couplinganalysisandcross-feedbackcontrolofthree-axis...

Research ArticleCoupling Analysis and Cross-Feedback Control of Three-AxisInertially Stabilized Platform with an Active MagneticBearing System

Tong Wen12 Biao Xiang 3 and Waion Wong3

1School of Instrument Science and Optoelectronics Engineering Beihang University Beijing 100191 China2Ningbo Institute of Technology Beihang University Ningbo Zhejiang 31500 China3Department of Mechanical Engineering +e Hong Kong Polytechnic University Kowloon China

Correspondence should be addressed to Biao Xiang thomasbiaogmailcom

Received 18 December 2019 Revised 17 June 2020 Accepted 6 July 2020 Published 22 July 2020

Academic Editor Fabio Minghini

Copyright copy 2020 Tong Wen et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An active magnetic bearing (AMB) system is used to suspend the yaw gimbal of three-axis inertially stabilized platform (ISP) tominimize the friction e dynamic functions of three gimbals in ISP are developed e base coupling at dynamic base plate isstronger than that at static base plate and the gimbal coupling among three gimbals increases with the number of unlockedgimbals erefore a cross-feedback control scheme is designed to minimize the base coupling and the gimbal coupling and thenthe multi-input multioutput system of three-axis ISP with coupling terms is simplified into three decoupled single-input single-output systems Experimental results verify that the yaw gimbal suspended by AMB system has better vibration isolation abilitythan the roll gimbal supported by mechanical bearing and the gimbal coupling and the base coupling are effectively suppressed bythe cross-feedback control scheme

1 Introduction

e inertially stabilized platform (ISP) is widely used in theremote sensing and imaging system on the airborneplatform [1ndash3] and the ship-based platform [4] to track theline of sight It could keep the stability of remote sensingand imaging payloads by connecting with the flight carrierso the external vibration disturbances such as the randomvibration of flight carrier would be isolated e remotesensing and imaging payloads are commonly installed onthe yaw gimbal of three-axis ISP and the stability of remotesensing and imaging system is therefore directly affected bythe attitude stabilization precision of yaw gimbal [5 6] Inthe normal three-axis ISP the roll pitch and yaw gimbalsare supported by mechanical bearings and gears [7] It ispossible that the friction of the mechanical bearing and gearaffects the attitude stabilization precision of supportedgimbals [8ndash11] In addition the disturbances could betransferred from the pitch and roll gimbals to the yaw

gimbal such that the attitude stabilization precision of yawgimbal is reduced [12ndash14] erefore in order to suppressthe vibration acting on yaw gimbal an active magneticbearing (AMB) [15ndash17] system was designed for sus-pending the yaw gimbal in three-axis ISP Since the AMBsystem provides a contactless suspension to yaw gimbal thevibration disturbances from other gimbals could be ef-fectively isolated and the friction between levitated gimbaland suspension elements would be minimized [18]Moreover the displacement of yaw gimbal is controllableby regulating the winding current of AMB system based ondisplacement feedback [19ndash23] and then the vibrationisolation could be realized with the active controllability ofan AMB system [24]

Although friction and vibration transmission of yawgimbal are suppressed by the AMB system the couplingsamong three gimbals in three-axis ISP still affect the attitudestabilization precision of yaw gimbal e coupling effectamong three gimbals and base plate was analyzed in [25]

HindawiShock and VibrationVolume 2020 Article ID 8290369 17 pageshttpsdoiorg10115520208290369

simulation results showed that attitude stabilization pre-cisions of three gimbals were affected by the coupling effectamong three gimbals and the base plate when the error ofaxial angular velocity increased from 005 rads to 03 radsA decoupling control method [26] was used in a differentialcable driving ISP to minimize the coupling effect amongthree gimbals and the base plate and the control precisionwas improved by suppressing the coupling terms Anotherdecoupling control method based on the feed-forwardcompensation was used to control the ISP [27] the peakvalue of pitch gimbalrsquos rotational angle was reduced from06deg to 02deg In addition the adaptive decoupling controlbased on neural network was used to improve the controlperformances of three gimbals in a three-axis ISP [28] thetrack error of azimuth gimbal was reduced by 80 esliding mode control based on the extended state observerin [29 30] was used to suppress various types of distur-bances in the ISP Simulation result showed that the un-known state and the lumped uncertainty of gimbal could beaccurately estimated and the maximum deflection of yawgimbal was declined from 05332deg to 00811deg In literature[31] a compound control method based on model refer-ence adaptive control and proportional-integral-derivative(PID) control was proposed to improve the tracking pre-cision and the robustness of ISP system It could suppressthe nonlinear unbalance torque caused by the time-varyingmass and the position precision of yaw gimbal was im-proved by 50

Even though the abovementioned control methods suchas the disturbance observer [32] robust control [33] andsliding mode control [34 35] were useful to improve thecontrol precisions of ISP gimbals the equations of motion ofthree gimbals were simplified as a decoupled system byneglecting coupling terms among three gimbals and the baseplate Moreover the coupling effect among three gimbalsand the base plate was not verified experimentally and itsinfluences on the attitude stabilization precision of eachgimbal were not tested More importantly the couplingeffect was not studied in a three-axis ISP with an AMBsystem integrating the control schemes of torque motors andAMB system

Above all the original works of this article are focused onthree parts to further improve the attitude stabilizationprecision of yaw gimbal and the novelties of this article arelisted as follows

(1) A novel structure of yaw gimbal with an AMB systemintegrating the axial and radial motion is firstlydesigned to minimize the friction and vibrationtransmission and dynamic functions of three gim-bals with coupling terms are developed

(2) e influences of coupling terms among threegimbals and the base plate are firstly quantified in theexperiment and the mathematic models of couplingterms are originally developed It proves that the basecoupling with dynamic base plate is greater than thatwith static base plate and the gimbal couplingamong three gimbals increases with the number ofunlocked gimbals

(3) e cross-feedback control scheme is proposed toimprove the attitude stabilization precision of yawgimbal with dynamic disturbances

Consequently the AMB system is used as a novel sus-pension system in three-axis ISP and this structure broadensthe application scope of AMB system and improves attitudestabilization precision of yaw gimbal

is article is organized as follows In Section 2 thestructure of three-axis ISP with an AMB system is intro-duced and equations of motion of three gimbals are de-veloped and then the coupling effects among three gimbalsand the base plate are analyzed In Section 3 the cross-feedback control scheme with angular displacements ofthree gimbals is used to suppress the coupling effect actingon the yaw gimbal and the experiments are conducted toverify the effectiveness of proposed cross-feedback controlscheme

2 Three-Axis ISP with an AMB System

21 Structure of +ree-Axis ISP with an AMB System emechanical structure of three-axis ISP with an AMB systemis illustrated in Figure 1 e three-axis ISP consists of threerotational gimbals (yaw pitch and roll gimbals) an AMBsystem and a base plate e yaw gimbal (the innermostgimbal) supports the sensing and imaging payloads and itrotates about zy axis as illustrated in Figure 2(c) As shownin Figure 1(b) the rotor part of AMB system is located on theyaw gimbal while the stator part is mounted on the pitchgimbal e AMB system generates magnetic forces tosuspend the yaw gimbal at the radial and axial equilibriumpositions and the position of yaw gimbal is controllable byregulating the winding current of AMB system based on thedisplacement feedbacke pitch gimbal connects to the rollgimbal by mechanical bearing and it locates outside of yawgimbal e torque motor of pitch gimbal controls its ro-tation about xp axis in Figure 2(b) As the external gimbalthe roll gimbal locates outside of pitch gimbal and thetorque motor on roll gimbal controls its rotation about yr

axis as shown in Figure 2(a) e bracket is used to connectthe roll gimbal with the base plate and the base plateconnects three-axis ISP with the swaying platform or flightcarriers

22Coordinate Systemof+ree-Axis ISPwithanAMBSysteme coordinate systems of three gimbals in the ISP are il-luminated in Figure 2 OXbYbZb is the coordinate system ofbase plate OXrYrZr is the coordinate system of roll gimbalOXpYpZp is the coordinate system of pitch gimbal andOXyyYyZy is the coordinate system of yaw gimbal θry θpxand θyz are rotational angles of three gimbals respectively_θry _θpx and _θyz are relative angular rates of three gimbalsrespectively ree torque motors on pitch roll and yawgimbals control the rotations of three-axis ISP about Xp Yrand Zy axes respectively erefore the rotations of three-axis ISP about three pivot axes could be expressed in terms ofangular rates of three gimbals

2 Shock and Vibration

e angular rates of roll gimbal around three axes couldbe derived and written as

ωrx

ωry

ωrz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ Crb

ωbx

ωby

ωbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_θry

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ωbx cos θry minus ωbz sin θry

ωby + _θry

ωbx cos θry + ωbz sin θry

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)

e coordinate transformation matrix from the baseplate to the roll gimbal is

Crb

cos θry 0 minussin θry

0 1 0

sin θry 0 cos θry

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (2)

e angular rates of pitch gimbal around three axescould be derived and written as

ωpx

ωpy

ωpz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ C

pr C

rb

ωbx

ωby

ωbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ + Cpr

0_θry

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

_θpx

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ωbx cos θry minus ωbz sin θry + _θpx

ωby + _θry1113872 1113873cos θpx + ωbx sin θry + ωbz cos θry1113872 1113873sin θpx

ωbx sin θry + ωbz cos θry1113872 1113873cos θpx minus ωby + _θry1113872 1113873sin θpx

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)

e coordinate transformation matrix from the rollgimbal to the pitch gimbal is C

pr

1 0 0

0 cos θpx sin θpx

0 minussin θpx cos θpx

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (4)

Yaw gimbal

Pitch gimbal

Roll gimbal

Base

Bracket

AMB

(a)

Axial AMB at high-end Yaw gimbal

Axialair gap

Radial air gapRadial AMB

Axial AMBat low end

(b)

Figure 1 (a) Configuration of the three-axis ISP (b) e structure of AMB system

Xb

Yb

Xr

Yr

ZbZr

O

θry

θryθmiddotry

(a)

Xp

Yp

Xr

Yr

ZpZr

O

θpx

θpx

θmiddotpx

(b)

Yp

Xp

Yy

ZpZy

θyz

θyz

Xy

O

θmiddotyz

(c)

Figure 2 Coordinate systems of (a) roll gimbal (b) pitch gimbal and (c) yaw gimbal in ISP

Shock and Vibration 3

e angular rates of yaw gimbal around three axes couldbe derived and written as

ωyx

ωyy

ωyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ C

ypC

pr C

rb

ωbx

ωby

ωbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ + CypC

pr

0_θry

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ + Cyp

_θpx

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+

0

0_θyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ωbx cos θry minus ωbz sin θry + _θpx1113872 1113873cos θyz + ωby + _θry1113872 1113873cos θpx + ωbx sin θry + ωbz cos θry1113872 1113873sin θyz1113960 1113961sin θyz

ωby + _θry1113872 1113873cos θpx + ωbx sin θry + ωbz cos θry1113872 1113873sin θyz1113960 1113961cos θyz minus ωbx cos θry minus ωbz sin θry + _θpx1113872 1113873sin θyz

ωbx sin θry + ωbz cos θry1113872 1113873cos θpx minus ωby + _θry1113872 1113873sin θpx + _θyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)

e coordinate transformation matrix from the pitchgimbal to the yaw gimbal is

Cyp

cos θyz sin θyz 0

minussin θyz cos θyz 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (6)

Using the equations from (1) to (6) the angular rates ofthree-axis ISP around three pivot axes can be written as

ωpx

ωry

ωyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


ωby + _θry


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0 1 0

cos θpx sin θry minussin θpx cos θpx cos θry

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦middot

ωbx

ωby

ωbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

1 0 0

0 1 0

0 minussin θpx 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦middot

_θpx

_θry

_θyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)

23 Dynamics of +ree Gimbals in +ree-Axis ISPAssuming each gimbal of three-axis ISP is a symmetricalrigid structure according to the NewtonndashEuler rotationalequation the torques of three gimbals are written as

T J _ω + ω times Jω (8)

For the roll gimbal the torque about each axis isTr [TrxTryTrz]T and

Trx Jrx _ωrx minus Jry minus Jrz1113872 1113873ωryωrz

Try Jry _ωry minus Jrx minus Jrz( 1113857ωrxωrz

Trz Jrz _ωrz minus Jrx minus Jry1113872 1113873ωrxωry

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(9)

where Jrx Jry and Jrz are moments of inertia about threeaxes of the roll gimbal respectively

For the pitch gimbal the torque acting on each axisis Tp [TpxTpyTpz]T and

Tpx Jpx _ωrx + euroθpx1113872 1113873 minus Jry minus Jrz1113872 1113873 ωrz sin θpx + ωry cos θpx1113872 1113873 ωrz cos θpx minus ωry sin θpx1113872 1113873

Tpy Jpy _ωry + _θpxωrz1113872 1113873cos θpx minus ωrz + _θpxωry1113872 1113873sin θpx1113872 1113873 minus Jrx minus Jrz( 1113857 ωrz cos θpx minus ωry sin θpx1113872 1113873 ωrx + _θpx1113872 1113873

Tpz Jpz minus _ωry sin θpx minus _θpxωry cos θpx + ωrz cos θpx minus _θpxωrz sin θpx1113872 1113873

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)

where Jpx Jpy and Jpz are moments of inertia about threeaxes of the pitch gimbal respectively

For the yaw gimbal the torque acting on each axis isTy [TyxTyyTyz]T and


Tyx Jyx

_ωrx + euroθpx1113872 1113873cos θyz minus _θyz ωrx + _θpx1113872 1113873sin θyz + _θyz ωry cos θpx + ωrz sin θpx1113872 1113873cos θyz

+ _ωry cos θpx minus _θpxωry sin θpx + _ωrz sin θpx + _θpxωrz cos θpx1113872 1113873sin θyz

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

minus Jyy minus Jyz1113872 1113873 ωrz cos θpx minus ωry sin θpx + _θyz1113872 1113873 ωrz sin θpx + ωry cos θpx1113872 1113873cos θyz minus ωrx + _θyz1113872 1113873sin θyz1113960 1113961

Tyy Jyy

minus _ωrx + euroθpx1113872 1113873sin θyz minus _θyz ωrx + _θpx1113872 1113873cos θyz minus _θyz ωry cos θpx + ωrz sin θpx1113872 1113873sin θyz

+ _ωry cos θpx minus _θpxωry sin θpx + _ωrz sin θpx + _θpxωrz cos θpx1113872 1113873cos θyz

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

minus Jyz minus Jyx1113872 1113873 ωrz cos θpx minus ωry sin θpx + _θyz1113872 1113873 ωrz sin θpx + ωry cos θpx1113872 1113873sin θyz + ωrx + _θyz1113872 1113873cos θyz1113960 1113961

Tyz Jyz minusωry sin θpx minus _θpxωry cos θpx + _ωrz cos θpx minus _θpxωrz sin θpx + euroθyz1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

where Jyx Jyy and Jyz are moments of inertia about threeaxes of the yaw gimbal respectively

erefore the torques acting on three pivot axes of thethree-axis ISP could be expressed as

Tx Tpx + Cyp1113872 1113873

minus 1Tyz

Ty Try + CypC

pr1113872 1113873

minus 1Tyz + C

pr1113872 1113873

minus 1Tpx

Tz Tyz

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(12)

e torques about three pivot axes of three-axis ISP TxTy and Tz are the synthesized torques about Xp axis of the

pitch gimbal Yr axis of the roll gimbal and Zy axis of theyaw gimbal respectively Tx is the summation of Tpx (torqueacting on the pivot axis of the pitch gimbal) and (C

yp)minus1Tyz is

the component of Tyz about Xp axis Similarly Ty is thesummation of Try (torque acting on pivot axis of the rollgimbal) (C

ypC

pr )minus1Tyz is the component of Tyz about Yr axis

and (Cpr )minus1Tpx is the component of Tpx about Yr axis

Furthermore substituting torques of three gimbals in(9)ndash(11) into (12) and using the coordinate transformationmatrices in (2) (4) and (6) we can write

Tx Jyx + Jpx1113872 1113873euroθpx + Jyz_θyz

_θry minus Jyz minus Jyy + Jrz minus Jry1113872 1113873 _θ2ryθpx

+ Jyz minus Jyy + Jrz minus Jry minus Jyx minus Jpx1113872 1113873 _θryωbz + Jyz_θyzωby

+ Jyx + Jpx1113872 1113873 _ωbx + Jyz minus Jyy + Jrz minus Jry1113872 1113873ωbyωbz

Ty Jyx + Jpy + Jry1113872 1113873euroθry minus Jyzθpxeuroθyz + Jyz + Jpz minus Jyx minus Jpy1113872 1113873 _θpx

_θryθpx minus Jyx_θpx

_θyz

+ Jyx + Jpy minus Jyz minus Jpz1113872 1113873 _θpx_θryωbx + Jyx + Jpy minus Jyz + Jpx1113872 1113873 _θpxωbz

minus Jyx_θyzωbx + Jyx + Jpy + Jry1113872 1113873 _ωby minus Jpz + Jrz minus Jpx minus Jrx1113872 1113873ωbxωbz

Tz Jyzeuroθyz minus Jyzθpx

euroθry minus Jyz_θpx

_θry minus Jyz_θpxωby + Jyz

_θryωbx + Jyz _ωbz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

ere are cross-coupling terms including angular ratesof the base plate and two gimbals in (13) e couplingtorque between the base plate and the gimbal is defined asthe base coupling torque and the coupling torque amongthree gimbals is defined as the gimbal coupling torque

For the pitch gimbal the gimbal coupling torque be-tween the pitch gimbal and other gimbals is

T(ry)p Jyz

_θyz_θry minus Jyz minus Jyy + Jrz minus Jry1113872 1113873 _θ

2ryθpx (14)

e base coupling torque between the pitch gimbal andthe base plate is

Tbp Jyz minus Jyy + Jrz minus Jry minus Jyx minus Jpx1113872 1113873 _θryωbz

+ Jyz_θyzωby + Jyx + Jpx1113872 1113873 _ωbx

+ Jyz minus Jyy + Jrz minus Jry1113872 1113873ωbyωbz

(15)

For the roll gimbal the gimbal coupling torque betweenthe roll gimbal and other gimbals is

T(py)r minusJyzθpx

euroθyz minus Jyx_θpx

_θyz

+ Jyz + Jpz minus Jyx minus Jpy1113872 1113873 _θpx_θryθpx

(16)

ebase coupling torque between the roll gimbal and thebase plate is

Tbr Jyx + Jpy minus Jyz minus Jpz1113872 1113873 _θpx

_θryωbx

+ Jyx + Jpy minus Jyz + Jpx1113872 1113873 _θpxωbz

minus Jyx_θyzωbx + Jyx + Jpy + Jry1113872 1113873 _ωby

minus Jpz + Jrz minus Jpx minus Jrx1113872 1113873ωbxωbz

(17)

For the yaw gimbal the gimbal coupling torque betweenthe yaw gimbal and other gimbals is


T(pr)y minusJyzθpx


_θry (18)

e base coupling torque between the yaw gimbal andthe base plate is

Tby minusJyz

_θpxωby + Jyz_θryωbx + Jyz _ωbz (19)

Using equations (14)ndash(19) equation (13) could be re-written into

Tx Jyx + Jpx1113872 1113873euroθpx + T(ry)p + Tb

p

Ty Jyx + Jpy + Jry1113872 1113873euroθry + T(py)r + Tb

r

Tz Jyzeuroθyz + T

(pr)y + Tb

y

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(20)

erefore the gimbal couplings among three gimbalsaffect the attitude stabilization precisions of three gimbalsand the gimbal coupling torque increases with the number ofcoupling gimbals Moreover the base coupling torque alsoaffects the attitude stabilization precisions of three gimbals

For the static base plate when the swaying platform andother carriers do not generate disturbances and conductrapid attitude maneuver there are

ωbx ωby ωbz 0

Tbp Tb

r Tby 0

⎧⎨

⎩ (21)

Substituting (21) into (20) we can write

Tx Jyx + Jpx1113872 1113873euroθpx + T(ry)p

Ty Jyx + Jpy + Jry1113872 1113873euroθry + T(py)r

Tz Jyzeuroθyz + T

(pr)y

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)

erefore the base coupling torque could be neglectedwhen the base plate works at static status because it does notaffect the attitude stabilization precisions of three gimbalsOnly when the base plate works at dynamic status the basecoupling torque would affect the attitude stabilizationprecisions of three gimbals

24 Single Gimbal Torque Motor of +ree-Axis ISP edriving torques generated by the torque motor control therotation of gimbal e control diagram of a gimbal motor ispresented in Figure 3 θ is the rotational angle of gimbalmotor u is the control voltage of gimbal motor L is thearmature inductance R is the armature resistance kt is thetorque coefficient N is gear ratio Tm is the driving torqueand ke is the back-EMF coefficient e driving moment ofgimbal motor is

Tm kt

Ls + Ru minus ke

_θ1113872 1113873 (23)

e electromechanical models of three gimbal motors inthree-axis ISP could be written as

Tx ktp

Lps + Rp

up minus kep_θpx1113872 1113873 Jyx + Jpx1113872 1113873euroθpx + T

(ry)p + T

bp

Ty ktr

Lrs + Rr

ur minus ker_θry1113872 1113873 Jyx + Jpy + Jry1113872 1113873euroθry + T

(py)r + T

br

Tz kty

Lys + Ry

uy minus key_θyz1113872 1113873 Jyz

euroθyz + T(pr)y + T

by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

If coupling torques including the base coupling torqueand the gimbal coupling torque are effectively compensated(24) could be simplified as

ktp

Lps + Rp

up minus kep_θpx1113872 1113873 Jyx + Jpx1113872 1113873euroθpx

ktr

Lrs + Rr

ur minus ker_θry1113872 1113873 Jyx + Jpy + Jry1113872 1113873euroθry

kty

Lys + Ry


euroθyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)

erefore the rotational controls around pivot axes ofthe three-axis ISP could be dominated by the torque motorsmounted on three gimbals

25 AMB System of +ree-Axis ISP

251 Axial AMB System e axial AMB system consists ofeight AMBs at low end and four AMBs at high end of yawgimbal as shown in Figure 4 e four pairs of axial AMBs(A1-B1 A2-B2 A3-B3 and A4-B4) at lower and upper endin Figure 4 control the translation of yaw gimbal in axialdirection and then the yaw gimbal is stably suspended at theaxial equilibrium position Based on [16] the suspensionforce in axial direction could be written as

fz kiziz minus kdzdz (26)

where kiz is the axial current stiffness and kdz is the axialdisplacement stiffness

e relationships amongst the axial suspension force theaxial displacement and the axial control current are plottedin Figures 5(a) and 5(b) respectivelyese two figures show


that the axial suspension force is proportional to bothcontrol current and control displacement e axial currentstiffness kiz is found to be 2000NA and the axial dis-placement stiffness kdz is found to be minus3000Nmm

252 Radial AMB System e radial AMB system has twopairs of radial AMBs as shown in Figure 6 One pair of radialAMBs (R2-R4) control the translation of yaw gimbal along Xaxis and another pair of radial AMBs (R1-R3) control thetranslation of yaw gimbal along Y axis Since the radial AMBsystem is symmetrical the current stiffness in X axis equalsthat in Y axis and the displacement stiffness in X axis equalsthat in Y axis e suspension forces along X and Y axes arelinearized and written as

fx kixix minus kdxdx

fy kiyiy minus kdydy

⎧⎨

⎩ (27)

where kix kiy is the radial current stiffness and kdx kdy isthe radial displacement stiffness

e radial suspension force is proportional to bothcontrol current and radial displacement in Figures 7(a) and7(b)e radial current stiffness kix is 300NA and the radialdisplacement stiffness kdx is minus480Nmm

253 Control of the AMB System in +ree-Axis ISP eforce analysis of the yaw gimbal with an AMB system isillustrated in Figure 8 the windings of radial and axial AMBs

u

ndash(1Ls + R) 1(Jl + N2Jm)s 1s

ke

kt

TmN θω

Figure 3 Control diagram of torque motor on gimbal

(a)

x

y

(b)

Figure 4 e axial AMB system (a) Axial AMB at lower end (b) Axial AMB at upper end and the single AMB

Axi

al su

spen

sion

forc

e (N

)

300

200

100

0

ndash100

ndash200

ndash01 ndash005Axial displacement (mm)

0 005 01ndash300

Measured valueFitted curve

(a)

01 015Axial control current (A)

02 025 03 035

Axi

al su

spen

sion

forc

e (N

)

800

700

600

500

400

300


(b)

Figure 5 (a) Axial suspension force versus control displacement (b) Axial suspension force versus control current


are mounted on the stator part of yaw gimbal and then therotor part of yaw gimbal is suspended by the suspensionforces of AMB system erefore the equations of yawgimbalrsquos motion in radial and axial directions could beexpressed into

meurodx fx+ minus fxminus

meurody fy+ minus fyminus

meurodz fz

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(28)

Considering that the rotational speed of yaw gimbalaround the axial pivot axis is slow compared with the high-speed rotational machine such as flywheel energy storagesystem [36] the gyroscopic effect of the yaw gimbal is notconsidered e translational control models of yaw gimbalin radial and axial directions are decoupled and thetranslational control of yaw gimbal along three axes issimilar so the control loop of radial translation is chosen asan example in Figure 9 e PD control model is appliedwith the proportional coefficient kPx the derivative coeffi-cient kDx ks is the sensitivity of eddy current displacement

Figure 6 Radial AMB system and its single AMB

Radi

al su

spen

sion

forc

e (N

)

60

40

20

0

ndash20

ndash40

ndash60

ndash80ndash02 ndash01

Radial displacement (mm)0 01 02


(a)

Radi

al su

spen

sion

forc

e (N

)

150

100

50

0

ndash50

ndash100


Radial control current (A)0 02 04 06


(b)

Figure 7 (a) Radial suspension force versus radial displacement (b) Radial suspension force versus control current

Axial AMB Rotor part of yaw gimbal

Stator part of yaw gimbal Radial AMB

zy

o

xy

fxndash

fx+

fyndashfz

fy+yy

Figure 8 Force analysis of the yaw gimbal with an AMB system


sensor and ka is the amplification coefficient e transferfunction of translation motion could be expressed as

G(s) kixkakPx + kixkakDxs

ms2 + kixkakskDxs + kixkakskPx minus kdx

(29)

e translational control loop of the yaw gimbal will bestable if the control model satisfies the following conditions

kDx ge 0

kixkakskPx minus kdx ge 01113896 (30)

erefore for the single closed loop of the AMB systemthe proportional coefficient could determine the stability ofthe yaw gimbal with an AMB system and the derivativecoefficient also affect the stability of the yaw gimbal with anAMB system

3 Experiment Verification

31 Control Diagram of+ree-Axis ISP with the AMB SystemAs illustrated in Figure 10 the whole control diagram ofthree-axis ISP consists of four parts including the controlmodule of torque motors the compensation module of basecoupling the compensation module of gimbal coupling andthe cross-feedbackmodule For the control module of torquemotors the control voltage of torque motor is the controlinput to drive three gimbals and the attitude control of thethree-axis ISP could be realizede angular displacement ofeach gimbal is the feedback signal to realize closed-loopcontrol about attitude of three-axis ISP and then the attitudestabilization precision of three gimbals would be kept atbalanced state Moreover the function of compensationmodule of base coupling in the red dotted-line diagram ofFigure 10 is to compensate the base coupling torques Tb

p in(15) Tb

r in (17) and Tby in (19) e angular rates of three

gimbals and the base plate are used as the feedback signal tocompute the compensation terms for minimizing the basecoupling torques In addition according to the compensa-tion module of gimbal coupling in the blue dotted-linediagram of Figure 10 the angular rates and angular dis-placements of three gimbals are used as feedback signals tocalculate the compensation amounts for the coupling tor-ques (T(ry)

p in (14) T(pr)y in (16) and T

(py)r in (18)) among

three gimbals So the coupling torques among three gimbalscould be mitigated Finally based on those calculatedcompensation terms for the base coupling torques and the

gimbal coupling torques the cross-feedback module willdistribute the compensation terms into different controlchannels of three-axis ISP erefore the attitude stabili-zation precisions of three gimbals will be improved by usingthe cross-feedback module for compensating base couplingtorques and gimbal coupling torques

32 Experimental Setup e whole experimental setup isshown in Figure 11 e self-designed main control unit(MCU) board of three gimbal motors and the AMB system isembedded in three-axis ISP It consists of a digital signalprocessor (DSP) TMS320F28335 with 12-bit AD converterand a digital PWM amplifier at 20 kHz and the samplingfrequency is chosen as 20 kHz e control frequency ofwhole system is 2 kHz and data output frequency sets at100Hz e eddy current displacement sensors are used tomeasure the dynamic displacements of yaw gimbal in axialand radial directions Moreover the position and orienta-tion system (POS) on the payload measures the attitudeinformation of three gimbals and the payload is mounted onthe yaw gimbal In addition the three-axis ISP is fixed on aswaying platform which can output relative disturbances onthe base plate Other parameters of three-axis ISP with anAMB system are listed in Table 1

As illustrated in Figure 12 to experimentally analyze thegimbal coupling and the base coupling the whole experi-ment includes three steps as follows

In the first step the suspension experiment is conductedto test the suspension stabilization and the active control-lability of yaw gimbal with an AMB system With torquemotors of three gimbals switched off and the AMB systemswitched on the suspension forces of the AMB system arechecked if it could make yaw gimbal stably suspend at theaxial and radial equilibrium positions e purpose of thisstep is to validate the active controllability of yaw gimbalwith the AMB system and to check if the yaw gimbalsuspended by the AMB system has better attitude stabili-zation precision than the pitch gimbal and the roll gimbalwhich are supported by mechanical bearings

Furthermore the base coupling effect is assessed withdifferent working statuses of three gimbals In the first casepitch gimbal and yaw gimbal are locked and roll gimbal isunlocked e torque motor of roll gimbal generates controltorque to keep roll gimbal stable e angular displacementsof unlocked roll gimbal and locked pitch gimbal are

Cont

rolle

r of

radi

al A

MB

dx0

ki

kskdx

1ms 1s

1ms 1siy

ix fx dx

dy

dx

dy

ndashndash

ndashndash

ndash

ndash

fy

ki

ks

kdy

ka kix

kiykady0

Figure 9 Control diagram of the magnetic suspension system in radial directions


measured and compared In the second case only pitchgimbal is locked and unlocked roll gimbal and yaw gimbalare controlled by torque motors e angular displacementsof three gimbals are measured and compared In the thirdcase three gimbals are unlocked and driven by torquemotors and then the angular displacements of three gimbals

are measured and compared Based on the comparison ofangular displacements of three gimbals at different operatingstatuses it is found that the gimbal coupling effect increaseswith the number of unlocked gimbals

Finally the cross-feedback control is used to compensatethe base coupling when the base plate works at static mode

Electromechanical model

Coupling model among three gimbals

Coupling model between gimbal and base plate

Inpu

t ang

le o

f tor

que m

otor

Ang

ular

acce

lera

tion

of b

ase p

late

Cros

s-fe

edba

ck co

ntro

l

Ang

ular

disp

lace

men

ts of

thre

e gim

bals

PID controller

PID controller

PID controller

ωbx

1sωby

ωbz

up

ur

θp

θr

θy

kp

kr

ky

uy

ktr

ker

kty

key

kep

ktp

ωbx

ωbyTb

p Tbr Tb

y

ωbz

1Ls + R

1Ls + R

1Ls + R

1 (Jyx + Jpx)

1(Jyx + Jpx + Jry)

1Jyz 1s 1s

1s1s

1s1s

θ yz

θ ry

θ px

θ ry θry

θyz

θ px θpx

θ yz

Tpry Tr

py Typr

ndash

ndash

ndash

ndash

ndash

ndash

+

+

+

+

+

+

Figure 10 Control diagram of three-axis ISP

POS MSISP

Swaying platform Power supply for MSISP

(a)

Power supply Control system Driving system Gimbal motor

Accelerometer

Rate Gyro

Data acquisition

(b)

Figure 11 Experimental setup (a) Experimental system (b) Control scheme of three-axis ISP


and dynamic mode respectively In the first case the baseplate mounted on the swaying platform is kept static reegimbals are driven by torque motors to keep them at steadystate and then the angular displacements of three gimbalsare measured In the second case the swaying platformworks at dynamic status ree gimbals are driven by torquemotors to stabilize three gimbals and the angular dis-placements of three gimbals are measured By comparing theangular displacements of three gimbals in the two cases theexperimental test proves the high effectiveness of cross-feedback control for the base coupling effect

33 Suspension Performance of Yaw Gimbal with an AMBSystem is suspension experiment is conducted to verify

the active controllability of yaw gimbal with an AMB systeme radial displacements of yaw gimbal are plotted inFigure 13(a) when the AMB system is switched on and thenthe radial displacements of yaw gimbal equal zero econtrol currents of the radial AMB are plotted inFigure 13(d) and they reach a stable amplitude 002A whenthe yaw gimbal is stably suspended at the radial equilibriumpoint ey show that the radial AMB system can stablysuspend the yaw gimbal at the radial equilibrium pointFigure 13(b) shows the axial displacements measured by theaxial displacement sensors (sensor Z1Z2 Z3 and Z4) and thecontrol currents of axial AMBs are shown in Figure 13(e)when the yaw gimbal is stably suspended at the axialequilibrium point It indicates that the axial AMB system cankeep the yaw gimbal stably suspend at the axial equilibrium

Table 1 Parameters of three-axis ISP with the AMB system

Symbol Quantify Valuekiz Axial current stiffness 2 middot 103 NAkdz Axial displacement stiffness minus3 middot 106 Nmminus

kdz Radial current stiffness 03 middot 103 NAkdx Radial displacement stiffness minus48 middot 105 NmJm Moment of inertia 3 middot 10minus 4 kgm2

ka Amplification coefficient 02AVkPx Proportion coefficient of radial AMB 28kPz Proportion coefficient of axial AMB 267ke Back-EMF coefficient 04V(rads)kt Torque constant 042 NmAm Mass of yaw gimbal 23 kgR Armature resistance 26ΩN Gear ratio 67ks Displacement sensitivity 03VmmkDx Derivative coefficient of radial AMB 07kDz Derivative coefficient of axial AMB 06

Suspension experiment of yaw gimbal with an AMB system

Axial suspension of yaw gimbal with an AMB system

Radial suspension of yaw gimbal with an AMB system

Rotational suspension of yaw gimbal with an AMB system

Is the yaw gimbal suspended stably

Base plate outputs disturbance

Yes

No

Tuning parameters

First stepsuspension performance of yaw gimbal with

an AMB system

Unlocked roll gimbal locked pitch gimbal locked yaw gimbal

Unlocked roll gimbal locked pitch gimbal unlocked yaw gimbal

Unlocked roll gimbal unlocked pitch gimbal unlocked yaw gimbal

Angle measurement

Base plate is static

Unlocked roll gimbalunlocked yaw gimbal

Coupling analysis

Using cross-feedback

Angle measurement

Second stepcoupling analysis of

yaw gimbal

ird stepcross-feedback control of

yaw gimbal

Base plate is dynamic

Angle comparison

Figure 12 Experimental steps


point as well Moreover the angular displacements of yawgimbal around Xy and Yy axes are depicted in Figure 13(c)the angular displacements are kept at the initial balancedstatus when the AMB system is turned off and the angulardisplacements will be forced back to final balanced status bythe suspension forces of AMB systeme root mean square(RMS) of angular displacement around Xy axis is 00192degand that aroundYy axis is 00198degerefore this suspensionexperiment proves that the suspension forces of AMBsystem can keep the yaw gimbal stably suspend at theequilibrium positions and it has the potential to isolate thevibration from other connected gimbals

34 Gimbal Coupling among+reeGimbals in+ree-Axis ISPis experimental test is conducted to analyze the gimbalcoupling among three gimbals e three-axis ISP ismounted on a swaying platformwhich outputs a disturbanceexpressed as r sin(04πt) e first test is to gauge the

gimbal coupling of one unlocked gimbal and then thegimbal coupling of two unlocked gimbals is measured In thesecond test the gimbal coupling of three unlocked gimbals isevaluated For the locked gimbal the angular displacement isnot affected by the dynamic base plate For unlocked gimbalit is driven by torque motor to keep it at the balanced statuse RMS of angular displacement represents the attitudestabilization precision A large RMS value refers to a lowattitude stabilization precision and vice versa

341 Gimbal Coupling of Single Unlocked Gimbal e rollgimbal is driven by the torque motor to keep it at balancedstatus e pitch gimbal and the yaw gimbal are locked withthe AMB system switched off When the harmonic vibrationis imposed on the base plate the angular displacements ofthree gimbals are measured and plotted in Figure 14 eangular displacement of unlocked roll gimbal is shown bythe red line in Figure 14(b) and the RMS is found to be

Disp

lace

men

t (m

m)

005

0

ndash0050 02 04 06

Time (s)

Radial sensor in Xy axisRadial sensor in Yy axis

08

(a)

Disp

lace

men

t (m

m)

02

01

0

ndash02

ndash01

Axial sensor Z1Axial sensor Z2Axial sensor Z3Axial sensor Z4

0 02 04Time (s)

06 08

(b)

Ang

ular

disp

lace

men

t (deg)

05

0

ndash05

Angle around Xy axisAngle around Yy axis

0 02 04Time (s)

06 08

(c)

Curr

ent (

A)

004

003

002

001

00 02 04 06

Time (s)

Radial AMB in Xy axisRadial AMB in Yy axis

08

(d)

Axial AMB A1Axial AMB A2Axial AMB A3Axial AMB A4

0 02 04 06Time (s)

08

Curr

ent (

mm

)

015

01

005

0

ndash005

(e)

Figure 13 Suspension traces and control currents of yaw gimbal with AMB system (a) Radial suspension traces (b) Axial suspensiontraces (c) Angular displacement around radial axes (d) Control currents of radial AMB (e) Control currents of axial AMB


00693deg because of the disturbance generated by the dynamicbase plate As shown in Figures 14(c) and 14(d) the pitchgimbal and yaw gimbal are kept at the balanced status whenthey are locked so the angular displacements of the pitchgimbal and the yaw gimbal are zeros

342 Gimbal Coupling of Two Unlocked Gimbals Whenonly the pitch gimbal is locked the roll gimbal and the yawgimbal work at unlocked mode and they are driven bygimbal motors With the AMB system of yaw gimbalswitched on the angular displacements of three gimbals aremeasured and plotted in Figure 15 e angular displace-ment of roll gimbal is plotted in Figure 15(b) and its RMS is00739deg e blue line in Figure 15(d) is the angular dis-placement of yaw gimbal and its RMS is found to be 00694dege RMS of pitch gimbalrsquos angular displacement is greaterthan that of yaw gimbal erefore this experimental resultvalidates the prediction of better performance on distur-bance isolation by using the yaw gimbal suspended by theAMB system than the roll gimbal supported by mechanicalbearings

343 Gimbal Coupling of +ree Unlocked Gimbals In thisexperiment all three gimbals are unlocked and driven bygimbal motors and the AMB system of yaw gimbal isswitched on e angular displacements of three gimbals areplotted in Figure 16 e RMS of roll gimbalrsquos angulardisplacement is 01349deg in Figure 16(b) e RMS of pitchgimbalrsquos angular displacement is found to be 00861deg inFigure 16(c) and the RMS of yaw gimbalrsquos angular dis-placement is 00725deg in Figure 16(d) Compared to theangular displacement of roll gimbal in Figure 15(b) the RMSof roll gimbalrsquos angular displacement increases from 00739degto 01349deg in this case It indicates that the attitude

stabilization precision of roll gimbal decreases with thenumber of unlocked gimbals so the gimbal coupling effectamong three gimbals increases with the number of unlockedgimbals

344 Comparison of Gimbal Coupling e comparison inTable 2 shows that the yaw gimbal suspended by an AMBsystem has better attitude stabilization precision than the rollgimbal and the pitch gimbal supported by mechanicalbearings so the yaw gimbal with an AMB system has betterperformance on isolating vibration than the pitch gimbaland the roll gimbal Moreover for the roll gimbal the RMSof angular displacement increases with the number ofunlocked gimbals so the gimbal coupling effect among threegimbals becomes more serious with increasing number ofunlocked gimbals is experimental result matches withanalysis results from (13) and (20)

35 Cross-Feedback Control for Gimbal Coupling and BaseCoupling is experiment is to validate that the cross-feedback control could effectively suppress the gimbalcoupling and the base coupling In the first test there is onlythe gimbal coupling when the base plate of three-axis ISPworks at static status three unlocked gimbals are driven bytorque motors with and without the cross-feedback controlappliede angular displacements of the roll gimbal and theyaw gimbal in the two cases are measured In the second testthe base plate of three-axis ISP works at dynamic status sothe gimbal coupling and the base coupling both exist reeunlocked gimbals are driven by torque motors to keep themat balanced status with and without cross-feedback controlappliede angular displacements of the roll gimbal and theyaw gimbal in the two cases are measured and compared

Ang

ular

disp

lace

men

t (deg)

2

0

0 10 20Time (s)

30

Base plate

40

ndash2

(a)

θ ry (deg

)

0 10 20Time (s)

30

Unlocked roll gimbal

40

05

0

ndash05

(b)

θ px (deg

)

0 10 20Time (s)

30

Locked pitch gimbal

40

1

0

ndash1

(c)

0 10 20Time (s)

30

Locked yaw gimbal

40

θ yz (deg

)

1

0

ndash1

(d)

Figure 14 Angular displacements of (a) base plate (b) roll gimbal (c) pitch gimba and (d) yaw gimbal with single unlocked gimbal


351 Cross-Feedback Control of Gimbals with Static BasePlate e respective angular displacements of the yawgimbal and the roll gimbal with and without the cross-feedback control are measured when the base plate of three-axis ISP is kept static e red line in Figure 17 is the angulardisplacement of the roll gimbal and the blue line is theangular displacement of the yaw gimbal When the cross-feedback control is used the RMS of roll gimbalrsquos angular

Ang

ular

disp

lace

men

t (deg)

2

0 10Time (s)

Base plate

20 30 40

0

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Locked pitch gimbal

20 30 40

θ px

(deg)

1

0

ndash1

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)

Figure 15 Angular displacements of (a) base plate (b) roll gimbal (c) pitch gimbal and (d) yaw gimbal with two unlocked gimbals

Ang

ular

disp

lace

men

t (deg)

2

0

0 10Time (s)

Base plate

20 30 40

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Unlocked pitch gimbal

20 30 40

θ px

(deg)

05

0

ndash05

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)

Figure 16 Angular displacements of (a) base plate (b) roll gimbal (c) pitch gimbal and (d) yaw gimbal with three unlocked gimbals

Table 2 Attitude stabilization precision with different numbers ofunlocked gimbals

Number of unlockedgimbals

RMS of angular displacementRoll gimbal Pitch gimbal Yaw gimbal

Single gimbal 00603deg 0 0Double gimbals 00739deg 0 00694degTriple gimbals 01349deg 00861deg 00725deg


displacements decreases from 00229deg to 00105deg in Table 3and the RMS of yaw gimbalrsquos angular displacements de-creases from 00129deg to 00049deg erefore the cross-feed-back control is effective in mitigating the gimbal couplingwhen the base plate works at static status

352 Cross-Feedback Control of Gimbals with Dynamic BasePlate e angular displacements of the roll gimbal and theyaw gimbal with the dynamic base plate are plotted inFigure 18 when the cross-feedback control is applied tosuppress the coupling effect e RMS of roll gimbalrsquos an-gular displacements decreases from 02134deg to 00526deg and

the RMS of yaw gimbalrsquos angular displacements decreasesfrom 00828deg to 00459deg in Table 4 erefore the attitudestabilization precisions of the roll gimbal and the yaw gimbal

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)

Unlocked roll gimbalUnlocked yaw gimbal

15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)

Figure 17 Angular displacements of roll and yaw gimbals with static base plate (a) without cross-feedback and (b) with cross-feedback

Table 3 Attitude stabilization precision of roll and yaw gimbal with static base plate

Without cross-feedback With cross-feedback ReductionRMS of roll gimbal 00229deg 00105deg 54RMS of yaw gimbal 00129deg 00049deg 62

Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)

Figure 18 Angular displacements of roll and yaw gimbals with dynamic base plate (a) without cross-feedback and (b) with cross-feedback

Table 4 Attitude stabilization precision of roll and yaw gimbalwith dynamic base plate

Without cross-feedback

With cross-feedback Reduction ()

RMS of rollgimbal 02134deg 00526deg 75

RMS of yawgimbal 00828deg 00459deg 44


are improved after using the cross-feedback so it proves thatthe cross-feedback control can suppress the base couplingeffect and the gimbal coupling effect when the base plateworks at dynamic status

4 Conclusion

In this article the complex dynamics of a three-axis ISP withan AMB system are analyzed By comparing the angulardisplacements of the roll gimbal with those of the yawgimbal the yaw gimbal suspended by an AMB system hasbetter performance on vibration isolation than the rollgimbal supported by mechanical bearing erefore theAMB system could be used in three-axis ISP to providebetter attitude stabilization precision to the suspendedgimbal so this structure expands the application range ofAMB system

In addition the gimbal coupling is firstly proven bycomparing angular displacements of different unlockedgimbals in the experiment e gimbal coupling is inten-sified with increasing number of unlocked gimbals when theRMS of the roll gimbalrsquos angular displacement increasesfrom 00603deg to 01349deg e base coupling effect with dy-namic base plate is more serious than that with static baseplate

Finally a cross-feedback control scheme is designed tomitigate the gimbal coupling and the base coupling Ex-perimental results demonstrate that the cross-feedbackcontrol scheme is useful to mitigate both the gimbal cou-pling and the base coupling For the static base plate theRMS of yaw gimbalrsquos angular displacements decreases from00129deg to 00049deg with applying cross-feedback control Forthe dynamic base plate the RMS of yaw gimbalrsquos angulardisplacements decreases from 00828deg to 00459deg

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] Z Hurak and M Rezac ldquoImage-based pointing and trackingfor inertially stabilized airborne camera platformrdquo IEEETransactions on Control Systems Technology vol 20 no 5pp 1146ndash1159 2012

[2] J-H Moon and S Y Jung ldquoImplementation of an imagestabilization system for a small digital camerardquo IEEETransactions on Consumer Electronics vol 54 no 2 2008

[3] J Hilkert ldquoInertially stabilized platform technology conceptsand principlesrdquo IEEE Control Systems vol 28 no 1pp 26ndash46 2008

[4] M F Reis G P Carvalho A F Neves and A J PeixotoldquoDynamic model and line of sight control of a 3-DOF inertialstabilization platform via feedback linearizationrdquo in Pro-ceedings of the 2018 Annual American Control Conference

(ACC) pp 1313ndash1318 IEEE Milwaukee WI USA August2018

[5] H F Mokbel L Q Ying A A Roshdy and C G HualdquoModeling and optimization of electro-optical dual axis in-ertially stabilized platformrdquo in Proceedings of the Interna-tional Conference on 2012 Optoelectronics andMicroelectronics (ICOM) pp 372ndash377 IEEE ChangchunChina August 2012

[6] P J Kennedy and R L Kennedy ldquoDirect versus indirect lineof sight (LOS) stabilizationrdquo IEEE Transactions on ControlSystems Technology vol 11 no 1 pp 3ndash15 2003

[7] J Hilkert and B Pautler ldquoA reduced-order disturbance ob-server applied to inertially stabilized line-of-sight controlrdquo inAcquisition Tracking Pointing and Laser Systems Technolo-gies XXV vol 8052 p 80520H International Society forOptics and Photonics Bellingham WA USA 2011

[8] X Zhou H Zhang and R Yu ldquoDecoupling control for two-axis inertially stabilized platform based on an inverse systemand internal model controlrdquo Mechatronics vol 24 no 8pp 1203ndash1213 2014

[9] A Safa and R Y Abdolmalaki ldquoRobust output feedbacktracking control for inertially stabilized platforms withmatched and unmatched uncertaintiesrdquo IEEE Transactions onControl Systems Technology vol 27 no 1 pp 118ndash131 2017

[10] Z Hurak and M Rezac ldquoControl design for image trackingwith an inertially stabilized airborne camera platformrdquo inProceedings of the Automatic Target Recognition XX Acqui-sition Tracking Pointing and Laser Systems TechnologiesXXIV and Optical Pattern Recognition XXI vol vol 7696p 76961H April 2010

[11] D Rajesh A Praveen and M R Pasumarthi ldquoDesign andanalysis of two-Axis seeker stabilization systemrdquo in Pro-ceedings of the International Conference on Modern Researchin Aerospace Engineering Springer Singapore Singaporepp 97ndash107 February 2018

[12] Y Zhang S Liu Y Xiong C Li andM Li ldquoStabilization loopmodeling of three-axis inertial stabilized platform with a newgyro installing moderdquo in Proceedings of the IEEE ChineseGuidance Navigation and Control Conference (CGNCC)pp 240ndash245 IEEE Yantai China August 2014

[13] S Li M Zhong and Y Zhao ldquoAccelerometer error estimationand compensation for three-axis gyro-stabilized cameramount based on proportional multiple-integral observerrdquoScience China Technological Sciences vol 57 no 12pp 2387ndash2395 2014

[14] F Konigseder W Kemmetmuller and A Kugi ldquoAttitudecontrol strategy for a camera stabilization platformrdquoMechatronics vol 46 pp 60ndash69 2017

[15] Z Lin and K Liu ldquoInertially stabilized line-of-sight controlsystem using a magnetic bearing with vernier gimbaling ca-pacityrdquo in Optical Design and Testing VI vol vol 9272p 92720Q International Society for Optics and PhotonicsBellingham WA USA 2014

[16] J Fang C Wang and T Wen ldquoDesign and optimization of aradial hybrid magnetic bearing with separate poles formagnetically suspended inertially stabilized platformrdquo IEEETransactions on Magnetics vol 50 no 5 pp 1ndash11 2014

[17] Q Mu G Liu and X Lei ldquoA RBFNN-based adaptive dis-turbance compensation approach applied to magnetic sus-pension inertially stabilized platformrdquo MathematicalProblems in Engineering vol 2014 pp 1ndash9 2014

[18] W Tong B Xiang and W Wong ldquoGimbal torque andcoupling torque of six degrees of freedom magnetically


suspended yaw gimbalrdquo International Journal of MechanicalSciences vol 168 2020

[19] J Tang B Xiang and Y Zhang ldquoDynamic characteristics ofthe rotor in a magnetically suspended control moment gy-roscope with active magnetic bearing and passive magneticbearingrdquo ISA Transactions vol 53 no 4 pp 1357ndash1365 2014

[20] B Xiang and J Tang ldquoSuspension and titling of vernier-gimballing magnetically suspended flywheel with conicalmagnetic bearing and Lorentz magnetic bearingrdquo Mecha-tronics vol 28 pp 46ndash54 2015

[21] L Zhuchong L Kun and Z Wei ldquoInertially stabilizedplatform for airborne remote sensing using magnetic bear-ingsrdquo IEEEASME Transactions on Mechatronics vol 21no 1 pp 288ndash301 2016

[22] M Ahrens L Kucera and R Larsonneur ldquoPerformance of amagnetically suspended flywheel energy storage devicerdquo IEEETransactions on Control Systems Technology vol 4 no 5pp 494ndash502 1996

[23] X Yao and Z Chen ldquoSliding mode control with deep learningmethod for rotor trajectory control of active magnetic bearingsystemrdquo Transactions of the Institute of Measurement andControl vol 41 no 5 pp 1383ndash1394 2019

[24] B Xiang and W Wong ldquoElectromagnetic vibration absorberfor torsional vibration in high speed rotational machinerdquoMechanical Systems and Signal Processing vol 140 2020

[25] Y-S Zhang R-Y Zhu and J-C Fang ldquoAnalysis on dynamicscoupling of inertial stabilized platform for aerial remotesensingrdquo Journal of Chinese Inertial Technology vol 5 2011

[26] H Liao Y Lu S Fan and D Fan ldquoDynamics analysis anddecoupling control for a differential cable drive inertiallystabilized platformrdquo Proceedings of the Institution of Me-chanical Engineers Part I Journal of Systems and ControlEngineering vol 229 no 7 pp 652ndash661 2015

[27] X Zhou G Gong J Li H Zhang and R Yu ldquoDecouplingcontrol for a three-axis inertially stabilized platform used foraerial remote sensingrdquo Transactions of the Institute of Mea-surement and Control vol 37 no 9 pp 1135ndash1145 2015

[28] J Fang R Yin and X Lei ldquoAn adaptive decoupling controlfor three-axis gyro stabilized platform based on neural net-worksrdquo Mechatronics vol 27 pp 38ndash46 2015

[29] J Mao J Yang Q Li and S Li ldquoExtended-state-observer-based output feedback sliding mode control of inertial sta-bilized platformrdquo in Proceedings of the IEEE ConferenceonIndustrial Electronics and Applications (ICIEA) pp 1528ndash1533 IEEE Siem Reap Cambodia February 2017

[30] F Dong X Lei andW Chou ldquoA dynamic model and controlmethod for a two-axis inertially stabilized platformrdquo IEEETransactions on Industrial Electronics vol 64 no 1pp 432ndash439 2016

[31] X Zhou C Yang and T Cai ldquoA model reference adaptivecontrolPID compound scheme on disturbance rejection foran aerial inertially stabilized platformrdquo Journal of Sensorsvol 2016 pp 1ndash11 2016

[32] F Wang R Wang E Liu and W Zhang ldquoStabilizationcontrol mothed for two-Axis inertially stabilized platformbased on active disturbance rejection control with noise re-duction disturbance observerrdquo IEEE Access vol 7pp 99521ndash99529 2019

[33] J Mao J Yang X Liu S Li and Q Li ldquoModeling and robustcontinuous TSM control for an inertially stabilized platformwith couplingsrdquo IEEE Transactions on Control SystemsTechnology pp 1ndash8 2019

[34] J Mao S Li Q Li and J Yang ldquoDesign and implementationof continuous finite-time sliding mode control for 2-DOF

inertially stabilized platform subject to multiple distur-bancesrdquo ISA Transactions vol 84 pp 214ndash224 2019

[35] X Zhou Y Shi L Li R Yu and L Zhao ldquoA high precisioncompound control scheme based on non-singular terminalsliding mode and extended state observer for an aerial in-ertially stabilized platformrdquo International Journal of ControlAutomation and Systems vol 18 no 6 pp 1498ndash1509 2020

[36] B Xiang and W O Wong ldquoVibration characteristics analysisof magnetically suspended rotor in flywheel energy storagesystemrdquo Journal of Sound and Vibration vol 444 pp 235ndash247 2019


simulation results showed that attitude stabilization pre-cisions of three gimbals were affected by the coupling effectamong three gimbals and the base plate when the error ofaxial angular velocity increased from 005 rads to 03 radsA decoupling control method [26] was used in a differentialcable driving ISP to minimize the coupling effect amongthree gimbals and the base plate and the control precisionwas improved by suppressing the coupling terms Anotherdecoupling control method based on the feed-forwardcompensation was used to control the ISP [27] the peakvalue of pitch gimbalrsquos rotational angle was reduced from06deg to 02deg In addition the adaptive decoupling controlbased on neural network was used to improve the controlperformances of three gimbals in a three-axis ISP [28] thetrack error of azimuth gimbal was reduced by 80 esliding mode control based on the extended state observerin [29 30] was used to suppress various types of distur-bances in the ISP Simulation result showed that the un-known state and the lumped uncertainty of gimbal could beaccurately estimated and the maximum deflection of yawgimbal was declined from 05332deg to 00811deg In literature[31] a compound control method based on model refer-ence adaptive control and proportional-integral-derivative(PID) control was proposed to improve the tracking pre-cision and the robustness of ISP system It could suppressthe nonlinear unbalance torque caused by the time-varyingmass and the position precision of yaw gimbal was im-proved by 50

Even though the abovementioned control methods suchas the disturbance observer [32] robust control [33] andsliding mode control [34 35] were useful to improve thecontrol precisions of ISP gimbals the equations of motion ofthree gimbals were simplified as a decoupled system byneglecting coupling terms among three gimbals and the baseplate Moreover the coupling effect among three gimbalsand the base plate was not verified experimentally and itsinfluences on the attitude stabilization precision of eachgimbal were not tested More importantly the couplingeffect was not studied in a three-axis ISP with an AMBsystem integrating the control schemes of torque motors andAMB system

Above all the original works of this article are focused onthree parts to further improve the attitude stabilizationprecision of yaw gimbal and the novelties of this article arelisted as follows

(1) A novel structure of yaw gimbal with an AMB systemintegrating the axial and radial motion is firstlydesigned to minimize the friction and vibrationtransmission and dynamic functions of three gim-bals with coupling terms are developed

(2) e influences of coupling terms among threegimbals and the base plate are firstly quantified in theexperiment and the mathematic models of couplingterms are originally developed It proves that the basecoupling with dynamic base plate is greater than thatwith static base plate and the gimbal couplingamong three gimbals increases with the number ofunlocked gimbals

(3) e cross-feedback control scheme is proposed toimprove the attitude stabilization precision of yawgimbal with dynamic disturbances

Consequently the AMB system is used as a novel sus-pension system in three-axis ISP and this structure broadensthe application scope of AMB system and improves attitudestabilization precision of yaw gimbal

is article is organized as follows In Section 2 thestructure of three-axis ISP with an AMB system is intro-duced and equations of motion of three gimbals are de-veloped and then the coupling effects among three gimbalsand the base plate are analyzed In Section 3 the cross-feedback control scheme with angular displacements ofthree gimbals is used to suppress the coupling effect actingon the yaw gimbal and the experiments are conducted toverify the effectiveness of proposed cross-feedback controlscheme

2 Three-Axis ISP with an AMB System

21 Structure of +ree-Axis ISP with an AMB System emechanical structure of three-axis ISP with an AMB systemis illustrated in Figure 1 e three-axis ISP consists of threerotational gimbals (yaw pitch and roll gimbals) an AMBsystem and a base plate e yaw gimbal (the innermostgimbal) supports the sensing and imaging payloads and itrotates about zy axis as illustrated in Figure 2(c) As shownin Figure 1(b) the rotor part of AMB system is located on theyaw gimbal while the stator part is mounted on the pitchgimbal e AMB system generates magnetic forces tosuspend the yaw gimbal at the radial and axial equilibriumpositions and the position of yaw gimbal is controllable byregulating the winding current of AMB system based on thedisplacement feedbacke pitch gimbal connects to the rollgimbal by mechanical bearing and it locates outside of yawgimbal e torque motor of pitch gimbal controls its ro-tation about xp axis in Figure 2(b) As the external gimbalthe roll gimbal locates outside of pitch gimbal and thetorque motor on roll gimbal controls its rotation about yr

axis as shown in Figure 2(a) e bracket is used to connectthe roll gimbal with the base plate and the base plateconnects three-axis ISP with the swaying platform or flightcarriers

22Coordinate Systemof+ree-Axis ISPwithanAMBSysteme coordinate systems of three gimbals in the ISP are il-luminated in Figure 2 OXbYbZb is the coordinate system ofbase plate OXrYrZr is the coordinate system of roll gimbalOXpYpZp is the coordinate system of pitch gimbal andOXyyYyZy is the coordinate system of yaw gimbal θry θpxand θyz are rotational angles of three gimbals respectively_θry _θpx and _θyz are relative angular rates of three gimbalsrespectively ree torque motors on pitch roll and yawgimbals control the rotations of three-axis ISP about Xp Yrand Zy axes respectively erefore the rotations of three-axis ISP about three pivot axes could be expressed in terms ofangular rates of three gimbals



ωrx

ωry

ωrz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ Crb

ωbx

ωby

ωbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_θry

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


ωby + _θry


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)


Crb


0 1 0

sin θry 0 cos θry

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (2)


ωpx

ωpy

ωpz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ C

pr C

rb

ωbx

ωby

ωbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ + Cpr

0_θry

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

_θpx

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)


pr

1 0 0

0 cos θpx sin θpx


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (4)

Yaw gimbal

Pitch gimbal

Roll gimbal

Base

Bracket

AMB

(a)


Axialair gap


Axial AMBat low end

(b)


Xb

Yb

Xr

Yr

ZbZr

O

θry

θryθmiddotry

(a)

Xp

Yp

Xr

Yr

ZpZr

O

θpx

θpx

θmiddotpx

(b)

Yp

Xp

Yy

ZpZy

θyz

θyz

Xy

O

θmiddotyz

(c)




ωyx

ωyy

ωyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ C

ypC

pr C

rb

ωbx

ωby

ωbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ + CypC

pr

0_θry

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ + Cyp

_θpx

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+

0

0_θyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)


Cyp

cos θyz sin θyz 0


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (6)


ωpx

ωry

ωyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


ωby + _θry


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0 1 0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦middot

ωbx

ωby

ωbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

1 0 0

0 1 0

0 minussin θpx 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦middot

_θpx

_θry

_θyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)







⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(9)






⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)




Tyx Jyx



⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦


Tyy Jyy



⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)



Tx Tpx + Cyp1113872 1113873

minus 1Tyz

Ty Try + CypC

pr1113872 1113873

minus 1Tyz + C

pr1113872 1113873

minus 1Tpx

Tz Tyz

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(12)



yp)minus1Tyz is


ypC










_θyz







⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)



T(ry)p Jyz


2ryθpx (14)





(15)


T(py)r minusJyzθpx


_θyz


(16)



_θryωbx




(17)



T(pr)y minusJyzθpx


_θry (18)


Tby minusJyz




p


r

Tz Jyzeuroθyz + T

(pr)y + Tb

y

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(20)



ωbx ωby ωbz 0

Tbp Tb

r Tby 0

⎧⎨

⎩ (21)




Tz Jyzeuroθyz + T

(pr)y

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



Tm kt

Ls + Ru minus ke

_θ1113872 1113873 (23)


Tx ktp

Lps + Rp


(ry)p + T

bp

Ty ktr

Lrs + Rr


(py)r + T

br

Tz kty

Lys + Ry



by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)


ktp

Lps + Rp


ktr

Lrs + Rr


kty

Lys + Ry


euroθyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)












⎧⎨

⎩ (27)




u


ke

kt

TmN θω


(a)

x

y

(b)


Axi

al su

spen

sion

forc

e (N

)

300

200

100

0

ndash100

ndash200


0 005 01ndash300


(a)


02 025 03 035

Axi

al su

spen

sion

forc

e (N

)

800

700

600

500

400

300


(b)






meurodz fz

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(28)



Radi

al su

spen

sion

forc

e (N

)

60

40

20

0

ndash20

ndash40

ndash60




(a)

Radi

al su

spen

sion

forc

e (N

)

150

100

50

0

ndash50

ndash100




(b)




zy

o

xy

fxndash

fx+

fyndashfz

fy+yy






(29)


kDx ge 0





p in(15) Tb











Cont

rolle

r of

radi

al A

MB

dx0

ki

kskdx

1ms 1s

1ms 1siy

ix fx dx

dy

dx

dy

ndashndash

ndashndash

ndash

ndash

fy

ki

ks

kdy

ka kix

kiykady0









Inpu

t ang

le o

f tor

que m

otor

Ang

ular

acce

lera

tion

of b

ase p

late

Cros

s-fe

edba

ck co

ntro

l

Ang

ular

disp

lace

men

ts of

thre

e gim

bals

PID controller

PID controller

PID controller

ωbx

1sωby

ωbz

up

ur

θp

θr

θy

kp

kr

ky

uy

ktr

ker

kty

key

kep

ktp

ωbx

ωbyTb

p Tbr Tb

y

ωbz

1Ls + R

1Ls + R

1Ls + R

1 (Jyx + Jpx)

1(Jyx + Jpx + Jry)

1Jyz 1s 1s

1s1s

1s1s

θ yz

θ ry

θ px

θ ry θry

θyz

θ px θpx

θ yz

Tpry Tr

py Typr

ndash

ndash

ndash

ndash

ndash

ndash

+

+

+

+

+

+


POS MSISP


(a)


Accelerometer

Rate Gyro

Data acquisition

(b)
















Yes

No

Tuning parameters


an AMB system




Angle measurement



Coupling analysis


Angle measurement


yaw gimbal


yaw gimbal


Angle comparison







Disp

lace

men

t (m

m)

005

0

ndash0050 02 04 06

Time (s)


08

(a)

Disp

lace

men

t (m

m)

02

01

0

ndash02

ndash01


0 02 04Time (s)

06 08

(b)

Ang

ular

disp

lace

men

t (deg)

05

0

ndash05


0 02 04Time (s)

06 08

(c)

Curr

ent (

A)

004

003

002

001

00 02 04 06

Time (s)


08

(d)


0 02 04 06Time (s)

08

Curr

ent (

mm

)

015

01

005

0

ndash005

(e)









Ang

ular

disp

lace

men

t (deg)

2

0

0 10 20Time (s)

30

Base plate

40

ndash2

(a)

θ ry (deg

)

0 10 20Time (s)

30


40

05

0

ndash05

(b)

θ px (deg

)

0 10 20Time (s)

30

Locked pitch gimbal

40

1

0

ndash1

(c)

0 10 20Time (s)

30

Locked yaw gimbal

40

θ yz (deg

)

1

0

ndash1

(d)




Ang

ular

disp

lace

men

t (deg)

2

0 10Time (s)

Base plate

20 30 40

0

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Locked pitch gimbal

20 30 40

θ px

(deg)

1

0

ndash1

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)


Ang

ular

disp

lace

men

t (deg)

2

0

0 10Time (s)

Base plate

20 30 40

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)


20 30 40

θ px

(deg)

05

0

ndash05

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)










Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)




Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)









4 Conclusion




Data Availability




References











































ωrx

ωry

ωrz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ Crb

ωbx

ωby

ωbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_θry

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


ωby + _θry


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)


Crb


0 1 0

sin θry 0 cos θry

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (2)


ωpx

ωpy

ωpz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ C

pr C

rb

ωbx

ωby

ωbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ + Cpr

0_θry

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

_θpx

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)


pr

1 0 0

0 cos θpx sin θpx


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (4)

Yaw gimbal

Pitch gimbal

Roll gimbal

Base

Bracket

AMB

(a)


Axialair gap


Axial AMBat low end

(b)


Xb

Yb

Xr

Yr

ZbZr

O

θry

θryθmiddotry

(a)

Xp

Yp

Xr

Yr

ZpZr

O

θpx

θpx

θmiddotpx

(b)

Yp

Xp

Yy

ZpZy

θyz

θyz

Xy

O

θmiddotyz

(c)




ωyx

ωyy

ωyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ C

ypC

pr C

rb

ωbx

ωby

ωbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ + CypC

pr

0_θry

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ + Cyp

_θpx

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+

0

0_θyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)


Cyp

cos θyz sin θyz 0


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (6)


ωpx

ωry

ωyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


ωby + _θry


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0 1 0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦middot

ωbx

ωby

ωbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

1 0 0

0 1 0

0 minussin θpx 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦middot

_θpx

_θry

_θyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)







⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(9)






⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)




Tyx Jyx



⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦


Tyy Jyy



⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)



Tx Tpx + Cyp1113872 1113873

minus 1Tyz

Ty Try + CypC

pr1113872 1113873

minus 1Tyz + C

pr1113872 1113873

minus 1Tpx

Tz Tyz

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(12)



yp)minus1Tyz is


ypC










_θyz







⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)



T(ry)p Jyz


2ryθpx (14)





(15)


T(py)r minusJyzθpx


_θyz


(16)



_θryωbx




(17)



T(pr)y minusJyzθpx


_θry (18)


Tby minusJyz




p


r

Tz Jyzeuroθyz + T

(pr)y + Tb

y

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(20)



ωbx ωby ωbz 0

Tbp Tb

r Tby 0

⎧⎨

⎩ (21)




Tz Jyzeuroθyz + T

(pr)y

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



Tm kt

Ls + Ru minus ke

_θ1113872 1113873 (23)


Tx ktp

Lps + Rp


(ry)p + T

bp

Ty ktr

Lrs + Rr


(py)r + T

br

Tz kty

Lys + Ry



by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)


ktp

Lps + Rp


ktr

Lrs + Rr


kty

Lys + Ry


euroθyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)












⎧⎨

⎩ (27)




u


ke

kt

TmN θω


(a)

x

y

(b)


Axi

al su

spen

sion

forc

e (N

)

300

200

100

0

ndash100

ndash200


0 005 01ndash300


(a)


02 025 03 035

Axi

al su

spen

sion

forc

e (N

)

800

700

600

500

400

300


(b)






meurodz fz

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(28)



Radi

al su

spen

sion

forc

e (N

)

60

40

20

0

ndash20

ndash40

ndash60




(a)

Radi

al su

spen

sion

forc

e (N

)

150

100

50

0

ndash50

ndash100




(b)




zy

o

xy

fxndash

fx+

fyndashfz

fy+yy






(29)


kDx ge 0





p in(15) Tb











Cont

rolle

r of

radi

al A

MB

dx0

ki

kskdx

1ms 1s

1ms 1siy

ix fx dx

dy

dx

dy

ndashndash

ndashndash

ndash

ndash

fy

ki

ks

kdy

ka kix

kiykady0









Inpu

t ang

le o

f tor

que m

otor

Ang

ular

acce

lera

tion

of b

ase p

late

Cros

s-fe

edba

ck co

ntro

l

Ang

ular

disp

lace

men

ts of

thre

e gim

bals

PID controller

PID controller

PID controller

ωbx

1sωby

ωbz

up

ur

θp

θr

θy

kp

kr

ky

uy

ktr

ker

kty

key

kep

ktp

ωbx

ωbyTb

p Tbr Tb

y

ωbz

1Ls + R

1Ls + R

1Ls + R

1 (Jyx + Jpx)

1(Jyx + Jpx + Jry)

1Jyz 1s 1s

1s1s

1s1s

θ yz

θ ry

θ px

θ ry θry

θyz

θ px θpx

θ yz

Tpry Tr

py Typr

ndash

ndash

ndash

ndash

ndash

ndash

+

+

+

+

+

+


POS MSISP


(a)


Accelerometer

Rate Gyro

Data acquisition

(b)
















Yes

No

Tuning parameters


an AMB system




Angle measurement



Coupling analysis


Angle measurement


yaw gimbal


yaw gimbal


Angle comparison







Disp

lace

men

t (m

m)

005

0

ndash0050 02 04 06

Time (s)


08

(a)

Disp

lace

men

t (m

m)

02

01

0

ndash02

ndash01


0 02 04Time (s)

06 08

(b)

Ang

ular

disp

lace

men

t (deg)

05

0

ndash05


0 02 04Time (s)

06 08

(c)

Curr

ent (

A)

004

003

002

001

00 02 04 06

Time (s)


08

(d)


0 02 04 06Time (s)

08

Curr

ent (

mm

)

015

01

005

0

ndash005

(e)









Ang

ular

disp

lace

men

t (deg)

2

0

0 10 20Time (s)

30

Base plate

40

ndash2

(a)

θ ry (deg

)

0 10 20Time (s)

30


40

05

0

ndash05

(b)

θ px (deg

)

0 10 20Time (s)

30

Locked pitch gimbal

40

1

0

ndash1

(c)

0 10 20Time (s)

30

Locked yaw gimbal

40

θ yz (deg

)

1

0

ndash1

(d)




Ang

ular

disp

lace

men

t (deg)

2

0 10Time (s)

Base plate

20 30 40

0

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Locked pitch gimbal

20 30 40

θ px

(deg)

1

0

ndash1

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)


Ang

ular

disp

lace

men

t (deg)

2

0

0 10Time (s)

Base plate

20 30 40

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)


20 30 40

θ px

(deg)

05

0

ndash05

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)










Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)




Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)









4 Conclusion




Data Availability




References











































ωyx

ωyy

ωyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ C

ypC

pr C

rb

ωbx

ωby

ωbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ + CypC

pr

0_θry

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ + Cyp

_θpx

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+

0

0_θyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)


Cyp

cos θyz sin θyz 0


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (6)


ωpx

ωry

ωyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


ωby + _θry


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0 1 0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦middot

ωbx

ωby

ωbz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

1 0 0

0 1 0

0 minussin θpx 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦middot

_θpx

_θry

_θyz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)







⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(9)






⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)




Tyx Jyx



⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦


Tyy Jyy



⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)



Tx Tpx + Cyp1113872 1113873

minus 1Tyz

Ty Try + CypC

pr1113872 1113873

minus 1Tyz + C

pr1113872 1113873

minus 1Tpx

Tz Tyz

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(12)



yp)minus1Tyz is


ypC










_θyz







⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)



T(ry)p Jyz


2ryθpx (14)





(15)


T(py)r minusJyzθpx


_θyz


(16)



_θryωbx




(17)



T(pr)y minusJyzθpx


_θry (18)


Tby minusJyz




p


r

Tz Jyzeuroθyz + T

(pr)y + Tb

y

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(20)



ωbx ωby ωbz 0

Tbp Tb

r Tby 0

⎧⎨

⎩ (21)




Tz Jyzeuroθyz + T

(pr)y

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



Tm kt

Ls + Ru minus ke

_θ1113872 1113873 (23)


Tx ktp

Lps + Rp


(ry)p + T

bp

Ty ktr

Lrs + Rr


(py)r + T

br

Tz kty

Lys + Ry



by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)


ktp

Lps + Rp


ktr

Lrs + Rr


kty

Lys + Ry


euroθyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)












⎧⎨

⎩ (27)




u


ke

kt

TmN θω


(a)

x

y

(b)


Axi

al su

spen

sion

forc

e (N

)

300

200

100

0

ndash100

ndash200


0 005 01ndash300


(a)


02 025 03 035

Axi

al su

spen

sion

forc

e (N

)

800

700

600

500

400

300


(b)






meurodz fz

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(28)



Radi

al su

spen

sion

forc

e (N

)

60

40

20

0

ndash20

ndash40

ndash60




(a)

Radi

al su

spen

sion

forc

e (N

)

150

100

50

0

ndash50

ndash100




(b)




zy

o

xy

fxndash

fx+

fyndashfz

fy+yy






(29)


kDx ge 0





p in(15) Tb











Cont

rolle

r of

radi

al A

MB

dx0

ki

kskdx

1ms 1s

1ms 1siy

ix fx dx

dy

dx

dy

ndashndash

ndashndash

ndash

ndash

fy

ki

ks

kdy

ka kix

kiykady0









Inpu

t ang

le o

f tor

que m

otor

Ang

ular

acce

lera

tion

of b

ase p

late

Cros

s-fe

edba

ck co

ntro

l

Ang

ular

disp

lace

men

ts of

thre

e gim

bals

PID controller

PID controller

PID controller

ωbx

1sωby

ωbz

up

ur

θp

θr

θy

kp

kr

ky

uy

ktr

ker

kty

key

kep

ktp

ωbx

ωbyTb

p Tbr Tb

y

ωbz

1Ls + R

1Ls + R

1Ls + R

1 (Jyx + Jpx)

1(Jyx + Jpx + Jry)

1Jyz 1s 1s

1s1s

1s1s

θ yz

θ ry

θ px

θ ry θry

θyz

θ px θpx

θ yz

Tpry Tr

py Typr

ndash

ndash

ndash

ndash

ndash

ndash

+

+

+

+

+

+


POS MSISP


(a)


Accelerometer

Rate Gyro

Data acquisition

(b)
















Yes

No

Tuning parameters


an AMB system




Angle measurement



Coupling analysis


Angle measurement


yaw gimbal


yaw gimbal


Angle comparison







Disp

lace

men

t (m

m)

005

0

ndash0050 02 04 06

Time (s)


08

(a)

Disp

lace

men

t (m

m)

02

01

0

ndash02

ndash01


0 02 04Time (s)

06 08

(b)

Ang

ular

disp

lace

men

t (deg)

05

0

ndash05


0 02 04Time (s)

06 08

(c)

Curr

ent (

A)

004

003

002

001

00 02 04 06

Time (s)


08

(d)


0 02 04 06Time (s)

08

Curr

ent (

mm

)

015

01

005

0

ndash005

(e)









Ang

ular

disp

lace

men

t (deg)

2

0

0 10 20Time (s)

30

Base plate

40

ndash2

(a)

θ ry (deg

)

0 10 20Time (s)

30


40

05

0

ndash05

(b)

θ px (deg

)

0 10 20Time (s)

30

Locked pitch gimbal

40

1

0

ndash1

(c)

0 10 20Time (s)

30

Locked yaw gimbal

40

θ yz (deg

)

1

0

ndash1

(d)




Ang

ular

disp

lace

men

t (deg)

2

0 10Time (s)

Base plate

20 30 40

0

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Locked pitch gimbal

20 30 40

θ px

(deg)

1

0

ndash1

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)


Ang

ular

disp

lace

men

t (deg)

2

0

0 10Time (s)

Base plate

20 30 40

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)


20 30 40

θ px

(deg)

05

0

ndash05

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)










Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)




Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)









4 Conclusion




Data Availability




References










































Tyx Jyx



⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦


Tyy Jyy



⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)



Tx Tpx + Cyp1113872 1113873

minus 1Tyz

Ty Try + CypC

pr1113872 1113873

minus 1Tyz + C

pr1113872 1113873

minus 1Tpx

Tz Tyz

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(12)



yp)minus1Tyz is


ypC










_θyz







⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)



T(ry)p Jyz


2ryθpx (14)





(15)


T(py)r minusJyzθpx


_θyz


(16)



_θryωbx




(17)



T(pr)y minusJyzθpx


_θry (18)


Tby minusJyz




p


r

Tz Jyzeuroθyz + T

(pr)y + Tb

y

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(20)



ωbx ωby ωbz 0

Tbp Tb

r Tby 0

⎧⎨

⎩ (21)




Tz Jyzeuroθyz + T

(pr)y

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



Tm kt

Ls + Ru minus ke

_θ1113872 1113873 (23)


Tx ktp

Lps + Rp


(ry)p + T

bp

Ty ktr

Lrs + Rr


(py)r + T

br

Tz kty

Lys + Ry



by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)


ktp

Lps + Rp


ktr

Lrs + Rr


kty

Lys + Ry


euroθyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)












⎧⎨

⎩ (27)




u


ke

kt

TmN θω


(a)

x

y

(b)


Axi

al su

spen

sion

forc

e (N

)

300

200

100

0

ndash100

ndash200


0 005 01ndash300


(a)


02 025 03 035

Axi

al su

spen

sion

forc

e (N

)

800

700

600

500

400

300


(b)






meurodz fz

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(28)



Radi

al su

spen

sion

forc

e (N

)

60

40

20

0

ndash20

ndash40

ndash60




(a)

Radi

al su

spen

sion

forc

e (N

)

150

100

50

0

ndash50

ndash100




(b)




zy

o

xy

fxndash

fx+

fyndashfz

fy+yy






(29)


kDx ge 0





p in(15) Tb











Cont

rolle

r of

radi

al A

MB

dx0

ki

kskdx

1ms 1s

1ms 1siy

ix fx dx

dy

dx

dy

ndashndash

ndashndash

ndash

ndash

fy

ki

ks

kdy

ka kix

kiykady0









Inpu

t ang

le o

f tor

que m

otor

Ang

ular

acce

lera

tion

of b

ase p

late

Cros

s-fe

edba

ck co

ntro

l

Ang

ular

disp

lace

men

ts of

thre

e gim

bals

PID controller

PID controller

PID controller

ωbx

1sωby

ωbz

up

ur

θp

θr

θy

kp

kr

ky

uy

ktr

ker

kty

key

kep

ktp

ωbx

ωbyTb

p Tbr Tb

y

ωbz

1Ls + R

1Ls + R

1Ls + R

1 (Jyx + Jpx)

1(Jyx + Jpx + Jry)

1Jyz 1s 1s

1s1s

1s1s

θ yz

θ ry

θ px

θ ry θry

θyz

θ px θpx

θ yz

Tpry Tr

py Typr

ndash

ndash

ndash

ndash

ndash

ndash

+

+

+

+

+

+


POS MSISP


(a)


Accelerometer

Rate Gyro

Data acquisition

(b)
















Yes

No

Tuning parameters


an AMB system




Angle measurement



Coupling analysis


Angle measurement


yaw gimbal


yaw gimbal


Angle comparison







Disp

lace

men

t (m

m)

005

0

ndash0050 02 04 06

Time (s)


08

(a)

Disp

lace

men

t (m

m)

02

01

0

ndash02

ndash01


0 02 04Time (s)

06 08

(b)

Ang

ular

disp

lace

men

t (deg)

05

0

ndash05


0 02 04Time (s)

06 08

(c)

Curr

ent (

A)

004

003

002

001

00 02 04 06

Time (s)


08

(d)


0 02 04 06Time (s)

08

Curr

ent (

mm

)

015

01

005

0

ndash005

(e)









Ang

ular

disp

lace

men

t (deg)

2

0

0 10 20Time (s)

30

Base plate

40

ndash2

(a)

θ ry (deg

)

0 10 20Time (s)

30


40

05

0

ndash05

(b)

θ px (deg

)

0 10 20Time (s)

30

Locked pitch gimbal

40

1

0

ndash1

(c)

0 10 20Time (s)

30

Locked yaw gimbal

40

θ yz (deg

)

1

0

ndash1

(d)




Ang

ular

disp

lace

men

t (deg)

2

0 10Time (s)

Base plate

20 30 40

0

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Locked pitch gimbal

20 30 40

θ px

(deg)

1

0

ndash1

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)


Ang

ular

disp

lace

men

t (deg)

2

0

0 10Time (s)

Base plate

20 30 40

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)


20 30 40

θ px

(deg)

05

0

ndash05

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)










Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)




Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)









4 Conclusion




Data Availability




References










































T(pr)y minusJyzθpx


_θry (18)


Tby minusJyz




p


r

Tz Jyzeuroθyz + T

(pr)y + Tb

y

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(20)



ωbx ωby ωbz 0

Tbp Tb

r Tby 0

⎧⎨

⎩ (21)




Tz Jyzeuroθyz + T

(pr)y

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)



Tm kt

Ls + Ru minus ke

_θ1113872 1113873 (23)


Tx ktp

Lps + Rp


(ry)p + T

bp

Ty ktr

Lrs + Rr


(py)r + T

br

Tz kty

Lys + Ry



by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)


ktp

Lps + Rp


ktr

Lrs + Rr


kty

Lys + Ry


euroθyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)












⎧⎨

⎩ (27)




u


ke

kt

TmN θω


(a)

x

y

(b)


Axi

al su

spen

sion

forc

e (N

)

300

200

100

0

ndash100

ndash200


0 005 01ndash300


(a)


02 025 03 035

Axi

al su

spen

sion

forc

e (N

)

800

700

600

500

400

300


(b)






meurodz fz

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(28)



Radi

al su

spen

sion

forc

e (N

)

60

40

20

0

ndash20

ndash40

ndash60




(a)

Radi

al su

spen

sion

forc

e (N

)

150

100

50

0

ndash50

ndash100




(b)




zy

o

xy

fxndash

fx+

fyndashfz

fy+yy






(29)


kDx ge 0





p in(15) Tb











Cont

rolle

r of

radi

al A

MB

dx0

ki

kskdx

1ms 1s

1ms 1siy

ix fx dx

dy

dx

dy

ndashndash

ndashndash

ndash

ndash

fy

ki

ks

kdy

ka kix

kiykady0









Inpu

t ang

le o

f tor

que m

otor

Ang

ular

acce

lera

tion

of b

ase p

late

Cros

s-fe

edba

ck co

ntro

l

Ang

ular

disp

lace

men

ts of

thre

e gim

bals

PID controller

PID controller

PID controller

ωbx

1sωby

ωbz

up

ur

θp

θr

θy

kp

kr

ky

uy

ktr

ker

kty

key

kep

ktp

ωbx

ωbyTb

p Tbr Tb

y

ωbz

1Ls + R

1Ls + R

1Ls + R

1 (Jyx + Jpx)

1(Jyx + Jpx + Jry)

1Jyz 1s 1s

1s1s

1s1s

θ yz

θ ry

θ px

θ ry θry

θyz

θ px θpx

θ yz

Tpry Tr

py Typr

ndash

ndash

ndash

ndash

ndash

ndash

+

+

+

+

+

+


POS MSISP


(a)


Accelerometer

Rate Gyro

Data acquisition

(b)
















Yes

No

Tuning parameters


an AMB system




Angle measurement



Coupling analysis


Angle measurement


yaw gimbal


yaw gimbal


Angle comparison







Disp

lace

men

t (m

m)

005

0

ndash0050 02 04 06

Time (s)


08

(a)

Disp

lace

men

t (m

m)

02

01

0

ndash02

ndash01


0 02 04Time (s)

06 08

(b)

Ang

ular

disp

lace

men

t (deg)

05

0

ndash05


0 02 04Time (s)

06 08

(c)

Curr

ent (

A)

004

003

002

001

00 02 04 06

Time (s)


08

(d)


0 02 04 06Time (s)

08

Curr

ent (

mm

)

015

01

005

0

ndash005

(e)









Ang

ular

disp

lace

men

t (deg)

2

0

0 10 20Time (s)

30

Base plate

40

ndash2

(a)

θ ry (deg

)

0 10 20Time (s)

30


40

05

0

ndash05

(b)

θ px (deg

)

0 10 20Time (s)

30

Locked pitch gimbal

40

1

0

ndash1

(c)

0 10 20Time (s)

30

Locked yaw gimbal

40

θ yz (deg

)

1

0

ndash1

(d)




Ang

ular

disp

lace

men

t (deg)

2

0 10Time (s)

Base plate

20 30 40

0

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Locked pitch gimbal

20 30 40

θ px

(deg)

1

0

ndash1

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)


Ang

ular

disp

lace

men

t (deg)

2

0

0 10Time (s)

Base plate

20 30 40

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)


20 30 40

θ px

(deg)

05

0

ndash05

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)










Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)




Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)









4 Conclusion




Data Availability




References














































⎧⎨

⎩ (27)




u


ke

kt

TmN θω


(a)

x

y

(b)


Axi

al su

spen

sion

forc

e (N

)

300

200

100

0

ndash100

ndash200


0 005 01ndash300


(a)


02 025 03 035

Axi

al su

spen

sion

forc

e (N

)

800

700

600

500

400

300


(b)






meurodz fz

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(28)



Radi

al su

spen

sion

forc

e (N

)

60

40

20

0

ndash20

ndash40

ndash60




(a)

Radi

al su

spen

sion

forc

e (N

)

150

100

50

0

ndash50

ndash100




(b)




zy

o

xy

fxndash

fx+

fyndashfz

fy+yy






(29)


kDx ge 0





p in(15) Tb











Cont

rolle

r of

radi

al A

MB

dx0

ki

kskdx

1ms 1s

1ms 1siy

ix fx dx

dy

dx

dy

ndashndash

ndashndash

ndash

ndash

fy

ki

ks

kdy

ka kix

kiykady0









Inpu

t ang

le o

f tor

que m

otor

Ang

ular

acce

lera

tion

of b

ase p

late

Cros

s-fe

edba

ck co

ntro

l

Ang

ular

disp

lace

men

ts of

thre

e gim

bals

PID controller

PID controller

PID controller

ωbx

1sωby

ωbz

up

ur

θp

θr

θy

kp

kr

ky

uy

ktr

ker

kty

key

kep

ktp

ωbx

ωbyTb

p Tbr Tb

y

ωbz

1Ls + R

1Ls + R

1Ls + R

1 (Jyx + Jpx)

1(Jyx + Jpx + Jry)

1Jyz 1s 1s

1s1s

1s1s

θ yz

θ ry

θ px

θ ry θry

θyz

θ px θpx

θ yz

Tpry Tr

py Typr

ndash

ndash

ndash

ndash

ndash

ndash

+

+

+

+

+

+


POS MSISP


(a)


Accelerometer

Rate Gyro

Data acquisition

(b)
















Yes

No

Tuning parameters


an AMB system




Angle measurement



Coupling analysis


Angle measurement


yaw gimbal


yaw gimbal


Angle comparison







Disp

lace

men

t (m

m)

005

0

ndash0050 02 04 06

Time (s)


08

(a)

Disp

lace

men

t (m

m)

02

01

0

ndash02

ndash01


0 02 04Time (s)

06 08

(b)

Ang

ular

disp

lace

men

t (deg)

05

0

ndash05


0 02 04Time (s)

06 08

(c)

Curr

ent (

A)

004

003

002

001

00 02 04 06

Time (s)


08

(d)


0 02 04 06Time (s)

08

Curr

ent (

mm

)

015

01

005

0

ndash005

(e)









Ang

ular

disp

lace

men

t (deg)

2

0

0 10 20Time (s)

30

Base plate

40

ndash2

(a)

θ ry (deg

)

0 10 20Time (s)

30


40

05

0

ndash05

(b)

θ px (deg

)

0 10 20Time (s)

30

Locked pitch gimbal

40

1

0

ndash1

(c)

0 10 20Time (s)

30

Locked yaw gimbal

40

θ yz (deg

)

1

0

ndash1

(d)




Ang

ular

disp

lace

men

t (deg)

2

0 10Time (s)

Base plate

20 30 40

0

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Locked pitch gimbal

20 30 40

θ px

(deg)

1

0

ndash1

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)


Ang

ular

disp

lace

men

t (deg)

2

0

0 10Time (s)

Base plate

20 30 40

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)


20 30 40

θ px

(deg)

05

0

ndash05

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)










Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)




Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)









4 Conclusion




Data Availability




References













































meurodz fz

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(28)



Radi

al su

spen

sion

forc

e (N

)

60

40

20

0

ndash20

ndash40

ndash60




(a)

Radi

al su

spen

sion

forc

e (N

)

150

100

50

0

ndash50

ndash100




(b)




zy

o

xy

fxndash

fx+

fyndashfz

fy+yy






(29)


kDx ge 0





p in(15) Tb











Cont

rolle

r of

radi

al A

MB

dx0

ki

kskdx

1ms 1s

1ms 1siy

ix fx dx

dy

dx

dy

ndashndash

ndashndash

ndash

ndash

fy

ki

ks

kdy

ka kix

kiykady0









Inpu

t ang

le o

f tor

que m

otor

Ang

ular

acce

lera

tion

of b

ase p

late

Cros

s-fe

edba

ck co

ntro

l

Ang

ular

disp

lace

men

ts of

thre

e gim

bals

PID controller

PID controller

PID controller

ωbx

1sωby

ωbz

up

ur

θp

θr

θy

kp

kr

ky

uy

ktr

ker

kty

key

kep

ktp

ωbx

ωbyTb

p Tbr Tb

y

ωbz

1Ls + R

1Ls + R

1Ls + R

1 (Jyx + Jpx)

1(Jyx + Jpx + Jry)

1Jyz 1s 1s

1s1s

1s1s

θ yz

θ ry

θ px

θ ry θry

θyz

θ px θpx

θ yz

Tpry Tr

py Typr

ndash

ndash

ndash

ndash

ndash

ndash

+

+

+

+

+

+


POS MSISP


(a)


Accelerometer

Rate Gyro

Data acquisition

(b)
















Yes

No

Tuning parameters


an AMB system




Angle measurement



Coupling analysis


Angle measurement


yaw gimbal


yaw gimbal


Angle comparison







Disp

lace

men

t (m

m)

005

0

ndash0050 02 04 06

Time (s)


08

(a)

Disp

lace

men

t (m

m)

02

01

0

ndash02

ndash01


0 02 04Time (s)

06 08

(b)

Ang

ular

disp

lace

men

t (deg)

05

0

ndash05


0 02 04Time (s)

06 08

(c)

Curr

ent (

A)

004

003

002

001

00 02 04 06

Time (s)


08

(d)


0 02 04 06Time (s)

08

Curr

ent (

mm

)

015

01

005

0

ndash005

(e)









Ang

ular

disp

lace

men

t (deg)

2

0

0 10 20Time (s)

30

Base plate

40

ndash2

(a)

θ ry (deg

)

0 10 20Time (s)

30


40

05

0

ndash05

(b)

θ px (deg

)

0 10 20Time (s)

30

Locked pitch gimbal

40

1

0

ndash1

(c)

0 10 20Time (s)

30

Locked yaw gimbal

40

θ yz (deg

)

1

0

ndash1

(d)




Ang

ular

disp

lace

men

t (deg)

2

0 10Time (s)

Base plate

20 30 40

0

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Locked pitch gimbal

20 30 40

θ px

(deg)

1

0

ndash1

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)


Ang

ular

disp

lace

men

t (deg)

2

0

0 10Time (s)

Base plate

20 30 40

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)


20 30 40

θ px

(deg)

05

0

ndash05

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)










Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)




Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)









4 Conclusion




Data Availability




References













































(29)


kDx ge 0





p in(15) Tb











Cont

rolle

r of

radi

al A

MB

dx0

ki

kskdx

1ms 1s

1ms 1siy

ix fx dx

dy

dx

dy

ndashndash

ndashndash

ndash

ndash

fy

ki

ks

kdy

ka kix

kiykady0









Inpu

t ang

le o

f tor

que m

otor

Ang

ular

acce

lera

tion

of b

ase p

late

Cros

s-fe

edba

ck co

ntro

l

Ang

ular

disp

lace

men

ts of

thre

e gim

bals

PID controller

PID controller

PID controller

ωbx

1sωby

ωbz

up

ur

θp

θr

θy

kp

kr

ky

uy

ktr

ker

kty

key

kep

ktp

ωbx

ωbyTb

p Tbr Tb

y

ωbz

1Ls + R

1Ls + R

1Ls + R

1 (Jyx + Jpx)

1(Jyx + Jpx + Jry)

1Jyz 1s 1s

1s1s

1s1s

θ yz

θ ry

θ px

θ ry θry

θyz

θ px θpx

θ yz

Tpry Tr

py Typr

ndash

ndash

ndash

ndash

ndash

ndash

+

+

+

+

+

+


POS MSISP


(a)


Accelerometer

Rate Gyro

Data acquisition

(b)
















Yes

No

Tuning parameters


an AMB system




Angle measurement



Coupling analysis


Angle measurement


yaw gimbal


yaw gimbal


Angle comparison







Disp

lace

men

t (m

m)

005

0

ndash0050 02 04 06

Time (s)


08

(a)

Disp

lace

men

t (m

m)

02

01

0

ndash02

ndash01


0 02 04Time (s)

06 08

(b)

Ang

ular

disp

lace

men

t (deg)

05

0

ndash05


0 02 04Time (s)

06 08

(c)

Curr

ent (

A)

004

003

002

001

00 02 04 06

Time (s)


08

(d)


0 02 04 06Time (s)

08

Curr

ent (

mm

)

015

01

005

0

ndash005

(e)









Ang

ular

disp

lace

men

t (deg)

2

0

0 10 20Time (s)

30

Base plate

40

ndash2

(a)

θ ry (deg

)

0 10 20Time (s)

30


40

05

0

ndash05

(b)

θ px (deg

)

0 10 20Time (s)

30

Locked pitch gimbal

40

1

0

ndash1

(c)

0 10 20Time (s)

30

Locked yaw gimbal

40

θ yz (deg

)

1

0

ndash1

(d)




Ang

ular

disp

lace

men

t (deg)

2

0 10Time (s)

Base plate

20 30 40

0

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Locked pitch gimbal

20 30 40

θ px

(deg)

1

0

ndash1

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)


Ang

ular

disp

lace

men

t (deg)

2

0

0 10Time (s)

Base plate

20 30 40

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)


20 30 40

θ px

(deg)

05

0

ndash05

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)










Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)




Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)









4 Conclusion




Data Availability




References
















































Inpu

t ang

le o

f tor

que m

otor

Ang

ular

acce

lera

tion

of b

ase p

late

Cros

s-fe

edba

ck co

ntro

l

Ang

ular

disp

lace

men

ts of

thre

e gim

bals

PID controller

PID controller

PID controller

ωbx

1sωby

ωbz

up

ur

θp

θr

θy

kp

kr

ky

uy

ktr

ker

kty

key

kep

ktp

ωbx

ωbyTb

p Tbr Tb

y

ωbz

1Ls + R

1Ls + R

1Ls + R

1 (Jyx + Jpx)

1(Jyx + Jpx + Jry)

1Jyz 1s 1s

1s1s

1s1s

θ yz

θ ry

θ px

θ ry θry

θyz

θ px θpx

θ yz

Tpry Tr

py Typr

ndash

ndash

ndash

ndash

ndash

ndash

+

+

+

+

+

+


POS MSISP


(a)


Accelerometer

Rate Gyro

Data acquisition

(b)
















Yes

No

Tuning parameters


an AMB system




Angle measurement



Coupling analysis


Angle measurement


yaw gimbal


yaw gimbal


Angle comparison







Disp

lace

men

t (m

m)

005

0

ndash0050 02 04 06

Time (s)


08

(a)

Disp

lace

men

t (m

m)

02

01

0

ndash02

ndash01


0 02 04Time (s)

06 08

(b)

Ang

ular

disp

lace

men

t (deg)

05

0

ndash05


0 02 04Time (s)

06 08

(c)

Curr

ent (

A)

004

003

002

001

00 02 04 06

Time (s)


08

(d)


0 02 04 06Time (s)

08

Curr

ent (

mm

)

015

01

005

0

ndash005

(e)









Ang

ular

disp

lace

men

t (deg)

2

0

0 10 20Time (s)

30

Base plate

40

ndash2

(a)

θ ry (deg

)

0 10 20Time (s)

30


40

05

0

ndash05

(b)

θ px (deg

)

0 10 20Time (s)

30

Locked pitch gimbal

40

1

0

ndash1

(c)

0 10 20Time (s)

30

Locked yaw gimbal

40

θ yz (deg

)

1

0

ndash1

(d)




Ang

ular

disp

lace

men

t (deg)

2

0 10Time (s)

Base plate

20 30 40

0

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Locked pitch gimbal

20 30 40

θ px

(deg)

1

0

ndash1

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)


Ang

ular

disp

lace

men

t (deg)

2

0

0 10Time (s)

Base plate

20 30 40

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)


20 30 40

θ px

(deg)

05

0

ndash05

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)










Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)




Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)









4 Conclusion




Data Availability




References























































Yes

No

Tuning parameters


an AMB system




Angle measurement



Coupling analysis


Angle measurement


yaw gimbal


yaw gimbal


Angle comparison







Disp

lace

men

t (m

m)

005

0

ndash0050 02 04 06

Time (s)


08

(a)

Disp

lace

men

t (m

m)

02

01

0

ndash02

ndash01


0 02 04Time (s)

06 08

(b)

Ang

ular

disp

lace

men

t (deg)

05

0

ndash05


0 02 04Time (s)

06 08

(c)

Curr

ent (

A)

004

003

002

001

00 02 04 06

Time (s)


08

(d)


0 02 04 06Time (s)

08

Curr

ent (

mm

)

015

01

005

0

ndash005

(e)









Ang

ular

disp

lace

men

t (deg)

2

0

0 10 20Time (s)

30

Base plate

40

ndash2

(a)

θ ry (deg

)

0 10 20Time (s)

30


40

05

0

ndash05

(b)

θ px (deg

)

0 10 20Time (s)

30

Locked pitch gimbal

40

1

0

ndash1

(c)

0 10 20Time (s)

30

Locked yaw gimbal

40

θ yz (deg

)

1

0

ndash1

(d)




Ang

ular

disp

lace

men

t (deg)

2

0 10Time (s)

Base plate

20 30 40

0

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Locked pitch gimbal

20 30 40

θ px

(deg)

1

0

ndash1

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)


Ang

ular

disp

lace

men

t (deg)

2

0

0 10Time (s)

Base plate

20 30 40

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)


20 30 40

θ px

(deg)

05

0

ndash05

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)










Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)




Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)









4 Conclusion




Data Availability




References














































Disp

lace

men

t (m

m)

005

0

ndash0050 02 04 06

Time (s)


08

(a)

Disp

lace

men

t (m

m)

02

01

0

ndash02

ndash01


0 02 04Time (s)

06 08

(b)

Ang

ular

disp

lace

men

t (deg)

05

0

ndash05


0 02 04Time (s)

06 08

(c)

Curr

ent (

A)

004

003

002

001

00 02 04 06

Time (s)


08

(d)


0 02 04 06Time (s)

08

Curr

ent (

mm

)

015

01

005

0

ndash005

(e)









Ang

ular

disp

lace

men

t (deg)

2

0

0 10 20Time (s)

30

Base plate

40

ndash2

(a)

θ ry (deg

)

0 10 20Time (s)

30


40

05

0

ndash05

(b)

θ px (deg

)

0 10 20Time (s)

30

Locked pitch gimbal

40

1

0

ndash1

(c)

0 10 20Time (s)

30

Locked yaw gimbal

40

θ yz (deg

)

1

0

ndash1

(d)




Ang

ular

disp

lace

men

t (deg)

2

0 10Time (s)

Base plate

20 30 40

0

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Locked pitch gimbal

20 30 40

θ px

(deg)

1

0

ndash1

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)


Ang

ular

disp

lace

men

t (deg)

2

0

0 10Time (s)

Base plate

20 30 40

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)


20 30 40

θ px

(deg)

05

0

ndash05

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)










Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)




Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)









4 Conclusion




Data Availability




References
















































Ang

ular

disp

lace

men

t (deg)

2

0

0 10 20Time (s)

30

Base plate

40

ndash2

(a)

θ ry (deg

)

0 10 20Time (s)

30


40

05

0

ndash05

(b)

θ px (deg

)

0 10 20Time (s)

30

Locked pitch gimbal

40

1

0

ndash1

(c)

0 10 20Time (s)

30

Locked yaw gimbal

40

θ yz (deg

)

1

0

ndash1

(d)




Ang

ular

disp

lace

men

t (deg)

2

0 10Time (s)

Base plate

20 30 40

0

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Locked pitch gimbal

20 30 40

θ px

(deg)

1

0

ndash1

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)


Ang

ular

disp

lace

men

t (deg)

2

0

0 10Time (s)

Base plate

20 30 40

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)


20 30 40

θ px

(deg)

05

0

ndash05

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)










Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)




Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)









4 Conclusion




Data Availability




References











































Ang

ular

disp

lace

men

t (deg)

2

0 10Time (s)

Base plate

20 30 40

0

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)

Locked pitch gimbal

20 30 40

θ px

(deg)

1

0

ndash1

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)


Ang

ular

disp

lace

men

t (deg)

2

0

0 10Time (s)

Base plate

20 30 40

ndash2

(a)

θ ry (

deg)

0 10Time (s)


20 30 40

05

0

ndash05

(b)

0 10Time (s)


20 30 40

θ px

(deg)

05

0

ndash05

(c)

0 10Time (s)

Unlocked yaw gimbal

20 30 40

θ yz (

deg)

05

0

ndash05

(d)










Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)




Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)









4 Conclusion




Data Availability




References













































Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(a)

Ang

ular

disp

lace

men

t (deg)

01

005

0

ndash005

ndash010 5 10

Time (s)


15

(b)




Ang

ular

disp

lace

men

t (deg)

210

ndash1

1

0

0 5 10Time (s)

15 20

0 5 10 15 20

0 5 10 15

Base plate


Unlocked yaw gimbal

20

ndash1

1

0

ndash1

(a)

Time (s)

Base plate


0 5 10 15 20

0 5 10 15 20

Unlocked yaw gimbal

0 5 10 15 20

Ang

ular

disp

lace

men

t (deg)

210

ndash1

05

0

ndash05

05

0

ndash05

(b)









4 Conclusion




Data Availability




References











































4 Conclusion




Data Availability




References










































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