dynamics of the space tug system with a short …...research article dynamics of the space tug...

Dynamics of the Space Tug System with a Short Tether

Liu J Cui N Shen F amp Rong S (2015) Dynamics of the Space Tug System with a Short TetherInternational Journal of Aerospace Engineering 2015 [740253] httpsdoiorg1011552015740253

Published inInternational Journal of Aerospace Engineering

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Download date15 Jul 2020

Research ArticleDynamics of the Space Tug System with a Short Tether

Jiafu Liu1 Naigang Cui2 Fan Shen2 and Siyuan Rong2

1Shenyang Aerospace University Shenyang 110136 China2Harbin Institute of Technology Harbin 150001 China

Correspondence should be addressed to Jiafu Liu liujiafuericking163com

Received 25 February 2015 Revised 12 June 2015 Accepted 2 July 2015

Academic Editor Paul Williams

Copyright copy 2015 Jiafu Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The dynamics of the space tug system with a short tether similar to the ROGER system during deorbiting is presented Thekinematical characteristic of this system is significantly different from the traditional tethered system as the tether is tensionaland tensionless alternately during the deorbiting processThe dynamics obtained based on the methods for the traditional tetheredsystem is not suitable for the space tug systemTherefore a novel method for deriving dynamics for the deorbiting system similar tothe ROGER system is proposed by adopting the orbital coordinates of the two spacecraft and the Euler angles of ROGER spacecraftas the generalized coordinates instead of in- and out-plane librations and the length of the tether and so forth Then the librationsof the system are equivalently obtained using the orbital positions of the two spacecraft At last the geostationary orbit (GEO) andthe orbit whose apogee is 300 km above GEO are chosen as the initial and target orbits respectively to perform the numericalsimulationsThe simulation results indicate that the dynamics can describe the characteristic of the tether-net system convenientlyand accurately and the deorbiting results are deeply affected by the initial conditions and parameters

1 Introduction

The number of flying objects including the controlled space-craft and the space debris (including the abandoned space-craft) in the low earth orbit (LEO) and the geostationaryorbit (GEO) increases sharply because of the frequent humanspace activities [1ndash3] It is reported by ESA that at the endof 2008 there were 1186 objects in GEO and only 32 werecontrolled spacecraft The collision risks will be posed by theuncontrolled spacecraft and space debris mentioned aboveTherefore it is necessary to monitor and clear the abandonedspacecraft and space debris

The RObotic GEostationary Orbit Restorer (ROGER)concept was proposed by ESA The researchers not onlyfocused on the economic consideration and future businessapplication but also focused on the orbital monitoring andclearance of the abandoned spacecraft [4 5] To make useof the space electrodynamic tethered system to deorbitspacecraft has been studied in [6 7] which utilizes themotion of the conductive tether relative to the magneticfield of the earth to produce electrodynamic forces It is alsoproposed that the abandoned spacecraft can be deorbited

by the solar radiation pressure experienced by the sailcraftattached to the spacecraft The sailcraft is initially folded andwill be deployed when the spacecraft is spent The feasibilityof the solar radiation pressure based deorbiting approach hasbeen demonstrated by the NanoSail-D project within NASAMoreover the cubeSail project in Surrey [8] and the projectusing solar pressure in Strathclyde [9] are also performed tostudy the spacecraft deorbiting approach by solar pressureIt is a practical method by using the space robots to capturethe noncooperative spacecraft or to deorbit the abandonedspacecraftThemethod can also be used in the spacemissionssuch as spacecraft repair and fuel charges [10 11]

The space tether based deorbiting system is the mostcomprehensively studied one among themethodsmentionedabove including the ROGER system and the space electrody-namic tethered system

The monitoring for the current orbit and forecasting forthe future orbit are studied in the ROGER project [5] Thefeasibility of the GEO service is also discussed from theview of economic and technical aspects The ROGER space-craft approaches captures and transports the abandonedspacecraft into the graveyard orbit The configuration of the

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2015 Article ID 740253 16 pageshttpdxdoiorg1011552015740253

2 International Journal of Aerospace Engineering

ROGER system including the ROGER spacecraft and thesubsystem used for capturing the abandoned spacecraft isstudied and developed in ESA The technical details of theROGER system mainly focus on the grappling equipmentattitude and orbit control devices and so forth Moreover theguidance navigation and control elements are also focusedon [4]

The typical electrodynamic tethered system for deorbit-ing is MXER which utilizes the electrodynamic force todeorbit the spacecraft The system design is presented in[12 13]The studies concerning space tethered systemmainlyfocus on tether models [14 15] tether vibrations [16ndash19]in-plane and out-plane librations [20ndash26] attitude motions[19 27ndash30] and orbital motions [31ndash33] The bead model isadopted to establish the dynamical equations in [14 15] InputShaping method is adopted to reduce the initial vibrations ofthe electrodynamic tether system in [34] The stability of in-plane and out-plane librations of the electrodynamic tetheredsystem in inclined and elliptic orbits with high eccentricityis studied in [25 26] and the chaotic librations can bestabilized to the periodicmotion by delayed feedback controlThe attitude control of the main satellite can be performedby the offset scheme [27] The vibration of the tether canalso be suppressed by the offset scheme [19] There are alarge number of literatures concerning with space tetheredsystem however few concerning with the system design anddynamics for the space tug system similar to the ROGERsystem during deorbiting It seems that the configurations ofthe tethered systems in [35 36] are analogous to the ROGERsystem however there exist differences between them

The paper is organized as follows Firstly the config-uration that the ROGER spacecraft in front of the aban-doned one is adopted the reference frames and coordinatetransformations are presented Secondly the conclusion thatthe traditional dynamic modeling strategy for the tetheredsystem is not suitable for the space tug system similarto the ROGER system in this paper is presented Thirdlythe dynamic model for the space short-tether system dur-ing deorbiting is presented The attitude motion of theROGER spacecraft the orbital motion of the two space-craft the librations of the system and the elasticity of thetether are all considered Finally the effectiveness of thedynamic model is verified by numerical simulations Thesuperior deorbiting conditions are summarized based onthe simulation results for various parameters and initialconditions

2 The Roger System and Reference Frames

21 The Configuration Simplification and Assumption Theconfiguration of the space tug system with a short tethersimilar to the ROGER system is presented in Figure 1 Asshown in Figure 1 the active (the ROGER) spacecraft andthe abandoned spacecraft are denoted as ldquo119860rdquo and ldquo119861rdquorespectively

The tether is tensional and tensionless alternatively dur-ing the deorbiting process because the ROGER spacecraft isin front of (not above) the abandoned one

A

B

B

A

Short tether

Short tether

(tensionless)

(tensional)

RO

EI

RS

Figure 1The schematic diagramof the space tug systemwith a shorttether similar to the ROGER system during deorbiting

Earth

Spacecraft

ZI

XI

XI

ZI1

ZI1

YI

YI

YI1

XI1

XI1 EI

EI

ZO

XO

f

Figure 2 The related reference frames

Some assumptions are presented as follows

(1) The attitude motion of the abandoned spacecraft isneglected and only the orbital motion is considered

(2) The mass of the short tether is neglected(3) Solar radiation pressure perturbation earth non-

spherical perturbation and other planets perturba-tion are neglected

22 Reference Frames and Transformations

221 Inertial Frame J2000 (E119868X119868Y119868Z119868 120587119868) 119864119868is the mass

center of the earth E119868X119868points to the vernal equinox

E119868Y119868points to the earthrsquos rotation axis perpendicular to the

equatorial plane and E119868X119868 E119868Y119868 and E

119868Z119868form a right-

handed orthogonal frame The basis of 120587119868is (i119868 j119868 k119868)119879

222 The ROGER Spacecraft Orbital (O119874

X119874

Y119874

Z119874

120587119874) and

Body (O119874

X119860

Y119860

Z119860

120587119860) Frames 119874

119874locates at the mass cen-

ter of ldquo119860rdquo O119874

Z119874points to 119864

119868 and O

119874X119874points to forward

direction located in the orbital plane and perpendicularto O119874

Z119874 O119874

X119874

O119874

Y119874 and O

119874Z119874

form a right-handedorthogonal frame The basis is (i

119874 j119874

k119874

)119879

The origin of 120587119860is at themass center of ldquo119860rdquo with the axes

pointing along the principal axes of inertia of ldquo119860rdquo The basisis (i119860

j119860

k119860

)119879

223 Inertial Frame (E119868X1198681

Y1198681

Z1198681

1205871198681) The detailed defini-

tion can be found in Figure 2 The basis is (i1198681

j1198681

k1198681

)119879

International Journal of Aerospace Engineering 3

224 Body Frame of the Tether (C119860

X119905Y119905Z119905 120587119905) 119862119860is the

attachment point with ldquo119860rdquo when there are no librations and120587119905coincides with120587

119874The in-plane and out-plane librations in

traditional tethered system are defined by the angles between120587119874and 120587

119905

225 The Transformation from 120587119860to 120587119874 A three-element

vector (120595119860

120593119860

120579119860

)119879 is used to describe the attitude of the

ROGER spacecraft with respect to the orbital frame Thetransformation from 120587

119874to 120587119860can be realized by 3-1-2 that

is 119911(120595119860

) rarr 119909(120593119860

) rarr 119910(120579119860

) rotation 120595119860

120593119860

120579119860are the

Euler angles of yaw-roll-pitch describing attitude motion forthe ROGER spacecraft

226TheTransformation from1205871198681to120587119874 Therotationmatrix

C1198741198681

is obtained by a 119910(minus119891 minus 1198911) rotation and the initial true

anomaly 1198911is selected as zero 119891 is the true anomaly C

1198741198681is

as follows

C1198741198681

=[[

[

cos (minus119891) 0 minus sin (minus119891)

0 1 0

sin (minus119891) 0 cos (minus119891)

]]

]

(1)

227 The Transformation from 120587119874

to 120587119905 A two-element

vector (120595 120579)119879 is used to describe the attitude of the tether

with respect to the orbital frame The transformation fromO119874

X119874

Y119874

Z119874to C119860

X119905Y119905Z119905 can be realized by a 119911(120595) rarr 119910(120579)

rotation The rotation matrix C119905119874

is as follows

C119905119874

=[[

[

cos 120579 0 minus sin 120579

0 1 0

sin 120579 0 cos 120579

]]

]

[[

[

cos120595 sin120595 0

minus sin120595 cos120595 0

0 0 1

]]

]

(2)

228 The Transformation from 120587119868to 120587119874 The constant rota-

tion matrix C1198681119868

is obtained by a 3-1 rotation presented asfollows

C1198681119868

=[[[

[

1 0 0

0 cos (minus90∘) sin (minus90

∘)

0 minus sin (minus90∘) cos (minus90

∘)

]]]

]

[[[

[

cos 90∘ sin 90

∘0

minus sin 90∘ cos 90

∘0

0 0 1

]]]

]

(3)

Thus the transformation from 120587119868to 120587119874is obtained as

C119874119868

= C1198741198681

C1198681119868

3 The Traditional Dynamic Modeling Methodfor the Roger System

31 The Selection of the Generalized Coordinates In order tostudy the dynamics of the ROGER system during deorbitingthe attitude and orbital motions of the ROGER spacecraftin-plane and out-plane librations of the system the lengthmotion of the tether and orbital motion of the abandonedspacecraft are studied in this paper When the tether is ten-sional the Euler angles 120595

119860 120593119860 and 120579

119860and orbital positions

119909119860 119910119860 and 119911

119860are chosen as the generalized coordinates

moreover the length of the tether 119897 and the in-plane andout-plane librations 120595 120579 are also chosen as the generalizedcoordinates When the tether is tensionless the strategy tochoose the generalized coordinates will be discussed later

32 Dynamic Modeling Consideration When the Tether IsTensional The Lagrange method is used to establish thedynamics for the ROGER system The kinetic energy 119879

119860and

gravitational potential energy 119880119860of the ROGER spacecraft

are as follows

119879119860

=1

2int119860

( r119860

+ 120596119860

times r) sdot ( r119860

+ 120596119860

times r) 119889119898

=1

2119898119860

1003816100381610038161003816r119860

1003816100381610038161003816

2+

1

2120596119860

sdot J119860

sdot 120596119860

119880119860

= minus120583119898119860

1003816100381610038161003816r1198601003816100381610038161003816

(4)

where r119860and r119860are the position and velocity vectors of point

119860 120596119860is the angular velocity of the ROGER spacecraft and r

and r119898are the position vectors of the infinitesimal mass 119889119898

with respect to point 119860 and 119864119868 respectively 119898

119860is the mass

J119860is the moment of inertia tensor 120583 is Earthrsquos gravitational

constant r119862119860

(constant in 120587119860) is the position vector of the

attachment point with respect to point 119860 r119862119860119861

is the positionvector of the abandoned spacecraft with respect to 119862

119860 In

this paper |rAB| is adopted as the length of the tether insteadof |r119862119860119861

| The kinetic energy 119879119861and gravitational potential

energy 119880119861are as follows

119879119861

=1

2119898119861

[ r119860

+∘

l (119905) + 120596119905times l (119905)]

sdot [ r119860

+∘

l (119905) + 120596119905times l (119905)]

119880119861

= minus120583119898119861

1003816100381610038161003816r119860 + l (119905)1003816100381610038161003816

(5)

The elastic potential energy of the tether is as follows

119880119890

=1

2119896 (119897 minus 119897

0)2

(6)

In the preceding three equations 119898119861and r119861indicate the

mass and velocity of the abandoned spacecraft l(119905) and∘

l (119905)

are the derivatives of l(119905) relative to 120587119868and 120587

119905 respectively

120596119905is the angular velocity of the tether 119896 is the coefficient of

elasticity of the tether 1198970is the original length of the tether

The Lagrange equation considering nonpotential force isas follows

119889

119889119905(

120597119871

120597q119895

) minus120597119871

120597q119895

= Q119895 (7)

where Q119895is the nonpotential force containing the thrust

force and the attitude control torques generated by theROGER spacecraft q

119895= (119909

119860 119910119860

119911119860

120595119860

120593119860

120579119860

120595 120579 119897)119879

is the vector of the generalized coordinates The dynamicmodeling procedure above can be used when the tether istensional


33 Dynamic Modeling Consideration When the Tether IsTensionless The ROGER and abandoned spacecraft becometwo free flying spacecraft orbiting the earth when thetether is tensionless Thus the orbital position coordinates(119909119860

119910119860

119911119860) and (119909

119861 119910119861 119911119861) of the ROGER and abandoned

spacecraft and the Euler angles (120595119860

120593119860

120579119860) of the ROGER

spacecraft are selected as the generalized coordinates whenthe tether is tensionless

34 Analysis of the Traditional Method The dynamic mod-eling method presented above is just the dynamic modelingstrategy for the traditional tethered system To select the in-plane and out-plane librations and the length of the tether asthe generalized coordinates is inappropriate for establishingthe dynamics of the ROGER system in this paper although itis easy to be understoodThe reasons are presented as follows

(1) The ROGER spacecraft is in front of (not above) theabandoned spacecraft during deorbiting thereforethe tether will be tensional and tensionless alternatelyThe in-plane and out-plane librations and the lengthof the tether can be selected as the generalizedcoordinates when the tether is tensional howeverthe librations and the length of the tether never existwhen the tether is tensionless

(2) It is assumed that the tether is tensional initially asshown in Figure 3 It is evident that the tether willbecome tensionless when 119897 = 30 and 119897 lt 0 Thedynamic model when the tether is tensionless willbe used And the tether will become tensional when119897 = 30 and 119897 gt 0 Accordingly the dynamic modelwhen the tether is tensional will be usedThe critical points are presented in Figure 4 It isevident that the tether will be tensionless at points 13 and 5 and tensional at points 2 4 and 6

(3) It is time-consuming and difficult to switch repeatedlybetween the two groups of equations that can bederived based on the ideas stated in Sections 32 and33 moreover the computation accuracy and real-time requirements cannot be satisfied during deor-biting Therefore it is essential to establish a uniquegroup of dynamic equation for the deorbiting processusing a unique group of generalized coordinates

Therefore the ROGER and abandoned spacecraft orbitalposition coordinates (119909

119860 119910119860

119911119860) and (119909

119861 119910119861 119911119861) and the

Euler angles (120595119860

120593119860

120579119860) of the ROGER spacecraft are

adopted as the generalized coordinates rather than the libra-tions and the length of the tether

4 The Dynamics of the Tether System

The orbital position coordinates of the ROGER and aban-doned spacecraft and the Euler angles of ROGER spacecraftare adopted as the generalized coordinates instead of thelibrations and the length of the tether because the followingrelation holds

(120595 120579 119897 119909119860

119910119860

119911119860

) lArrrArr (119909119861 119910119861 119911119861 119909119860

119910119860

119911119860

) (8)

A

dm

r B

CA

rC119860

rmrA

EI

rAB

rB

rC119860B

ang120575 asymp 0

Short tether

Figure 3 The detailed configuration of the ROGER spacecraft

tO

1

2 3 4 5

6l0

l(t)

Figure 4 The length of the tether with the critical points (1ndash6)

The kinetic energy of the ROGER system is as follows

119879 = 119879119860

+ 119879119861

119879119860

=1

2119898119860

(2

119860+ 1199102

119860+ 2

119860) +

1

2120596119860

sdot J119860

sdot 120596119860

119879119861

=1

2119898119861

(2

119861+ 1199102

119861+ 2

119861)

(9)

The potential energy is as follows

119880 = 119880119892

+ 119880119890

119880119892

= 119880119892119860

+ 119880119892119861

119880119892119860

= minus120583119898119860

radic1199092

119860+ 1199102

119860+ 1199112

119860

119880119892119861

= minus120583119898119861

radic1199092

119861+ 1199102

119861+ 1199112

119861

(10)

The elastic potential energy of the tether is as follows byreferring to Figure 3

119880119890

=

010038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816le 1198970

1

2119896Δ2 10038161003816100381610038161003816

r119862119860119861

10038161003816100381610038161003816gt 1198970

Δ =10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816minus 1198970

119897 = r119862119860119861

= r119861

minus r119860

minus r119862119860

(11)

The related position vectors in the preceding equation canbe observed in Figure 3 All the position vectors are easy toobtain since r

119862119860is constant in 120587

119860


TheLagrange function of theROGER system is as follows

119871 = 119879 minus 119880 = 119879119860

+ 119879119861

minus 119880119892119860

minus 119880119892119861

minus 119880119890

=1

2119898119860

(2

119860+ 1199102

119860+ 2

119860) +

1

2120596119860

sdot J119860

sdot 120596119860

+1

2119898119861

(2

119861+ 1199102

119861+ 2

119861) minus (minus

120583119898119860

radic1199092

119860+ 1199102

119860+ 1199112

119860

)

minus (minus120583119898119861

radic1199092

119861+ 1199102

119861+ 1199112

119861

) minus 119880119890

(12)

The dynamic equations will be obtained when 119871 issubstituted into the Lagrange equations

41 Dynamic Equations for Tensional State Firstly the orbitalmotion of the ROGER spacecraft is given as follows

119860

= minus120583119909119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+119896Δ (119909

119861minus 119909119860

minus 119903119909)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119909

119910119860

= minus120583119910119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+

119896Δ (119910119861

minus 119910119860

minus 119903119910)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119910

119860

= minus120583119911119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+119896Δ (119911119861

minus 119911119860

minus 119903119911)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119911

(13)

where 119903119909 119903119910 and 119903

119911are the components of r

119862119860in 120587119868

Secondly the attitude motion equation of the ROGERspacecraft is as follows

J119860

sdot 119860

+ 120596119860

times J119860

sdot 120596119860

= T119888

+ T119889 (14)

where T119888is the control torque and T

119889is the disturbance

torque caused by the offset of the thrust force the librationsof the tether combining the tension of the tether T

119889119891and T

119889119905

are used to represent the former and the latter disturbancetorques respectively

As shown in Figure 5 the offset vector AP the compo-nents of the thrust force and T

119889119905are as follows

AP = r119901

= 119903119901119910

j119860

+ 119903119901119911

k119860

F = 119865119909

i119860

T119889119905

= r119901

times F

(15)

We can obtain T119889119891

as T119889119891

= r119862119860

times T r119862119860

and T can becalculated during deorbiting

The procedure to obtain the orbital motion of the aban-doned spacecraft is similar to the ROGER spacecraft

FP

A

A

T

BrC119860

CA

XA

YA

ZA

Tension

ROGER

Figure 5 The disturbance torques

42 The Dynamic Equations regardless of the State (Tensionalor Tensionless) of the Tether Theorbital and attitude dynamicequations of the ROGER spacecraft are as follows

119860

= minus120583119909119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119909

+

119896Δ (119909119861

minus 119909119860

minus 119903119909)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)

119910119860

= minus120583119910119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119910

+

119896Δ (119910119861

minus 119910119860

minus 119903119910)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)

119860

= minus120583119911119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119911

+

119896Δ (119911119861

minus 119911119860

minus 119903119911)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)

J sdot + 120596 times J sdot 120596 =

T119888

+ T119889119891

+ T119889119905

(119897 ge 1198970)

T119888

+ T119889119891

(119897 lt 1198970)

(16)

TheGaussian perturbation equations are used to performthe dynamics simulations

119889119886

119889119905=

2

119899radic1 minus 1198902(120590119911119890 sin119891 + 120590

119909

119901

119903)

119889Ω

119889119905=

120590119910119903 sin 119906

1198991198862radic1 minus 1198902 sin 119894

119889119894

119889119905=

120590119910119903 cos 119906

1198991198862radic1 minus 1198902


119889119890119909

119889119905=

radic1 minus 1198902

119899119886120590119911sin 119906

+ 120590119909

[(1 +119903

119901) cos 119906 +

119903

119901119890119909] +

119889Ω

119889119905119890119910cos 119894

119889119890119910

119889119905=


119899119886minus120590119911cos 119906

+ 120590119909

[(1 +119903

119901) sin 119906 +

119903

119901119890119910] minus

119889Ω

119889119905119890119909cos 119894

119889120593

119889119905= 119899 minus

1

119899119886[120590119911

(2119903

119886+


1 + radic1 minus 1198902119890 cos119891)

minus 120590119909

(1 +119903

119901)


1 + radic1 minus 1198902119890 sin119891]

minus

120590119910119903 cos 119894 sin 119906

1198991198862radic1 minus 1198902 sin 119894

(17)where (119886 Ω 119894 119890

119909 119890119910 120593) are six independent orbital elements

for the ROGER and abandoned spacecraft In the paper 119886Ω and 119894 denote the semimajor axis longitude of ascendingnode of the orbit and orbital inclination Please note that119890119909

= 119890 cos120596 119890119910

= 119890 sin120596 and 120593 (sum of mean anomalyand argument of perigee) are used instead of the orbitaleccentricity 119890 argument of perigee 120596 and true anomaly ]to avoid singularity when 119890 approaches zero 119899 119901 119903 and119906 are the mean orbital rate semilatus rectum of an orbitgeocentric radius and sum of true anomaly and argument ofperigee respectively 120590

119909 120590119910 and 120590

119911are the components of the

perturbative accelerations along 119909- 119910- and 119911-axis of 120587119874

43 Equivalent Expression of Librations It is evident that theROGER system will never librate when the tether is ten-sionless and tensional alternately during deorbiting whichis quite different from the traditional space tethered systemA novel strategy to equivalently express the librations of theROGER system has been proposed based on the positions ofthe ROGER and abandoned spacecraft

As in Figure 6(a) the direction cosines of line section ABin E119868X119868Y119868Z119868are expressed as follows

1205821

= cos (AB i119868) =

(119909119860

minus 119909119861)

|AB|

1205822

= cos (AB j119868) =

(119910119860

minus 119910119861)

|AB|

1205823

= cos (AB k119868) =

(119911119860

minus 119911119861)

|AB|

(18)

The direction cosines in O119874

X119874

Y119874

Z119874are expressed as

follows

[[

[

120574119909

120574119910

120574119911

]]

]119874

= C119874119868

[[

[

1205821

1205822

1205823

]]

]119868

(19)

B

E

C

F

H

G

YO

A(O)

120579998400

120579

XO

ZO

120595

(a)

t

F

T

F

T998400

(b)

Figure 6 The librations (a) and the thrust force (b)

where 120574119909 120574119910 and 120574

119911are the components of the direction

cosines in O119874

X119874

Y119874

Z119874 The out-plane (120595) and in-plane

(120579) librations can be obtained later And the coordinatetransformation matrix C

119874119868has been derived before

As shown in Figure 6 line sections 119861119867 and 119861119864 areperpendicular to planes 119860X

119874Z119874and 119860X

119874Y119874 respectively

And the in-plane and out-plane librations can be expressedas follows

120595 = arcsin(119861119864

|AB|) = 120574119910

120579 = arcsin(119861119867

|AB|) = 120574119911

(20)

The definition above will be better than the traditionalone as there is no singularity

44 Nonpotential Force of the ROGER Spacecraft The pur-pose of this paper is to transport the abandoned spacecraftinto the orbit whose apogee is 300 km above GEOThe thrustforce direction is n = minusr times (r times v)|r times (r times v)| wheren r and v are the unit vectors of the thrust force theposition and velocity of the ROGER spacecraft respectivelyThe amplitude period and duty cycle of the thrust force canbe obtained by referring to Figure 6 119865 and 120578 = 119879

1015840119879 are the

amplitude and duty cycle of the thrust forceThe vector of thethrust force can be written as F = 119865

119909i119860

+ 119865119910

j119860

+ 119865119911k119860

The components of T119862in O119874

X119860

Y119860

Z119860can be written as

T119862

= 119879119909

i119860

+ 119879119910

j119860

+ 119879119911k119860


Table 1 The cases with different parameters

Rm0 Rs0 119896 119879 120578

Case 1 [42114203 minus 03 0 minus02]119879 [42114203 + 12 minus292 15]119879 5000 05 50Case 2 [42114203 0 0]119879 [42114203 minus2928788 0]119879 5000 05 50Case 3 [42114203 minus 03 0 minus02]119879 [42114203 + 12 minus30 15]119879 5000 05 50Case 4 [42114203 minus 03 0 minus02]119879 [42114203 + 12 minus292 15]119879 20000 05 50Case 5 [42114203 minus 03 0 minus02]119879 [42114203 + 12 minus292 15]119879 5000 05 25 9999Case 6 [42114203 minus 03 0 minus02]119879 [42114203 + 12 minus292 15]119879 5000 4 01 50

The attitude control torque for the ROGER spacecraftduring deorbiting process is as follows

T119862

= (119870119875120593119860

+ 119870119863

119860

) i119860

+ (119870119875120579119860

+ 119870119863

120579119860

) j119860

+ (119870119875120595119860

+ 119870119863

119860

) k119860

(21)

119870119875and 119870

119863are the proportional and differential coeffi-

cients

5 Numerical Simulations

International System of Units is adopted except specialstatements The initial roll pitch and yaw angles are 35734573 and 5573 respectively and the corresponding Eulerangular velocities are zeros The tensor of moment of inertiaof the ROGER spacecraft is as follows

J119860

=[[

[

1000 minus49 32

minus49 1000 43

32 43 1000

]]

]

(22)

The initial orbital positions and velocities of the ROGERand abandoned spacecraft are denoted by Rm

0 Rs0 Vm0

and Vs0 respectively Vm

0and Vs

0are both selected as

[0 30765 0]119879 in this paper And the radius of the earth is

6378145 and the gravitational constant is 120583 = 39860044 times

1014 The amplitude of the thrust is 20 The parameters

mentioned above will be used throughout the paperThe deorbiting results in Case 1 adopting the certain

parameters will be regarded as the benchmark In Case 2Rm0and Rs

0are changed to reduce the librations dramati-

cally The deorbiting results with comparatively large (Case1) librations and few (Case 2) librations will be presented andcompared

In Case 3 Rm0and Rs

0are changed to increase the initial

tension force in the tetherThe deorbiting results with smallerforce (Case 1) larger force (Case 3) will be presented andcompared

In Case 4 119896 is selected as 20000 to see how the deorbitingresults are affected by the elasticity of the tether In Cases 5

and 6 120578 and 119879 are selected as 25 9999 and 4 01 to seehow the deorbiting results are affected by the two parameters

The parameters in all cases can be found in Table 1The deorbiting results for Case 1 are presented in Figures

7ndash12The tensionless and tensional states are alternate during

the deorbiting process by observing Figure 7 moreover

1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)Δ

(m)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

) [0 2929] [1581 3004]

Figure 7The tensional (1) and tensionless (0) states the elongationof the tether (Δ) and the distance between two spacecraft (119897) duringdeorbiting

Rs (m

)

t (s)

5

0

minus50 500 1000 1500 2000

times107

Z(m

)

Y (m)X (m)

times106

times107

Forward direction

42242

418minus5

0

5minus100

0

100

RsxRsyRsz

Figure 8 The trajectory of the abandoned spacecraft


120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s)t (s)

t (s)

t (s) t (s)

t (s)

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus50 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000

1

0

minus1

410 420 430 440

Figure 9 The equivalent expression of in-plane (120579) and out-plane (120595) librations

Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

50

0

minus50

minus100

01

0

minus010 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Rmz

Rsz

Vmz

VszFigure 10 The out-plane positions and velocities of ROGER (m)and abandoned (s) spacecraft

the tether is tensionless most of the time And the distanceis between 30 and 29m it can be seen that the deorbitingprocess is stable and the collision between the two spacecraftis avoided

The 3D trajectory of the abandoned spacecraft is givenin Figure 8 It is evident that the out-plane displacement of

the abandoned spacecraft is negative during the deorbitingprocess

The in-plane (120579) and out-plane (120595) librations are pre-sented in Figure 9 based on the orbital position coordinatesof the ROGER and abandoned spacecraft The librationscorresponding to tensional and tensionless states are plottedby sparse dots and almost continuous curves respectivelyAnd it can be seen that the librations (or we call it the relativeorientation relation of the two spacecraft) are stable and thetether is tensionless most of time

The out-plane positions and velocities of the ROGER andabandoned spacecraft are given in Figure 10 The out-planepositions are negative during deorbiting and the out-planemotion is stable

The components of the tension force and disturbingtorques caused by the tether are given in Figure 11 Thedisturbing torques can be utilized as the control torques forthe traditional tethered system

The angular rate and angle errors during deorbiting areplotted in Figure 12 It can be seen that the attitude motionaffected by the disturbance torques is stable It is concludedthat the deorbiting process is acceptable by analyzing Figures7ndash12

The deorbiting results for Case 2 are presented in Figures13ndash15

The tensional and tensionless states the elongation andthe distance showed in Figure 13 are almost the same as inFigure 7 The distance during deorbiting is hardly affectedby librations when adopting the same parameters and otherinitial conditions But the attitude motion of the ROGER

International Journal of Aerospace Engineering 9F x

(N)

F y(N

)F z

(N)

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)

t (s)t (s)

t (s) t (s)

t (s) t (s)

300

200

100

0

20

0

minus20

20

0

minus20

20

0

minus20

002

001

0

minus001

10

0

minus100 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000

Figure 11 The components of the tension force (119865119909 119865119910 and 119865

119911) and the disturbing torque (119879

119909 119879119910 and 119879

119911) in 120587

119860caused by the tether

t (s)

t (s)

1

0

minus1

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt

Figure 12 The angular rate and angle of the ROGER spacecraft

1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)

Δ(m

)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

)

Figure 13The tensional (1) and tensionless (0) states the elongationof the tether (Δ) and the distance (119897) between the two spacecraft

spacecraft will be deeply affected by the librations by observ-ing Figures 14-15


t (s)

t (s)

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt

1

05

0

minus05

minus1


F x(N

)F y

(N)

F z(N

)

t (s)t (s)

t (s) t (s)

t (s)t (s)

400

200

0

5

0

minus5

10

0

minus10

10

0

minus10

002

0

minus002

2

0

minus20 1000 2000

0 1000 2000

0 1000 20000 1000 2000

0 1000 2000

0 1000 2000

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587



1

05

0

Δ(m

)

t (s)

t (s)

t (s)

l(m

)

4

2

0

40

20

00 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

[0 3009] [2016 3143]

Figure 16 The tensional (1) and tensionless (0) states the elonga-tion of the tether (Δ) and the distance (119897) between the two spacecraft

In order to study the attitude motion affected by thelibrations the same control torque vector in Figure 12 isused to obtain the attitude motion in Figure 14 The attitudemotions are more stable than the results in Figure 12 asthe attitude motions are affected by the librations deeplyIt also can be demonstrated by comparing the disturbingtorques in Figure 11 with the torques in Figure 15 The largerthe deorbiting system librates the larger the disturbancetorques will generate and the more unstable the systemwill be from the view of the ROGER spacecraft attitudemotion


It can be seen that the deorbiting results are very unsat-isfactory in Figure 16 by comparing Figure 16 with Figure 7as the initial tension force exists for the deorbiting processThe probability of collision sharply increases for this case Itis suggested that the initial tension should be small or zero

It can be seen from Figure 17 that the out-plane orbitalmotion direction is along positive 119911-axis as observed inFigure 17

The simulations in Figure 18 indicate that the libra-tions are completely unstable and show bad behaviors Themaximum in-plane and out-plane angles both nearly reach100 deg respectively This will be bad for deorbiting becauseof the great possibility of collision

The time histories of the out-plane motions of theROGER and abandoned spacecraft vary very sharply byobserving Figure 19 Thus the deorbiting results under thiscondition areworse by comparing the results in Figure 10Theout-plane motions are unacceptable for a safe deorbiting

The amplitude of the tension force is too large to assurethe deorbiting process safety The disturbing torques are solarge that it is too difficult to control the attitude of ROGERspacecraft Thus the parameters and initial conditions arecompletely unsuitable in this case

Z(m

)

Y (m)X (m)

t (s)

500

0

minus50010

5

0 41542

425

5

0

minus50 500 1000 1500 2000 2500

times106times107

times107

Forward direction

Rs (m

)

RsxRsyRsz

Figure 17 The deorbiting trajectory for abandoned spacecraft

The deorbiting results for Case 4 are presented inFigure 21

It can be seen that the deorbiting results in Figure 21 arealmost the same as in Figure 7 Thus it is concluded that thedeorbiting results are hardly affected by the parameter ldquo119896rdquo

The deorbiting results for Case 4 are presented in Figures22-23

The deorbiting results can be observed in Figure 22 underthe condition of 9999 duty cycle


It can be seen that the deorbiting results are bad when120578 = 9999 on the contrary the deorbiting process is goodwhen 120578 = 25Thus it is concluded that 120578 should not be thatlarge in order to make the deorbiting safe


The deorbiting results can be observed in Figure 24 underthe condition of 4 s period

The deorbiting results are as follows under the conditionof 01 s period

It can be concluded that deorbiting results are hardlyaffected by the period But the extreme large period shouldnot be adopted as the stability of the distance between the twospacecraft affected by the large period is studied deeply

6 Conclusions and Future Work

The universal dynamic equations for ROGER system arederived during the entire deorbiting process The simulationresults and conclusions are obtained by adopting variousparameters and initial conditions to study the deorbiting


120579(d

eg)

120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s) t (s)

t (s)

t (s)t (s)

t (s)100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus1000 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

500

0

minus500

5

0

minus5

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

Rmz

Rsz

Vmz

Vsz

Figure 19The out-plane positions and velocities of the ROGER (m)and abandoned (s) spacecraft

results affected by these according parameters and initialconditions The detailed conclusions are as follows

(1) Two groups of dynamic equations are obtainedaccording to the tensional and tensionless states of thetether respectively Moreover the reasons the equa-tions cannot be adopted in this paper are analyzed

A simple and practical dynamic modeling strategyfor entire deorbiting process is proposed and thelibrations are also equivalently expressed

(2) Most of the time the tether is tensionlessThedeorbit-ing effects affected by initial librations initial distance(initial tension force) between the two spacecraft thecoefficient of elasticity of the tether the duty ratio andperiod of the thrust are analyzed It is concluded asfollows

(a) The distance between the two spacecraft ishardly affected by librations under the conditionof the same initial distance and other parame-ters However the few librations do benefit forattitude motion of the ROGER spacecraft

(b) The initial distance should be within a rea-sonable range If the initial tension force isexcessively large when the tether is tensionalthen the deorbiting results are bad as the leastdistance during deorbiting is too small to avoidcollision and the librations are nearly unstablemoreover the attitude motion of the ROGERspacecraft is badly affected by this large initialtension force

(c) The deorbiting process is hardly affected by theelasticity of the tether

(d) The condition of 25 duty cycle is better thanthat of 9999 thus the more the duty cycleof the thrust force is the worse the deorbitingresults are


F x(N

)F y

(N)

F z(N

)

t (s) t (s)

t (s)t (s)

t (s) t (s)

2

0

minus2

2

0

minus2

2

0

minus2

50

0

minus50

1

0

minus1

1

0

minus1

0 1000 2000 30000 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 30000 1000 2000 3000

times104times104

times104 times104

times104

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

0

002

001

0

32

30

28

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000

Figure 21The tensional and tensionless states the elongation of thetether and the distance between the two spacecraft

(e) The deorbiting results under the conditions of4 s 01 s and 05 s period of the thrust force

are almost the same thus within a reasonablerange of period the deorbiting results are hardlyaffected by the period of the thrust force

(3) It can be seen that the tether is tensionless most ofthe time the tension force and the torques causedby librations and the tension force are so complexthus the torques are difficult to be utilized as controltorques in this paper

(4) The librations equivalently expressed by the positionvectors of the two spacecraft never exist singularityand thus show better advantages than the traditionaldefinition

The further studies are as follows

(1) The more complex dynamic equations based onthe modeling strategy in this paper considering theoffset of the tether the attitude and orbital motionsof the two spacecraft and the movable attachmentbetween abandoned spacecraft and the tether shouldbe established

(2) More simulations should be carried out to studythe deorbiting results affected by libations distance

14 International Journal of Aerospace EngineeringΔ

(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0


between the two spacecraft and the coefficient ofelasticity of the tether and so forth Moreover thesuperior deorbiting parameters and initial conditionsshould be summarized based on the simulations

(3) The stability conditions for librations should beproved theoretically The stability of librations underthe conditions of small initial librations is merelyspeculated through simulations and will be hard to beassured if the initial libations are large

Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


(4) How to utilize the torques caused by the offset andtension force of the tether as attitude control torquesis worthy of being studied

(5) The paper merely considers the dynamic modelingproblems and the optimal orbital transfer problemsshould be considered in detail

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Acknowledgments

This work was supported partially by the National Nat-ural Science Foundation of China (Project nos 1130213411272101) The authors highly appreciate the above financialsupport The authors would like to thank the reviewers andthe editor for their comments and constructive suggestionsthat helped to improve the paper significantly

References

[1] W Xu B Liang B Li and Y Xu ldquoA universal on-orbit servicingsystem used in the geostationary orbitrdquo Advances in SpaceResearch vol 48 no 1 pp 95ndash119 2011

[2] D A Smith C Martin M Kassebom et al ldquoA mission topreserve the geostationary regionrdquo Advances in Space Researchvol 34 no 5 pp 1214ndash1218 2004

[3] F Piergentili and F Graziani ldquoSIRDARIA a low-costautonomous deorbiting system for microsatellitesrdquo TechRep IAC-06-B6407 2006

[4] M Kassebom D Koebel C Tobehn et al ldquoROGERmdashanadvanced solution for a geostationary service satelliterdquo inProceedings of the 26th American Control Conference ChicagoIll USA June 2000

[5] B Bischof L Kerstein J Starke et al ldquoRoger-robotic geo-stationary orbit restorerrdquo in Proceedings of the 54th Interna-tional Astronautical Congress of the International AstronauticalFederation (IAC rsquo03) pp 1365ndash1373 IAF Bremen GermanySeptember-October 2003

[6] C Pardini T Hanada and P H Krisko ldquoBenefits and risksof using electrodynamic tethers to de-orbit spacecraftrdquo ActaAstronautica vol 64 no 5-6 pp 571ndash588 2009

[7] C Pardini T Hanada P H Krisko L Anselmo and HHirayama ldquoAre de-orbiting missions possible using electrody-namic tethers Task review from the space debris perspectiverdquoActa Astronautica vol 60 no 10-11 pp 916ndash929 2007

[8] V Lappas L Visagie N Adeli T Theodorou J Ferndnez andW H Steyn ldquoCubeSail a low cost small cubesat mission forsolar sailing and deorbiting LEO objectsrdquo in Proceedings of the2nd International Symposium on Solar Sailing (ISSS rsquo10) NewYork NY USA July 2010

[9] C Lucking C Colombo and C Mcinnes ldquoA passive highlatitude deorbiting strategyrdquo in Proceedings of the 25th AnnualIAAUSU Conference on Small Satellites Logan Utah USAAugust 2011

[10] G Zhai Y Qiu B Liang and C Li ldquoSystem dynamicsand feedforward control for tether-net space robot systemrdquoInternational Journal of Advanced Robotic Systems vol 6 no 2pp 137ndash144 2009

[11] W Xu B Liang C Li Y Liu and X Wang ldquoA modelling andsimulation system of space robot for capturing non-cooperativetargetrdquo Mathematical and Computer Modelling of DynamicalSystems vol 15 no 4 pp 371ndash393 2009

[12] J A Bonometti K F Sorensen J W Dankanich and K LFrame ldquoStatus of the momentum eXchange electrodynamic re-boost (MXER) tether developmentrdquo in Proceedings of the 42ndAIAAASMESAEASEE Joint Propulsion Conference amp ExhibitSacramento Calif USA July 2006

[13] G Khazanov A L Huntsville E Krivorutsky and K SorensenldquoAnalyses of bare-tether systems as a thruster for MXERstudiesrdquo in Proceedings of the 41st AIAAASMESAEASEE Joint

Propulsion Conference and Exhibit Tucson Ariz USA July2005

[14] Y Ishige S Kawamoto and S Kibe ldquoStudy on electrodynamictether system for space debris removalrdquo Acta Astronautica vol55 no 11 pp 917ndash929 2004

[15] E Kim and S R Vadali ldquoModeling issues related to retrieval offlexible tethered satellite systemsrdquo Journal of Guidance Controland Dynamics vol 18 no 5 pp 1169ndash1176 1995

[16] S Bergamaschi F Bonon and M Legnami ldquoSpectral analysisof tethered satellite system-mission 1 vibrationsrdquo Journal ofGuidance Control and Dynamics vol 18 no 3 pp 618ndash6241995

[17] S Bergamaschi P Zanetti and C Zottarel ldquoNonlinear vibra-tions in the tethered satellite system-mission 1rdquo Journal ofGuidance Control and Dynamics vol 19 no 2 pp 289ndash2961996

[18] M Pasca M Pignataro and A Luongot ldquoThree-dimensionalvibrations of tethered satellite systemrdquo in Proceedings of the 3rdInternational Conference on Tethers in Space-Toward Flight SanFrancisco Calif USA May 1989

[19] S Pradhan V J Modi and A K Misra ldquoSimultaneous controlof platform attitude and tether vibration using offset strategyrdquoin Proceedings of the AIAAAAS Astrodynamics Conference SanDiego Calif USA July 1996

[20] P Williams C Blanksby P Trivailo and H A Fujii ldquoLibrationcontrol of flexible tethers using electromagnetic forces andmoveable attachmentrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit Austin TexUSA August 2003

[21] P Williams ldquoEnergy rate feedback for libration control ofelectrodynamic tethersrdquo Journal of Guidance Control andDynamics vol 29 no 1 pp 221ndash223 2006

[22] PWilliams ldquoElectrodynamic tethers under forced-current vari-ations part 1 periodic solutions for tether librationsrdquo Journal ofSpacecraft and Rockets vol 47 no 2 pp 308ndash319 2010

[23] PWilliams ldquoElectrodynamic tethers under forced-current vari-ations Part 2 Flexible-tether estimation and controlrdquo Journal ofSpacecraft and Rockets vol 47 no 2 pp 320ndash333 2010

[24] M Inarrea and J Pelaez ldquoLibration control of electrodynamictethers using the extended time-delayed autosynchronizationmethodrdquo Journal of Guidance Control and Dynamics vol 33no 3 pp 923ndash933 2010

[25] H Kojima and T Sugimoto ldquoStability analysis of in-plane andout-of-plane periodic motions of electrodynamic tether systemin inclined elliptic orbitrdquo Acta Astronautica vol 65 no 3-4 pp477ndash488 2009

[26] H Kojima and T Sugimoto ldquoStability analysis of periodicmotion of electrodynamic tether system in elliptic orbitrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit Honolulu Hawaii USA August 2008

[27] S Pradhan V J Modi and A K Misra ldquoTether-platformcoupled controlrdquo Acta Astronautica vol 44 no 5-6 pp 243ndash256 1999

[28] K Kumar and K D Kumar ldquoPitch and roll attitude maneuverof twin-satellite systems through short tethersrdquo Journal ofSpacecraft and Rockets vol 37 no 2 pp 287ndash290 2000

[29] K D Kumar and K Kumar ldquoSatellite pitch and roll attitudemaneuvers through very short tethersrdquo Acta Astronautica vol44 no 5-6 pp 257ndash265 1999

[30] S Pradhan V J Modi and A K Misra ldquoOn the offset controlof flexible nonautonomous tethered two-body systemsrdquo ActaAstronautica vol 38 no 10 pp 783ndash801 1996


[31] R E Stevens and W P Baker ldquoOptimal control of a librat-ing electrodynamic tether performing a multirevolution orbitchangerdquo Journal of Guidance Control and Dynamics vol 32no 5 pp 1497ndash1507 2009

[32] E L M Lanoix A K Misra V J Modi and G Tyc ldquoEffectof electrodynamic forces on the orbital dynamics of tetheredsatellitesrdquo Journal of Guidance Control and Dynamics vol 28no 6 pp 1309ndash1315 2005

[33] P Williams ldquoOptimal orbital transfer with electrodynamictetherrdquo Journal of Guidance Control and Dynamics vol 28 no2 pp 369ndash372 2005

[34] T Watanabe T Makida H A Fujii H Kojima and WSinghose ldquoAn application of input shaping for electrodynamictether systemrdquo in Proceedings of the AIAAAAS AstrodynamicsSpecialist Conference andExhibit pp 1442ndash1453 Providence RIUSA August 2004

[35] H J Pernicka M Dancer and A Abrudan ldquoSimulation of thedynamics of a short tethered satellite systemrdquo in Proceedings ofthe AIAAAAS Astrodynamics Specialist Conference and ExhibitProvidence RI USA August 2004

[36] J Valverde J L Escalona JMayo and J Domınguez ldquoDynamicanalysis of a light structure in outer space short electrodynamictetherrdquo Multibody System Dynamics vol 10 no 1 pp 125ndash1462003

International Journal of

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Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


Research ArticleDynamics of the Space Tug System with a Short Tether

Jiafu Liu1 Naigang Cui2 Fan Shen2 and Siyuan Rong2

1Shenyang Aerospace University Shenyang 110136 China2Harbin Institute of Technology Harbin 150001 China

Correspondence should be addressed to Jiafu Liu liujiafuericking163com

Received 25 February 2015 Revised 12 June 2015 Accepted 2 July 2015

Academic Editor Paul Williams

Copyright copy 2015 Jiafu Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The dynamics of the space tug system with a short tether similar to the ROGER system during deorbiting is presented Thekinematical characteristic of this system is significantly different from the traditional tethered system as the tether is tensionaland tensionless alternately during the deorbiting processThe dynamics obtained based on the methods for the traditional tetheredsystem is not suitable for the space tug systemTherefore a novel method for deriving dynamics for the deorbiting system similar tothe ROGER system is proposed by adopting the orbital coordinates of the two spacecraft and the Euler angles of ROGER spacecraftas the generalized coordinates instead of in- and out-plane librations and the length of the tether and so forth Then the librationsof the system are equivalently obtained using the orbital positions of the two spacecraft At last the geostationary orbit (GEO) andthe orbit whose apogee is 300 km above GEO are chosen as the initial and target orbits respectively to perform the numericalsimulationsThe simulation results indicate that the dynamics can describe the characteristic of the tether-net system convenientlyand accurately and the deorbiting results are deeply affected by the initial conditions and parameters

1 Introduction

The number of flying objects including the controlled space-craft and the space debris (including the abandoned space-craft) in the low earth orbit (LEO) and the geostationaryorbit (GEO) increases sharply because of the frequent humanspace activities [1ndash3] It is reported by ESA that at the endof 2008 there were 1186 objects in GEO and only 32 werecontrolled spacecraft The collision risks will be posed by theuncontrolled spacecraft and space debris mentioned aboveTherefore it is necessary to monitor and clear the abandonedspacecraft and space debris

The RObotic GEostationary Orbit Restorer (ROGER)concept was proposed by ESA The researchers not onlyfocused on the economic consideration and future businessapplication but also focused on the orbital monitoring andclearance of the abandoned spacecraft [4 5] To make useof the space electrodynamic tethered system to deorbitspacecraft has been studied in [6 7] which utilizes themotion of the conductive tether relative to the magneticfield of the earth to produce electrodynamic forces It is alsoproposed that the abandoned spacecraft can be deorbited

by the solar radiation pressure experienced by the sailcraftattached to the spacecraft The sailcraft is initially folded andwill be deployed when the spacecraft is spent The feasibilityof the solar radiation pressure based deorbiting approach hasbeen demonstrated by the NanoSail-D project within NASAMoreover the cubeSail project in Surrey [8] and the projectusing solar pressure in Strathclyde [9] are also performed tostudy the spacecraft deorbiting approach by solar pressureIt is a practical method by using the space robots to capturethe noncooperative spacecraft or to deorbit the abandonedspacecraftThemethod can also be used in the spacemissionssuch as spacecraft repair and fuel charges [10 11]

The space tether based deorbiting system is the mostcomprehensively studied one among themethodsmentionedabove including the ROGER system and the space electrody-namic tethered system

The monitoring for the current orbit and forecasting forthe future orbit are studied in the ROGER project [5] Thefeasibility of the GEO service is also discussed from theview of economic and technical aspects The ROGER space-craft approaches captures and transports the abandonedspacecraft into the graveyard orbit The configuration of the

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2015 Article ID 740253 16 pageshttpdxdoiorg1011552015740253








A

B

B

A

Short tether

Short tether

(tensionless)

(tensional)

RO

EI

RS


Earth

Spacecraft

ZI

XI

XI

ZI1

ZI1

YI

YI

YI1

XI1

XI1 EI

EI

ZO

XO

f














X119874

Y119874

Z119874

120587119874) and

Body (O119874

X119860

Y119860

Z119860

120587119860) Frames 119874



Z119874points to 119864

119868 and O



Z119874 O119874

X119874

O119874

Y119874 and O

119874Z119874


119874 j119874

k119874

)119879



j119860

k119860

)119879


Y1198681

Z1198681



j1198681

k1198681

)119879



X119905Y119905Z119905 120587119905) 119862119860is the




119905


vector (120595119860

120593119860

120579119860




is 119911(120595119860

) rarr 119909(120593119860

) rarr 119910(120579119860

) rotation 120595119860

120593119860

120579119860are the



C1198741198681



1198741198681is

as follows

C1198741198681

=[[

[


0 1 0


]]

]

(1)





X119874

Y119874

Z119874to C119860



is as follows

C119905119874

=[[

[

cos 120579 0 minus sin 120579

0 1 0

sin 120579 0 cos 120579

]]

]

[[

[

cos120595 sin120595 0


0 0 1

]]

]

(2)




C1198681119868

=[[[

[

1 0 0


∘)


∘)

]]]

]

[[[

[

cos 90∘ sin 90

∘0


∘0

0 0 1

]]]

]

(3)


C119874119868

= C1198741198681

C1198681119868



119860 120593119860 and 120579


119909119860 119910119860 and 119911




119860and


are as follows

119879119860

=1

2int119860

( r119860

+ 120596119860


+ 120596119860

times r) 119889119898

=1

2119898119860

1003816100381610038161003816r119860

1003816100381610038161003816

2+

1

2120596119860

sdot J119860

sdot 120596119860

119880119860

= minus120583119898119860

1003816100381610038161003816r1198601003816100381610038161003816

(4)





119860is the mass






119860 In




119879119861

=1

2119898119861

[ r119860

+∘

l (119905) + 120596119905times l (119905)]

sdot [ r119860

+∘

l (119905) + 120596119905times l (119905)]

119880119861

= minus120583119898119861

1003816100381610038161003816r119860 + l (119905)1003816100381610038161003816

(5)


119880119890

=1

2119896 (119897 minus 119897

0)2

(6)



l (119905)


119905 respectively




119889

119889119905(

120597119871

120597q119895

) minus120597119871

120597q119895

= Q119895 (7)



119895= (119909

119860 119910119860

119911119860

120595119860

120593119860

120579119860

120595 120579 119897)119879




119910119860

119911119860) and (119909



120593119860

120579119860) of the ROGER







119860 119910119860

119911119860) and (119909

119861 119910119861 119911119861) and the


120593119860





(120595 120579 119897 119909119860

119910119860

119911119860

) lArrrArr (119909119861 119910119861 119911119861 119909119860

119910119860

119911119860

) (8)

A

dm

r B

CA

rC119860

rmrA

EI

rAB

rB

rC119860B

ang120575 asymp 0

Short tether


tO

1

2 3 4 5

6l0

l(t)



119879 = 119879119860

+ 119879119861

119879119860

=1

2119898119860

(2

119860+ 1199102

119860+ 2

119860) +

1

2120596119860

sdot J119860

sdot 120596119860

119879119861

=1

2119898119861

(2

119861+ 1199102

119861+ 2

119861)

(9)


119880 = 119880119892

+ 119880119890

119880119892

= 119880119892119860

+ 119880119892119861

119880119892119860

= minus120583119898119860

radic1199092

119860+ 1199102

119860+ 1199112

119860

119880119892119861

= minus120583119898119861

radic1199092

119861+ 1199102

119861+ 1199112

119861

(10)


119880119890

=

010038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816le 1198970

1

2119896Δ2 10038161003816100381610038161003816

r119862119860119861

10038161003816100381610038161003816gt 1198970

Δ =10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816minus 1198970

119897 = r119862119860119861

= r119861

minus r119860

minus r119862119860

(11)


119862119860is constant in 120587

119860



119871 = 119879 minus 119880 = 119879119860

+ 119879119861

minus 119880119892119860

minus 119880119892119861

minus 119880119890

=1

2119898119860

(2

119860+ 1199102

119860+ 2

119860) +

1

2120596119860

sdot J119860

sdot 120596119860

+1

2119898119861

(2

119861+ 1199102

119861+ 2


120583119898119860

radic1199092

119860+ 1199102

119860+ 1199112

119860

)

minus (minus120583119898119861

radic1199092

119861+ 1199102

119861+ 1199112

119861

) minus 119880119890

(12)



119860

= minus120583119909119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+119896Δ (119909

119861minus 119909119860

minus 119903119909)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119909

119910119860

= minus120583119910119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+

119896Δ (119910119861

minus 119910119860

minus 119903119910)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119910

119860

= minus120583119911119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+119896Δ (119911119861

minus 119911119860

minus 119903119911)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119911

(13)

where 119903119909 119903119910 and 119903


119862119860in 120587119868


J119860

sdot 119860

+ 120596119860

times J119860

sdot 120596119860

= T119888

+ T119889 (14)




119889119891and T

119889119905




AP = r119901

= 119903119901119910

j119860

+ 119903119901119911

k119860

F = 119865119909

i119860

T119889119905

= r119901

times F

(15)


as T119889119891

= r119862119860

times T r119862119860



FP

A

A

T

BrC119860

CA

XA

YA

ZA

Tension

ROGER



119860

= minus120583119909119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119909

+

119896Δ (119909119861

minus 119909119860

minus 119903119909)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)

119910119860

= minus120583119910119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119910

+

119896Δ (119910119861

minus 119910119860

minus 119903119910)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)

119860

= minus120583119911119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119911

+

119896Δ (119911119861

minus 119911119860

minus 119903119911)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)


T119888

+ T119889119891

+ T119889119905

(119897 ge 1198970)

T119888

+ T119889119891

(119897 lt 1198970)

(16)


119889119886

119889119905=

2

119899radic1 minus 1198902(120590119911119890 sin119891 + 120590

119909

119901

119903)

119889Ω

119889119905=

120590119910119903 sin 119906

1198991198862radic1 minus 1198902 sin 119894

119889119894

119889119905=

120590119910119903 cos 119906

1198991198862radic1 minus 1198902


119889119890119909

119889119905=


119899119886120590119911sin 119906

+ 120590119909

[(1 +119903

119901) cos 119906 +

119903

119901119890119909] +

119889Ω

119889119905119890119910cos 119894

119889119890119910

119889119905=


119899119886minus120590119911cos 119906

+ 120590119909

[(1 +119903

119901) sin 119906 +

119903

119901119890119910] minus

119889Ω

119889119905119890119909cos 119894

119889120593

119889119905= 119899 minus

1

119899119886[120590119911

(2119903

119886+


1 + radic1 minus 1198902119890 cos119891)

minus 120590119909

(1 +119903

119901)


1 + radic1 minus 1198902119890 sin119891]

minus

120590119910119903 cos 119894 sin 119906

1198991198862radic1 minus 1198902 sin 119894

(17)where (119886 Ω 119894 119890



= 119890 cos120596 119890119910


119909 120590119910 and 120590





1205821

= cos (AB i119868) =

(119909119860

minus 119909119861)

|AB|

1205822

= cos (AB j119868) =

(119910119860

minus 119910119861)

|AB|

1205823

= cos (AB k119868) =

(119911119860

minus 119911119861)

|AB|

(18)


X119874

Y119874


follows

[[

[

120574119909

120574119910

120574119911

]]

]119874

= C119874119868

[[

[

1205821

1205822

1205823

]]

]119868

(19)

B

E

C

F

H

G

YO

A(O)

120579998400

120579

XO

ZO

120595

(a)

t

F

T

F

T998400

(b)


where 120574119909 120574119910 and 120574


cosines in O119874

X119874

Y119874





119874Z119874and 119860X



120595 = arcsin(119861119864

|AB|) = 120574119910

120579 = arcsin(119861119867

|AB|) = 120574119911

(20)



1015840119879 are the


119909i119860

+ 119865119910

j119860

+ 119865119911k119860


X119860

Y119860


T119862

= 119879119909

i119860

+ 119879119910

j119860

+ 119879119911k119860



Rm0 Rs0 119896 119879 120578



T119862

= (119870119875120593119860

+ 119870119863

119860

) i119860

+ (119870119875120579119860

+ 119870119863

120579119860

) j119860

+ (119870119875120595119860

+ 119870119863

119860

) k119860

(21)

119870119875and 119870


cients



J119860

=[[

[

1000 minus49 32

minus49 1000 43

32 43 1000

]]

]

(22)


0 Rs0 Vm0


0and Vs









In Case 3 Rm0and Rs








1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)Δ

(m)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

) [0 2929] [1581 3004]


Rs (m

)

t (s)

5

0

minus50 500 1000 1500 2000

times107

Z(m

)

Y (m)X (m)

times106

times107

Forward direction

42242

418minus5

0

5minus100

0

100

RsxRsyRsz



120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s)t (s)

t (s)

t (s) t (s)

t (s)

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus50 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000

1

0

minus1

410 420 430 440


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

50

0

minus50

minus100

01

0

minus010 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Rmz

Rsz

Vmz












(N)

F y(N

)F z

(N)

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)

t (s)t (s)

t (s) t (s)

t (s) t (s)

300

200

100

0

20

0

minus20

20

0

minus20

20

0

minus20

002

001

0

minus001

10

0

minus100 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000



119909 119879119910 and 119879

119911) in 120587


t (s)

t (s)

1

0

minus1

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt


1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)

Δ(m

)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

)




t (s)

t (s)

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt

1

05

0

minus05

minus1


F x(N

)F y

(N)

F z(N

)

t (s)t (s)

t (s) t (s)

t (s)t (s)

400

200

0

5

0

minus5

10

0

minus10

10

0

minus10

002

0

minus002

2

0

minus20 1000 2000

0 1000 2000

0 1000 20000 1000 2000

0 1000 2000

0 1000 2000

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587



1

05

0

Δ(m

)

t (s)

t (s)

t (s)

l(m

)

4

2

0

40

20

00 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

[0 3009] [2016 3143]









Z(m

)

Y (m)X (m)

t (s)

500

0

minus50010

5

0 41542

425

5

0

minus50 500 1000 1500 2000 2500

times106times107

times107

Forward direction

Rs (m

)

RsxRsyRsz















120579(d

eg)

120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s) t (s)

t (s)

t (s)t (s)

t (s)100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus1000 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

500

0

minus500

5

0

minus5

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

Rmz

Rsz

Vmz

Vsz











F x(N

)F y

(N)

F z(N

)

t (s) t (s)

t (s)t (s)

t (s) t (s)

2

0

minus2

2

0

minus2

2

0

minus2

50

0

minus50

1

0

minus1

1

0

minus1

0 1000 2000 30000 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 30000 1000 2000 3000

times104times104

times104 times104

times104

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

0

002

001

0

32

30

28

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000










(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0




Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000







Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















A

B

B

A

Short tether

Short tether

(tensionless)

(tensional)

RO

EI

RS


Earth

Spacecraft

ZI

XI

XI

ZI1

ZI1

YI

YI

YI1

XI1

XI1 EI

EI

ZO

XO

f














X119874

Y119874

Z119874

120587119874) and

Body (O119874

X119860

Y119860

Z119860

120587119860) Frames 119874



Z119874points to 119864

119868 and O



Z119874 O119874

X119874

O119874

Y119874 and O

119874Z119874


119874 j119874

k119874

)119879



j119860

k119860

)119879


Y1198681

Z1198681



j1198681

k1198681

)119879



X119905Y119905Z119905 120587119905) 119862119860is the




119905


vector (120595119860

120593119860

120579119860




is 119911(120595119860

) rarr 119909(120593119860

) rarr 119910(120579119860

) rotation 120595119860

120593119860

120579119860are the



C1198741198681



1198741198681is

as follows

C1198741198681

=[[

[


0 1 0


]]

]

(1)





X119874

Y119874

Z119874to C119860



is as follows

C119905119874

=[[

[

cos 120579 0 minus sin 120579

0 1 0

sin 120579 0 cos 120579

]]

]

[[

[

cos120595 sin120595 0


0 0 1

]]

]

(2)




C1198681119868

=[[[

[

1 0 0


∘)


∘)

]]]

]

[[[

[

cos 90∘ sin 90

∘0


∘0

0 0 1

]]]

]

(3)


C119874119868

= C1198741198681

C1198681119868



119860 120593119860 and 120579


119909119860 119910119860 and 119911




119860and


are as follows

119879119860

=1

2int119860

( r119860

+ 120596119860


+ 120596119860

times r) 119889119898

=1

2119898119860

1003816100381610038161003816r119860

1003816100381610038161003816

2+

1

2120596119860

sdot J119860

sdot 120596119860

119880119860

= minus120583119898119860

1003816100381610038161003816r1198601003816100381610038161003816

(4)





119860is the mass






119860 In




119879119861

=1

2119898119861

[ r119860

+∘

l (119905) + 120596119905times l (119905)]

sdot [ r119860

+∘

l (119905) + 120596119905times l (119905)]

119880119861

= minus120583119898119861

1003816100381610038161003816r119860 + l (119905)1003816100381610038161003816

(5)


119880119890

=1

2119896 (119897 minus 119897

0)2

(6)



l (119905)


119905 respectively




119889

119889119905(

120597119871

120597q119895

) minus120597119871

120597q119895

= Q119895 (7)



119895= (119909

119860 119910119860

119911119860

120595119860

120593119860

120579119860

120595 120579 119897)119879




119910119860

119911119860) and (119909



120593119860

120579119860) of the ROGER







119860 119910119860

119911119860) and (119909

119861 119910119861 119911119861) and the


120593119860





(120595 120579 119897 119909119860

119910119860

119911119860

) lArrrArr (119909119861 119910119861 119911119861 119909119860

119910119860

119911119860

) (8)

A

dm

r B

CA

rC119860

rmrA

EI

rAB

rB

rC119860B

ang120575 asymp 0

Short tether


tO

1

2 3 4 5

6l0

l(t)



119879 = 119879119860

+ 119879119861

119879119860

=1

2119898119860

(2

119860+ 1199102

119860+ 2

119860) +

1

2120596119860

sdot J119860

sdot 120596119860

119879119861

=1

2119898119861

(2

119861+ 1199102

119861+ 2

119861)

(9)


119880 = 119880119892

+ 119880119890

119880119892

= 119880119892119860

+ 119880119892119861

119880119892119860

= minus120583119898119860

radic1199092

119860+ 1199102

119860+ 1199112

119860

119880119892119861

= minus120583119898119861

radic1199092

119861+ 1199102

119861+ 1199112

119861

(10)


119880119890

=

010038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816le 1198970

1

2119896Δ2 10038161003816100381610038161003816

r119862119860119861

10038161003816100381610038161003816gt 1198970

Δ =10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816minus 1198970

119897 = r119862119860119861

= r119861

minus r119860

minus r119862119860

(11)


119862119860is constant in 120587

119860



119871 = 119879 minus 119880 = 119879119860

+ 119879119861

minus 119880119892119860

minus 119880119892119861

minus 119880119890

=1

2119898119860

(2

119860+ 1199102

119860+ 2

119860) +

1

2120596119860

sdot J119860

sdot 120596119860

+1

2119898119861

(2

119861+ 1199102

119861+ 2


120583119898119860

radic1199092

119860+ 1199102

119860+ 1199112

119860

)

minus (minus120583119898119861

radic1199092

119861+ 1199102

119861+ 1199112

119861

) minus 119880119890

(12)



119860

= minus120583119909119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+119896Δ (119909

119861minus 119909119860

minus 119903119909)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119909

119910119860

= minus120583119910119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+

119896Δ (119910119861

minus 119910119860

minus 119903119910)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119910

119860

= minus120583119911119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+119896Δ (119911119861

minus 119911119860

minus 119903119911)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119911

(13)

where 119903119909 119903119910 and 119903


119862119860in 120587119868


J119860

sdot 119860

+ 120596119860

times J119860

sdot 120596119860

= T119888

+ T119889 (14)




119889119891and T

119889119905




AP = r119901

= 119903119901119910

j119860

+ 119903119901119911

k119860

F = 119865119909

i119860

T119889119905

= r119901

times F

(15)


as T119889119891

= r119862119860

times T r119862119860



FP

A

A

T

BrC119860

CA

XA

YA

ZA

Tension

ROGER



119860

= minus120583119909119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119909

+

119896Δ (119909119861

minus 119909119860

minus 119903119909)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)

119910119860

= minus120583119910119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119910

+

119896Δ (119910119861

minus 119910119860

minus 119903119910)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)

119860

= minus120583119911119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119911

+

119896Δ (119911119861

minus 119911119860

minus 119903119911)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)


T119888

+ T119889119891

+ T119889119905

(119897 ge 1198970)

T119888

+ T119889119891

(119897 lt 1198970)

(16)


119889119886

119889119905=

2

119899radic1 minus 1198902(120590119911119890 sin119891 + 120590

119909

119901

119903)

119889Ω

119889119905=

120590119910119903 sin 119906

1198991198862radic1 minus 1198902 sin 119894

119889119894

119889119905=

120590119910119903 cos 119906

1198991198862radic1 minus 1198902


119889119890119909

119889119905=


119899119886120590119911sin 119906

+ 120590119909

[(1 +119903

119901) cos 119906 +

119903

119901119890119909] +

119889Ω

119889119905119890119910cos 119894

119889119890119910

119889119905=


119899119886minus120590119911cos 119906

+ 120590119909

[(1 +119903

119901) sin 119906 +

119903

119901119890119910] minus

119889Ω

119889119905119890119909cos 119894

119889120593

119889119905= 119899 minus

1

119899119886[120590119911

(2119903

119886+


1 + radic1 minus 1198902119890 cos119891)

minus 120590119909

(1 +119903

119901)


1 + radic1 minus 1198902119890 sin119891]

minus

120590119910119903 cos 119894 sin 119906

1198991198862radic1 minus 1198902 sin 119894

(17)where (119886 Ω 119894 119890



= 119890 cos120596 119890119910


119909 120590119910 and 120590





1205821

= cos (AB i119868) =

(119909119860

minus 119909119861)

|AB|

1205822

= cos (AB j119868) =

(119910119860

minus 119910119861)

|AB|

1205823

= cos (AB k119868) =

(119911119860

minus 119911119861)

|AB|

(18)


X119874

Y119874


follows

[[

[

120574119909

120574119910

120574119911

]]

]119874

= C119874119868

[[

[

1205821

1205822

1205823

]]

]119868

(19)

B

E

C

F

H

G

YO

A(O)

120579998400

120579

XO

ZO

120595

(a)

t

F

T

F

T998400

(b)


where 120574119909 120574119910 and 120574


cosines in O119874

X119874

Y119874





119874Z119874and 119860X



120595 = arcsin(119861119864

|AB|) = 120574119910

120579 = arcsin(119861119867

|AB|) = 120574119911

(20)



1015840119879 are the


119909i119860

+ 119865119910

j119860

+ 119865119911k119860


X119860

Y119860


T119862

= 119879119909

i119860

+ 119879119910

j119860

+ 119879119911k119860



Rm0 Rs0 119896 119879 120578



T119862

= (119870119875120593119860

+ 119870119863

119860

) i119860

+ (119870119875120579119860

+ 119870119863

120579119860

) j119860

+ (119870119875120595119860

+ 119870119863

119860

) k119860

(21)

119870119875and 119870


cients



J119860

=[[

[

1000 minus49 32

minus49 1000 43

32 43 1000

]]

]

(22)


0 Rs0 Vm0


0and Vs









In Case 3 Rm0and Rs








1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)Δ

(m)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

) [0 2929] [1581 3004]


Rs (m

)

t (s)

5

0

minus50 500 1000 1500 2000

times107

Z(m

)

Y (m)X (m)

times106

times107

Forward direction

42242

418minus5

0

5minus100

0

100

RsxRsyRsz



120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s)t (s)

t (s)

t (s) t (s)

t (s)

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus50 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000

1

0

minus1

410 420 430 440


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

50

0

minus50

minus100

01

0

minus010 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Rmz

Rsz

Vmz












(N)

F y(N

)F z

(N)

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)

t (s)t (s)

t (s) t (s)

t (s) t (s)

300

200

100

0

20

0

minus20

20

0

minus20

20

0

minus20

002

001

0

minus001

10

0

minus100 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000



119909 119879119910 and 119879

119911) in 120587


t (s)

t (s)

1

0

minus1

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt


1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)

Δ(m

)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

)




t (s)

t (s)

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt

1

05

0

minus05

minus1


F x(N

)F y

(N)

F z(N

)

t (s)t (s)

t (s) t (s)

t (s)t (s)

400

200

0

5

0

minus5

10

0

minus10

10

0

minus10

002

0

minus002

2

0

minus20 1000 2000

0 1000 2000

0 1000 20000 1000 2000

0 1000 2000

0 1000 2000

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587



1

05

0

Δ(m

)

t (s)

t (s)

t (s)

l(m

)

4

2

0

40

20

00 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

[0 3009] [2016 3143]









Z(m

)

Y (m)X (m)

t (s)

500

0

minus50010

5

0 41542

425

5

0

minus50 500 1000 1500 2000 2500

times106times107

times107

Forward direction

Rs (m

)

RsxRsyRsz















120579(d

eg)

120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s) t (s)

t (s)

t (s)t (s)

t (s)100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus1000 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

500

0

minus500

5

0

minus5

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

Rmz

Rsz

Vmz

Vsz











F x(N

)F y

(N)

F z(N

)

t (s) t (s)

t (s)t (s)

t (s) t (s)

2

0

minus2

2

0

minus2

2

0

minus2

50

0

minus50

1

0

minus1

1

0

minus1

0 1000 2000 30000 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 30000 1000 2000 3000

times104times104

times104 times104

times104

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

0

002

001

0

32

30

28

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000










(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0




Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000







Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











X119905Y119905Z119905 120587119905) 119862119860is the




119905


vector (120595119860

120593119860

120579119860




is 119911(120595119860

) rarr 119909(120593119860

) rarr 119910(120579119860

) rotation 120595119860

120593119860

120579119860are the



C1198741198681



1198741198681is

as follows

C1198741198681

=[[

[


0 1 0


]]

]

(1)





X119874

Y119874

Z119874to C119860



is as follows

C119905119874

=[[

[

cos 120579 0 minus sin 120579

0 1 0

sin 120579 0 cos 120579

]]

]

[[

[

cos120595 sin120595 0


0 0 1

]]

]

(2)




C1198681119868

=[[[

[

1 0 0


∘)


∘)

]]]

]

[[[

[

cos 90∘ sin 90

∘0


∘0

0 0 1

]]]

]

(3)


C119874119868

= C1198741198681

C1198681119868



119860 120593119860 and 120579


119909119860 119910119860 and 119911




119860and


are as follows

119879119860

=1

2int119860

( r119860

+ 120596119860


+ 120596119860

times r) 119889119898

=1

2119898119860

1003816100381610038161003816r119860

1003816100381610038161003816

2+

1

2120596119860

sdot J119860

sdot 120596119860

119880119860

= minus120583119898119860

1003816100381610038161003816r1198601003816100381610038161003816

(4)





119860is the mass






119860 In




119879119861

=1

2119898119861

[ r119860

+∘

l (119905) + 120596119905times l (119905)]

sdot [ r119860

+∘

l (119905) + 120596119905times l (119905)]

119880119861

= minus120583119898119861

1003816100381610038161003816r119860 + l (119905)1003816100381610038161003816

(5)


119880119890

=1

2119896 (119897 minus 119897

0)2

(6)



l (119905)


119905 respectively




119889

119889119905(

120597119871

120597q119895

) minus120597119871

120597q119895

= Q119895 (7)



119895= (119909

119860 119910119860

119911119860

120595119860

120593119860

120579119860

120595 120579 119897)119879




119910119860

119911119860) and (119909



120593119860

120579119860) of the ROGER







119860 119910119860

119911119860) and (119909

119861 119910119861 119911119861) and the


120593119860





(120595 120579 119897 119909119860

119910119860

119911119860

) lArrrArr (119909119861 119910119861 119911119861 119909119860

119910119860

119911119860

) (8)

A

dm

r B

CA

rC119860

rmrA

EI

rAB

rB

rC119860B

ang120575 asymp 0

Short tether


tO

1

2 3 4 5

6l0

l(t)



119879 = 119879119860

+ 119879119861

119879119860

=1

2119898119860

(2

119860+ 1199102

119860+ 2

119860) +

1

2120596119860

sdot J119860

sdot 120596119860

119879119861

=1

2119898119861

(2

119861+ 1199102

119861+ 2

119861)

(9)


119880 = 119880119892

+ 119880119890

119880119892

= 119880119892119860

+ 119880119892119861

119880119892119860

= minus120583119898119860

radic1199092

119860+ 1199102

119860+ 1199112

119860

119880119892119861

= minus120583119898119861

radic1199092

119861+ 1199102

119861+ 1199112

119861

(10)


119880119890

=

010038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816le 1198970

1

2119896Δ2 10038161003816100381610038161003816

r119862119860119861

10038161003816100381610038161003816gt 1198970

Δ =10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816minus 1198970

119897 = r119862119860119861

= r119861

minus r119860

minus r119862119860

(11)


119862119860is constant in 120587

119860



119871 = 119879 minus 119880 = 119879119860

+ 119879119861

minus 119880119892119860

minus 119880119892119861

minus 119880119890

=1

2119898119860

(2

119860+ 1199102

119860+ 2

119860) +

1

2120596119860

sdot J119860

sdot 120596119860

+1

2119898119861

(2

119861+ 1199102

119861+ 2


120583119898119860

radic1199092

119860+ 1199102

119860+ 1199112

119860

)

minus (minus120583119898119861

radic1199092

119861+ 1199102

119861+ 1199112

119861

) minus 119880119890

(12)



119860

= minus120583119909119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+119896Δ (119909

119861minus 119909119860

minus 119903119909)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119909

119910119860

= minus120583119910119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+

119896Δ (119910119861

minus 119910119860

minus 119903119910)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119910

119860

= minus120583119911119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+119896Δ (119911119861

minus 119911119860

minus 119903119911)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119911

(13)

where 119903119909 119903119910 and 119903


119862119860in 120587119868


J119860

sdot 119860

+ 120596119860

times J119860

sdot 120596119860

= T119888

+ T119889 (14)




119889119891and T

119889119905




AP = r119901

= 119903119901119910

j119860

+ 119903119901119911

k119860

F = 119865119909

i119860

T119889119905

= r119901

times F

(15)


as T119889119891

= r119862119860

times T r119862119860



FP

A

A

T

BrC119860

CA

XA

YA

ZA

Tension

ROGER



119860

= minus120583119909119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119909

+

119896Δ (119909119861

minus 119909119860

minus 119903119909)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)

119910119860

= minus120583119910119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119910

+

119896Δ (119910119861

minus 119910119860

minus 119903119910)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)

119860

= minus120583119911119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119911

+

119896Δ (119911119861

minus 119911119860

minus 119903119911)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)


T119888

+ T119889119891

+ T119889119905

(119897 ge 1198970)

T119888

+ T119889119891

(119897 lt 1198970)

(16)


119889119886

119889119905=

2

119899radic1 minus 1198902(120590119911119890 sin119891 + 120590

119909

119901

119903)

119889Ω

119889119905=

120590119910119903 sin 119906

1198991198862radic1 minus 1198902 sin 119894

119889119894

119889119905=

120590119910119903 cos 119906

1198991198862radic1 minus 1198902


119889119890119909

119889119905=


119899119886120590119911sin 119906

+ 120590119909

[(1 +119903

119901) cos 119906 +

119903

119901119890119909] +

119889Ω

119889119905119890119910cos 119894

119889119890119910

119889119905=


119899119886minus120590119911cos 119906

+ 120590119909

[(1 +119903

119901) sin 119906 +

119903

119901119890119910] minus

119889Ω

119889119905119890119909cos 119894

119889120593

119889119905= 119899 minus

1

119899119886[120590119911

(2119903

119886+


1 + radic1 minus 1198902119890 cos119891)

minus 120590119909

(1 +119903

119901)


1 + radic1 minus 1198902119890 sin119891]

minus

120590119910119903 cos 119894 sin 119906

1198991198862radic1 minus 1198902 sin 119894

(17)where (119886 Ω 119894 119890



= 119890 cos120596 119890119910


119909 120590119910 and 120590





1205821

= cos (AB i119868) =

(119909119860

minus 119909119861)

|AB|

1205822

= cos (AB j119868) =

(119910119860

minus 119910119861)

|AB|

1205823

= cos (AB k119868) =

(119911119860

minus 119911119861)

|AB|

(18)


X119874

Y119874


follows

[[

[

120574119909

120574119910

120574119911

]]

]119874

= C119874119868

[[

[

1205821

1205822

1205823

]]

]119868

(19)

B

E

C

F

H

G

YO

A(O)

120579998400

120579

XO

ZO

120595

(a)

t

F

T

F

T998400

(b)


where 120574119909 120574119910 and 120574


cosines in O119874

X119874

Y119874





119874Z119874and 119860X



120595 = arcsin(119861119864

|AB|) = 120574119910

120579 = arcsin(119861119867

|AB|) = 120574119911

(20)



1015840119879 are the


119909i119860

+ 119865119910

j119860

+ 119865119911k119860


X119860

Y119860


T119862

= 119879119909

i119860

+ 119879119910

j119860

+ 119879119911k119860



Rm0 Rs0 119896 119879 120578



T119862

= (119870119875120593119860

+ 119870119863

119860

) i119860

+ (119870119875120579119860

+ 119870119863

120579119860

) j119860

+ (119870119875120595119860

+ 119870119863

119860

) k119860

(21)

119870119875and 119870


cients



J119860

=[[

[

1000 minus49 32

minus49 1000 43

32 43 1000

]]

]

(22)


0 Rs0 Vm0


0and Vs









In Case 3 Rm0and Rs








1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)Δ

(m)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

) [0 2929] [1581 3004]


Rs (m

)

t (s)

5

0

minus50 500 1000 1500 2000

times107

Z(m

)

Y (m)X (m)

times106

times107

Forward direction

42242

418minus5

0

5minus100

0

100

RsxRsyRsz



120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s)t (s)

t (s)

t (s) t (s)

t (s)

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus50 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000

1

0

minus1

410 420 430 440


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

50

0

minus50

minus100

01

0

minus010 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Rmz

Rsz

Vmz












(N)

F y(N

)F z

(N)

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)

t (s)t (s)

t (s) t (s)

t (s) t (s)

300

200

100

0

20

0

minus20

20

0

minus20

20

0

minus20

002

001

0

minus001

10

0

minus100 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000



119909 119879119910 and 119879

119911) in 120587


t (s)

t (s)

1

0

minus1

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt


1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)

Δ(m

)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

)




t (s)

t (s)

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt

1

05

0

minus05

minus1


F x(N

)F y

(N)

F z(N

)

t (s)t (s)

t (s) t (s)

t (s)t (s)

400

200

0

5

0

minus5

10

0

minus10

10

0

minus10

002

0

minus002

2

0

minus20 1000 2000

0 1000 2000

0 1000 20000 1000 2000

0 1000 2000

0 1000 2000

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587



1

05

0

Δ(m

)

t (s)

t (s)

t (s)

l(m

)

4

2

0

40

20

00 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

[0 3009] [2016 3143]









Z(m

)

Y (m)X (m)

t (s)

500

0

minus50010

5

0 41542

425

5

0

minus50 500 1000 1500 2000 2500

times106times107

times107

Forward direction

Rs (m

)

RsxRsyRsz















120579(d

eg)

120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s) t (s)

t (s)

t (s)t (s)

t (s)100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus1000 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

500

0

minus500

5

0

minus5

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

Rmz

Rsz

Vmz

Vsz











F x(N

)F y

(N)

F z(N

)

t (s) t (s)

t (s)t (s)

t (s) t (s)

2

0

minus2

2

0

minus2

2

0

minus2

50

0

minus50

1

0

minus1

1

0

minus1

0 1000 2000 30000 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 30000 1000 2000 3000

times104times104

times104 times104

times104

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

0

002

001

0

32

30

28

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000










(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0




Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000







Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119910119860

119911119860) and (119909



120593119860

120579119860) of the ROGER







119860 119910119860

119911119860) and (119909

119861 119910119861 119911119861) and the


120593119860





(120595 120579 119897 119909119860

119910119860

119911119860

) lArrrArr (119909119861 119910119861 119911119861 119909119860

119910119860

119911119860

) (8)

A

dm

r B

CA

rC119860

rmrA

EI

rAB

rB

rC119860B

ang120575 asymp 0

Short tether


tO

1

2 3 4 5

6l0

l(t)



119879 = 119879119860

+ 119879119861

119879119860

=1

2119898119860

(2

119860+ 1199102

119860+ 2

119860) +

1

2120596119860

sdot J119860

sdot 120596119860

119879119861

=1

2119898119861

(2

119861+ 1199102

119861+ 2

119861)

(9)


119880 = 119880119892

+ 119880119890

119880119892

= 119880119892119860

+ 119880119892119861

119880119892119860

= minus120583119898119860

radic1199092

119860+ 1199102

119860+ 1199112

119860

119880119892119861

= minus120583119898119861

radic1199092

119861+ 1199102

119861+ 1199112

119861

(10)


119880119890

=

010038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816le 1198970

1

2119896Δ2 10038161003816100381610038161003816

r119862119860119861

10038161003816100381610038161003816gt 1198970

Δ =10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816minus 1198970

119897 = r119862119860119861

= r119861

minus r119860

minus r119862119860

(11)


119862119860is constant in 120587

119860



119871 = 119879 minus 119880 = 119879119860

+ 119879119861

minus 119880119892119860

minus 119880119892119861

minus 119880119890

=1

2119898119860

(2

119860+ 1199102

119860+ 2

119860) +

1

2120596119860

sdot J119860

sdot 120596119860

+1

2119898119861

(2

119861+ 1199102

119861+ 2


120583119898119860

radic1199092

119860+ 1199102

119860+ 1199112

119860

)

minus (minus120583119898119861

radic1199092

119861+ 1199102

119861+ 1199112

119861

) minus 119880119890

(12)



119860

= minus120583119909119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+119896Δ (119909

119861minus 119909119860

minus 119903119909)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119909

119910119860

= minus120583119910119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+

119896Δ (119910119861

minus 119910119860

minus 119903119910)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119910

119860

= minus120583119911119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+119896Δ (119911119861

minus 119911119860

minus 119903119911)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119911

(13)

where 119903119909 119903119910 and 119903


119862119860in 120587119868


J119860

sdot 119860

+ 120596119860

times J119860

sdot 120596119860

= T119888

+ T119889 (14)




119889119891and T

119889119905




AP = r119901

= 119903119901119910

j119860

+ 119903119901119911

k119860

F = 119865119909

i119860

T119889119905

= r119901

times F

(15)


as T119889119891

= r119862119860

times T r119862119860



FP

A

A

T

BrC119860

CA

XA

YA

ZA

Tension

ROGER



119860

= minus120583119909119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119909

+

119896Δ (119909119861

minus 119909119860

minus 119903119909)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)

119910119860

= minus120583119910119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119910

+

119896Δ (119910119861

minus 119910119860

minus 119903119910)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)

119860

= minus120583119911119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119911

+

119896Δ (119911119861

minus 119911119860

minus 119903119911)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)


T119888

+ T119889119891

+ T119889119905

(119897 ge 1198970)

T119888

+ T119889119891

(119897 lt 1198970)

(16)


119889119886

119889119905=

2

119899radic1 minus 1198902(120590119911119890 sin119891 + 120590

119909

119901

119903)

119889Ω

119889119905=

120590119910119903 sin 119906

1198991198862radic1 minus 1198902 sin 119894

119889119894

119889119905=

120590119910119903 cos 119906

1198991198862radic1 minus 1198902


119889119890119909

119889119905=


119899119886120590119911sin 119906

+ 120590119909

[(1 +119903

119901) cos 119906 +

119903

119901119890119909] +

119889Ω

119889119905119890119910cos 119894

119889119890119910

119889119905=


119899119886minus120590119911cos 119906

+ 120590119909

[(1 +119903

119901) sin 119906 +

119903

119901119890119910] minus

119889Ω

119889119905119890119909cos 119894

119889120593

119889119905= 119899 minus

1

119899119886[120590119911

(2119903

119886+


1 + radic1 minus 1198902119890 cos119891)

minus 120590119909

(1 +119903

119901)


1 + radic1 minus 1198902119890 sin119891]

minus

120590119910119903 cos 119894 sin 119906

1198991198862radic1 minus 1198902 sin 119894

(17)where (119886 Ω 119894 119890



= 119890 cos120596 119890119910


119909 120590119910 and 120590





1205821

= cos (AB i119868) =

(119909119860

minus 119909119861)

|AB|

1205822

= cos (AB j119868) =

(119910119860

minus 119910119861)

|AB|

1205823

= cos (AB k119868) =

(119911119860

minus 119911119861)

|AB|

(18)


X119874

Y119874


follows

[[

[

120574119909

120574119910

120574119911

]]

]119874

= C119874119868

[[

[

1205821

1205822

1205823

]]

]119868

(19)

B

E

C

F

H

G

YO

A(O)

120579998400

120579

XO

ZO

120595

(a)

t

F

T

F

T998400

(b)


where 120574119909 120574119910 and 120574


cosines in O119874

X119874

Y119874





119874Z119874and 119860X



120595 = arcsin(119861119864

|AB|) = 120574119910

120579 = arcsin(119861119867

|AB|) = 120574119911

(20)



1015840119879 are the


119909i119860

+ 119865119910

j119860

+ 119865119911k119860


X119860

Y119860


T119862

= 119879119909

i119860

+ 119879119910

j119860

+ 119879119911k119860



Rm0 Rs0 119896 119879 120578



T119862

= (119870119875120593119860

+ 119870119863

119860

) i119860

+ (119870119875120579119860

+ 119870119863

120579119860

) j119860

+ (119870119875120595119860

+ 119870119863

119860

) k119860

(21)

119870119875and 119870


cients



J119860

=[[

[

1000 minus49 32

minus49 1000 43

32 43 1000

]]

]

(22)


0 Rs0 Vm0


0and Vs









In Case 3 Rm0and Rs








1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)Δ

(m)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

) [0 2929] [1581 3004]


Rs (m

)

t (s)

5

0

minus50 500 1000 1500 2000

times107

Z(m

)

Y (m)X (m)

times106

times107

Forward direction

42242

418minus5

0

5minus100

0

100

RsxRsyRsz



120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s)t (s)

t (s)

t (s) t (s)

t (s)

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus50 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000

1

0

minus1

410 420 430 440


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

50

0

minus50

minus100

01

0

minus010 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Rmz

Rsz

Vmz












(N)

F y(N

)F z

(N)

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)

t (s)t (s)

t (s) t (s)

t (s) t (s)

300

200

100

0

20

0

minus20

20

0

minus20

20

0

minus20

002

001

0

minus001

10

0

minus100 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000



119909 119879119910 and 119879

119911) in 120587


t (s)

t (s)

1

0

minus1

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt


1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)

Δ(m

)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

)




t (s)

t (s)

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt

1

05

0

minus05

minus1


F x(N

)F y

(N)

F z(N

)

t (s)t (s)

t (s) t (s)

t (s)t (s)

400

200

0

5

0

minus5

10

0

minus10

10

0

minus10

002

0

minus002

2

0

minus20 1000 2000

0 1000 2000

0 1000 20000 1000 2000

0 1000 2000

0 1000 2000

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587



1

05

0

Δ(m

)

t (s)

t (s)

t (s)

l(m

)

4

2

0

40

20

00 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

[0 3009] [2016 3143]









Z(m

)

Y (m)X (m)

t (s)

500

0

minus50010

5

0 41542

425

5

0

minus50 500 1000 1500 2000 2500

times106times107

times107

Forward direction

Rs (m

)

RsxRsyRsz















120579(d

eg)

120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s) t (s)

t (s)

t (s)t (s)

t (s)100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus1000 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

500

0

minus500

5

0

minus5

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

Rmz

Rsz

Vmz

Vsz











F x(N

)F y

(N)

F z(N

)

t (s) t (s)

t (s)t (s)

t (s) t (s)

2

0

minus2

2

0

minus2

2

0

minus2

50

0

minus50

1

0

minus1

1

0

minus1

0 1000 2000 30000 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 30000 1000 2000 3000

times104times104

times104 times104

times104

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

0

002

001

0

32

30

28

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000










(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0




Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000







Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119871 = 119879 minus 119880 = 119879119860

+ 119879119861

minus 119880119892119860

minus 119880119892119861

minus 119880119890

=1

2119898119860

(2

119860+ 1199102

119860+ 2

119860) +

1

2120596119860

sdot J119860

sdot 120596119860

+1

2119898119861

(2

119861+ 1199102

119861+ 2


120583119898119860

radic1199092

119860+ 1199102

119860+ 1199112

119860

)

minus (minus120583119898119861

radic1199092

119861+ 1199102

119861+ 1199112

119861

) minus 119880119890

(12)



119860

= minus120583119909119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+119896Δ (119909

119861minus 119909119860

minus 119903119909)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119909

119910119860

= minus120583119910119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+

119896Δ (119910119861

minus 119910119860

minus 119903119910)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119910

119860

= minus120583119911119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+119896Δ (119911119861

minus 119911119860

minus 119903119911)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

+ 119865119911

(13)

where 119903119909 119903119910 and 119903


119862119860in 120587119868


J119860

sdot 119860

+ 120596119860

times J119860

sdot 120596119860

= T119888

+ T119889 (14)




119889119891and T

119889119905




AP = r119901

= 119903119901119910

j119860

+ 119903119901119911

k119860

F = 119865119909

i119860

T119889119905

= r119901

times F

(15)


as T119889119891

= r119862119860

times T r119862119860



FP

A

A

T

BrC119860

CA

XA

YA

ZA

Tension

ROGER



119860

= minus120583119909119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119909

+

119896Δ (119909119861

minus 119909119860

minus 119903119909)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)

119910119860

= minus120583119910119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119910

+

119896Δ (119910119861

minus 119910119860

minus 119903119910)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)

119860

= minus120583119911119860

(1199092

119860+ 1199102

119860+ 1199112

119860)32

+ 119865119911

+

119896Δ (119911119861

minus 119911119860

minus 119903119911)

119898119860

10038161003816100381610038161003816r119862119860119861

10038161003816100381610038161003816

(119897 ge 1198970)

0 (119897 lt 1198970)


T119888

+ T119889119891

+ T119889119905

(119897 ge 1198970)

T119888

+ T119889119891

(119897 lt 1198970)

(16)


119889119886

119889119905=

2

119899radic1 minus 1198902(120590119911119890 sin119891 + 120590

119909

119901

119903)

119889Ω

119889119905=

120590119910119903 sin 119906

1198991198862radic1 minus 1198902 sin 119894

119889119894

119889119905=

120590119910119903 cos 119906

1198991198862radic1 minus 1198902


119889119890119909

119889119905=


119899119886120590119911sin 119906

+ 120590119909

[(1 +119903

119901) cos 119906 +

119903

119901119890119909] +

119889Ω

119889119905119890119910cos 119894

119889119890119910

119889119905=


119899119886minus120590119911cos 119906

+ 120590119909

[(1 +119903

119901) sin 119906 +

119903

119901119890119910] minus

119889Ω

119889119905119890119909cos 119894

119889120593

119889119905= 119899 minus

1

119899119886[120590119911

(2119903

119886+


1 + radic1 minus 1198902119890 cos119891)

minus 120590119909

(1 +119903

119901)


1 + radic1 minus 1198902119890 sin119891]

minus

120590119910119903 cos 119894 sin 119906

1198991198862radic1 minus 1198902 sin 119894

(17)where (119886 Ω 119894 119890



= 119890 cos120596 119890119910


119909 120590119910 and 120590





1205821

= cos (AB i119868) =

(119909119860

minus 119909119861)

|AB|

1205822

= cos (AB j119868) =

(119910119860

minus 119910119861)

|AB|

1205823

= cos (AB k119868) =

(119911119860

minus 119911119861)

|AB|

(18)


X119874

Y119874


follows

[[

[

120574119909

120574119910

120574119911

]]

]119874

= C119874119868

[[

[

1205821

1205822

1205823

]]

]119868

(19)

B

E

C

F

H

G

YO

A(O)

120579998400

120579

XO

ZO

120595

(a)

t

F

T

F

T998400

(b)


where 120574119909 120574119910 and 120574


cosines in O119874

X119874

Y119874





119874Z119874and 119860X



120595 = arcsin(119861119864

|AB|) = 120574119910

120579 = arcsin(119861119867

|AB|) = 120574119911

(20)



1015840119879 are the


119909i119860

+ 119865119910

j119860

+ 119865119911k119860


X119860

Y119860


T119862

= 119879119909

i119860

+ 119879119910

j119860

+ 119879119911k119860



Rm0 Rs0 119896 119879 120578



T119862

= (119870119875120593119860

+ 119870119863

119860

) i119860

+ (119870119875120579119860

+ 119870119863

120579119860

) j119860

+ (119870119875120595119860

+ 119870119863

119860

) k119860

(21)

119870119875and 119870


cients



J119860

=[[

[

1000 minus49 32

minus49 1000 43

32 43 1000

]]

]

(22)


0 Rs0 Vm0


0and Vs









In Case 3 Rm0and Rs








1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)Δ

(m)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

) [0 2929] [1581 3004]


Rs (m

)

t (s)

5

0

minus50 500 1000 1500 2000

times107

Z(m

)

Y (m)X (m)

times106

times107

Forward direction

42242

418minus5

0

5minus100

0

100

RsxRsyRsz



120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s)t (s)

t (s)

t (s) t (s)

t (s)

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus50 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000

1

0

minus1

410 420 430 440


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

50

0

minus50

minus100

01

0

minus010 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Rmz

Rsz

Vmz












(N)

F y(N

)F z

(N)

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)

t (s)t (s)

t (s) t (s)

t (s) t (s)

300

200

100

0

20

0

minus20

20

0

minus20

20

0

minus20

002

001

0

minus001

10

0

minus100 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000



119909 119879119910 and 119879

119911) in 120587


t (s)

t (s)

1

0

minus1

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt


1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)

Δ(m

)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

)




t (s)

t (s)

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt

1

05

0

minus05

minus1


F x(N

)F y

(N)

F z(N

)

t (s)t (s)

t (s) t (s)

t (s)t (s)

400

200

0

5

0

minus5

10

0

minus10

10

0

minus10

002

0

minus002

2

0

minus20 1000 2000

0 1000 2000

0 1000 20000 1000 2000

0 1000 2000

0 1000 2000

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587



1

05

0

Δ(m

)

t (s)

t (s)

t (s)

l(m

)

4

2

0

40

20

00 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

[0 3009] [2016 3143]









Z(m

)

Y (m)X (m)

t (s)

500

0

minus50010

5

0 41542

425

5

0

minus50 500 1000 1500 2000 2500

times106times107

times107

Forward direction

Rs (m

)

RsxRsyRsz















120579(d

eg)

120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s) t (s)

t (s)

t (s)t (s)

t (s)100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus1000 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

500

0

minus500

5

0

minus5

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

Rmz

Rsz

Vmz

Vsz











F x(N

)F y

(N)

F z(N

)

t (s) t (s)

t (s)t (s)

t (s) t (s)

2

0

minus2

2

0

minus2

2

0

minus2

50

0

minus50

1

0

minus1

1

0

minus1

0 1000 2000 30000 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 30000 1000 2000 3000

times104times104

times104 times104

times104

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

0

002

001

0

32

30

28

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000










(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0




Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000







Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119889119890119909

119889119905=


119899119886120590119911sin 119906

+ 120590119909

[(1 +119903

119901) cos 119906 +

119903

119901119890119909] +

119889Ω

119889119905119890119910cos 119894

119889119890119910

119889119905=


119899119886minus120590119911cos 119906

+ 120590119909

[(1 +119903

119901) sin 119906 +

119903

119901119890119910] minus

119889Ω

119889119905119890119909cos 119894

119889120593

119889119905= 119899 minus

1

119899119886[120590119911

(2119903

119886+


1 + radic1 minus 1198902119890 cos119891)

minus 120590119909

(1 +119903

119901)


1 + radic1 minus 1198902119890 sin119891]

minus

120590119910119903 cos 119894 sin 119906

1198991198862radic1 minus 1198902 sin 119894

(17)where (119886 Ω 119894 119890



= 119890 cos120596 119890119910


119909 120590119910 and 120590





1205821

= cos (AB i119868) =

(119909119860

minus 119909119861)

|AB|

1205822

= cos (AB j119868) =

(119910119860

minus 119910119861)

|AB|

1205823

= cos (AB k119868) =

(119911119860

minus 119911119861)

|AB|

(18)


X119874

Y119874


follows

[[

[

120574119909

120574119910

120574119911

]]

]119874

= C119874119868

[[

[

1205821

1205822

1205823

]]

]119868

(19)

B

E

C

F

H

G

YO

A(O)

120579998400

120579

XO

ZO

120595

(a)

t

F

T

F

T998400

(b)


where 120574119909 120574119910 and 120574


cosines in O119874

X119874

Y119874





119874Z119874and 119860X



120595 = arcsin(119861119864

|AB|) = 120574119910

120579 = arcsin(119861119867

|AB|) = 120574119911

(20)



1015840119879 are the


119909i119860

+ 119865119910

j119860

+ 119865119911k119860


X119860

Y119860


T119862

= 119879119909

i119860

+ 119879119910

j119860

+ 119879119911k119860



Rm0 Rs0 119896 119879 120578



T119862

= (119870119875120593119860

+ 119870119863

119860

) i119860

+ (119870119875120579119860

+ 119870119863

120579119860

) j119860

+ (119870119875120595119860

+ 119870119863

119860

) k119860

(21)

119870119875and 119870


cients



J119860

=[[

[

1000 minus49 32

minus49 1000 43

32 43 1000

]]

]

(22)


0 Rs0 Vm0


0and Vs









In Case 3 Rm0and Rs








1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)Δ

(m)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

) [0 2929] [1581 3004]


Rs (m

)

t (s)

5

0

minus50 500 1000 1500 2000

times107

Z(m

)

Y (m)X (m)

times106

times107

Forward direction

42242

418minus5

0

5minus100

0

100

RsxRsyRsz



120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s)t (s)

t (s)

t (s) t (s)

t (s)

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus50 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000

1

0

minus1

410 420 430 440


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

50

0

minus50

minus100

01

0

minus010 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Rmz

Rsz

Vmz












(N)

F y(N

)F z

(N)

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)

t (s)t (s)

t (s) t (s)

t (s) t (s)

300

200

100

0

20

0

minus20

20

0

minus20

20

0

minus20

002

001

0

minus001

10

0

minus100 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000



119909 119879119910 and 119879

119911) in 120587


t (s)

t (s)

1

0

minus1

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt


1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)

Δ(m

)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

)




t (s)

t (s)

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt

1

05

0

minus05

minus1


F x(N

)F y

(N)

F z(N

)

t (s)t (s)

t (s) t (s)

t (s)t (s)

400

200

0

5

0

minus5

10

0

minus10

10

0

minus10

002

0

minus002

2

0

minus20 1000 2000

0 1000 2000

0 1000 20000 1000 2000

0 1000 2000

0 1000 2000

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587



1

05

0

Δ(m

)

t (s)

t (s)

t (s)

l(m

)

4

2

0

40

20

00 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

[0 3009] [2016 3143]









Z(m

)

Y (m)X (m)

t (s)

500

0

minus50010

5

0 41542

425

5

0

minus50 500 1000 1500 2000 2500

times106times107

times107

Forward direction

Rs (m

)

RsxRsyRsz















120579(d

eg)

120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s) t (s)

t (s)

t (s)t (s)

t (s)100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus1000 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

500

0

minus500

5

0

minus5

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

Rmz

Rsz

Vmz

Vsz











F x(N

)F y

(N)

F z(N

)

t (s) t (s)

t (s)t (s)

t (s) t (s)

2

0

minus2

2

0

minus2

2

0

minus2

50

0

minus50

1

0

minus1

1

0

minus1

0 1000 2000 30000 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 30000 1000 2000 3000

times104times104

times104 times104

times104

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

0

002

001

0

32

30

28

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000










(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0




Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000







Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Rm0 Rs0 119896 119879 120578



T119862

= (119870119875120593119860

+ 119870119863

119860

) i119860

+ (119870119875120579119860

+ 119870119863

120579119860

) j119860

+ (119870119875120595119860

+ 119870119863

119860

) k119860

(21)

119870119875and 119870


cients



J119860

=[[

[

1000 minus49 32

minus49 1000 43

32 43 1000

]]

]

(22)


0 Rs0 Vm0


0and Vs









In Case 3 Rm0and Rs








1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)Δ

(m)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

) [0 2929] [1581 3004]


Rs (m

)

t (s)

5

0

minus50 500 1000 1500 2000

times107

Z(m

)

Y (m)X (m)

times106

times107

Forward direction

42242

418minus5

0

5minus100

0

100

RsxRsyRsz



120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s)t (s)

t (s)

t (s) t (s)

t (s)

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus50 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000

1

0

minus1

410 420 430 440


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

50

0

minus50

minus100

01

0

minus010 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Rmz

Rsz

Vmz












(N)

F y(N

)F z

(N)

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)

t (s)t (s)

t (s) t (s)

t (s) t (s)

300

200

100

0

20

0

minus20

20

0

minus20

20

0

minus20

002

001

0

minus001

10

0

minus100 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000



119909 119879119910 and 119879

119911) in 120587


t (s)

t (s)

1

0

minus1

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt


1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)

Δ(m

)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

)




t (s)

t (s)

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt

1

05

0

minus05

minus1


F x(N

)F y

(N)

F z(N

)

t (s)t (s)

t (s) t (s)

t (s)t (s)

400

200

0

5

0

minus5

10

0

minus10

10

0

minus10

002

0

minus002

2

0

minus20 1000 2000

0 1000 2000

0 1000 20000 1000 2000

0 1000 2000

0 1000 2000

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587



1

05

0

Δ(m

)

t (s)

t (s)

t (s)

l(m

)

4

2

0

40

20

00 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

[0 3009] [2016 3143]









Z(m

)

Y (m)X (m)

t (s)

500

0

minus50010

5

0 41542

425

5

0

minus50 500 1000 1500 2000 2500

times106times107

times107

Forward direction

Rs (m

)

RsxRsyRsz















120579(d

eg)

120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s) t (s)

t (s)

t (s)t (s)

t (s)100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus1000 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

500

0

minus500

5

0

minus5

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

Rmz

Rsz

Vmz

Vsz











F x(N

)F y

(N)

F z(N

)

t (s) t (s)

t (s)t (s)

t (s) t (s)

2

0

minus2

2

0

minus2

2

0

minus2

50

0

minus50

1

0

minus1

1

0

minus1

0 1000 2000 30000 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 30000 1000 2000 3000

times104times104

times104 times104

times104

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

0

002

001

0

32

30

28

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000










(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0




Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000







Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s)t (s)

t (s)

t (s) t (s)

t (s)

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus5

5

0

minus50 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000

1

0

minus1

410 420 430 440


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

50

0

minus50

minus100

01

0

minus010 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Rmz

Rsz

Vmz












(N)

F y(N

)F z

(N)

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)

t (s)t (s)

t (s) t (s)

t (s) t (s)

300

200

100

0

20

0

minus20

20

0

minus20

20

0

minus20

002

001

0

minus001

10

0

minus100 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000



119909 119879119910 and 119879

119911) in 120587


t (s)

t (s)

1

0

minus1

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt


1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)

Δ(m

)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

)




t (s)

t (s)

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt

1

05

0

minus05

minus1


F x(N

)F y

(N)

F z(N

)

t (s)t (s)

t (s) t (s)

t (s)t (s)

400

200

0

5

0

minus5

10

0

minus10

10

0

minus10

002

0

minus002

2

0

minus20 1000 2000

0 1000 2000

0 1000 20000 1000 2000

0 1000 2000

0 1000 2000

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587



1

05

0

Δ(m

)

t (s)

t (s)

t (s)

l(m

)

4

2

0

40

20

00 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

[0 3009] [2016 3143]









Z(m

)

Y (m)X (m)

t (s)

500

0

minus50010

5

0 41542

425

5

0

minus50 500 1000 1500 2000 2500

times106times107

times107

Forward direction

Rs (m

)

RsxRsyRsz















120579(d

eg)

120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s) t (s)

t (s)

t (s)t (s)

t (s)100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus1000 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

500

0

minus500

5

0

minus5

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

Rmz

Rsz

Vmz

Vsz











F x(N

)F y

(N)

F z(N

)

t (s) t (s)

t (s)t (s)

t (s) t (s)

2

0

minus2

2

0

minus2

2

0

minus2

50

0

minus50

1

0

minus1

1

0

minus1

0 1000 2000 30000 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 30000 1000 2000 3000

times104times104

times104 times104

times104

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

0

002

001

0

32

30

28

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000










(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0




Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000







Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(N)

F y(N

)F z

(N)

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)

t (s)t (s)

t (s) t (s)

t (s) t (s)

300

200

100

0

20

0

minus20

20

0

minus20

20

0

minus20

002

001

0

minus001

10

0

minus100 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000



119909 119879119910 and 119879

119911) in 120587


t (s)

t (s)

1

0

minus1

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt


1

05

00 200 400 600 800 1000 1200 1400 1600

t (s)005

00 200 400 600 800 1000 1200 1400 1600

t (s)

Δ(m

)

31

30

290 200 400 600 800 1000 1200 1400 1600

t (s)

l(m

)




t (s)

t (s)

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt

1

05

0

minus05

minus1


F x(N

)F y

(N)

F z(N

)

t (s)t (s)

t (s) t (s)

t (s)t (s)

400

200

0

5

0

minus5

10

0

minus10

10

0

minus10

002

0

minus002

2

0

minus20 1000 2000

0 1000 2000

0 1000 20000 1000 2000

0 1000 2000

0 1000 2000

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587



1

05

0

Δ(m

)

t (s)

t (s)

t (s)

l(m

)

4

2

0

40

20

00 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

[0 3009] [2016 3143]









Z(m

)

Y (m)X (m)

t (s)

500

0

minus50010

5

0 41542

425

5

0

minus50 500 1000 1500 2000 2500

times106times107

times107

Forward direction

Rs (m

)

RsxRsyRsz















120579(d

eg)

120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s) t (s)

t (s)

t (s)t (s)

t (s)100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus1000 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

500

0

minus500

5

0

minus5

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

Rmz

Rsz

Vmz

Vsz











F x(N

)F y

(N)

F z(N

)

t (s) t (s)

t (s)t (s)

t (s) t (s)

2

0

minus2

2

0

minus2

2

0

minus2

50

0

minus50

1

0

minus1

1

0

minus1

0 1000 2000 30000 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 30000 1000 2000 3000

times104times104

times104 times104

times104

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

0

002

001

0

32

30

28

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000










(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0




Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000







Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










t (s)

t (s)

5

0

minus50 200 400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Ang

le (d

eg)

Ang

ular

rate

(deg

s)

120593A120579A120595A

d120593Adt

d120579Adt

d120595Adt

1

05

0

minus05

minus1


F x(N

)F y

(N)

F z(N

)

t (s)t (s)

t (s) t (s)

t (s)t (s)

400

200

0

5

0

minus5

10

0

minus10

10

0

minus10

002

0

minus002

2

0

minus20 1000 2000

0 1000 2000

0 1000 20000 1000 2000

0 1000 2000

0 1000 2000

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587



1

05

0

Δ(m

)

t (s)

t (s)

t (s)

l(m

)

4

2

0

40

20

00 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

[0 3009] [2016 3143]









Z(m

)

Y (m)X (m)

t (s)

500

0

minus50010

5

0 41542

425

5

0

minus50 500 1000 1500 2000 2500

times106times107

times107

Forward direction

Rs (m

)

RsxRsyRsz















120579(d

eg)

120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s) t (s)

t (s)

t (s)t (s)

t (s)100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus1000 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

500

0

minus500

5

0

minus5

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

Rmz

Rsz

Vmz

Vsz











F x(N

)F y

(N)

F z(N

)

t (s) t (s)

t (s)t (s)

t (s) t (s)

2

0

minus2

2

0

minus2

2

0

minus2

50

0

minus50

1

0

minus1

1

0

minus1

0 1000 2000 30000 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 30000 1000 2000 3000

times104times104

times104 times104

times104

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

0

002

001

0

32

30

28

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000










(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0




Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000







Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

05

0

Δ(m

)

t (s)

t (s)

t (s)

l(m

)

4

2

0

40

20

00 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

[0 3009] [2016 3143]









Z(m

)

Y (m)X (m)

t (s)

500

0

minus50010

5

0 41542

425

5

0

minus50 500 1000 1500 2000 2500

times106times107

times107

Forward direction

Rs (m

)

RsxRsyRsz















120579(d

eg)

120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s) t (s)

t (s)

t (s)t (s)

t (s)100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus1000 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

500

0

minus500

5

0

minus5

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

Rmz

Rsz

Vmz

Vsz











F x(N

)F y

(N)

F z(N

)

t (s) t (s)

t (s)t (s)

t (s) t (s)

2

0

minus2

2

0

minus2

2

0

minus2

50

0

minus50

1

0

minus1

1

0

minus1

0 1000 2000 30000 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 30000 1000 2000 3000

times104times104

times104 times104

times104

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

0

002

001

0

32

30

28

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000










(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0




Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000







Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120579(d

eg)

120579(d

eg)

120579(d

eg)

120579(d

eg)

120595(d

eg)

120595(d

eg)

t (s) t (s)

t (s)

t (s)t (s)

t (s)100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus100

100

0

minus1000 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 3000

0 1000 2000 3000


Zve

loci

ty (m

s)

Zdi

spla

cem

ent (

m)

t (s)

t (s)

500

0

minus500

5

0

minus5

0 500 1000 1500 2000 2500

0 500 1000 1500 2000 2500

Rmz

Rsz

Vmz

Vsz











F x(N

)F y

(N)

F z(N

)

t (s) t (s)

t (s)t (s)

t (s) t (s)

2

0

minus2

2

0

minus2

2

0

minus2

50

0

minus50

1

0

minus1

1

0

minus1

0 1000 2000 30000 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 30000 1000 2000 3000

times104times104

times104 times104

times104

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

0

002

001

0

32

30

28

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000










(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0




Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000







Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










F x(N

)F y

(N)

F z(N

)

t (s) t (s)

t (s)t (s)

t (s) t (s)

2

0

minus2

2

0

minus2

2

0

minus2

50

0

minus50

1

0

minus1

1

0

minus1

0 1000 2000 30000 1000 2000 3000

0 1000 2000 3000 0 1000 2000 3000

0 1000 2000 30000 1000 2000 3000

times104times104

times104 times104

times104

T x(N

lowastm)

T y(N

lowastm)

T z(N

lowastm)



119909 119879119910 and 119879

119911) in 120587


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

0

002

001

0

32

30

28

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000










(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0




Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000







Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(m)

t (s)

t (s)

t (s)

l(m

)

1

05

1

05

0

0

40

20

0

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

004

002

0

0

31

30

29

0 1000 2000 40003000

1000 2000 40003000

1000 2000 40003000

0

0




Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000


Δ(m

)

t (s)

t (s)

t (s)

l(m

)

1

05

005

0

0

31

30

29

0 500 1000 1500 2000

0 500 1000 1500 2000

0 500 1000 1500 2000







Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









dynamics of the space tug system with a short …...research article dynamics of the space tug...

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