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Acta Astronautica

Acta Astronautica 94 (2014) 563–576

0094-57http://d

n Tel.:E-m

journal homepage: www.elsevier.com/locate/actaastro

Chaos and its avoidance in spinup dynamicsof an axial dual-spin spacecraft

Anton V. Doroshin n

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 14 April 2013Received in revised form6 September 2013Accepted 11 September 2013Available online 27 September 2013

Keywords:Dual-spin spacecraftDual-spin satelliteGyrostatHomoclinic chaosPolyharmonic perturbationsNutation restoring/overturning torquesMelnikov functionPoincaré map

65/$ - see front matter & 2013 IAA. Publishex.doi.org/10.1016/j.actaastro.2013.09.003

þ7 92 7656 3036ail addresses: doran@inbox.ru, doroshin@ssa

a b s t r a c t

Attitude dynamics of a dual-spin spacecraft (DSSC) and a torque-free angular motion of acoaxial bodies system are considered. Some regimes of the heteroclinic chaos are described.The local chaotization of the DSSC is investigated at the presence of polyharmonicperturbations and small nutation restoring/overturning torques on the base of the Melnikovmethod and Poincaré Maps. Reasons of the chaotic regimes initiation at the spinupmaneuver realization are studied. An approach for the local heteroclinic chaos escape/avoidance at the modification of the classical spinup maneuver is suggested.

& 2013 IAA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

The study of the angular motion of coaxial rigid bodiessystems and dual-spin spacecraft (DSSC) attitude dynamics isone of the main problems of rigid body systems mechanics[1–41].

Tasks of the analysis/synthesis of the coaxial bodies'motion have important applications in the spaceflightdynamics and corresponding space missions which includeorbital path sections with the attitude stabilization ofthe DSSC [8–16]. The DSSC consists of two coaxial bodies,rotating about a common axis (freely or at the presence ofthe internal engine torques).

The dual-spin construction-scheme is quite useful inthe practice during the history of the space flights realiza-tion; and it is possible to present some examples of the

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DSSC, which was used in real space-programs (most ofthem are communications dual-spin satellites and obser-ving geostationary satellites).

This is the long-continued and well successful project“Intelsat” (the Intelsat II series of satellites first launched in1966) including the 8th generation of geostationary com-munications satellites and the Intelsat VI (1991) designedand built by Hughes Aircraft Company. Also one of thefamous Hughes' DSSC is the experimental tactical commu-nications dual-spin satellite TACSAT-I which was launchedinto synchronous orbit in 1969. The “Meteosat”- projectby European Space Research Organization (initiated withthe Meteosat-1 in 1977 and operated until 2007 with theMeteosat-7) also used the dual-spin configuration space-craft. The spin-stabilized spacecraft with mechanicallydespun antennas was applied in the framework of GEOTAIL(the collaborative mission of Japan JAXA/ISAS and NASA,within the program “International Solar-Terrestrial Phy-sics”) launched in 1992; the GEOTAIL spacecraft and itspayload continues to operate in 2013. Analogously the

reserved.

A.V. Doroshin / Acta Astronautica 94 (2014) 563–576564

construction scheme with the despun antenna was selectedfor Chinese communications satellites DFH-2 (STW-3, 1988;STW-4, 1988; STW-55, 1990). The well-known Galileomission's spacecraft (the fifth spacecraft to visit Jupiter,launched on October 19, 1989) was designed by the dual-spin scheme. Of course, we should indicate one of theworld's most-purchased commercial communications satel-lite models such as Hughes/Boeing HS-376/BSS-376 (forexample, the “Satellite Business Systems”with projects SBS-1, 2, 3, 4, 5, 6/HGS-5, etc.): they have spun sectionscontaining propulsion systems, solar drums, and despunsections containing the satellite's communications payloadsand antennas. Also very popular and versatile dual-spinmodels are the Hughes' HS-381 (the Leasat project), HS-389(the Intelsat project), HS-393 (the JCSat project).

The DSSC usually is used for the attitude stabilizationby the partial twist method: only one of the DSSC's coaxialbodies (the «rotor»-body) has rotation at the «quiescence»of the second body (the «platform»-body)—it allows toplace into the «platform»-body some exploratory equip-ment and to perform of space-mission tests withoutrotational disturbances. Moreover, one of the widespreadtypes of the DSSC is the axial DSSC, where the axisym-metric rotor is aligned with the principal axis of theplatform-body.

The investigation of the motion chaotic regimes is alsothe significant part of the research into DSSC attitudedynamics. Here we can indicate many works, e.g. [18–27].Analysis of the angular motion of the coaxial bodies withthe variable structure was carried out in [19,36–38]. In [18]the motion chaotization of a rigid body with a rotorattachment was studied based on the Melnikov method'smodification using the classical heteroclinic solutions forthe single rigid body torque-free angular motion. In [12–16]DSSC tilting motion's evolutions at the rotor's spinupmaneuver realization was in details investigated with thehelp of direct integration of dynamical equations.

As it is described in previous works, e.g. [15], the DSSC isusually placed into the orbit with zero relative rotor angularvelocity, and then a spinup engine provides an internaltorque from the «platform»-body to the «rotor»-body alongthe common rotation axis of the DSSC's coaxial bodies. Theeffect of the axial torque is to spin-up the rotor and despinthe platform, thereby transferring all or most of the angularmomentum to the rotor. Usually the spinup torque isconstant; also this torque is switched off right after theangular momentum transferring. However, for maintainingof the constancies of the rotor's relative angular velocity theDSSC's control system can form the corresponding stabiliz-ing internal torque. This scheme of the angular momentumtranslation (from the platform-body to the rotor-body) iscalled as the classical spinup maneuver.

During this maneuver the DSSC's longitudinal axis canbe captured in the tilting precessional motion with thecomplex time-dependence of the nutation angle—it corre-sponds to the space mission disruption. Here we also notethat at this spinup maneuver the «platform»-body movefrom quite high-velocity rotations to slow oscillations. Bythis reason the DSSC's dynamical system passes throughthe separatrix-trajectory in the corresponding phase-space[12–16].

In an extension of the indicated works [9–38] in thisresearch on the base of the Melnikov method we considerthe complex tilting precessional motion's initiation as therealization of the chaotic regimes close to the heteroclinicseparatrix-trajectory at the presence of small internalperturbations in the rotor's spinup-engine. Also the newscheme of the spinup maneuver realization is suggested—this scheme can immediately translate the gyroscopicstabilizing angular momentum to the rotor-body withoutchaotic regimes arising.

At the end of Section 1 we can shortly indicate someimportant facts about the DSSC and its possible chaoticbehavior. So, the dual-spin satellites usually are used incommunication systems. The average lifetime (the designservice life) of the DSSC is 10 years, but some practical casesare known which exceeded the contractual lifetime by morethan 20 years. The dual-spin satellites, as well as other typesof satellites, are affected by many external and internalinfluences, so the various failures are possible. The appro-priate analysis of the failures rate/source was made bydifferent organizations and authors, e.g. [42–44]. The fail-ures/anomalies have been reclassified [43] into six cate-gories: (1) Mission Failure, (2) Random Part Failure, (3)Degraded Performance, (4) Phantom Commands, (5) Spur-ious Signals, and (6) Command Errors. In the framework ofthe stated problem of the chaotic DSSC modes investigationthe most interesting anomalies are connected with phantomcommands and spurious signals. It is quite possible thatsome of the listed implementations of “the phantom com-mands” (uncommanded reconfigurations of the vehicles)[43] can be considered as the realizations of the unidentifiedchaotic regimes in the dual-spin satellites' dynamics. Theseare “loss of earth lock and spin-up” (DSCS-2), “despunplatform spun up” (INTELSAT-3), “loss of despin controlcaused despin platform to drive off the earth” (INTELSAT-4), “despin electronics control switched automatically” (TACCOMSAT), also anomalies into the attitude control electronics[44] in cases of such DSSC as the AUSSAT-A3 and the SBS-1.All of the indicated and similar accidents constitute theproblem of the chaotic DSSC motion which is partiallyconsidered in this article.

2. The DSSC angular motion equations

Let us consider the torques-free angular motion of theaxial DSSC (the body #1 is the rotor; the body #2 is theplatform). The platform-body has the general inertiatensor and the rotor-body is dynamically symmetrical.The DSSC motion is considered in the inertial coordinatesystem OXYZ (Fig. 1). We can describe correspondingattitude dynamics in terms of the Euler angles [1,2] and/or on the base of well-known Andoyer–Deprit (also calledas Serret–Andoyer) variables [3–5]. We assume the follow-ing inertia moments ratio A24B24C24A14C1:

The system's motion equation can be written in theform [26]

A _pþðC2�BÞqrþqΔ¼ 0; B_qþðA�C2Þpr�pΔ¼ 0;

C2 _rþ _ΔþðB�AÞpq¼ 0; _Δ¼MΔ; ð1:1Þ

Fig. 1. The coaxial bodies (the axial DSSC).

A.V. Doroshin / Acta Astronautica 94 (2014) 563–576 565

where ½p; q; r�T are components of the absolute angularvelocity of the main body (the platform-body) in the con-nected frame Ox2y2z2, s the relative angular velocity of therotor-body; Δ¼ C1ðrþsÞ; A¼ A1þA2; B¼ A1þB2;

diag½A1;A1;C1�; diag½A2;B2;C2� are the inertia tensors of thecoaxial bodies (##i¼1, 2) in the corresponding connectedframes Oxiyizi, MΔ the internal torque of the rotor-engine.

The Andoyer–Deprit variables [3,4] are defined asfollows (Fig. 1):

L¼ ∂T∂_l

¼KUk; I2 ¼∂T∂ _φ2

¼KUs¼ Kj j ¼ K;

I3 ¼∂T∂ _φ3

¼KUk′; Lr I2

Kx2 ¼ Ap¼ffiffiffiffiffiffiffiffiffiffiffiffiffiI22�L2

qsin l;

Ky2 ¼ Bq¼ffiffiffiffiffiffiffiffiffiffiffiffiffiI22�L2

qcos l; Kz2 ¼ C2rþΔ¼ L; ð1:2Þ

where K is the vector of the DSSC angular momentum.In the Andoyer–Deprit variables we have the well-knownHamiltonian form:

H¼H0þεH1; H0 ¼ T ¼ I22�L2

2sin 2l

A1þA2þ cos 2lA1þB2

" #þ12

Δ2

C1þðL�ΔÞ

C2

2" #

;

ð1:3Þwhere T is the system kinetic energy, H1 the perturbedpart of the Hamiltonian and ε the small parameter corre-sponding to perturbations.

The coordinates φ2; φ3 miss in the Hamiltonian (1.3),then the corresponding conjugate momentums are constant,and the dynamical system reduced to the one-degree-of-freedom subsystem l; L

� �[26,27,32,33]:

_L¼ f Lðl; LÞþεgLðtÞ; _l¼ f lðl; LÞþεglðtÞf Lðl; LÞ ¼ �∂H0

∂l¼ αðI22�L2Þ sin l cos l

f lðl; LÞ ¼∂H0

∂L¼ L

1C2

� sin 2lðA1þA2Þ

� cos 2lðA1þB2Þ

" #� Δ

C2

gL ¼ �∂H1

∂l; gl ¼

∂H1

∂Lð1:4Þ

where gl; gL are the perturbations, andα¼ ðA1þB2Þ�1�ðA1þA2Þ�1:

3. Perturbations of the DSSC attitude motion

Let us consider the DSSC motion at the initiation of thesmall polyharmonic perturbation between the coaxialbodies after the spinup maneuver's implementationðΔ¼ Δ¼ constÞ

_Δ¼MΔ ¼ ε ∑N

n ¼ 0½an sin ntþbn cos nt� ð2:1Þ

With the help of the polyharmonic form (2.1) we cansimulate arbitrary decomposable in finite Fourier series(with N expansion terms) internal torques correspondingto the complex impulse signals in the DSSC control system,including spurious currents at the presence of latency inthe rotor angular velocity sensors.

After Eq. (2.1) integration we obtain the polyharmonicsolution form

Δ¼ Δþε ∑N

n ¼ 0½an sin ntþbn cos nt� ð2:2Þ

The substitution of (2.2) into Eq. (1.4) gives us theperturbed dynamical system

_L¼ f Lðl; LÞþεgLðtÞ; _l¼ f lðl; LÞþεglðtÞ;f Lðl; LÞ ¼ αðI22�L2Þ sin l cos l;

f lðl; LÞ ¼ L1C2

� sin 2lðA1þA2Þ

� cos 2lðA1þB2Þ

" #� Δ

C2;

gLðtÞ ¼ 0; glðtÞ ¼ �η ∑N

n ¼ 0½an sin ntþbn cos nt�; η¼ 1

C2

ð2:3ÞAlso the perturbation form (2.3) is actual at the

presence of the small potential (P) corresponding to smallexternal “nutation” restoring/overturning torques with thepolyharmonic amplitude ζ tð Þ:

ζðtÞ ¼ ε ∑N

n ¼ 0½an sin ntþbn cos nt�;

P ¼ ζðtÞ cos θ¼ ε ∑N

n ¼ 0½an sin ntþbn cos nt� cos θ;

H1 ¼ cos θ ∑N

n ¼ 0½an sin ntþbn cos nt�;

gLðtÞ ¼ 0; glðtÞ ¼ �∂H1

∂L

Here we note that the nutation angle θ (the angle betweenlongitudinal DSSC's axis and the selected direction in theinertial space) can be counted out from the system angularmoments' direction (Fig. 1) [26,34], and then we will havethe following expressions

cos θ¼ LI2; gLðtÞ ¼ 0;

glðtÞ ¼ � ~η ∑N

n ¼ 0½an sin ntþbn cos nt�; ~η ¼ 1

I2

4. The Melnikov function evaluation

As it was indicated in the Section 1, the spinup maneuverincludes the passing through the separatrix-trajectory in the

Fig. 2. The Poincaré section: A2 ¼ 15; B2 ¼ 8; C2 ¼ 6; A1 ¼ 5; C1 ¼ 4; I2 ¼ 20; Δ¼ 3: (a) ε¼ 0 and (b) ε¼ 0:3; η¼ 1=6; a3 ¼ 1.

A.V. Doroshin / Acta Astronautica 94 (2014) 563–576566

system's phase space. During this process the DSSC's long-itudinal axis can be captured in tilting precessional regimeswith complex time-dependences of the nutation angle. Thiseffect can be explained on the basis of the local chaotization ofthe DSSC motion close to the separatrix-trajectory.

First of all, we present examples of the unperturbedphase portrait (Fig. 2a) and the phase portrait (Fig. 2b) ofthe system at the perturbations (2.3) in the Andoyer–Deprit phase space corresponding to the Poincaré section.

The presented (Fig. 2) examples of the Poincaré sec-tions were plotted in the case of the single-harmonicperturbation ðaj ¼ bj ¼ 0 8 ja3Þ on the basis of the fol-lowing “oscillations-phase-repetition” section-condition:ð3t mod 2πÞ ¼ 0. As can we see, at the presence of theperturbation the “chaotic layer” formed in the region closeto the heteroclinic separatrix-trajectory. This chaotic layeris generated as a result of multiple intersections of thestable and unstable split manifolds of the heteroclinicseparatrix-trajectory. Inside the chaotic layer phase trajec-tories can performed complex “chaotic” evolutions includ-ing complicated alternations of rotating and oscillatingregimes with variable characteristics—it is the main reasonof the DSSC capture in the complex tilting motion.

To analyze the indicated possibility of the local motionchaotization we can use the well-known Melnikov method[17]. For using this method it is needed to obtain corre-sponding exact explicit heteroclinic dependences. In thisresearch we will use the heteroclinic solutions obtained inthe work [26]:

pðtÞ ¼ 7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2ðB�C2ÞAðA�BÞ

syðtÞ;

qðtÞ ¼ 7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2�k2ðyðtÞþΔβÞ2

q; rðtÞ ¼ yðtÞþ Δ

B�C2;

yðtÞ ¼ 4a0Eðy70 Þexpð8ðM ffiffiffiffiffi

a0p

=k2Þt Þ½Eðy7

0 Þexpð8ðMtffiffiffiffiffia0

p=k2ÞÞ�a1�2�4a2a0

; ð3:1Þ

where

T ¼ kinetic energy level ; Δ¼ const40; a2 ¼ �k2;

a1 ¼ �2Δβk2; a0 ¼ s2�k2Δ2β2;

y70 ¼ 7

sk�Δβ; β¼ ðB�C2Þ�1�ðA�C2Þ�1;

EðzÞ ¼ 2a0þa1zþ2ffiffiffiffiffia0

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2z2þa1zþa0

pz

;

s2 ¼ HBðA�BÞ; k2 ¼ C2ðA�C2Þ

BðA�BÞ ;

H¼ 2TðA� ~DÞþΔ2b; ~D ¼ Δ2a2T

þB;

M¼ ðA�C2ÞB

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2ðB�C2ÞAðA�BÞ

s; a¼ C1C2þðB�C2ÞðC1�BÞ

ðB�C2ÞC1;

b¼ C2C1þðA�C2ÞðC1�AÞðA�C2ÞC1

:

The Melnikov function in the considered case (for thesystem (2.3)) has the form:

Mðt0Þ ¼ ε

Z þ1

�1f LðpðtÞ; qðtÞ; rðtÞÞglðtþt0Þdt ð3:2Þ

where taking into account expressions (1.4) and with thehelp of (1.2) we can rewrite the function fL

f LðpðtÞ; qðtÞ; rðtÞÞ ¼ f Lðl; LÞ ¼ αðI22�L2Þ sin l cos l¼ αApðtÞBqðtÞThen the Melnikov function is evaluated as follows:

Mðt0Þ ¼ εαη

Z þ1

�1ApðtÞBqðtÞ ∑

N

n ¼ 0½an sin nðtþt0Þ

þbn cos nðtþt0Þ�dt

¼ εαη ∑N

n ¼ 0½JðnÞs fan cos nt0�bn sin nt0g

1 At this and other following figures we have for the phase portraitsthe dimensionless system axes: the abscissa corresponds to the angle l[rad], and the ordinate corresponds to the dimensionless L/I2-ratio(whereas (4.1) this coordinate plane {l, L/I2} is conformed to {φ, cos θ};the phase structure corresponds to the nutation-angle changing). So, wewill not write notations of coordinate systems' axes at the phase-portrait-figures.

A.V. Doroshin / Acta Astronautica 94 (2014) 563–576 567

þ JðnÞc fan sin nt0þbn cos nt0g�;

where

JðnÞs ¼Z þ1

�1gðtÞ sin ðntÞdt; JðnÞc ¼

Z þ1

�1gðtÞ cos ðntÞdt;

ð3:3Þ

gðtÞ ¼ ABpðtÞqðtÞ ¼ 7AB

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2ðB�C2ÞAðA�BÞ

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2�k2ðyðtÞþΔβÞ2

qyðtÞ

It is easy to show that the function gðtÞ is an odd-functionexponentially damped to the zero-value at t-71. So, bythis reasonwe have convergent improper integrals (3.3) withthe following corresponding constant values:

JðnÞc ¼ 0; JðnÞs ¼ constn ð3:4ÞHaving in the mind constants (3.4) we obtain the polyhar-monic form of the Melnikov function:

Mðt0Þ ¼ εαη ∑N

n ¼ 0JðnÞs fan cos nt0�bn sin nt0g ð3:5Þ

The polyharmonic form (3.5) allows us to conclude that theMelnikov function almost always has the infinity quantity ofthe simple zero-roots (with the exception of some separatecases of the parameters fan; bn; JðnÞs gn ¼ 1…N combinations). Itproves the fact of multiple intersections of the stable andunstable split manifolds of the heteroclinic separatrix-trajectory and, therefore, the fact of the local heteroclinicchaos initiation.

Thus, as showed with the help of the Melnikov func-tion, practically in all cases of perturbations the DSSCdynamics is liable to the local heteroclinic chaotization,which is the main reason of the tilting motion with thecorresponding disruption of the space mission.

5. Additional aspects of the realization of thespinup maneuver

To understanding of the DSSC chaotization's featureswe need to consider some details of the spinup maneu-ver's realization [12–16].

First of all, the aim of the maneuver is the creation ofthe DSSC gyroscopic momentum for the attitude stabiliza-tion with the help of the rotation of only the rotor-body(“the partial twist” [12–16,35–38]). The stabilized directionof the DSSC (its longitudinal axis) in ideal conditionscoincides with the direction of the angular momentum'svector of the system. In real conditions, certainly, somesmall misalignments of the angular momentum directionand the DSSC's longitudinal axis take place (the angularmomentum vector is the main direction in the space)—thisdifference corresponds to the nutation angle: θ¼ ∠ k; sð Þ(Fig. 1). So, the DSSC's partial-twist-stabilization providesits attitude orientation with the small defined nutationangle ðθÞ. Also we have to indicate, that in this case thefollowing correspondences between classical Euler's anglesand Andoyers–Deprit's variables take place [34]:

cos θ¼ L=I2; l¼ φ ð4:1Þwhere φ is the platform-body's intrinsic rotation angle.

In the second place, during this maneuver the angularmomentum is transporting from the platform-body to therotor-body, and by this reason the absolute longitudinalangular momentum of the platform-body is reducing tothe zero-value ðC2r0-0Þ, and at the end of the maneuverthe rotor-body receives all the system initial absolutelongitudinal angular momentum ðC1r0-L0Þ.

Let us consider the spinup maneuver's realization asthe series of small separated steps with the gracefulincrement of the Δ-value, which occurs under the actionof the internal torque of the rotor-spinup-engine. In thisaspect of the maneuver consideration we take the series ofthe systems' independent phase portraits (the frames set),corresponding to the series of Δ-values. We can showsome important frames of the phase portraits1 variation(Fig. 3). Here we note that the defined stabilizing value ofthe nutation angle (and the corresponding particularsolution) is depicted as the red curve (Fig. 3).

As can we see, during the Δ-value increasing theseparatrix-region (the saddle points and correspondingheteroclinic separatrices) gradually moves up at the phaseplane (Fig. 3: frames a–e). Herewith, the separatrix saddlepoints coincides with the stabilizing-nutation-angle parti-cular solution (frame d), and, at the continued increasingof the Δ-value the separatrix-region breaks through (bot-tom-to-up) the stabilizing particular solution (frame e).Following further, the separatrix-region rises up, and thesaddle-points coincide with the upper level of the phasespace (frame f)—this corresponds to the «critical» Δn-value[26,27]. After this the rising up of the separatrix-regionis continued (frame g) with the overcritical Δ-values, and,finally, the separatrix-region fully vanishes (frame h) at thesecondary critical Δnn-value. Here we note that the follow-ing magnitudes correspond to the critical values:

Δn ¼ KðB�C2Þ=B¼ Kβ; Δnn ¼ KðA�C2Þ=A¼ Kα

α¼ 1�C2

A; β¼ 1�C2

B; 0oβoαo1;

0oΔnoΔnnoK ¼ I2 ð4:2Þ

We can conclude that the separatrix-region inevitablyintersects the particular solution of the stabilizing-nutation-angle [16]; this intersection occurs between frames c–e(Fig. 3). By this reasonwe have the inevitable passage of thestabilizing-nutation-angle solution through the local het-eroclinic chaotic area (Fig. 3, frames a′–g′). This fact con-firms the guaranteed capture of the DSSC into the localchaotic regime (in [16] is indicated that there are prob-ability phenomena of various outcomes of the DSSC evolu-tion at the separatrix crossing). It is worth to note that inthe case of small perturbations this chaotic regime can beexpressed in a very weak form, which hides its fundamentalpresence in the DSSC dynamics, especially in the practicalimplementation of well-proven space missions (with

Fig. 3. The frames of the phase portrait (start) and the frames of the phase portrait (finish). (For interpretation of the references to color in this figure, thereader is referred to the web version of this article).

A.V. Doroshin / Acta Astronautica 94 (2014) 563–576568

minimal errors and perturbations). So, the chaotizationaspects in the DSSC's perturbed dynamics always takeplace, and it can always become the main reason for theDSSC's attitude disorientation and, moreover, for the dis-ruption of the space mission.

Separately we need to describe the phase portraitsconnected with the frames h and h′ (Fig. 3). As it wasindicated, the heteroclinic separatrix-region fully vanishes atthe values ΔZΔnn this circumstance eliminates all reasons forthe local heteroclinic chaos generation (though some compli-cation of the phase portrait's (Fig. 3h′) structure occurs). Thiscase is most favorable for the DSSC attitude stabilization, but it

can be implemented only at the end of the classical spinupmaneuver's realization. It is also necessary to note, that thesmaller the internal torque of the rotor's spinup, the moregradual is the transition to the finite case (h/h′), and themore stable is the chaotic regime. If the spinup maneuverwere discrete and could instantly translate the whole long-itudinal angular momentum of the platform-body to therotor-body (the instantaneous transition to the case h/h′),then the reasons for the local heteroclinic chaos would befully eliminated.

For the numerical examples (Fig. 3) of the system (2.3)behavior we used the following parameters (Δ-frame-values

Fig. 3. (continued)

A.V. Doroshin / Acta Astronautica 94 (2014) 563–576 569

are presented in Table 1):

A2 ¼ 15; B2 ¼ 8; C2 ¼ 6; A1 ¼ 5; C1 ¼ 4½kg m2�;

I2 ¼ 20 ½kg m2=s�;

η¼ 1=6 ½ðkgm2Þ�1�;

a1 ¼ 1; a2…N ¼ b0…N ¼ 0 ½kgm2=s�; cos θ¼ 0:8:

The small parameter: the left column corresponds to ε¼ 0;the right column ε¼ 0:3:

The Poincaré sections' condition: ðt mod 2πÞ ¼ 0:

Also it is worth to underscore some differences in thechaos regimes, which arise during the spinup maneuver.The indicated differences are presented in Fig. 4.

First of all, at the relatively “small” Δ-values ð0oΔoΔnÞwe see (Fig. 4a) the broad chaotic layer spreading on thewide zone of the nutation oscillations (cos θ is changinginside the interval, which includes both negative and near-unit values, therefore the nutation-angle can oscillatewith big amplitudes from near-zero magnitudes up tothe values above π/2). In this case the longitudinal DSSC-axis performs chaotic tumbling-evolutions. We can definethis chaotic regime as the “oscillating chaos”.

Fig. 3. (continued)

A.V. Doroshin / Acta Astronautica 94 (2014) 563–576570

The second type of the chaos regimes (Fig. 4b) correspondsto the overcritical Δ-values ðΔnrΔoΔnnÞ, when the motionoccurs in the “upper” zone of the phase spaceð0o cos θr1Þ. In this case the rotation of the DSSC long-itudinal axis about angular momentum's direction preferablytakes place, when the chaotic regime resembles the gyroscoperegular precession (certainly with the chaotic amplitude). Wecan define this chaotic regime as the “rotating chaos”. Thiscase is less dangerous for the DSSC's attitude stabilization thanthe oscillating chaos.

The third case of the chaos regimes is possible. This isthe heteroclinic chaos close to the secondary heteroclinicstructures arising in addition to the main separatrix-region

at the presence of perturbations (Fig. 4c). These secondaryheteroclinic bundles usually are quite narrow (along theordinate-direction) and have properties similar to the rota-tional chaos. The figure (Fig. 4c) demonstrates the narrowsecondary chaotic bundle at the value Δnn; when the mainseparatrix-region is already vanished. This type of thechaotic regime we can define as the “secondary chaos”.

Also as can we see (Fig. 4d), the absence of any chaoticregimes is possible at large Δ-values comparable with thevalue of the system angular momentum (Δ�0.9I2). It is quiteclear, because in this case the greater part of the system'sangular momentum is transferred to the dynamically-symmetrical rotor-body and the platform-body with the

Fig. 3. (continued)

Table 1

Frame a, a′ b, b′ c, c′ d, d′ e, e′ f, f′ g, g′ h, h′

Δ, kg m2/s 0 5 8 8.62 9 Δn¼10.77 13 Δnn¼14

A.V. Doroshin / Acta Astronautica 94 (2014) 563–576 571

three-axial inertia tensor is almost motionless in the inertialspace, and therefore the DSSC dynamics is determined by thesimple dynamics of the dynamically-symmetrical rotor. Thiscase of the motion is most preferred to the DSSC's attitudestabilization. The implementation of the “chaos-free” phase

portrait (Fig. 4d) is possible if the initial DSSC's attitudeorientation is quite precise and the nutation-angle's initialvalue is quite small—then after the spinup maneuver thelongitudinal angular momentum will be fully transferred tothe rotor-body with the appropriate large value of the

A.V. Doroshin / Acta Astronautica 94 (2014) 563–576572

angular momentum Δ (in the considered Fig. 4d example thenutation-angle corresponds to the condition:cos θ¼ Δ=I2 ¼ 0:9).

In addition to the Poincaré-maps (Figs. 3, 4) we presentillustrations of the chaotic motion with the help of the time-history of the angular velocity's components (Fig. 5a), theperturbed polhode (in the p–q–r-space of the angularmoment's ellipsoid) and the corresponding Poincaré-map(Fig. 5b), the time-history of the nutation angle (Fig. 5c), thetime-history of the intrinsic rotation angle (Fig. 5d), the sixPoincaré-maps-images of the unperturbed heteroclinic

Fig. 4. Cases of chaos (in system (2.3)): (a) the oscillating chaos Δ¼ 4ð Þ; (b) – the(d) the disappearance of chaos ðΔ¼ 0:9I2Þ A2 ¼ 15; B2 ¼ 8; C2 ¼ 7; A1 ¼ 5; C1 ¼ 4

separatrix in the forward-time-direction t-þ1 (Fig. 5e),the heteroclinic net (Fig. 5f) as the set of the six Poincaré-map-images [26] of the unperturbed heteroclinic separatrix inthe forward-time-direction and in the back-time-direction t-�1 (six Poincaré-map-preimage). All of the simulations(Fig. 5) correspond to the oscillating chaos (Fig. 4a) with thesame DSSC parameters and the motion's initial conditions. So,the illustrations (Fig. 5) show us the appropriate features ofthe chaotic regime: the irregularity of the motion parameters(their time-history), the tangled polhode, and complicatedheteroclinic nets.

rotating chaos ðΔ¼ ΔnÞ; (c) the secondary heteroclinic chaos ðΔ¼ ΔnnÞ; and; I2 ¼ 20; a1 ¼ 1; a2::N ¼ b0::N ¼ 0; η¼ 1=7; ε¼ 0:6.

Fig. 5. The oscillating chaos' features.

A.V. Doroshin / Acta Astronautica 94 (2014) 563–576 573

Fig. 6. The three-body-DSSC.

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6. The new approach of the realization of the chaos-freespinup maneuver

As it was indicated in the previous section, in order tothe local heteroclinic chaos avoidance during the spinupmaneuver we need to instantly translate the whole long-itudinal angular momentum of the platform-body to therotor-body (the instantaneous transition from the phaseportrait Fig. 3a/a′ straight to the frame Fig. 3h/h′).

For the realization of this “jumping” change of the phaseportraits' form Fig. 3a–h we can use the method of theconjugated rotors spinup-captures [39–41], which were devel-oped for the multi-rotor spacecraft's attitude reorientation.

For application of this method to the chaos-free spinupmaneuver we can add to the DSSC-two-body-coaxial-sys-tem the “opposite” rotor #3 (Fig. 6), which is conjugated tothe rotor #1. Then it is possible to perform the “conjugatespinup”—this is the process of the spinup of the conjugatedrotors in opposite directions up to desired values of therelative angular velocity with the help of internal torques;the summarized angular momentum of the conjugatedrotors during the conjugate spinup is constant. Then, forthe instantaneous transition of the angular momentum ofone of the conjugate rotors to the main body (to compen-sate the angular momentum of the main body, and to stopit) we can make the «rotor capture»—it is the immediatestopping-deceleration of the rotor relative angular velocitywith the help of internal torques. So, the rotor capturemeans the “instantaneous freezing” of the rotor withrespect to the main body. The capture can be performedwith the help of the gear meshing, large frictions or othermethods.

To perform the spinup maneuver based on the newscheme we execute the following steps:

(1)

The DSSC is placed into the orbit in the “monobody”form (with the fixed rotors ##1 and 3) with the initialvalue of the longitudinal angular velocity r0.

(2)

The rotors-restrictions are removed and the rotors areexempted for the relative rotation.

(3)

The conjugated spinup of the rotors ##1 and 3 isrealized with the help of the following constant internalengines' torques:

Mi1 ¼

ðC2þC3Þr0Ts

; Mi3 ¼ �ðC2þC3Þr0

Tsð5:1Þ

where C3 is the axial inertia moment of the rotor #3; Mi1

the internal torque of the rotor's #1 spinup engine; Mi3

the internal torque of the rotor's #3 spinup engine; Ts isthe duration (time-moment) of the rotor's spinup. At theend of the spinup with the torques (5.1) the rotor #3 andthe main body (#2) have equal magnitudes of the long-itudinal angular momentum, but with opposite signs.

(4)

The capture of the rotor #3 is performed, then thelongitudinal angular momentums of the body #2 andthe rotor #3 immediately compensate each other—itstops the rotation of both bodies (##2 and 3). After thecapture of the rotor #3 its relative rotation is prohibited,and from this time-moment we can consider the bodies#2 and #3 as the new coupled platform-body.

The indicated four steps present the new spinup maneu-ver's scheme with the instant translation of the stabilizinglongitudinal angular momentum to the rotor-body (rotor #1).This scheme avoids the heteroclinic chaos in the DSSCdynamics during the spinup maneuver realization.

Based on the material [39] we can reduce the motionequations of the multi-rotor system to the considered three-body case:

A _pþðC�BÞqrþqΔ¼ 0;B_qþðA�CÞrp�pΔ¼ 0;C_rþ _ΔþðB�AÞpq¼ 0;

8><>:

_Δ1 ¼Mi1; _Δ3 ¼Mi

3;_Δ¼ _Δ1þ _Δ3;

(ð5:2Þ

where

Δ¼ Δ1þΔ3; Δ1 ¼ C1ðrþs1Þ; Δ3 ¼ C3ðrþs3ÞA¼ A1þA2þA3; B¼ A1þB2þA3; C ¼ C2 ð5:3Þdiag½A3;A3;C3� the inertia tensor of the rotor #3 in thecorresponding connected frames Ox3y3z3; s1 the rotor #1relative angular velocity, s3 the rotor #3 relative angularvelocity; Δ the rotors' summarized absolute longitudinalangular momentum.

Here it is needed to note that the internal torques of therotors engines correspond to the two actions: the rotorsspinup within time Ts, and the rotor #3 capture at the time-moment Tc. We can use in this task the following form ofthe engine-torques based on the Heaviside function:

Mi1 ¼

ðC2þC3Þr0Ts

ðHðtÞ�Hðt�TsÞÞ;

Mi3 ¼ �ðC2þC3Þr0

TsðHðtÞ�Hðt�TsÞÞ�νs3Hðt�TcÞ; ð5:4Þ

where ν is the coefficient of the large viscous friction for thefast capture modeling, H(t) is the Heaviside function.

The modeling results are presented in Fig. 7. As we cansee, the “jumping” change of the summarized longitudinalangular momentum Δ takes place at the time-moment Tc(Fig. 7a), and the body-platform angular velocity r alsoimmediately takes the zero-value (Fig. 7b).

Here it is worth to note that the rotors ##1 and 3 canbe different in geometrical and inertia-mass parameters.

Fig. 7. The DSSC′s spinup maneuver modeling:

A2 ¼ 15; B2 ¼ 8; C2 ¼ 6; A1 ¼ A3 ¼ 5; C1 ¼ C3 ¼ 4 ½kgm2�; ν¼ 200 ½kgm2=s�; r0 ¼ 1 ½1=s�; Ts ¼ 0:5; Tc ¼ 1 ½s�

A.V. Doroshin / Acta Astronautica 94 (2014) 563–576 575

This difference does not change the principle of theimplementation of DSSC spinup maneuver. So, the rotor's#3 parameters can be selected based on the DSSC'sconstructive appropriateness.

The considered spinup maneuver's scheme with instanttranslation of the stabilizing longitudinal angular momen-tum to the rotor-body (rotor #1) allows to avoid theheteroclinic chaos in the DSSC dynamics. This is possibledue to the instant change of the phase portraits' form withthe separatrix-region disappearance.

7. Conclusion

In the article the attitude dynamics of the dual-spinspacecraft was considered during the classical spinup man-euver realization, including the investigation of the regimesof the heteroclinic chaos arising at the presence of poly-harmonic perturbations. Also the classification of the possi-ble cases of the DSSC chaotic regimes was implemented.

As it was shown, the main reason of the chaoticregimes' initiation is the qualitative changing of the phaseportraits' structure during the classical spinup maneuver,when the gradually increasing of the Δ-value inside theinterval ½C1r0; L0 ¼ ðC1þC2Þr0� takes place. In this processthe separatrix-region is moved up to upper border of thephase portrait, and, hence, the zone of the heteroclinicchaos (in its evolution) inevitably passes through theparticular solution corresponding to the DSSC's attitudeposition, which is stabilized by the partial twist. So this“stabilized solution” is inevitably involved in the chaoticregime, and, therefore, the attitude orientation/stabiliza-tion can be lost.

Here we underline that the failures/anomalies factsindicated in the introduction-section correspond to the

real “experimental data” [42–44] illustrating the irregularmotion cases and phantom commands realization; but thecomparison of this data with the theoretical results isproblematic because the indicated failures were not beingmonitored. So, the special experiments for the chaosphenomena study were not conducted in the frameworkof the known DSSC space programs. The space programs'engineers/implementators ought to plan and conductthese interesting experiments in the near future.

To avoid the chaotic regimes initiation we suggested thenew method of the spinup maneuver realization, which isbased on the three-body-DSSC scheme with the executionof the conjugated rotors spinup-captures. This modifiedspinup maneuver provides the instant translation (the jump)between the phase portrait's forms with the separatrix-regiondisappearance, and, therefore, without any chaotic regimearising.

Acknowledgments

This work was supported by the Russian Foundation forBasic Research (RFBR #11-08-00794-a).

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