well-known dynamicalsystems with strange attractors: Lorenz, Rossler, Newton–Leipnikand Sprott systems. Cases for perturbed gyrostats motion withvariable periodical inertia moments and with periodical internalrotor relative angular moment was investigated.

In [36] Heteroclinic dynamics of the torque-free dual-spinspacecraft was examined, and analytical solutions for heteroclinicorbits, corresponded to the polhodes-separatrices in the space ofthe angular moment components, were obtained. Analysis ofpossibility of the motion chaotization with the help of Melnikovmethod was conducted on the basis of these analytical hetero-clinic solutions.

In works [37–40] interesting recent results for gyrostat’s typesystems can be found. These results connected with investigationof new aspects of stabilization and synchronization of electro-mechanical gyrostat systems (in regular and chaotic cases),dynamics of gyrostats in the gravitational field, and gyrostats’reorientation problems.

In the framework of gyrostats dynamics we need to emphasizethe analytical study problem. Analytical exact solutions for gyrostatsmotion have an important value as intrinsic mathematical andmechanical problem, and also these results may be consideredas unperturbed generating dependences at investigation of the

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A.V. Doroshin / International Journal of Non-Linear Mechanics 50 (2013) 68–74 69

perturbed motion’s dynamics in the presence of small external andinternal torques, such as gravity gradient influence, geomagnetic fieldtorques, aerodynamic moments, interactions of DSSC bodies, etc.

The main analytical investigations and exact solutions for theparameters of balanced gyrostat motion are presented in works[1–4,7]; also explicit solution for gyrostat was recently found inRef. [8]. Obtaining analytical results for non-Kelvin-type gyrostats anddual-spin spacecraft motion was considered in Refs. [9–12]. Hereparticularly we need to underscore the exact explicit solutions forDSSC angular motion presented in Ref. [10] and replicated in Refs.[11,12]—these important solutions were obtained for the Andoyer–Deprit canonical variables in closed form in Jacobi elliptic functions.

So, in this paper on the base of Euler’s dynamical equations weobtain a new form of the exact explicit solutions for the coaxialsystem and DSSC in the space of angular velocity components.

The paper is organized as follows. In Section 2 the mathematicalmodel of the coaxial bodies system, GS and DSSC is presented.In Section 3 the explicit analytical solutions for the angular velocity(angular moment) components are obtained. In Section 4 analyticalsolutions for Euler’s angles are found.

2. The motion equations of the coaxial bodies, GS and DSSC

Let us consider the torques free attitude dynamics of the coaxialbodies, GS and DSSC which was stated in Ref. [36]. GS and DSSCconsist of two coaxial bodies (body ]1 is a rotor; body ]2 is a main/core/carrier body). We assume that the main body has triaxialinertia tensor and the rotor is dynamically symmetrical body. Let usintroduce following coordinate frames (Fig. 1): OXYZ is the inertialsystem of coordinates, Ox2y2z2—the connected principal system ofcoordinates of the carrier body, and Ox1y1z1—the connected princi-pal system of coordinates of the rotor body. The axes Oz1 and Oz2 ofthe connected systems are identical to the common rotation axis ofthe coaxial bodies.

The system motion can be described on the basis of Eulerdynamical equations [3,4], and with the help of Andoyer–Depritcanonical variables [13,14]. The dynamical Euler’s equation of thetorque-free motion of the coaxial system can be written, forexample [34,36], as:

A _pþ C�Bð ÞqrþqC1s¼ 0

B _qþ A�Cð Þpr�pC1s¼ 0

C _rþC1 _sþ B�Að Þpq¼ 0

C1 _rþ _sð Þ ¼MD ¼ 0

8>>>><>>>>:

ð1:1Þ

Fig. 1. DSSC (coaxial bodies system) and the coordinate frames.

or in the following form:

A _pþ C2�Bð ÞqrþqD¼ 0; B _qþ A�C2ð Þpr�pD¼ 0

C2 _rþ _Dþ B�Að Þpq¼ 0; _D ¼MD ¼ 0 ð1:2Þ

where {p,q,r} are components of the carrier body’s angularvelocity which represented in projections onto the axes of theOx2y2z2 frame; s� the rotor angular velocity relatively the carrierbody; I2¼diag[A2,B2,C2] is the triaxial inertia tensor of the carrierbody in the connected frame Ox2y2z2; I1¼diag[A1,A1,C1] is theinertia tensor of the dynamically symmetrical rotor in the con-nected frame Ox1y1z1; A¼A1þA2, B¼A1þB2, C¼C1þC2 are themain inertia moments of the coaxial bodies system in the frameOx2y2z2 (including rotor); MD� is the internal torque of thecoaxial bodies interaction (assume MD¼0); D¼C1(rþs)� thelongitudinal angular moment of the rotor along Oz1; C1s¼ hz1

�

the rotor relative angular moment in the carrier bodyframe Ox2y2z2. We assume following conditions A24B24C24A14C1, D¼ const40:

Also we need to add the well-known kinematical equations forEuler’s angles:

_c ¼ 1sin y psin jþqcos j

� �_y ¼ pcos j�qsin j_j ¼ r� cos y

sin y psin jþqcos j� �

_d ¼ s

8>>>><>>>>:

ð1:3Þ

Here we note that the equation system (1.1) corresponds tothe torque-free motion of the coaxial bodies and the unbalancedgyrostat with non-constant relative angular moment of rotor(hz1¼ C1saconst,even if MD¼0). In this case results of analysis

for Kelvin-type gyrostats [1–4, 7,8] are not applicable.Also we can use the Hamiltonian form of equations in the

Andoyer–Deprit canonical variables. The Andoyer–Deprit variables[5,9,10,33,36] can be expressed with the help of the coaxial system’sangular moment K (Fig. 1):

L¼@T

@_l¼KUk; I2 ¼

@T

@ _j2

¼KUs¼ 9K9¼ K; I3 ¼@T

@ _j3

¼KUk0; Lr I2

Kx2¼ Ap¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiI22�L2

qsin l; Ky2

¼ Bq¼ffiffiffiffiffiffiffiffiffiffiffiffiffiI22�L2

qcos l; Kz2

¼ C2rþD¼ L:

ð1:4Þ

The system Hamiltonian [9,33,36] in the Andoyer–Depritphase space takes the form:

H¼I22�L2

2

sin2l

A1þA2þ

cos2l

A1þB2

!þ

1

2

D2

C1þ

L�Dð Þ

C2

2 !

þeH1, ð1:5Þ

where T—is the system kinetic energy; eH1� is the small per-turbed part of the Hamiltonian, connected with such disturbancesas gravity gradient, geomagnetic field, etc. and correspondedsmall potentials. In this research we focus only on the unper-turbed (e¼0) generating solutions obtaining and, therefore,effects of small perturbations can be eliminated.

Using the Hamiltonian (1.5), we can write correspondeddynamical system:

_L ¼� @H@l ¼ a I2

2�L2� �

sinlcosl;

_l ¼ @H@L ¼ L 1

C2� sin2 l

A1þA2ð Þ� cos2 l

A1þB2ð Þ

h i� D

C2;

8><>:

8><>:_I2 ¼�

@H@j2¼ 0) I2 ¼ const;

_j2 ¼@H@I2¼ I2

sin2 lA1þA2

þ cos2 lA1þB2

� �;

8><>:

A.V. Doroshin / International Journal of Non-Linear Mechanics 50 (2013) 68–7470

_I3 ¼�@H@j3¼ 0) I2 ¼ const;

_j3 ¼@H@I3¼ 0) j3 ¼ const;

_D ¼� @H@d ¼ 0) D¼ const;

_d ¼ s¼ @H@D ¼

DC1�

L�Dð Þ

C2

8<:

8<: ð1:6Þ

where a¼(A1þB2)�1�(A1þA2)�1.

As we can see, the system (1.6) consists of four subsystems forevery pair of canonical Andoyer–Deprit variables {{l,L},{j2,I2},{j3,I3},{d,D}}. The subsystem {l,L} is independent and can beintegrated separately. This integration in elliptical quadratureswas performed in paper [10].

We need to note that truncation of the subsystems{j2,I2},{j3,I3},{d,D} inevitably leads to information loss. Forexample, comparison of the third-order differential equationssystem (1.2) (D¼ const) with the second-order subsystem {l,L}(1.6) is identical to the comparison of 3D geometrical object andits plane projection – for our mechanical task dynamical systemorbit represents 3D-polhode, and it should be described with thehelp of three parameters in 3D-space {p,q,r} instead of theconsideration based on its projection on the Andoyer–Depritphase plane. Therefore, reduction {p,q,r}-{l,L} is easily execu-table, but inverse transformation {l,L}-{p,q,r} without informa-tion loss is impossible (it will be shown at the end of the section 2).Moreover, to give complete description of the motion we need totake into account such kinematical parameters as Euler’s angles.Thus, the full dynamical system corresponds to the mechanicalsystem for coaxial bodies (GS, DSSC) with four degree of freedom(4-DOF).

In the next section we will obtain analytical solutions forangular velocity components of the coaxial bodies system.

3. Explicit analytical solutions for angular momentcomponents

The term ‘‘polhode’’ is well-known [3]. The polhode is the fourth-order curve in 3D-space (Fig. 2) which corresponds to the intersec-tion of a kinetic energy ellipsoid and an angular moment ellipsoid,

Fig. 2. Projections and 3D-imag

which are defined with the help of the following expressions [3]:

Ap2þBq2þC2r2þD2

C1¼ 2T ð2:1Þ

A2p2þB2q2þ½C2rþD�2 ¼ K2¼ 2DT ð2:2Þ

D¼K2

2Tð2:3Þ

We can write polhode equation [3,36] on the basis of expres-sions (2.1) and (2.2) combinations. So, multiplication of Eq. (2.1)by A and deduction of Eq. (2.2) gives

B A�Bð Þq2þC2 A�C2ð Þr2þC1 A�C1ð Þ rþsð Þ2

�2C1C2r rþsð Þ ¼ 2T A�Dð Þ ð2:4Þ

Multiplication of Eq. (2.1) by C1 and deduction of Eq. (2.2)gives

A C1�Að Þp2þB C1�Bð Þq2þC2 C1�C2ð Þr2

�2C1C2r rþsð Þ ¼ 2T C1�Dð Þ ð2:5Þ

Multiplication of Eq. (2.1) by B and deduction of Eq. (2.2) gives

A B�Að Þp2þB C2r2þC1½rþs�2� �

� C2rþC1½rþs�ð Þ2¼ 2T B�Dð Þ ð2:6Þ

Taking into account perfect square, equation of hyperbolae (onthe coordinate plane Opr in Fig. 2) follows from Eq. (2.6)

�A A�Bð Þp2þC2 B�C2ð Þ r�D

B�C2

� �2

¼ F ð2:7Þ

where F ¼ 2T B�Dð ÞþD2a, a¼ C1C2þ B�C2ð Þ C1�Bð Þ½ � B�C2ð ÞC1½ �=

From expression (2.4) equation of ellipses (on the coordinateplane Oqr (Fig. 2)) follows:

B A�Bð Þq2þC2 A�C2ð Þ r�D

A�C2

� �2

¼H

b¼C2C1þ A�C2ð Þ C1�Að Þ

A�C2ð ÞC1H¼ 2T A�Dð ÞþD2b ð2:8Þ

e of the polhodes ellipsoid.


Eq. (2.7) can be rewritten as

p¼ 7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2 B�C2ð Þ r� D

B�C2

� �2�F

A A�Bð Þ

vuutð2:9Þ

From Eq. (2.8) we can get

r�D

A�C2¼ 7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH�B A�Bð Þq2

C2 A�C2ð Þ

sð2:10Þ

With the help of Eq. (2.10) we can write

r�D

B�C2¼ 7


C2 A�C2ð Þ

s�Db ð2:11Þ

where b¼(B�C2)�1�(A�C2)�1.

On the basis of Eqs. (2.11) and (2.9) we can rewrite the secondEq. (1.2) in the form

B _q ¼ 8

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2 B�C2ð Þ

A A�Bð Þ7


C2 A�C2ð Þ

s�Db

!2

�F

A A�Bð Þ

vuuut U

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA�C2

C2H�B A�Bð Þq2 s

ð2:12Þ

We can make the change of variables (case 1)

x¼ þ


C2 A�C2ð Þ

s�Db ð2:13Þ

Then from Eq. (2.13) we obtain

xþDbð Þ2¼

1

C2 A�C2ð ÞH�B A�Bð Þq2

ð2:14Þ

q¼ 7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH�C2 A�C2ð Þ xþDbð Þ

2

B A�Bð Þ

s;

dq¼ 8C2 A�C2ð Þ

B A�Bð Þ

xþDbð ÞdxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH�C2 A�C2ð Þ xþDbð Þ

2

B A�Bð Þ

q ð2:15Þ

Also we can make the following change of variables (case 2)

x¼�


C2 A�C2ð Þ

s�Db ð2:16Þ

Similar to previous case (case of change (2.13)) from Eq. (2.16)expressions (2.14) and (2.15) follow again. Thus, for both changeswe have interconnected equation:

dt¼ 7M dxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

H�a xþbð Þ2

q ffiffiffiffiffiffiffiffiffiffiffiffiffifficx2�Gp ð2:17Þ

where

M¼ C2

ffiffiffiffiffiffiffiffiffiffiB

A�B

r; G¼

F

A A�Bð Þ;

a¼ C2 A�C2ð Þ; b¼Db; c¼C2 B�C2ð Þ

A A�Bð Þ

It is needed to note that differences between initial valuescorresponded to cases (i¼1, 2):

x t0ð Þ ¼ xinii ¼ �1ð Þ

iþ1


0

C2 A�C2ð Þ

s�Db, i¼ 1,2

Let us rewrite Eq. (2.17) in the form with differences of squares

dt¼ 7Mffiffiffiffiffiacp

dxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHa

q� �2

� xþbð Þ2

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2�

ffiffiffiGc

q� �2r ð2:18Þ

Now we make the next change of variables

z¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR x�eð Þ

P xþeð Þ

s

R¼�b�dþe; P¼�b�d�e

d¼ffiffiffiffiffiffiffiffiffiH=a

p; e¼

ffiffiffiffiffiffiffiffiG=c

pð2:19Þ

Then we have the following expressions

x¼�R=P�z2

�R=Pþz2e; dx¼

R

P

4ez dz

�R=Pþz2� �2

ð2:20Þ

Substitution of Eq. (2.20) into Eq. (2.18) gives us

dt¼ 74eMR

Pffiffiffiffiffiacp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH

az2�

R

P

� �2

� e�R

P�z2

� �þb z2�

R

P

� �� 2s0

@

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2�R

P�z2

� �2

�G

cz2�

R

P

� �2s 1

A�1

zdz

After some transformations with expansion of difference ofsquares last equation takes the form

dt¼ 72eM

ffiffiffiffiffiffiffiffiR=P

pffiffiffiffiffiffiaGp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis1z2�s2

� �s3z2�s4

� �q� ��1

dz

s1 ¼

ffiffiffiffiH

a

rþe�b; s2 ¼

R

P

ffiffiffiffiH

a

r�e�b

!

s3 ¼

ffiffiffiffiH

a

r�eþb; s4 ¼

R

P

ffiffiffiffiH

a

rþeþb

!ð2:21Þ

Now Eq. (2.21) can be rewritten as

dt¼ 72eM



ffiffiffiffiffiffiffiffiffis2s4p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

z2

c21

!1�

z2

c22

!vuut24

35�1

dz ð2:22Þ

where c21 ¼ s2=s1; c2

2 ¼ s4=s3.Here we should consider two cases of reduction of Eq. (2.22) to

elliptic integral of the first kind:

(1)
If c1oc2 then the following substitution is efficient: z¼c1y
and, moreover, k¼c1/c2o1.
(2) If c14c2 then: z¼c2y and k¼c2/c1o1.
The Eq. (2.22) takes the form

dt¼ 72eM



cjdyffiffiffiffiffiffiffiffiffis2s4p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�y2� �

1�k2y2� �r ; j¼ 1,2 ð2:23Þ

where index j corresponds to number of reduction case.Integration gives us

7 N t�t0ð Þþ I0½ � ¼

Z y

0

dyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�y2� �

1�k2y2� �r

N¼ 2eMcj


pffiffiffiffiffiffiaGp ffiffiffiffiffiffiffiffiffi

s2s4p

!�1

; I0 ¼

Z y0

0

dyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�y2� �

1�k2y2� �r ¼ const

ð2:24Þ

Inversion of elliptic integral gives the explicit analytical solu-tion

yðtÞ ¼ sn½7 N t�t0ð Þþ I0Þ,k�ð ð2:25Þ


where sn(u,k) is the Jacobi elliptic sine function with classicaldefinition:

u¼F j� �¼

Z j

0

dWffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�k2sin2

ðWÞq ;

j¼jðuÞ ¼F�1ðuÞ; sn u,kð Þ ¼ sin jðu

� �Þ;

After inverse transformations we obtain the exact explicitanalytical solutions for all angular velocity components:

q tð Þ ¼ 7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

B A�Bð ÞH�fxðtÞþDbg2C2 A�C2ð Þ

h ir

p tð Þ ¼ 7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1A A�Bð Þ

C2 B�C2ð Þx2 tð Þ�F q

r tð Þ ¼ DA�C2

7 x tð ÞþDbð Þ

s tð Þ ¼ DC1�r tð Þ ¼ A�C

C1 A�C2ð ÞD8 x tð ÞþDbð Þ

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

ð2:26Þ

where

x tð Þ ¼ eR=Pþsn2½7 ðN t�t0ð Þþ I0Þ,k�

R=P�sn2½7 N t�t0ð Þþ I0Þ,k�ðð2:27Þ

Fig. 3 demonstrates the validity of solutions (2.26)—we seecomprehensive coincidence of the analytical (points) and numer-ical integration results (lines).

It is well-known fact that solutions for gyrostats (coaxialbodies) generalize results for the free rigid body in Euler’s caseof motion [1–8]. In contrast to Refs. [1–8], in this paper theessential motion of the coaxial system (DSSC) was investigatedwithout assumption of constancy of the gyrostatic moment(hz ¼ C1saconst).

In Ref. [10] analytical solutions for DSSC were obtained interms of Jacobi elliptic functions for the Andoyer–Deprit canonicalvariables. With the help of Ref. [10] and based on kinematicexpressions (1.4) we can also write the dependences {p(t),q(t),r(t)}as functions of Deprit’s variable L. In this case expressions shapeincludes compositions of Jacobi elliptic functions, radicals, trigo-nometric functions and inverse trigonometric functions at thesame time – this structural complexity connected with informa-tion loss at the transformation of 3D-polhodes to their 2D-images

Fig. 3. Numerical (lines) and analytical (points) integration results. A2¼15; B2¼8;

C2¼6; A1¼5; C1¼4; p0¼5; q0¼5; r0¼10; D¼5.

on the {l,L}�plane:

p tð Þ ¼I2

A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�s2ðtÞ

qsin 7

1

2arccos

a0 þb0�2� �

s2ðtÞþ4d0sðtÞþ4h0�a0�b0

1�s2ðt� �

Þ b0�a0� �

!" #

q tð Þ ¼I2

B

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�s2ðtÞ

qcos 7

1

2arccos

a0 þb0�2� �

s2ðtÞþ4d0sðtÞþ4h0�a0�b0

1�s2ðt� �

Þ b0�a0� �

!" #

r tð Þ ¼ L tð Þ�Dð Þ=C2; L tð Þ ¼ s tð ÞI2;

s tð Þ ¼s02s031�s03s021sn2 ot,kð Þ

s031�s021sn2 ot,kð Þ; t¼ tUI2=C1; a0,b0,d0,h0,s0is

0ij�const

ð2:28Þ

We additionally note that results (2.28) follow directly fromthe work [10].

Reduction of the solutions (2.28) to the new form (2.26) byanalytical transformations is problematic. Vice versa, based onEq. (2.26) we can write simple solutions for Andoyer–Depritvariables:

L tð Þ ¼ C2r tð ÞþD; l tð Þ ¼ arcsinApðtÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

K2� C2rðtð ÞþDÞ2

q¼ arccos

BqðtÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2� C2rðtð ÞþDÞ2

q ð2:29Þ

4. Explicit analytical solutions for Euler’s angles

Let us consider angular motion of the coaxial system (DSSC)with respect to the initial frame OXYZ (Fig. 1) in the case whenvector of angular moment K is directed along OZ. It always can berealized by the changing of coordinate frames. In this casenutation angle (angle between OZ and Oz1,2 axes) and intrinsicrotation angle (j) can be found using the classical method [3,5]from expressions:

Kx2¼ Ap¼ Ksin ysin j

Ky2¼ Bq¼ Ksin ycos j

Kz2¼ C2rþD¼ Kcos y

8><>: ð3:1Þ

So, taking into account analytical solutions for angular velocitycomponents (2.26) we obtain exact solutions for angles

cos y tð Þ ¼ C2r tð ÞþD

=K; tgj tð Þ ¼ApðtÞ

BqðtÞð3:2Þ

In the considered case (K¼ I2¼ I3) with the help of comparisonof expressions (3.1) and (1.4) we can write the correspondencesbetween Euler and Andoyer–Deprit variables

cos y¼ L=I2; l¼j ð3:3Þ

Solutions for precession (c) and relative rotation angle (d)follow from the first and the fourth equations (1.3) and expres-sions (3.1)

c tð Þ�c0 ¼

Z t

t0

KAp2ðtÞþBq2ðtÞ

A2p2ðtÞþB2q2ðtÞdt; d tð Þ�d0 ¼

Z t

t0

DC1�r tð Þ

� �dt

ð3:4Þ

Fig. 4 demonstrates the validity of solution (3.2) for nutationangle (y); Fig. 5 shows the validity of analytical solution forintrinsic rotation angle (j) (3.2), for precession angle (c) (3.4)and for relative rotation angle (d) (3.4). We note that all angles(Figs. 4 and 5) were calculated in radians.

We need to note that magnitude (analytical result) of intrinsicrotation angle (Fig. 5) is located into the interval �p=2

� �r

�jr p=2

� �Þ in compliance with actual range of arctangent-func-

tion: j¼ arctg ApðtÞ=BqðtÞ� �

Fig. 4. Numerical (lines) and analytical (points) integration results for nutation.

A2¼15; B2¼8; C2¼6; A1¼5; C1¼4; p0¼5; q0¼5; r0¼10; D¼5.

Fig. 5. Numerical (lines) and analytical (points) integration results for. intrinsic

rotation, precession and relative rotation angle. A2¼15; B2¼8; C2¼6; A1¼5;

C1¼4; p0¼5; q0¼5; r0¼10; D¼5.


By this reason we ought to add p to j-value every rotationalperiod.

Thus, all explicit exact analytical solutions are found fortorque-free angular motion of 4-DOF {p,q,r,s,c,y,j,d} coaxialbodies system.

These unperturbed generating solutions ((3.2) and (3.4)) canbe used for investigation of perturbed motion problems, such asthe angular motion of DSSC with electromagnetic equipment ingeomagnetic field, orbital motion of large DSSC taking intoaccount influence of the gravity gradient, and also attitudedynamics DSSC in the perturbed environment and with chaoticbehavior at presence of small external/internal disturbances.

5. Conclusion

Dynamics of the torque-free coaxial bodies system (GS, DSSC)has been examined in the space of angular velocity (angularmoment) components. The new analytical solutions for all angu-lar moment components have been obtained in terms of Jacobielliptic functions. Also analytical solutions for Euler’s angles havebeen found. These solutions can be used for dual-spin spacecraftand gyrostat-satellites attitude motion analysis and synthesis.

Acknowledgments

This research was supported by Russian Foundation for BasicResearch (RFBR## 11-08-00794-a, 12-08-09202-mob-z) and theRussian Federation Presidential Program for the Support ofRussian Scientists and Leading Scientific Schools (MK-1497.2010.8).

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Exact solutions for angular motion of coaxial bodies and attitude dynamics of gyrostat-satellites

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