research article error analysis and adaptive-robust control...

Research ArticleError Analysis and Adaptive-Robust Control ofa 6-DoF Parallel Robot with Ball-Screw Drive Actuators

Navid Negahbani1 Hermes Giberti2 and Enrico Fiore1

1Dipartimento di Meccanica Politecnico di Milano Campus Bovisa Sud Via La Masa 1 20156 Milano Italy2Dipartimento di Ingegneria Industriale e dellrsquoInformazione Universita degli Studi di Pavia Via A Ferrata 5 27100 Pavia Italy

Correspondence should be addressed to Hermes Giberti hermesgibertipolimiit

Received 29 November 2015 Revised 21 February 2016 Accepted 10 March 2016

Academic Editor Andrew A Goldenberg

Copyright copy 2016 Navid Negahbani et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Parallel kinematic machines (PKMs) are commonly used for tasks that require high precision and stiffness In this sense therigidity of the drive system of the robot which is composed of actuators and transmissions plays a fundamental role In thispaper ball-screw drive actuators are considered and a 6-degree of freedom (DoF) parallel robot with prismatic actuated joints isused as application case A mathematical model of the ball-screw drive is proposed considering the most influencing sources ofnonlinearity sliding-dependent flexibility backlash and friction Using this model themost critical poses of the robot with respectto the kinematic mapping of the error from the joint- to the task-space are systematically investigated to obtain the workspacepositional and rotational resolution apart from control issues Finally a nonlinear adaptive-robust control algorithm for trajectorytracking based on the minimization of the tracking error is described and simulated

1 Introduction

Different drive systems are used for moving PKMs Electricallinear motors ball-screw-driven units and belt-driven unitsare the three most commonly used solutions to actuate pris-matic joints Linear motors directly provide the thrust forcewhile ball-screw- and belt-driven units convert a rotationalmotion into a linear one Ball-screw-driven units have goodstiffness and good precision for short and medium travel [1]Belt-driven units can be used for high speed and long travelstrokes [2 3] but with a low accuracy because of the flexibilityof the belt Linear motors put together the advantageousfeatures of ball-screw and belt zero backlash high stiffnesshigh velocities and acceleration Nevertheless this is paidfor in terms of higher costs lower energy efficiency smallerforce-to-size ratio and higher constructive complexity [4]Moreover linear motors are not available as ready-to-useunits especially when a large actuation force is requiredIn this case custom-made solutions are necessary includingan adequate liquid cooling system Accordingly a ball-screwdrive actuator often represents the best compromise solution

Many researchers have modeled ball-screw-driven trans-mission by considering its structural flexibility Flexible ball-screw models range from fairly simple ones consisting oflumped masses connected by springs such as those in [56] to more complicated ones with distributed parameteremploying FEM such as those in [7 8] Hybrid methodsare also proposed taking into account distributed stiffnessand inertia of the screw while modeling other componentsas lumped masses connected by springs as set out in [9 10]However only lumped-masses equivalent models are usedfor control purposes and frequently 2 DoFs are consideredenough a rotational DoF and a translational one [11ndash13] Insupport of this approach Frey et al in [14] show that anaccurate representation of the screw elastic behaviour can beobtained through a simplified lumped model Complemen-tary other researchers such as Han and Lee in [15] model arigid ball-screw-driven system by considering backlash andfriction In this work flexibility backlash and friction aretaken into account at the same time

The control strategies for a PKM can be largely dividedinto two schemes joint-space control and task-space controlThe joint-space control scheme can be readily implemented

Hindawi Publishing CorporationJournal of RoboticsVolume 2016 Article ID 4938562 15 pageshttpdxdoiorg10115520164938562

2 Journal of Robotics

as a collection of multiple independent single-input single-output (SISO) control systems using only information oneach actuator displacement [16] On the other hand the task-space control implements direct control of the 6 DoFs bymeasuring or estimating them solving the forward kinemat-ics In addition robot manipulator control strategies canbe classified into model-based control and performance-based control [17] Model-based control requires an accuratedynamic model of the robot while performance-based con-trol adjusts the control parameters according to the trackingerror as has been done in this work with an estimated model

Due to the closed-loop kinematic chains of PKMs stronginteractions between different actuators appear This com-binedwith high speedmotionmakes the dynamics of parallelmanipulators highly nonlinear In order to perform suitablecompensation for these sources of nonlinearity the parame-ters of the motion equations should be known exactly [17]Since PKMs require generally a relatively small workspacecompared to their overall size even if this characteristicrepresents a drawback compared to serial manipulators theconfiguration-dependent coefficient matrices of the dynamicequations may be approximated as constant ones withoutintroducing large modeling errors Based on these constantmatrices calculation of the approximated inverse dynam-ics becomes much simpler than solving the full inversedynamics However to prevent deterioration of trackingperformance that may cause this approximation with acontroller based on the inverse dynamics [18] performance-based control methods can be used The most importantof these methods are (1) sliding mode control (2) robustcontrol (3) adaptive control and (4) adaptive-robust control

In spite of the properties of robustness claimed the realimplementation of sliding mode control techniques presentsa major drawback that is the so-called chattering effectwith dangerous high-frequency vibrations of the controlledsystemThis phenomenon is due to the fact that in real appli-cations it is not reasonable to assume that the control signalcan switch at infinite frequency Robust control means todesign a controller such that some level of performance of thecontrolled system is guaranteed independently from changesin the plant dynamics In the robust approach the controllerhas a fixed structure that yields acceptable performance fora class of plants which include the one in question Robustcontrol is widely used for 6-DoF parallel robot especiallyfor Gough-Stewart platform [16 18 19] A disadvantage ofcommonly used robust controllers is that they do not have theability to learn This inability implicates conservative designand the stability of the system is achieved at the cost of per-formance Adaptive controlmeasures performance indexes ofthe controlled system using the input the state the outputand the known disturbances [20] From the comparisonbetween themeasured performance indexes and the requiredones the adaptation mechanism modifies the parameters ofthe adjustable controller andor generates an auxiliary controlinput According to [21] we can classify adaptive controllersin three schemes adaptive control by a computed-torqueapproach adaptive control by an inertia-related approachand adaptive control based on passivity Honegger et al useadaptive feed-forward control method in [22 23] because

it is simpler to implement than other algorithms and it isrobust to noise and leads to similar or better results [22]Disadvantages of adaptive control are limited to nonlinearsystems in which the uncertain parameters appear linearlyand adaptive controls often exhibit poor transients [24] Tosimultaneously achieve the advantages of both adaptive androbust control these methods can be used together such asin [25]

In this paper a systematic methodology to find theerror of a 6-DoF robotic device with parallel kinematicand Hexaglide architecture [22] is presented This robot wasdesigned to equip the Politecnico di Milano wind tunnelwith a motion simulator for Hardware-in-the-Loop (HIL)large-amplitude dynamic test In particular this works asan emulator to reproduce the hydrodynamic interactionbetween floating bodies and sea water in the case of floatingoffshore wind turbines [26] and sailing boat scale models Inprevious works [27] the drive system is mechanically sizedwithout considering flexibility backlash and friction

The experiments that will be conducted in the civilenvironmental chamber of the wind tunnel require turbineand boats scale models to be aeroelastic In order to obtainconsistent results these models are designed to have specificmodal shapes by the way the coupling with the robot andthe transmission unitsmight affect the desired behaviour Forthese reasons the manipulator has to be as rigid as possibleand in addition all the worsening effects coming fromthe transmission units (backlashes flexibility and friction)should be properly modeled and compensated by means of asuitable control algorithm otherwise the experimental resultswould be invalidated

Amathematicalmodel of the ball-screw drive is proposedconsidering the most influential sources of nonlinearitysliding-dependent flexibility backlash and friction Usingthis model the most critical poses of the robot with respectto the kinematic mapping of the error from the joint tothe task-space are systematically investigated to obtain theworkspace positional and rotational resolution apart fromcontrol issues Finally a nonlinear adaptive-robust controlalgorithm for trajectory tracking based on the minimizationof the tracking error is described and simulated This deviceis developed for the experimental simulation of the dynamicworking conditions of the above-mentioned hydroaeroelasticstructures in our wind tunnel (specially for wind turbines)

This paper is organized as follows Section 2 introducesthe inverse kinematics and dynamics of the Hexaglide Sec-tion 3 sets out the equations of motion of a ball-screw driveactuator Section 4 describes a systematic error evaluationmethodology Section 5 presents adaptive-robust controlmethod for trajectory tracking whose performances aresimulated and compared with other methods

2 Hexaglide Robot

Among the PKMs that can provide 6DoFs attention has beenfocused on Hexaglide which is also reported in the literatureas 6-PUS kinematic architecture with parallel linear guide-ways Its linkage consists of six closed-loop kinematic chainswhich connect the fixed base to the mobile platform with

Journal of Robotics 3

TCPx998400

z998400

y998400

bi Bi

pli

di

q i

Ox

z

siy

ni

Ai

Figure 1 119894th closed-loop kinematic chain of the Hexaglide

the same sequence of joints actuated prism (P) universal(U) and spherical (S)The links have a fixed length while theactuation takes place through linear guideways which are notnecessarily on the same plane but lie on parallel planes

21 Inverse Kinematics To perform the inverse kinematicsanalysis of the Hexaglide means to find each slider position119902119894from TCP position and mobile platform orientation Θ =

120572 120573 120574 [28] where 120572 is the rotation around 119909-axis 120573 is therotation around 119910-axis and 120574 is the rotation around 119911-axisWith reference to the quantities shown in Figure 1 the inversekinematics of the Hexaglide is solved as described in [26] Inparticular it is possible to write

l119894= d119894+ 119902119894u119894

with d119894= p + Rb1015840

119894minus s119894 (1)

where u119894is a unitary vector aligned with the 119894th guide axis

s119894is the position of the same guide with respect to the

fixed frame l119894is the vector aligned with the 119894th link p is

the vector containing the position and orientation of themobile platform R is the rotation matrix and b1015840

119894expresses

the position of the 119894th platform joint with respect to the TCPin the relative frame while 119902

119894represents the position of the 119894th

slider along the corresponding guide axis After some simplemathematical passages the following expression is found

119902119894= 119889119894119909plusmn radic1198972119894minus 1198892119894119910minus 1198892119894119911 (2)

The velocity and acceleration of the each slider can bederived from (2) as

119894= 119894119909∓119894119910119889119894119910+ 119894119911119889119894119911

radic1198972119894minus 1198892119894119910minus 1198892119894119911

119894= 119894119909

∓(119894119910119889119894119910+ 2

119894119910+ 119894119911119889119894119911+ 2

119894119911)Δ + (

119894119910119889119894119910+ 119894119911119889119894119911)2

ΔradicΔ

(3)

where Δ = 1198972119894minus 1198892119894119910minus 1198892119894119911 di = [119894119909 119894119910 119894119911]

119879

= k119862+ 120596b119894

and di = [119894119909

119894119910

119894119911]119879

= a119862+ Ξb119894+ 120596(120596b

119894) whereas

k119862and a119862are TCP velocity and acceleration while 120596 and Ξ

are absolute angular velocity and acceleration of the platformIt must be noted that the components of the angular velocityand acceleration of the platformdo not coincidewith the timederivative of the angular coordinates

120596119909 120596119910 120596119911 =

120572119909 120572119910 120572119911 =

(4)

120596 and Ξ are defined as skew-symmetric matrices

120596 =[[[

[

0 minus120596119911120596119910

120596119911

0 minus120596119909

minus120596119910120596119909

0

]]]

]

Ξ =[[[

[

0 minus120572119911120572119910

120572119911

0 minus120572119909

minus120572119910120572119909

0

]]]

]

(5)

22 Inverse Dynamics Inverse dynamics is the calculationof forces and torques on the robot actuated joints in orderto produce the required motion of the mobile platformA multibody model is implemented in Simulink using theSimMechanics library Inverse dynamics is solved taking intoconsideration two payloads a scale model of a sailing boatand a scale model of an offshore wind turbine In orderto perform the simulation the required slider motions arecalculated using (2) with first required platform motions asinput parameters Subsequently required slider motions areapplied to the SimMechanics robot model and the forcesat the robot joints are computed Figure 2 shows the blockscheme of the inverse dynamics model in SimMechanics

3 Dynamic Model of Ball-Screw Drive

Typical ball-screw drive consists of a motor and reducercoupling ball-screw table and end bearings In this sectionthe dynamic model of ball-screw drive is studied takinginto account flexibility (in gearbox coupling screw-ball andbearing) backlash (in gearbox and between screw and nut)and friction Figure 3 illustrates the lumped-mass-springmodel of ball-screw drive

The translational movement of the nut was modeled bytaking into account two elements the ball-screw transmis-sion ratio 119877 and the axial elastic deformation of the screwIn particular the first contribution is easily described by therelation 119877120579 where 119877 = ℎ

1199012120587 is the ball-screw transmission

ratio and 120579 is the rotational movement of the screw whilethe second one is represented by 119909bs that is the translationmovement due to axial elastic deformation of screw

In addition ball-screw has torsional flexibility (119870120579is ball-

screw torsional stiffness) and axial flexibility (119870eq is ball-screw axial stiffness)


x TCP

y TCP

z TCP

120572

120573

120574

x TCP

y TCP

z TCP

120572

120573

120574

Desired platform motions Inverse kinematics

Platform

Hexaglide robot

Base of thepayload

Weld

Payload

B F

q1 qp1 qpp1

q2 qp2 qpp2

q3 qp3 qpp3

q4 qp4 qpp4

q5 qp5 qpp5 q5 qp5 qpp5

q6 qp6 qpp6

q1 qp1 qpp1

q2 qp2 qpp2

q3 qp3 qpp3

q4 qp4 qpp4

q6 qp6 qpp6

Figure 2 SimMechanics robot model for inverse dynamic solution

120579m Tm

Jm

1205791 Tg

kg

G 120578

JgKc

Ceq

Keq

xbs

Jbs120579K120579

Knut

Cnut

Mc

qi

Motor Gearbox Coupling

Ball-screw

Jc

120579c

Figure 3 Lumped mass-spring model of ball-screw drive

According to Figure 3 the kinetic energy (119864119896) potential

energy (119881) dissipative function (119863) and virtual work of theexternal forces (120575119882ext) can be respectively expressed as

119864119896

=1

2[1198691198982

119898+ 1198691198922

1+ 1198691198882

119888+ 119869bs

2

+1198721198882

119894+119872bs

2

bs] (6)

119881

=1

2[119896119888(120579119888minus 1205791)2+ 119896120579(119902119894) (120579 minus 120579

119888)2+ 119896eq (119902119894) 119909

2

bs] (7)

119863 =1

2[119862nut (119877 + bs minus 119894)

2

+ 119862eq2

bs] (8)

120575119882ext

= 119879119898120575120579119898+ 119879119892(1205751205791minus1

120578119866120575120579119898)

+ 119865nut (120575119902119894 minus 120575119909bs minus 119877120575120579) minus 119865119891120575119902119894 + 119865119860119894119909120575119902119894

(9)

where119879119898 119869119898 119869119892 and119872

119888are themotor torque motor inertia

gearbox inertia and slider mass respectively Also 119865119891and

119865119860119894119909

are the slider friction force and reaction force betweenthe 119894th slider and the 119894th robot link respectively The terms120579119898and 120579

1are the angular position of driver motor and the

gearbox angle whereas119872bs and 119869bs are themass andmomentof inertia of screw (119869bs = (119872bs119889

2

screw)8 where 119889screw is thediameter of screw shaft)The backlash torque 119879

119892is calculated

by means of (10) whereas 120578 is gearbox mechanical efficiencyand 119866 is the gearbox ratio 120596in120596out The angular position ofcoupling and the moment of inertial of coupling are 120579

119888and

119869119888 respectively Finally 119865nut is the backlash force in nut and

119870119888119862eq and119862nut are coupling stiffness screw axial equivalent

damping and nut damping respectivelyThe following systems of equations allow evaluating 119879

119892

and 119865nut

119879119892= 119896119892

120579119898

119866minus 1205791minus 120579119887

120579119898

119866minus 1205791ge 120579119887

0 minus120579119887le120579119898

119866minus 1205791le 120579119887

120579119898

119866minus 1205791+ 120579119887

120579119898

119866minus 1205791le minus120579119887

(10)


L1 + qi

(a)

Kbearing Kscrew

Mbs

xbs

(b)

Figure 4 (a) Ball-screw shaft free at one end and (b) mass-spring model of the axial deformation of the ball-screw

L1 + qi L2 minus qi

L

(a)

Kbearing Kscrew1

Mbs

xbs

KbearingKscrew2

(b)

Figure 5 (a) Ball-screw shaft fixed at both ends and (b) mass-spring model of the axial deformation of the ball-screw

119865nut

= 119870nut

119877120579 + 119909bs minus 119902119894 minus Δ 119877120579 + 119909bs minus 119902119894 ge Δ

0 minusΔ le 119877120579 + 119909bs minus 119902119894 le Δ

119877120579 + 119909bs minus 119902119894 + Δ 119877120579 + 119909bs minus 119902119894 le minusΔ

(11)

where 119870nut is the stiffness between nut and screw 120579119887is the

backlash in the gearbox and Δ is the backlash between nutand screw

Torsional stiffness of the screw 119870120579 is calculated as

explained in [29] and shown below

119870120579(119902119894) =

119866screw119869screw1198711+ 119902119894

(12)

where 119866screw is the shear module of the screw 1198711is the

distance between the home position and the bearing near tothe motor and 119869screw is the polar moment of the screw Axialstiffness of the screwdepends on type of bearingwhich is usedin the ball-screw drive As a result axial stiffness of the screwand119870eq are found in the two conditions set out as follows

Ball-Screw Shaft Fixed at One End Figure 4 shows this casewhere the screw shaft has axial force only between the axial-fixed bearing and the nut Therefore only this part of thescrew shaft has axial deformation and can be modeled witha spring Also the axial-fixed bearing can be modeled by aspring that is series with an equivalent spring model of theshaft screw Equivalent stiffness is calculated by the followingequation

1

119870eq=

1

119870bearing+

1

119870screw1 (119902119894)

119870screw1 (119902119894) =119864119860

1198711+ 119902119894

(13)

According to [9 12] stiffness of the screw in this condition isfunction of the table position

Ball-Screw Shaft Fixed at Both Ends In this case there aretwo springs connected in series on each side of the shaft asshown in Figure 5Therefore equivalent stiffness is calculated


by using the formula of a parallel spring as shown in thefollowing equation

119870eq = (1

119870bearing+

1

119870screw1 (119902119894))

minus1

+ (1

119870bearing+

1

119870screw2 (119902119894))

minus1

119870screw1 (119902119894) =119864119860

1198711+ 119902119894

119870screw2 (119902119894) =119864119860

1198712minus 119902119894

(14)

The friction force is estimated using this exponentialform

119865119891= (119872119888119892 minus 119865119894119911) [120583119896+ (120583119904minus 120583119896) 119890minus(119894119881119904)2

] sign (119894)

+ 119862V119894

(15)

where 120583119904is the static coefficient of friction 120583

119896is the kinetic

coefficient of friction 119865119894119911

is the normal reaction forcebetween 119894th slider and 119894th robot link119862V is the viscous frictionparameter and119881

119904is the characteristic velocity of the Stribeck

friction Ball-screw drive equations ofmotion are resolved viaSimulink and the results are used in Hexaglide model

Considering (6) (7) (8) and (9) and applying Lagrangemethod the following equations are obtained

119869119898119898= 119879119898minus

119879119892

(120578119866)

1198691198921+ 119896119888(120579119888minus 1205791) minus 119879119892= 0

119869119888119888+ 119896119888(1205791minus 120579119888) + 119896120579(119902119894) (120579119888minus 120579) = 0

119869bs + 119896120579 (119902119894) (120579 minus 120579119888) + 119877119865nut

+ 119862nut119877 (119877 + bs minus 119894) = 0

119872bsbs + 119865nut + 119862nut (119877 + bs minus 119894) + 119896eq (119902119894) 119909bs

+ 119862eqbs = 0

119872119888119894minus 119865nut minus 119862nut (119877 + bs minus 119894) + 119865119891 = 119865119860119894119909

(16)

4 Error Evaluation

To evaluate the positioning error of the TCP in the task-spaceamethod based on a kinematics analysis has been performedFirst of all the critical poses of the Hexaglide workspace arefound via kinematic mapping of the error taking into accountas a source of error only the slider position Subsequentlythe critical poses found are used as the initial positions andorientations of the TCP from which to begin the dynamicsimulation of the behaviour of the robot The aim of thesesteps is to detect the point which has the worst condition in

minus02 minus01 0 01 02045

05

055

06

065

07

075

08

085

y (m)

z (m

) 39

93

93

49

94

94

95

95

95

96

96

96

97

97

97

98

98

98

99

99

99

10

10

10

101

101

101

102

102

102

103

103

103

104

104

104

105

105

105

106

106

106

107

107

107

108

108

108

109

109

109

11

11

11

111

111111

111 112

112112

112

113

113

114

114

115

115611116 117

117

WSd

0

5

10

15

totoWS 120579joint = plusmn30∘


Figure 6 Error of TCP position in 119910 direction

terms of the maximum error This method is explained inmore depth hereafter

From kinematic analysis of the Hexaglide we know thatW = [119869]q where [119869] is a Jacobianmatrix q = [

1 2

6]119879

is the sliders velocity vector andW = [ 120596119909 120596119910 120596119911]119879 is

the velocity vector of the platform For small variations it iscorrect to write

ΔX = [119869] Δq (17)

where X = [119909 119910 119911 120572 120573 120574]119879 is the robot pose If all the

actuators and transmissions are equal and the robot isconsidered rigid with ideal joints the drive systems are theonly source of errors It is reasonable to assume that theerrors in the sliders positions are limited by amaximumvalueΔ119902max (infinity norm is the best suited norm when it comesto representing this situation Δ119902

infinle Δ119902max) Considering

that all the sliders have the same errors Δ119902max the maximumerrors of TCP position and platform orientations are definedby

ΔXmax = [119869]

1

1

1

6times1

Δ119902max (18)

Figure 6 shows an example of the density distributionof error contours in the workspace of the Hexaglide in 119910direction Critical TCP positions are summarised in Table 1furthermore critical orientations are plusmn10∘ for sailboats andare plusmn75∘ for wind turbines


Desired TCPmotions

Inversekinematics

Convert tomotor

rotation

Reaction forcesbetween eachlink and slider

Robot(SimMechanics)

Lineartransmission

model(Simulink)

X

X

d qd q

eX

120579m

+ minus

Figure 7 Block scheme to evaluate the error pose

Table 1 Critical points of initial position for each movement

Movement TCP position [m]1199090

1199100

1199110

119909mov 0 01 08119910mov 0 0 07119911mov 0 01 07120572mov 0 01 06120573mov 0 01 06120574mov 0 01 06

Error of TCP position and platform orientation is calcu-lated by means of

eX = Xd minus X (19)

where Xd and eX are the required pose of the robot and thepose error respectivelyThen the slider position setpoints arecalculated from the required TCP position using the inversekinematic equation (2) This analysis has been performedtaking into consideration each degree of freedom of the TCP(119909 119910 119911 120572 (rotation around 119909-axis) 120573 (rotation around 119910-axis) and 120574 (rotation around 119911-axis)) separately and themotion law used for each DoF is sinusoidal In order to setthe correct initial conditions of the simulation and to preventimpulse forces in the robot joints a five-order polynomialfunction has been used to fade in the sinusoidal functionIn this way the simulation begins with zero velocity andacceleration

Thus the required platform motions are defined by thefollowing equation

119883119894= 119883119900119894+

11988651199055 + 11988641199054 + 11988631199053 119905 le 119905

119888

119860119894sin (2120587119891 (119905 minus 119905

119889)) 119905 gt 119905

119888

(20)

where X119900is the initial platform pose (Table 1) whereas 119860

119894

and119891 are amplitude and frequency ofmovement respectively(Table 2)

For the ideal situation (linear transmission rigid andgearbox without backlash) the angular motor positions are

Table 2 Maximum amplitudes and frequencies of the desiredmovements

Movement Amplitude Frequency (Hz)Wind turbine Sailboat Wind turbine Sailboat

119909mov 025m mdash 07 12119910mov 015m mdash 07 12119911mov 015m mdash 07 12120572mov 75

∘10∘ 07 12

120573mov 75∘ 10∘ 07 12120574mov 75∘ 10∘ 07 12

calculated from the slider positions by 120579119898ref = 119902

119894ref119866119877and the dynamic behaviour of the sliders is evaluated bythe equations shown in (16) Figure 7 summarises the mainsteps necessary to obtain the TCP position and the platformorientation for the nonideal system using the SimMechanicsmodel earlier described As can be see the error betweenideal system and the systemwhich includes backlash flexibil-ity and friction is calculated The worst operating conditionin otherwords themaximumerror is found by comparing theresults achieved by each DoF under different motion laws

The previous analysis has been performed using dataof a Rexroth Bosch CKK 25-200 ball-screw drive A ball-screw drive with shaft free at one end has been investigatedIn fact this condition has a lower stiffness with respect tothe situation in which the shaft is fixed at both ends soit represents the worst situation On the other hand theball-screw configuration with both ends fixed in general isnot used because any gradient of temperature on the screwcan generate high stress reducing the screw and bearingslife Instead backlash is a real problem in precise controltherefore screw with preload must be chosen to reduce itBelow parameters are used in the simulations

119870nut = 1371198906Nm

119870bearing = 1131198906Nm

Δ = 005mm


Table 3 Values of the maximum error and RMS of error in thedesired robot workspace

Position or orientation MaximumSailboat Wind turbine

Maximum error

119909 068mm 070mm119910 068mm 070mm119911 04mm 036mm120572 017

∘016∘

120573 010∘ 010∘

120574 005∘ 005∘

RMS error

119909 035mm 037mm119910 036mm 038mm119911 028mm 026mm120572 009∘ 008∘

120573 006∘

006∘

120574 003∘ 003∘

120579119887= 4 arcmin

ℎ119901= 32mm

119889screw = 32mm

119870119888= 39543N sdotmrad

119896119892= 752871198904N sdotmrad

119866 = 2

119871 = 16m(21)

Note that the ball-screw damping (119862eq and119862nut) has to beidentified bymeans of experimental tests on the realmachinebut it does not have too much significance in predicting theelastic deformation [12] Consequently values of119862eq and119862nutare chosen from the literature [12 30] The same approach isused to choose the values of the friction parameters in (15) inparticular by the cited reference Okwudire [31]

Figure 8 shows maximum and RMS error for TCPposition and platform orientation in each movement whenthe requirements are defined for sailboats simulations Table 3summarises maximum value of TCP positioning and ori-entation errors while Table 4 shows in which poses thesemaximum error conditions have been achieved

According to Tables 3 and 4 and Figure 8 the criticalcondition is found when 120572 is moving and 120573 = minus10

∘ and120574 = minus10∘ are the initial platform orientation for the sailboatsimulations When 120572 is moving and 120573 = minus75∘ and 120574 = minus75∘are the initial platform orientation we have the maximumerror in the case of wind turbine simulation

5 Control of the Hexaglide

The design of a control system of a six-degree of freedomparallel kinematic machine is a very difficult task Usuallyit is very expensive to measure the end effector position ofa 6-DoF robot Instead of the pose of the robot platform

the position of each slider or angular position of each motoris measured and the pose of the end effector is estimatedusing the direct kinematics By the way this method can onlybe applied if the machine is properly calibrated and if allthe machine components are realized respecting very stricttolerances otherwise the estimation of the pose of themobileplatform would be erroneous If these requirements cannotbe met it is necessary to use specific measurements deviceslike CMM or 3D positioning systems as the ones describedin [32 33] Furthermore 6-DoF PKM robot has complexdynamics without an analytical solution made more complexby the nonlinearity of the actuation systems Therefore tosetup a control strategy it is necessary to pass through a linearform of dynamic equations as described in [22 23]

A promising approach for developing a control algorithmin these conditions is the adaptive and robust nonlinearcontrol as presented in [34 35] In this paper starting froma literature review a PID adaptive-robust control for theHexaglide is developed Figure 9 shows the block diagram ofthe control proposed

We can write the dynamic equation of the robot includ-ing the actuator dynamics as follows

Tm = M (q) q + f (q q) (22)

where Tm is the vector of motor torques and q is the vectorof the joint position The manipulator mass matrix M(q) issymmetric and positive definiteThe vector f(q q) representstorque or force arising from centrifugal Coriolis gravity andfriction forces

The control action can be obtained by means of a suitableinput motor torque defined in this way

Tm = Ψ (q q qr qr) p + KDs + KI int s 119889119905

+ 120578119904119886119905 (Φminus1s)

(23)

In this expression Ψ is a matrix containing nonlinearequation and p is a vector containing dynamical parameterswhereas KD KI 120578 and Φ are positive diagonal matricesPosition error vector of the sliders is defined as

e = qd minus q (24)

where qd is the desired position of slider found from thedesired platform pose via the inverse kinematics The vectors represents the combined error and it is defined in a similarway to the sliding control approach

s = e + Λe (25)

whereΛ is a positive and diagonal matrixThe vector s can bealso defined as

s = qr minus q (26)

where qr is called reference value of q and it is obtained bymodifying qd according to the tracking error qr is defined as

qr = qd + Λe (27)


Table 4 Conditions of maximum error and maximum RMS of the error

Position or orientation ConditionSailboat Wind turbine

Maximum error

119909 120573mov 120572 = 10∘ 120574 = minus10∘ 120573mov 120572 = minus75∘ 120574 = 75∘

119910 120572mov 120573 = minus10∘ 120574 = 10∘ 120573mov 120572 = minus75∘ 120574 = 75∘

119911 120572mov 120573 = minus10∘ 120574 = minus10∘ 120574mov 120572 = 75∘ 120573 = minus75∘

120572 120572mov 120573 = minus10∘ 120574 = minus10∘ 120572mov 120573 = minus75∘ 120574 = minus75∘



RMS error

119909 120573mov 120572 = minus10∘ 120574 = 10∘ 119909-mov 120572 120573 = 75∘ 120574 = minus75∘


119911 120574mov 120572 = 10∘ 120573 = minus10∘ 120574mov 120572 = 75∘ 120573 = minus75∘



120574 120572mov 120573 = minus10∘ 120574 = 10∘ 120572mov 120573 = minus75∘ 120574 = minus75∘

In order to simplify the computational aspects of thiscontrol structure it is possible to omit the dynamic behaviourof the six links In this way the model of the robot ismade up of only seven bodies six linear transmission driveservomechanisms and one mobile platform The functionΨ(q q qr qr)p can be modified in the following way

Trobot = [Ψ1sdotsdotsdot6 Ψb Ψ7][[

[

p1sdotsdotsdot6pbp7

]]

]

(28)

where Ψ1sdotsdotsdot6 and p1sdotsdotsdot6 describe the dynamics of the six lineartransmission drive servomechanisms Ψb and pb describeCoulomb frictionwithin the linear transmissions andΨ7 andp7 describe the dynamics of the platformThe termsΨ1sdotsdotsdot6 andp1sdotsdotsdot6 are defined by

Ψ1sdotsdotsdot6 =[[[

[

119903111990311199021

0d d d

0 119903611990361199026

]]]

]

p1sdotsdotsdot6 = [1198981 1198881 1198961 sdot sdot sdot 1198986 1198886 1198966]119879

(29)

where 1198981 1198881 1198961 119898

6 1198886 1198966are estimated mass damper

and spring coefficient respectively for each linear transmis-sion whereasΨb and pb are defined by

Ψb =[[[

[

sign (1199031) 0

d

0 sign (1199036)

]]]

]

pb =[[[[

[

1198871

1198876

]]]]

]

(30)

where 1198871 119887

6are Coulomb friction coefficients in each

linear transmission The definition of Ψ7 and p7 is more

complicated The vector of dynamical parameters is set outby

p7 = [1198987 1198987119903119909 1198987119903119910 1198987119903119911 119868119909119909 119868119910119910 119868119911119911]119879

(31)

It is made up of themass of the platform the payload andthe inertia moments 119868

119909119909 119868119910119910 and 119868

119911119911 The frame connected to

the TCP is supposed to be oriented in order not to considerthe inertia moments 119868

119909119910 119868119909119911 and 119868

119910119911 The matrixΨ7 is given

by

Ψ7 = J[a7

RΩ7 03times3

01times3

minusa7R R1205951205967

] (32)

where J is the Jacobian matrix and R is the rotation matrixThe acceleration a

7and the skew-symmetric matrix a7 corre-

sponding to the cross product are defined as follows

a7=[[

[

119889

119889

119889+ 119892

]]

]

a7 =[[

[

0 minus119889minus 119892

119889

119889+ 119892 0 minus

119889

minus119889

119889

0

]]

]

(33)

The matricesΩ7 and 1205951205967 are defined as

Ω7 =[[[

[

minus1205962119910minus 1205962119911

minus119911+ 120596119909120596119910119910+ 120596119909120596119911

119911+ 120596119909120596119910

minus1205962119909minus 1205962119911

minus119909+ 120596119911120596119910

minus119910+ 120596119911120596119909119909+ 120596119911120596119910

minus1205962119909minus 1205962119910

]]]

]

1205951205967=[[[

[

119909

minus120596119911120596119910120596119910120596119911

120596119909120596119911

119910

minus120596119911120596119909

minus120596119910120596119909120596119909120596119910

119911

]]]

]

(34)

where 120596 is the absolute angular velocity of the platform


x y z0

01

02

03

04

05

06

07M

axim

um T

CP p

ositi

on er

ror (

mm

)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10

120574 movement 120573 = minus10 120572 = minus10

120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



(a) Maximum error of TCP position

0002004006008

01012014016018

Max

imum

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574

(b) Maximum error of mobile platform orientation

x y z0

005

01

015

02

025

03

035

04

RMS

TCP

posit

ion

erro

r (m

m)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



(c) RMS error of TCP position

0001002003004005006007008009

RMS

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574

(d) RMS error of mobile platform orientation

Figure 8 Maximum and RMS of pose error when sailboat is installed on platform

Given the desired trajectory qd (we shall assume that thedesired position velocity and acceleration are all bounded)and with some or all manipulator parameters unknown theadaptive controller design problem is to obtain a control lawfor the actuator torques and an estimation of the unknown

parameters in such a way that the manipulator follows therequired trajectory in the best way possible

To do that we define a function to estimate the parametererror p = p minus p as a difference between a vector of unknownparameters describing the manipulatorrsquos mass properties and


Inversekinematics

Adaptivecontrol

Parameterestimation

Lineartransmission

driveHexaglide

robot

HIL

Aerodynamicforces

qd

Xd

qTm

p

Figure 9 Block diagram of the adaptive-robust control for controlling the Hexaglide

Table 5 Position errors of the sliders

Slider 1 Slider 2 Slider 3 Slider 4 Slider 5 Slider 6Maximum error [mm] 057 040 093 030 030 038Error percentage 023 100 026 030 073 022RMS error [mm] 023 018 038 010 012 013

its estimate By considering as a candidate the Lyapunovfunction

119881 (119905) =1

2(s119879Ms + p119879Γminus1p + (int s 119889119905)

119879

KI int s 119889119905) (35)

where Γ is a symmetric positive definite matrix Differentiat-ing and using (22) (23) and (26) yield

(119905) = s119879Ψp + p119879Γminus1p minus s119879KDs minus s119879120578119904119886119905 (Φminus1s) (36)

Updating the parameter estimates p according to the correla-tion integrals [35]

p = ΓΨ119879s (37)

and (23) changes to


+ 120578119904119886119905 (Φminus1s)

(38)

By choosing KD = 100I KI = 100I and Λ = 200Irobot is controlled for tracking six-degree of freedom sinu-soidal movements ((20) and Table 2) in its task-spacesimultaneously The position error of each slider is shown inFigure 10 whereas Table 5 presents themaximum and RMS ofthe position error of sliders

Figure 11 shows the tracking errors of the platformTable 6presents the maximum tracking error and the RMS trackingerror According to this table maximum percentage of thetracking error occurs in 120574 orientation with 129 of the 120574

movement amplitude According to these results adaptive-robust control has shown a good performance

To highlight the efficiency of the controller designedtwo different control methods are analysed dual PID controlpresented in [36] and PD adaptive control shown in [23]Figure 12 shows the comparison of the results achieved withthese two control strategies with the one described in thispaper PID adaptive-robust control method has minor error

6 Conclusions

In this paper a systematic methodology to find the error of a6-DoF robotic device with parallel kinematic and Hexaglidearchitecture is presented This robot works as an emulatorto reproduce the hydrodynamic interaction between floatingbodies and sea water for aerodynamic tests in wind tunnel

A systematic error evaluation methodology is based onaccuratemodeling of the behaviour of the linear transmissionactuators that move the robot and by means of a mappingof the robot working volume in order to identify the worstwork conditions The critical poses of the end effector inthe workspace for each desired movement have been foundthrough a kinematic analysis whereas the dynamic analysisof Hexaglide actuated by ball-screw linear transmissions hasbeen performed in these critical poses for obtaining theworst cases The robot has been simulated into Simulink-SimMechanics environment and an adaptive-robust controlstrategy has been designed to control the end effector positionin order to track spatial complex trajectory Finally thecontrol strategy performances have been compared withother control methods


Table 6 Poses errors of the platform when adaptive-robust control is used

119909

[mm]119910

[mm]119911

[mm]120572

[degree]120573

[degree]120574

[degree]Maximumerror 064 068 031 006 009 010

Errorpercentage 043 068 031 081 120 129

RMS error 018 016 011 002 002 002

0 05 1 15 2 25 3 35 4 45 5minus04minus03minus02minus01

0010203040506

t (s)

q er

ror (

mm

)

Slider 1 error

(a) Slider 1

t (s)0 05 1 15 2 25 3 35 4 45 5

minus04minus03minus02minus01

001020304

Slider 2 error

q er

ror (

mm

)

(b) Slider 2

t (s)0 05 1 15 2 25 3 35 4 45 5

minus06minus04minus02

002040608

1Slider 3 error

q er

ror (

mm

)

(c) Slider 3

t (s)0 05 1 15 2 25 3 35 4 45 5


minus0050

00501

01502

02503

Slider 4 error

q er

ror (

mm

)

(d) Slider 4

t (s)0 05 1 15 2 25 3 35 4 45 5

minus04

minus03

minus02

minus01

0

01

02

03Slider 5 error

q er

ror (

mm

)

(e) Slider 5

t (s)0 05 1 15 2 25 3 35 4 45 5

minus03

minus02

minus01

0

01

02

03

04Slider 6 error

q er

ror (

mm

)

(f) Slider 6

Figure 10 Position errors of the sliders when adaptive-robust control is used


minus08

minus06

minus04

minus02

0

02

04

06TCP x position error

t (s)0 05 1 15 2 25 3 35 4 45 5

xer

ror (

mm

)

(a) 119909 direction

t (s)

minus08

minus06

minus04

minus02

0

02

04

06TCP y position error

0 05 1 15 2 25 3 35 4 45 5

yer

ror (

mm

)

(b) 119910 direction

t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

zer

ror (

mm

)

TCP z position error

(c) 119911 direction

minus006

minus004

minus002

0

002

004

006

008TCP 120572 orientation error

t (s)0 05 1 15 2 25 3 35 4 45 5

120572er

ror (

deg

)

(d) 120572 orientation


0002004006008

TCP 120573 orientation error

t (s)0 05 1 15 2 25 3 35 4 45 5

120573er

ror (

deg

)

(e) 120573 orientation

minus01minus008minus006minus004minus002

0002004006008

01

t (s)


0 05 1 15 2 25 3 35 4 45 5

120574er

ror (

deg

)

(f) 120574 orientation

Figure 11 Pose error of the platform when adaptive-robust control is used

x y z02468

10121416

PID adaptive-robustPD adaptiveDual PID

120572 120573 120574

Erro

r (

) (er

ror lowast

100

am

plitu

de)

Figure 12 Pose error percentage in three types of the control method


The results of the work demonstrate that the ball-screwlinear actuator used to move the Hexaglide architecturedeveloped and the PID adaptive-robust control allows one toachieve accuracy of approximately 07mm in TCP positionand of 017 degrees in platform orientation These results arein line with our required performance and consolidate thedesign choices with respect to the actuation system and thealgorithm control strategy

Competing Interests

The authors declare that they have no competing interests

References

[1] V Scheinman and JMMcCarthy ldquoMechanisms and actuationrdquoin Springer Handbook of Robotics B Siciliano and O KhatibEds chapter 3 pp 67ndash86 2008

[2] A S Kulkarni and M A El-Sharkawi ldquoIntelligent precisionposition control of elastic drive systemsrdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 26ndash31 2001

[3] A Hace K Jezernik and A Sabanovic ldquoSMC with disturbanceobserver for a linear belt driverdquo IEEE Transactions on IndustrialElectronics vol 54 no 6 pp 3402ndash3412 2007

[4] D Tosi G Legnani N Pedrocchi P Righettini and H GibertildquoCheope a new reconfigurable redundant manipulatorrdquoMech-anism and Machine Theory vol 45 no 4 pp 611ndash626 2010

[5] J-S Chen Y-K Huang and C-C Cheng ldquoMechanical modeland contouring analysis of high-speed ball-screw drive systemswith compliance effectrdquo International Journal of AdvancedManufacturing Technology vol 24 no 3-4 pp 241ndash250 2004

[6] L Liu Z Wu and H Liu ldquoModeling and analysis of thecrossfeed servo system of a heavy-duty lathe with frictionrdquoMechanics Based Design of Structures and Machines vol 41 no1 pp 1ndash20 2013

[7] E Schafers J Denk and J Hamann ldquoMechatronic modelingand analysis of machine toolsrdquo in Proceedings of the 2ndInternational Conference on High Performance Cutting (CIRP-HPC rsquo06) Vancouver Canada June 2006

[8] S J Ma G Liu G Qiao and X J Fu ldquoThermo-mechanicalmodel and thermal analysis of hollow cylinder planetary rollerscrew mechanismrdquo Mechanics Based Design of Structures andMachines vol 43 no 3 pp 359ndash381 2015

[9] K K Varanasi and S A Nayfeh ldquoThe dynamics of lead-screw drives low-order modeling and experimentsrdquo Journal ofDynamic Systems Measurement and Control vol 126 no 2 pp388ndash396 2004

[10] D A Vicente R L Hecker F J Villegas and G M FloresldquoModeling and vibration mode analysis of a ball screw driverdquoInternational Journal of Advanced Manufacturing Technologyvol 58 no 1ndash4 pp 257ndash265 2012

[11] C Okwudire and Y Altintas ldquoMinimum tracking error controlof flexible ball screw drives using a discrete-time sliding modecontrollerrdquo Journal of Dynamic Systems Measurement andControl vol 131 no 5 pp 1ndash12 2009

[12] A Kamalzadeh D J Gordon and K Erkorkmaz ldquoRobustcompensation of elastic deformations in ball screw drivesrdquoInternational Journal ofMachine Tools andManufacture vol 50no 6 pp 559ndash574 2010

[13] L Dong and W C Tang ldquoAdaptive backstepping slidingmode control of flexible ball screw drives with time-varying

parametric uncertainties and disturbancesrdquo ISA Transactionsvol 53 no 1 pp 125ndash133 2014

[14] S Frey A Dadalau and A Verl ldquoExpedient modeling of ballscrew feed drivesrdquoProduction Engineering vol 6 no 2 pp 205ndash211 2012

[15] S I Han and J M Lee ldquoAdaptive dynamic surface control withsliding mode control and RWNN for robust positioning of alinear motion stagerdquo Mechatronics vol 22 no 2 pp 222ndash2382012

[16] S K Hag M C Young and L Kyo ldquoRobust nonlinear taskspace control for 6 DOF parallel manipulatorrdquo Automaticavol 41 no 9 pp 1591ndash1600 2005

[17] J F He H Z Jiang D C Cong Z M Ye and J W HanldquoA survey on control of parallel manipulatorrdquo Key EngineeringMaterials vol 339 pp 307ndash313 2007

[18] S-H Lee J-B Song W-C Choi and D Hong ldquoPositioncontrol of a Stewart platform using inverse dynamics controlwith approximate dynamicsrdquo Mechatronics vol 13 no 6 pp605ndash619 2003

[19] H Abdellatif and B Heimann ldquoAdvanced model-based controlof a 6-DOF hexapod robot a case studyrdquo IEEEASME Transac-tions on Mechatronics vol 15 no 2 pp 269ndash279 2010

[20] Z Ma Y Hu J Huang et al ldquoA novel design of in pipe robot forinner surface inspection of large size pipesrdquo Mechanics BasedDesign of Structures and Machines vol 35 no 4 pp 447ndash4652007

[21] F L Lewis D M Dawson and T A Chaouki Robot Manipula-tor Control Theory and Practice Marcel Dekker New York NYUSA 2nd edition 2004

[22] M Honegger A Codourey and E Burdet ldquoAdaptive control ofthe Hexaglide a 6 dof parallel manipulatorrdquo in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA rsquo97) vol 1 pp 543ndash548 Albuquerque NM USA April1997

[23] M Honegger R Brega and G Schweitzer ldquoApplication of anonlinear adaptive controller to a 6 dof parallel manipulatorrdquoin Proceedings of the IEEE International Conference on RoboticsandAutomation (ICRA rsquo00) pp 1930ndash1935 San Francisco CalifUSA April 2000

[24] G Song R W Longman R Mukherjee and J Zhang ldquoInte-grated sliding-mode adaptive-robust controlrdquo in Proceedings ofthe IEEE International Conference on Control Applications pp656ndash661 Dearborn Mich USA September 1996

[25] X Zhu G Tao B Yao and J Cao ldquoAdaptive robust posturecontrol of a parallel manipulator driven by pneumatic musclesrdquoAutomatica vol 44 no 9 pp 2248ndash2257 2008

[26] I Bayati M Belloli D Ferrari F Fossati and H GibertildquoDesign of a 6-dof robotic platform for wind tunnel tests offloatingwind turbinesrdquoEnergy Procedia Journal vol 53 pp 313ndash323 2014

[27] H Giberti and D Ferrari ldquoDrive system sizing of a 6-Dofparallel robotic platformrdquo in Proceedings of ASME 12th BiennialConference on Engineering Systems Design and Analysis (ESDArsquo14) pp 25ndash27 Copenhagen Denmark June 2014

[28] M Valles M Dıaz-Rodrıguez A Valera V Mata and A PageldquoMechatronic development and dynamic control of a 3-DOFparallel manipulatorrdquoMechanics Based Design of Structures andMachines vol 40 no 4 pp 434ndash452 2012

[29] K K Varanasi and S Nayfeh ldquoThe dynamics of lead-screwdrives low-order modeling and experimentsrdquo Journal of Dyna-mic Systems Measurement and Control vol 126 no 2 pp 388ndash396 2004


[30] A Kamalzadeh and K Erkorkmaz ldquoCompensation of axialvibrations in ball screw drivesrdquo CIRP AnnalsmdashManufacturingTechnology vol 56 no 1 pp 373ndash378 2007

[31] C Okwudire Finite element modeling of ballscrew feed drivesystems for control purposes [Master of Applied Science thesis]The University of British Columbia 2005

[32] M Verner F Xi and C Mechefske ldquoOptimal calibration ofparallel kinematic machinesrdquo Journal of Mechanical Design vol127 no 1 pp 62ndash69 2005

[33] YM Zhao Y Lin F Xi and S Guo ldquoCalibration-based iterativelearning control for path tracking of industrial robotsrdquo IEEETransactions on Industrial Electronics vol 62 no 5 pp 2921ndash2929 2015

[34] J Craig P Hsu and S S Sastry ldquoAdaptive control of mechanicalmanipulatorsrdquo in Proceedings of the IEEE International Confer-ence on Robotics and Automation Slotine April 1986

[35] J-J Slotine and W Li Applied Nonlinear Control Prentice-HallInternational New York NY USA 1991

[36] B Ding RM Stanley B S Cazzolato and J J Costi ldquoReal-timeFPGA control of a hexapod robot for 6-DOF biomechanicaltestingrdquo in Proceedings of the 37th Annual Conference of theIEEE Industrial Electronics Society (IECON rsquo11) pp 252ndash257Melbourne Australia November 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



as a collection of multiple independent single-input single-output (SISO) control systems using only information oneach actuator displacement [16] On the other hand the task-space control implements direct control of the 6 DoFs bymeasuring or estimating them solving the forward kinemat-ics In addition robot manipulator control strategies canbe classified into model-based control and performance-based control [17] Model-based control requires an accuratedynamic model of the robot while performance-based con-trol adjusts the control parameters according to the trackingerror as has been done in this work with an estimated model

Due to the closed-loop kinematic chains of PKMs stronginteractions between different actuators appear This com-binedwith high speedmotionmakes the dynamics of parallelmanipulators highly nonlinear In order to perform suitablecompensation for these sources of nonlinearity the parame-ters of the motion equations should be known exactly [17]Since PKMs require generally a relatively small workspacecompared to their overall size even if this characteristicrepresents a drawback compared to serial manipulators theconfiguration-dependent coefficient matrices of the dynamicequations may be approximated as constant ones withoutintroducing large modeling errors Based on these constantmatrices calculation of the approximated inverse dynam-ics becomes much simpler than solving the full inversedynamics However to prevent deterioration of trackingperformance that may cause this approximation with acontroller based on the inverse dynamics [18] performance-based control methods can be used The most importantof these methods are (1) sliding mode control (2) robustcontrol (3) adaptive control and (4) adaptive-robust control

In spite of the properties of robustness claimed the realimplementation of sliding mode control techniques presentsa major drawback that is the so-called chattering effectwith dangerous high-frequency vibrations of the controlledsystemThis phenomenon is due to the fact that in real appli-cations it is not reasonable to assume that the control signalcan switch at infinite frequency Robust control means todesign a controller such that some level of performance of thecontrolled system is guaranteed independently from changesin the plant dynamics In the robust approach the controllerhas a fixed structure that yields acceptable performance fora class of plants which include the one in question Robustcontrol is widely used for 6-DoF parallel robot especiallyfor Gough-Stewart platform [16 18 19] A disadvantage ofcommonly used robust controllers is that they do not have theability to learn This inability implicates conservative designand the stability of the system is achieved at the cost of per-formance Adaptive controlmeasures performance indexes ofthe controlled system using the input the state the outputand the known disturbances [20] From the comparisonbetween themeasured performance indexes and the requiredones the adaptation mechanism modifies the parameters ofthe adjustable controller andor generates an auxiliary controlinput According to [21] we can classify adaptive controllersin three schemes adaptive control by a computed-torqueapproach adaptive control by an inertia-related approachand adaptive control based on passivity Honegger et al useadaptive feed-forward control method in [22 23] because

it is simpler to implement than other algorithms and it isrobust to noise and leads to similar or better results [22]Disadvantages of adaptive control are limited to nonlinearsystems in which the uncertain parameters appear linearlyand adaptive controls often exhibit poor transients [24] Tosimultaneously achieve the advantages of both adaptive androbust control these methods can be used together such asin [25]

In this paper a systematic methodology to find theerror of a 6-DoF robotic device with parallel kinematicand Hexaglide architecture [22] is presented This robot wasdesigned to equip the Politecnico di Milano wind tunnelwith a motion simulator for Hardware-in-the-Loop (HIL)large-amplitude dynamic test In particular this works asan emulator to reproduce the hydrodynamic interactionbetween floating bodies and sea water in the case of floatingoffshore wind turbines [26] and sailing boat scale models Inprevious works [27] the drive system is mechanically sizedwithout considering flexibility backlash and friction

The experiments that will be conducted in the civilenvironmental chamber of the wind tunnel require turbineand boats scale models to be aeroelastic In order to obtainconsistent results these models are designed to have specificmodal shapes by the way the coupling with the robot andthe transmission unitsmight affect the desired behaviour Forthese reasons the manipulator has to be as rigid as possibleand in addition all the worsening effects coming fromthe transmission units (backlashes flexibility and friction)should be properly modeled and compensated by means of asuitable control algorithm otherwise the experimental resultswould be invalidated

Amathematicalmodel of the ball-screw drive is proposedconsidering the most influential sources of nonlinearitysliding-dependent flexibility backlash and friction Usingthis model the most critical poses of the robot with respectto the kinematic mapping of the error from the joint tothe task-space are systematically investigated to obtain theworkspace positional and rotational resolution apart fromcontrol issues Finally a nonlinear adaptive-robust controlalgorithm for trajectory tracking based on the minimizationof the tracking error is described and simulated This deviceis developed for the experimental simulation of the dynamicworking conditions of the above-mentioned hydroaeroelasticstructures in our wind tunnel (specially for wind turbines)

This paper is organized as follows Section 2 introducesthe inverse kinematics and dynamics of the Hexaglide Sec-tion 3 sets out the equations of motion of a ball-screw driveactuator Section 4 describes a systematic error evaluationmethodology Section 5 presents adaptive-robust controlmethod for trajectory tracking whose performances aresimulated and compared with other methods

2 Hexaglide Robot

Among the PKMs that can provide 6DoFs attention has beenfocused on Hexaglide which is also reported in the literatureas 6-PUS kinematic architecture with parallel linear guide-ways Its linkage consists of six closed-loop kinematic chainswhich connect the fixed base to the mobile platform with


TCPx998400

z998400

y998400

bi Bi

pli

di

q i

Ox

z

siy

ni

Ai





l119894= d119894+ 119902119894u119894

with d119894= p + Rb1015840

119894minus s119894 (1)





119894expresses






119894= 119894119909∓119894119910119889119894119910+ 119894119911119889119894119911


119894= 119894119909

∓(119894119910119889119894119910+ 2

119894119910+ 119894119911119889119894119911+ 2

119894119911)Δ + (

119894119910119889119894119910+ 119894119911119889119894119911)2

ΔradicΔ

(3)


119879

= k119862+ 120596b119894

and di = [119894119909

119894119910

119894119911]119879

= a119862+ Ξb119894+ 120596(120596b

119894) whereas



120596119909 120596119910 120596119911 =

120572119909 120572119910 120572119911 =

(4)


120596 =[[[

[

0 minus120596119911120596119910

120596119911

0 minus120596119909

minus120596119910120596119909

0

]]]

]

Ξ =[[[

[

0 minus120572119911120572119910

120572119911

0 minus120572119909

minus120572119910120572119909

0

]]]

]

(5)










x TCP

y TCP

z TCP

120572

120573

120574

x TCP

y TCP

z TCP

120572

120573

120574


Platform

Hexaglide robot

Base of thepayload

Weld

Payload

B F

q1 qp1 qpp1

q2 qp2 qpp2

q3 qp3 qpp3

q4 qp4 qpp4


q6 qp6 qpp6

q1 qp1 qpp1

q2 qp2 qpp2

q3 qp3 qpp3

q4 qp4 qpp4

q6 qp6 qpp6


120579m Tm

Jm

1205791 Tg

kg

G 120578

JgKc

Ceq

Keq

xbs

Jbs120579K120579

Knut

Cnut

Mc

qi


Ball-screw

Jc

120579c




119864119896

=1

2[1198691198982

119898+ 1198691198922

1+ 1198691198882

119888+ 119869bs

2

+1198721198882

119894+119872bs

2

bs] (6)

119881

=1

2[119896119888(120579119888minus 1205791)2+ 119896120579(119902119894) (120579 minus 120579

119888)2+ 119896eq (119902119894) 119909

2

bs] (7)

119863 =1

2[119862nut (119877 + bs minus 119894)

2

+ 119862eq2

bs] (8)

120575119882ext

= 119879119898120575120579119898+ 119879119892(1205751205791minus1

120578119866120575120579119898)


(9)

where119879119898 119869119898 119869119892 and119872



119865119860119894119909




2


119892is calculated


119888and




119892

and 119865nut

119879119892= 119896119892

120579119898

119866minus 1205791minus 120579119887

120579119898

119866minus 1205791ge 120579119887

0 minus120579119887le120579119898

119866minus 1205791le 120579119887

120579119898

119866minus 1205791+ 120579119887

120579119898

119866minus 1205791le minus120579119887

(10)


L1 + qi

(a)

Kbearing Kscrew

Mbs

xbs

(b)


L1 + qi L2 minus qi

L

(a)

Kbearing Kscrew1

Mbs

xbs

KbearingKscrew2

(b)


119865nut

= 119870nut




(11)





119870120579(119902119894) =

119866screw119869screw1198711+ 119902119894

(12)




1

119870eq=

1

119870bearing+

1

119870screw1 (119902119894)

119870screw1 (119902119894) =119864119860

1198711+ 119902119894

(13)





119870eq = (1

119870bearing+

1

119870screw1 (119902119894))

minus1

+ (1

119870bearing+

1

119870screw2 (119902119894))

minus1

119870screw1 (119902119894) =119864119860

1198711+ 119902119894

119870screw2 (119902119894) =119864119860

1198712minus 119902119894

(14)



] sign (119894)

+ 119862V119894

(15)








119869119898119898= 119879119898minus

119879119892

(120578119866)

1198691198921+ 119896119888(120579119888minus 1205791) minus 119879119892= 0

119869119888119888+ 119896119888(1205791minus 120579119888) + 119896120579(119902119894) (120579119888minus 120579) = 0

119869bs + 119896120579 (119902119894) (120579 minus 120579119888) + 119877119865nut

+ 119862nut119877 (119877 + bs minus 119894) = 0


+ 119862eqbs = 0


(16)

4 Error Evaluation


minus02 minus01 0 01 02045

05

055

06

065

07

075

08

085

y (m)

z (m

) 39

93

93

49

94

94

95

95

95

96

96

96

97

97

97

98

98

98

99

99

99

10

10

10

101

101

101

102

102

102

103

103

103

104

104

104

105

105

105

106

106

106

107

107

107

108

108

108

109

109

109

11

11

11

111

111111

111 112

112112

112

113

113

114

114

115

115611116 117

117

WSd

0

5

10

15






1 2

6]119879



ΔX = [119869] Δq (17)





ΔXmax = [119869]

1

1

1

6times1

Δ119902max (18)



Desired TCPmotions

Inversekinematics

Convert tomotor

rotation


Robot(SimMechanics)

Lineartransmission

model(Simulink)

X

X

d qd q

eX

120579m

+ minus




1199100

1199110






119883119894= 119883119900119894+

11988651199055 + 11988641199054 + 11988631199053 119905 le 119905

119888

119860119894sin (2120587119891 (119905 minus 119905

119889)) 119905 gt 119905

119888

(20)


119894






∘10∘ 07 12

120573mov 75∘ 10∘ 07 12120574mov 75∘ 10∘ 07 12




119870nut = 1371198906Nm

119870bearing = 1131198906Nm

Δ = 005mm




Maximum error

119909 068mm 070mm119910 068mm 070mm119911 04mm 036mm120572 017

∘016∘

120573 010∘ 010∘

120574 005∘ 005∘

RMS error

119909 035mm 037mm119910 036mm 038mm119911 028mm 026mm120572 009∘ 008∘

120573 006∘

006∘

120574 003∘ 003∘

120579119887= 4 arcmin

ℎ119901= 32mm

119889screw = 32mm

119870119888= 39543N sdotmrad

119896119892= 752871198904N sdotmrad

119866 = 2

119871 = 16m(21)










Tm = M (q) q + f (q q) (22)




+ 120578119904119886119905 (Φminus1s)

(23)


e = qd minus q (24)


s = e + Λe (25)


s = qr minus q (26)


qr = qd + Λe (27)




Maximum error







RMS error









[

p1sdotsdotsdot6pbp7

]]

]

(28)



[

119903111990311199021

0d d d

0 119903611990361199026

]]]

]


(29)

where 1198981 1198881 1198961 119898



Ψb =[[[

[

sign (1199031) 0

d

0 sign (1199036)

]]]

]

pb =[[[[

[

1198871

1198876

]]]]

]

(30)

where 1198871 119887




p7 = [1198987 1198987119903119909 1198987119903119910 1198987119903119911 119868119909119909 119868119910119910 119868119911119911]119879

(31)


119909119909 119868119910119910 and 119868



119909119910 119868119909119911 and 119868


by

Ψ7 = J[a7

RΩ7 03times3

01times3

minusa7R R1205951205967

] (32)




a7=[[

[

119889

119889

119889+ 119892

]]

]

a7 =[[

[


119889

119889+ 119892 0 minus

119889

minus119889

119889

0

]]

]

(33)


Ω7 =[[[

[

minus1205962119910minus 1205962119911

minus119911+ 120596119909120596119910119910+ 120596119909120596119911

119911+ 120596119909120596119910

minus1205962119909minus 1205962119911

minus119909+ 120596119911120596119910

minus119910+ 120596119911120596119909119909+ 120596119911120596119910

minus1205962119909minus 1205962119910

]]]

]

1205951205967=[[[

[

119909

minus120596119911120596119910120596119910120596119911

120596119909120596119911

119910

minus120596119911120596119909

minus120596119910120596119909120596119909120596119910

119911

]]]

]

(34)



x y z0

01

02

03

04

05

06

07M

axim

um T

CP p

ositi

on er

ror (

mm

)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0002004006008

01012014016018

Max

imum

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574


x y z0

005

01

015

02

025

03

035

04

RMS

TCP

posit

ion

erro

r (m

m)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0001002003004005006007008009

RMS

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574







Inversekinematics

Adaptivecontrol

Parameterestimation

Lineartransmission

driveHexaglide

robot

HIL

Aerodynamicforces

qd

Xd

qTm

p





119881 (119905) =1


119879

KI int s 119889119905) (35)




p = ΓΨ119879s (37)

and (23) changes to


+ 120578119904119886119905 (Φminus1s)

(38)





6 Conclusions





119909

[mm]119910

[mm]119911

[mm]120572

[degree]120573

[degree]120574



RMS error 018 016 011 002 002 002


0010203040506

t (s)

q er

ror (

mm

)

Slider 1 error

(a) Slider 1

t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

Slider 2 error

q er

ror (

mm

)

(b) Slider 2

t (s)0 05 1 15 2 25 3 35 4 45 5


002040608

1Slider 3 error

q er

ror (

mm

)

(c) Slider 3

t (s)0 05 1 15 2 25 3 35 4 45 5


minus0050

00501

01502

02503

Slider 4 error

q er

ror (

mm

)

(d) Slider 4

t (s)0 05 1 15 2 25 3 35 4 45 5

minus04

minus03

minus02

minus01

0

01

02

03Slider 5 error

q er

ror (

mm

)

(e) Slider 5

t (s)0 05 1 15 2 25 3 35 4 45 5

minus03

minus02

minus01

0

01

02

03

04Slider 6 error

q er

ror (

mm

)

(f) Slider 6



minus08

minus06

minus04

minus02

0

02

04


t (s)0 05 1 15 2 25 3 35 4 45 5

xer

ror (

mm

)


t (s)

minus08

minus06

minus04

minus02

0

02

04


0 05 1 15 2 25 3 35 4 45 5

yer

ror (

mm

)


t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

zer

ror (

mm

)



minus006

minus004

minus002

0

002

004

006


t (s)0 05 1 15 2 25 3 35 4 45 5

120572er

ror (

deg

)



0002004006008


t (s)0 05 1 15 2 25 3 35 4 45 5

120573er

ror (

deg

)



0002004006008

01

t (s)


0 05 1 15 2 25 3 35 4 45 5

120574er

ror (

deg

)



x y z02468

10121416


120572 120573 120574

Erro

r (

) (er

ror lowast

100

am

plitu

de)




Competing Interests


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










TCPx998400

z998400

y998400

bi Bi

pli

di

q i

Ox

z

siy

ni

Ai





l119894= d119894+ 119902119894u119894

with d119894= p + Rb1015840

119894minus s119894 (1)





119894expresses






119894= 119894119909∓119894119910119889119894119910+ 119894119911119889119894119911


119894= 119894119909

∓(119894119910119889119894119910+ 2

119894119910+ 119894119911119889119894119911+ 2

119894119911)Δ + (

119894119910119889119894119910+ 119894119911119889119894119911)2

ΔradicΔ

(3)


119879

= k119862+ 120596b119894

and di = [119894119909

119894119910

119894119911]119879

= a119862+ Ξb119894+ 120596(120596b

119894) whereas



120596119909 120596119910 120596119911 =

120572119909 120572119910 120572119911 =

(4)


120596 =[[[

[

0 minus120596119911120596119910

120596119911

0 minus120596119909

minus120596119910120596119909

0

]]]

]

Ξ =[[[

[

0 minus120572119911120572119910

120572119911

0 minus120572119909

minus120572119910120572119909

0

]]]

]

(5)










x TCP

y TCP

z TCP

120572

120573

120574

x TCP

y TCP

z TCP

120572

120573

120574


Platform

Hexaglide robot

Base of thepayload

Weld

Payload

B F

q1 qp1 qpp1

q2 qp2 qpp2

q3 qp3 qpp3

q4 qp4 qpp4


q6 qp6 qpp6

q1 qp1 qpp1

q2 qp2 qpp2

q3 qp3 qpp3

q4 qp4 qpp4

q6 qp6 qpp6


120579m Tm

Jm

1205791 Tg

kg

G 120578

JgKc

Ceq

Keq

xbs

Jbs120579K120579

Knut

Cnut

Mc

qi


Ball-screw

Jc

120579c




119864119896

=1

2[1198691198982

119898+ 1198691198922

1+ 1198691198882

119888+ 119869bs

2

+1198721198882

119894+119872bs

2

bs] (6)

119881

=1

2[119896119888(120579119888minus 1205791)2+ 119896120579(119902119894) (120579 minus 120579

119888)2+ 119896eq (119902119894) 119909

2

bs] (7)

119863 =1

2[119862nut (119877 + bs minus 119894)

2

+ 119862eq2

bs] (8)

120575119882ext

= 119879119898120575120579119898+ 119879119892(1205751205791minus1

120578119866120575120579119898)


(9)

where119879119898 119869119898 119869119892 and119872



119865119860119894119909




2


119892is calculated


119888and




119892

and 119865nut

119879119892= 119896119892

120579119898

119866minus 1205791minus 120579119887

120579119898

119866minus 1205791ge 120579119887

0 minus120579119887le120579119898

119866minus 1205791le 120579119887

120579119898

119866minus 1205791+ 120579119887

120579119898

119866minus 1205791le minus120579119887

(10)


L1 + qi

(a)

Kbearing Kscrew

Mbs

xbs

(b)


L1 + qi L2 minus qi

L

(a)

Kbearing Kscrew1

Mbs

xbs

KbearingKscrew2

(b)


119865nut

= 119870nut




(11)





119870120579(119902119894) =

119866screw119869screw1198711+ 119902119894

(12)




1

119870eq=

1

119870bearing+

1

119870screw1 (119902119894)

119870screw1 (119902119894) =119864119860

1198711+ 119902119894

(13)





119870eq = (1

119870bearing+

1

119870screw1 (119902119894))

minus1

+ (1

119870bearing+

1

119870screw2 (119902119894))

minus1

119870screw1 (119902119894) =119864119860

1198711+ 119902119894

119870screw2 (119902119894) =119864119860

1198712minus 119902119894

(14)



] sign (119894)

+ 119862V119894

(15)








119869119898119898= 119879119898minus

119879119892

(120578119866)

1198691198921+ 119896119888(120579119888minus 1205791) minus 119879119892= 0

119869119888119888+ 119896119888(1205791minus 120579119888) + 119896120579(119902119894) (120579119888minus 120579) = 0

119869bs + 119896120579 (119902119894) (120579 minus 120579119888) + 119877119865nut

+ 119862nut119877 (119877 + bs minus 119894) = 0


+ 119862eqbs = 0


(16)

4 Error Evaluation


minus02 minus01 0 01 02045

05

055

06

065

07

075

08

085

y (m)

z (m

) 39

93

93

49

94

94

95

95

95

96

96

96

97

97

97

98

98

98

99

99

99

10

10

10

101

101

101

102

102

102

103

103

103

104

104

104

105

105

105

106

106

106

107

107

107

108

108

108

109

109

109

11

11

11

111

111111

111 112

112112

112

113

113

114

114

115

115611116 117

117

WSd

0

5

10

15






1 2

6]119879



ΔX = [119869] Δq (17)





ΔXmax = [119869]

1

1

1

6times1

Δ119902max (18)



Desired TCPmotions

Inversekinematics

Convert tomotor

rotation


Robot(SimMechanics)

Lineartransmission

model(Simulink)

X

X

d qd q

eX

120579m

+ minus




1199100

1199110






119883119894= 119883119900119894+

11988651199055 + 11988641199054 + 11988631199053 119905 le 119905

119888

119860119894sin (2120587119891 (119905 minus 119905

119889)) 119905 gt 119905

119888

(20)


119894






∘10∘ 07 12

120573mov 75∘ 10∘ 07 12120574mov 75∘ 10∘ 07 12




119870nut = 1371198906Nm

119870bearing = 1131198906Nm

Δ = 005mm




Maximum error

119909 068mm 070mm119910 068mm 070mm119911 04mm 036mm120572 017

∘016∘

120573 010∘ 010∘

120574 005∘ 005∘

RMS error

119909 035mm 037mm119910 036mm 038mm119911 028mm 026mm120572 009∘ 008∘

120573 006∘

006∘

120574 003∘ 003∘

120579119887= 4 arcmin

ℎ119901= 32mm

119889screw = 32mm

119870119888= 39543N sdotmrad

119896119892= 752871198904N sdotmrad

119866 = 2

119871 = 16m(21)










Tm = M (q) q + f (q q) (22)




+ 120578119904119886119905 (Φminus1s)

(23)


e = qd minus q (24)


s = e + Λe (25)


s = qr minus q (26)


qr = qd + Λe (27)




Maximum error







RMS error









[

p1sdotsdotsdot6pbp7

]]

]

(28)



[

119903111990311199021

0d d d

0 119903611990361199026

]]]

]


(29)

where 1198981 1198881 1198961 119898



Ψb =[[[

[

sign (1199031) 0

d

0 sign (1199036)

]]]

]

pb =[[[[

[

1198871

1198876

]]]]

]

(30)

where 1198871 119887




p7 = [1198987 1198987119903119909 1198987119903119910 1198987119903119911 119868119909119909 119868119910119910 119868119911119911]119879

(31)


119909119909 119868119910119910 and 119868



119909119910 119868119909119911 and 119868


by

Ψ7 = J[a7

RΩ7 03times3

01times3

minusa7R R1205951205967

] (32)




a7=[[

[

119889

119889

119889+ 119892

]]

]

a7 =[[

[


119889

119889+ 119892 0 minus

119889

minus119889

119889

0

]]

]

(33)


Ω7 =[[[

[

minus1205962119910minus 1205962119911

minus119911+ 120596119909120596119910119910+ 120596119909120596119911

119911+ 120596119909120596119910

minus1205962119909minus 1205962119911

minus119909+ 120596119911120596119910

minus119910+ 120596119911120596119909119909+ 120596119911120596119910

minus1205962119909minus 1205962119910

]]]

]

1205951205967=[[[

[

119909

minus120596119911120596119910120596119910120596119911

120596119909120596119911

119910

minus120596119911120596119909

minus120596119910120596119909120596119909120596119910

119911

]]]

]

(34)



x y z0

01

02

03

04

05

06

07M

axim

um T

CP p

ositi

on er

ror (

mm

)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0002004006008

01012014016018

Max

imum

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574


x y z0

005

01

015

02

025

03

035

04

RMS

TCP

posit

ion

erro

r (m

m)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0001002003004005006007008009

RMS

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574







Inversekinematics

Adaptivecontrol

Parameterestimation

Lineartransmission

driveHexaglide

robot

HIL

Aerodynamicforces

qd

Xd

qTm

p





119881 (119905) =1


119879

KI int s 119889119905) (35)




p = ΓΨ119879s (37)

and (23) changes to


+ 120578119904119886119905 (Φminus1s)

(38)





6 Conclusions





119909

[mm]119910

[mm]119911

[mm]120572

[degree]120573

[degree]120574



RMS error 018 016 011 002 002 002


0010203040506

t (s)

q er

ror (

mm

)

Slider 1 error

(a) Slider 1

t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

Slider 2 error

q er

ror (

mm

)

(b) Slider 2

t (s)0 05 1 15 2 25 3 35 4 45 5


002040608

1Slider 3 error

q er

ror (

mm

)

(c) Slider 3

t (s)0 05 1 15 2 25 3 35 4 45 5


minus0050

00501

01502

02503

Slider 4 error

q er

ror (

mm

)

(d) Slider 4

t (s)0 05 1 15 2 25 3 35 4 45 5

minus04

minus03

minus02

minus01

0

01

02

03Slider 5 error

q er

ror (

mm

)

(e) Slider 5

t (s)0 05 1 15 2 25 3 35 4 45 5

minus03

minus02

minus01

0

01

02

03

04Slider 6 error

q er

ror (

mm

)

(f) Slider 6



minus08

minus06

minus04

minus02

0

02

04


t (s)0 05 1 15 2 25 3 35 4 45 5

xer

ror (

mm

)


t (s)

minus08

minus06

minus04

minus02

0

02

04


0 05 1 15 2 25 3 35 4 45 5

yer

ror (

mm

)


t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

zer

ror (

mm

)



minus006

minus004

minus002

0

002

004

006


t (s)0 05 1 15 2 25 3 35 4 45 5

120572er

ror (

deg

)



0002004006008


t (s)0 05 1 15 2 25 3 35 4 45 5

120573er

ror (

deg

)



0002004006008

01

t (s)


0 05 1 15 2 25 3 35 4 45 5

120574er

ror (

deg

)



x y z02468

10121416


120572 120573 120574

Erro

r (

) (er

ror lowast

100

am

plitu

de)




Competing Interests


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x TCP

y TCP

z TCP

120572

120573

120574

x TCP

y TCP

z TCP

120572

120573

120574


Platform

Hexaglide robot

Base of thepayload

Weld

Payload

B F

q1 qp1 qpp1

q2 qp2 qpp2

q3 qp3 qpp3

q4 qp4 qpp4


q6 qp6 qpp6

q1 qp1 qpp1

q2 qp2 qpp2

q3 qp3 qpp3

q4 qp4 qpp4

q6 qp6 qpp6


120579m Tm

Jm

1205791 Tg

kg

G 120578

JgKc

Ceq

Keq

xbs

Jbs120579K120579

Knut

Cnut

Mc

qi


Ball-screw

Jc

120579c




119864119896

=1

2[1198691198982

119898+ 1198691198922

1+ 1198691198882

119888+ 119869bs

2

+1198721198882

119894+119872bs

2

bs] (6)

119881

=1

2[119896119888(120579119888minus 1205791)2+ 119896120579(119902119894) (120579 minus 120579

119888)2+ 119896eq (119902119894) 119909

2

bs] (7)

119863 =1

2[119862nut (119877 + bs minus 119894)

2

+ 119862eq2

bs] (8)

120575119882ext

= 119879119898120575120579119898+ 119879119892(1205751205791minus1

120578119866120575120579119898)


(9)

where119879119898 119869119898 119869119892 and119872



119865119860119894119909




2


119892is calculated


119888and




119892

and 119865nut

119879119892= 119896119892

120579119898

119866minus 1205791minus 120579119887

120579119898

119866minus 1205791ge 120579119887

0 minus120579119887le120579119898

119866minus 1205791le 120579119887

120579119898

119866minus 1205791+ 120579119887

120579119898

119866minus 1205791le minus120579119887

(10)


L1 + qi

(a)

Kbearing Kscrew

Mbs

xbs

(b)


L1 + qi L2 minus qi

L

(a)

Kbearing Kscrew1

Mbs

xbs

KbearingKscrew2

(b)


119865nut

= 119870nut




(11)





119870120579(119902119894) =

119866screw119869screw1198711+ 119902119894

(12)




1

119870eq=

1

119870bearing+

1

119870screw1 (119902119894)

119870screw1 (119902119894) =119864119860

1198711+ 119902119894

(13)





119870eq = (1

119870bearing+

1

119870screw1 (119902119894))

minus1

+ (1

119870bearing+

1

119870screw2 (119902119894))

minus1

119870screw1 (119902119894) =119864119860

1198711+ 119902119894

119870screw2 (119902119894) =119864119860

1198712minus 119902119894

(14)



] sign (119894)

+ 119862V119894

(15)








119869119898119898= 119879119898minus

119879119892

(120578119866)

1198691198921+ 119896119888(120579119888minus 1205791) minus 119879119892= 0

119869119888119888+ 119896119888(1205791minus 120579119888) + 119896120579(119902119894) (120579119888minus 120579) = 0

119869bs + 119896120579 (119902119894) (120579 minus 120579119888) + 119877119865nut

+ 119862nut119877 (119877 + bs minus 119894) = 0


+ 119862eqbs = 0


(16)

4 Error Evaluation


minus02 minus01 0 01 02045

05

055

06

065

07

075

08

085

y (m)

z (m

) 39

93

93

49

94

94

95

95

95

96

96

96

97

97

97

98

98

98

99

99

99

10

10

10

101

101

101

102

102

102

103

103

103

104

104

104

105

105

105

106

106

106

107

107

107

108

108

108

109

109

109

11

11

11

111

111111

111 112

112112

112

113

113

114

114

115

115611116 117

117

WSd

0

5

10

15






1 2

6]119879



ΔX = [119869] Δq (17)





ΔXmax = [119869]

1

1

1

6times1

Δ119902max (18)



Desired TCPmotions

Inversekinematics

Convert tomotor

rotation


Robot(SimMechanics)

Lineartransmission

model(Simulink)

X

X

d qd q

eX

120579m

+ minus




1199100

1199110






119883119894= 119883119900119894+

11988651199055 + 11988641199054 + 11988631199053 119905 le 119905

119888

119860119894sin (2120587119891 (119905 minus 119905

119889)) 119905 gt 119905

119888

(20)


119894






∘10∘ 07 12

120573mov 75∘ 10∘ 07 12120574mov 75∘ 10∘ 07 12




119870nut = 1371198906Nm

119870bearing = 1131198906Nm

Δ = 005mm




Maximum error

119909 068mm 070mm119910 068mm 070mm119911 04mm 036mm120572 017

∘016∘

120573 010∘ 010∘

120574 005∘ 005∘

RMS error

119909 035mm 037mm119910 036mm 038mm119911 028mm 026mm120572 009∘ 008∘

120573 006∘

006∘

120574 003∘ 003∘

120579119887= 4 arcmin

ℎ119901= 32mm

119889screw = 32mm

119870119888= 39543N sdotmrad

119896119892= 752871198904N sdotmrad

119866 = 2

119871 = 16m(21)










Tm = M (q) q + f (q q) (22)




+ 120578119904119886119905 (Φminus1s)

(23)


e = qd minus q (24)


s = e + Λe (25)


s = qr minus q (26)


qr = qd + Λe (27)




Maximum error







RMS error









[

p1sdotsdotsdot6pbp7

]]

]

(28)



[

119903111990311199021

0d d d

0 119903611990361199026

]]]

]


(29)

where 1198981 1198881 1198961 119898



Ψb =[[[

[

sign (1199031) 0

d

0 sign (1199036)

]]]

]

pb =[[[[

[

1198871

1198876

]]]]

]

(30)

where 1198871 119887




p7 = [1198987 1198987119903119909 1198987119903119910 1198987119903119911 119868119909119909 119868119910119910 119868119911119911]119879

(31)


119909119909 119868119910119910 and 119868



119909119910 119868119909119911 and 119868


by

Ψ7 = J[a7

RΩ7 03times3

01times3

minusa7R R1205951205967

] (32)




a7=[[

[

119889

119889

119889+ 119892

]]

]

a7 =[[

[


119889

119889+ 119892 0 minus

119889

minus119889

119889

0

]]

]

(33)


Ω7 =[[[

[

minus1205962119910minus 1205962119911

minus119911+ 120596119909120596119910119910+ 120596119909120596119911

119911+ 120596119909120596119910

minus1205962119909minus 1205962119911

minus119909+ 120596119911120596119910

minus119910+ 120596119911120596119909119909+ 120596119911120596119910

minus1205962119909minus 1205962119910

]]]

]

1205951205967=[[[

[

119909

minus120596119911120596119910120596119910120596119911

120596119909120596119911

119910

minus120596119911120596119909

minus120596119910120596119909120596119909120596119910

119911

]]]

]

(34)



x y z0

01

02

03

04

05

06

07M

axim

um T

CP p

ositi

on er

ror (

mm

)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0002004006008

01012014016018

Max

imum

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574


x y z0

005

01

015

02

025

03

035

04

RMS

TCP

posit

ion

erro

r (m

m)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0001002003004005006007008009

RMS

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574







Inversekinematics

Adaptivecontrol

Parameterestimation

Lineartransmission

driveHexaglide

robot

HIL

Aerodynamicforces

qd

Xd

qTm

p





119881 (119905) =1


119879

KI int s 119889119905) (35)




p = ΓΨ119879s (37)

and (23) changes to


+ 120578119904119886119905 (Φminus1s)

(38)





6 Conclusions





119909

[mm]119910

[mm]119911

[mm]120572

[degree]120573

[degree]120574



RMS error 018 016 011 002 002 002


0010203040506

t (s)

q er

ror (

mm

)

Slider 1 error

(a) Slider 1

t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

Slider 2 error

q er

ror (

mm

)

(b) Slider 2

t (s)0 05 1 15 2 25 3 35 4 45 5


002040608

1Slider 3 error

q er

ror (

mm

)

(c) Slider 3

t (s)0 05 1 15 2 25 3 35 4 45 5


minus0050

00501

01502

02503

Slider 4 error

q er

ror (

mm

)

(d) Slider 4

t (s)0 05 1 15 2 25 3 35 4 45 5

minus04

minus03

minus02

minus01

0

01

02

03Slider 5 error

q er

ror (

mm

)

(e) Slider 5

t (s)0 05 1 15 2 25 3 35 4 45 5

minus03

minus02

minus01

0

01

02

03

04Slider 6 error

q er

ror (

mm

)

(f) Slider 6



minus08

minus06

minus04

minus02

0

02

04


t (s)0 05 1 15 2 25 3 35 4 45 5

xer

ror (

mm

)


t (s)

minus08

minus06

minus04

minus02

0

02

04


0 05 1 15 2 25 3 35 4 45 5

yer

ror (

mm

)


t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

zer

ror (

mm

)



minus006

minus004

minus002

0

002

004

006


t (s)0 05 1 15 2 25 3 35 4 45 5

120572er

ror (

deg

)



0002004006008


t (s)0 05 1 15 2 25 3 35 4 45 5

120573er

ror (

deg

)



0002004006008

01

t (s)


0 05 1 15 2 25 3 35 4 45 5

120574er

ror (

deg

)



x y z02468

10121416


120572 120573 120574

Erro

r (

) (er

ror lowast

100

am

plitu

de)




Competing Interests


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










L1 + qi

(a)

Kbearing Kscrew

Mbs

xbs

(b)


L1 + qi L2 minus qi

L

(a)

Kbearing Kscrew1

Mbs

xbs

KbearingKscrew2

(b)


119865nut

= 119870nut




(11)





119870120579(119902119894) =

119866screw119869screw1198711+ 119902119894

(12)




1

119870eq=

1

119870bearing+

1

119870screw1 (119902119894)

119870screw1 (119902119894) =119864119860

1198711+ 119902119894

(13)





119870eq = (1

119870bearing+

1

119870screw1 (119902119894))

minus1

+ (1

119870bearing+

1

119870screw2 (119902119894))

minus1

119870screw1 (119902119894) =119864119860

1198711+ 119902119894

119870screw2 (119902119894) =119864119860

1198712minus 119902119894

(14)



] sign (119894)

+ 119862V119894

(15)








119869119898119898= 119879119898minus

119879119892

(120578119866)

1198691198921+ 119896119888(120579119888minus 1205791) minus 119879119892= 0

119869119888119888+ 119896119888(1205791minus 120579119888) + 119896120579(119902119894) (120579119888minus 120579) = 0

119869bs + 119896120579 (119902119894) (120579 minus 120579119888) + 119877119865nut

+ 119862nut119877 (119877 + bs minus 119894) = 0


+ 119862eqbs = 0


(16)

4 Error Evaluation


minus02 minus01 0 01 02045

05

055

06

065

07

075

08

085

y (m)

z (m

) 39

93

93

49

94

94

95

95

95

96

96

96

97

97

97

98

98

98

99

99

99

10

10

10

101

101

101

102

102

102

103

103

103

104

104

104

105

105

105

106

106

106

107

107

107

108

108

108

109

109

109

11

11

11

111

111111

111 112

112112

112

113

113

114

114

115

115611116 117

117

WSd

0

5

10

15






1 2

6]119879



ΔX = [119869] Δq (17)





ΔXmax = [119869]

1

1

1

6times1

Δ119902max (18)



Desired TCPmotions

Inversekinematics

Convert tomotor

rotation


Robot(SimMechanics)

Lineartransmission

model(Simulink)

X

X

d qd q

eX

120579m

+ minus




1199100

1199110






119883119894= 119883119900119894+

11988651199055 + 11988641199054 + 11988631199053 119905 le 119905

119888

119860119894sin (2120587119891 (119905 minus 119905

119889)) 119905 gt 119905

119888

(20)


119894






∘10∘ 07 12

120573mov 75∘ 10∘ 07 12120574mov 75∘ 10∘ 07 12




119870nut = 1371198906Nm

119870bearing = 1131198906Nm

Δ = 005mm




Maximum error

119909 068mm 070mm119910 068mm 070mm119911 04mm 036mm120572 017

∘016∘

120573 010∘ 010∘

120574 005∘ 005∘

RMS error

119909 035mm 037mm119910 036mm 038mm119911 028mm 026mm120572 009∘ 008∘

120573 006∘

006∘

120574 003∘ 003∘

120579119887= 4 arcmin

ℎ119901= 32mm

119889screw = 32mm

119870119888= 39543N sdotmrad

119896119892= 752871198904N sdotmrad

119866 = 2

119871 = 16m(21)










Tm = M (q) q + f (q q) (22)




+ 120578119904119886119905 (Φminus1s)

(23)


e = qd minus q (24)


s = e + Λe (25)


s = qr minus q (26)


qr = qd + Λe (27)




Maximum error







RMS error









[

p1sdotsdotsdot6pbp7

]]

]

(28)



[

119903111990311199021

0d d d

0 119903611990361199026

]]]

]


(29)

where 1198981 1198881 1198961 119898



Ψb =[[[

[

sign (1199031) 0

d

0 sign (1199036)

]]]

]

pb =[[[[

[

1198871

1198876

]]]]

]

(30)

where 1198871 119887




p7 = [1198987 1198987119903119909 1198987119903119910 1198987119903119911 119868119909119909 119868119910119910 119868119911119911]119879

(31)


119909119909 119868119910119910 and 119868



119909119910 119868119909119911 and 119868


by

Ψ7 = J[a7

RΩ7 03times3

01times3

minusa7R R1205951205967

] (32)




a7=[[

[

119889

119889

119889+ 119892

]]

]

a7 =[[

[


119889

119889+ 119892 0 minus

119889

minus119889

119889

0

]]

]

(33)


Ω7 =[[[

[

minus1205962119910minus 1205962119911

minus119911+ 120596119909120596119910119910+ 120596119909120596119911

119911+ 120596119909120596119910

minus1205962119909minus 1205962119911

minus119909+ 120596119911120596119910

minus119910+ 120596119911120596119909119909+ 120596119911120596119910

minus1205962119909minus 1205962119910

]]]

]

1205951205967=[[[

[

119909

minus120596119911120596119910120596119910120596119911

120596119909120596119911

119910

minus120596119911120596119909

minus120596119910120596119909120596119909120596119910

119911

]]]

]

(34)



x y z0

01

02

03

04

05

06

07M

axim

um T

CP p

ositi

on er

ror (

mm

)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0002004006008

01012014016018

Max

imum

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574


x y z0

005

01

015

02

025

03

035

04

RMS

TCP

posit

ion

erro

r (m

m)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0001002003004005006007008009

RMS

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574







Inversekinematics

Adaptivecontrol

Parameterestimation

Lineartransmission

driveHexaglide

robot

HIL

Aerodynamicforces

qd

Xd

qTm

p





119881 (119905) =1


119879

KI int s 119889119905) (35)




p = ΓΨ119879s (37)

and (23) changes to


+ 120578119904119886119905 (Φminus1s)

(38)





6 Conclusions





119909

[mm]119910

[mm]119911

[mm]120572

[degree]120573

[degree]120574



RMS error 018 016 011 002 002 002


0010203040506

t (s)

q er

ror (

mm

)

Slider 1 error

(a) Slider 1

t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

Slider 2 error

q er

ror (

mm

)

(b) Slider 2

t (s)0 05 1 15 2 25 3 35 4 45 5


002040608

1Slider 3 error

q er

ror (

mm

)

(c) Slider 3

t (s)0 05 1 15 2 25 3 35 4 45 5


minus0050

00501

01502

02503

Slider 4 error

q er

ror (

mm

)

(d) Slider 4

t (s)0 05 1 15 2 25 3 35 4 45 5

minus04

minus03

minus02

minus01

0

01

02

03Slider 5 error

q er

ror (

mm

)

(e) Slider 5

t (s)0 05 1 15 2 25 3 35 4 45 5

minus03

minus02

minus01

0

01

02

03

04Slider 6 error

q er

ror (

mm

)

(f) Slider 6



minus08

minus06

minus04

minus02

0

02

04


t (s)0 05 1 15 2 25 3 35 4 45 5

xer

ror (

mm

)


t (s)

minus08

minus06

minus04

minus02

0

02

04


0 05 1 15 2 25 3 35 4 45 5

yer

ror (

mm

)


t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

zer

ror (

mm

)



minus006

minus004

minus002

0

002

004

006


t (s)0 05 1 15 2 25 3 35 4 45 5

120572er

ror (

deg

)



0002004006008


t (s)0 05 1 15 2 25 3 35 4 45 5

120573er

ror (

deg

)



0002004006008

01

t (s)


0 05 1 15 2 25 3 35 4 45 5

120574er

ror (

deg

)



x y z02468

10121416


120572 120573 120574

Erro

r (

) (er

ror lowast

100

am

plitu

de)




Competing Interests


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119870eq = (1

119870bearing+

1

119870screw1 (119902119894))

minus1

+ (1

119870bearing+

1

119870screw2 (119902119894))

minus1

119870screw1 (119902119894) =119864119860

1198711+ 119902119894

119870screw2 (119902119894) =119864119860

1198712minus 119902119894

(14)



] sign (119894)

+ 119862V119894

(15)








119869119898119898= 119879119898minus

119879119892

(120578119866)

1198691198921+ 119896119888(120579119888minus 1205791) minus 119879119892= 0

119869119888119888+ 119896119888(1205791minus 120579119888) + 119896120579(119902119894) (120579119888minus 120579) = 0

119869bs + 119896120579 (119902119894) (120579 minus 120579119888) + 119877119865nut

+ 119862nut119877 (119877 + bs minus 119894) = 0


+ 119862eqbs = 0


(16)

4 Error Evaluation


minus02 minus01 0 01 02045

05

055

06

065

07

075

08

085

y (m)

z (m

) 39

93

93

49

94

94

95

95

95

96

96

96

97

97

97

98

98

98

99

99

99

10

10

10

101

101

101

102

102

102

103

103

103

104

104

104

105

105

105

106

106

106

107

107

107

108

108

108

109

109

109

11

11

11

111

111111

111 112

112112

112

113

113

114

114

115

115611116 117

117

WSd

0

5

10

15






1 2

6]119879



ΔX = [119869] Δq (17)





ΔXmax = [119869]

1

1

1

6times1

Δ119902max (18)



Desired TCPmotions

Inversekinematics

Convert tomotor

rotation


Robot(SimMechanics)

Lineartransmission

model(Simulink)

X

X

d qd q

eX

120579m

+ minus




1199100

1199110






119883119894= 119883119900119894+

11988651199055 + 11988641199054 + 11988631199053 119905 le 119905

119888

119860119894sin (2120587119891 (119905 minus 119905

119889)) 119905 gt 119905

119888

(20)


119894






∘10∘ 07 12

120573mov 75∘ 10∘ 07 12120574mov 75∘ 10∘ 07 12




119870nut = 1371198906Nm

119870bearing = 1131198906Nm

Δ = 005mm




Maximum error

119909 068mm 070mm119910 068mm 070mm119911 04mm 036mm120572 017

∘016∘

120573 010∘ 010∘

120574 005∘ 005∘

RMS error

119909 035mm 037mm119910 036mm 038mm119911 028mm 026mm120572 009∘ 008∘

120573 006∘

006∘

120574 003∘ 003∘

120579119887= 4 arcmin

ℎ119901= 32mm

119889screw = 32mm

119870119888= 39543N sdotmrad

119896119892= 752871198904N sdotmrad

119866 = 2

119871 = 16m(21)










Tm = M (q) q + f (q q) (22)




+ 120578119904119886119905 (Φminus1s)

(23)


e = qd minus q (24)


s = e + Λe (25)


s = qr minus q (26)


qr = qd + Λe (27)




Maximum error







RMS error









[

p1sdotsdotsdot6pbp7

]]

]

(28)



[

119903111990311199021

0d d d

0 119903611990361199026

]]]

]


(29)

where 1198981 1198881 1198961 119898



Ψb =[[[

[

sign (1199031) 0

d

0 sign (1199036)

]]]

]

pb =[[[[

[

1198871

1198876

]]]]

]

(30)

where 1198871 119887




p7 = [1198987 1198987119903119909 1198987119903119910 1198987119903119911 119868119909119909 119868119910119910 119868119911119911]119879

(31)


119909119909 119868119910119910 and 119868



119909119910 119868119909119911 and 119868


by

Ψ7 = J[a7

RΩ7 03times3

01times3

minusa7R R1205951205967

] (32)




a7=[[

[

119889

119889

119889+ 119892

]]

]

a7 =[[

[


119889

119889+ 119892 0 minus

119889

minus119889

119889

0

]]

]

(33)


Ω7 =[[[

[

minus1205962119910minus 1205962119911

minus119911+ 120596119909120596119910119910+ 120596119909120596119911

119911+ 120596119909120596119910

minus1205962119909minus 1205962119911

minus119909+ 120596119911120596119910

minus119910+ 120596119911120596119909119909+ 120596119911120596119910

minus1205962119909minus 1205962119910

]]]

]

1205951205967=[[[

[

119909

minus120596119911120596119910120596119910120596119911

120596119909120596119911

119910

minus120596119911120596119909

minus120596119910120596119909120596119909120596119910

119911

]]]

]

(34)



x y z0

01

02

03

04

05

06

07M

axim

um T

CP p

ositi

on er

ror (

mm

)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0002004006008

01012014016018

Max

imum

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574


x y z0

005

01

015

02

025

03

035

04

RMS

TCP

posit

ion

erro

r (m

m)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0001002003004005006007008009

RMS

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574







Inversekinematics

Adaptivecontrol

Parameterestimation

Lineartransmission

driveHexaglide

robot

HIL

Aerodynamicforces

qd

Xd

qTm

p





119881 (119905) =1


119879

KI int s 119889119905) (35)




p = ΓΨ119879s (37)

and (23) changes to


+ 120578119904119886119905 (Φminus1s)

(38)





6 Conclusions





119909

[mm]119910

[mm]119911

[mm]120572

[degree]120573

[degree]120574



RMS error 018 016 011 002 002 002


0010203040506

t (s)

q er

ror (

mm

)

Slider 1 error

(a) Slider 1

t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

Slider 2 error

q er

ror (

mm

)

(b) Slider 2

t (s)0 05 1 15 2 25 3 35 4 45 5


002040608

1Slider 3 error

q er

ror (

mm

)

(c) Slider 3

t (s)0 05 1 15 2 25 3 35 4 45 5


minus0050

00501

01502

02503

Slider 4 error

q er

ror (

mm

)

(d) Slider 4

t (s)0 05 1 15 2 25 3 35 4 45 5

minus04

minus03

minus02

minus01

0

01

02

03Slider 5 error

q er

ror (

mm

)

(e) Slider 5

t (s)0 05 1 15 2 25 3 35 4 45 5

minus03

minus02

minus01

0

01

02

03

04Slider 6 error

q er

ror (

mm

)

(f) Slider 6



minus08

minus06

minus04

minus02

0

02

04


t (s)0 05 1 15 2 25 3 35 4 45 5

xer

ror (

mm

)


t (s)

minus08

minus06

minus04

minus02

0

02

04


0 05 1 15 2 25 3 35 4 45 5

yer

ror (

mm

)


t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

zer

ror (

mm

)



minus006

minus004

minus002

0

002

004

006


t (s)0 05 1 15 2 25 3 35 4 45 5

120572er

ror (

deg

)



0002004006008


t (s)0 05 1 15 2 25 3 35 4 45 5

120573er

ror (

deg

)



0002004006008

01

t (s)


0 05 1 15 2 25 3 35 4 45 5

120574er

ror (

deg

)



x y z02468

10121416


120572 120573 120574

Erro

r (

) (er

ror lowast

100

am

plitu

de)




Competing Interests


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Desired TCPmotions

Inversekinematics

Convert tomotor

rotation


Robot(SimMechanics)

Lineartransmission

model(Simulink)

X

X

d qd q

eX

120579m

+ minus




1199100

1199110






119883119894= 119883119900119894+

11988651199055 + 11988641199054 + 11988631199053 119905 le 119905

119888

119860119894sin (2120587119891 (119905 minus 119905

119889)) 119905 gt 119905

119888

(20)


119894






∘10∘ 07 12

120573mov 75∘ 10∘ 07 12120574mov 75∘ 10∘ 07 12




119870nut = 1371198906Nm

119870bearing = 1131198906Nm

Δ = 005mm




Maximum error

119909 068mm 070mm119910 068mm 070mm119911 04mm 036mm120572 017

∘016∘

120573 010∘ 010∘

120574 005∘ 005∘

RMS error

119909 035mm 037mm119910 036mm 038mm119911 028mm 026mm120572 009∘ 008∘

120573 006∘

006∘

120574 003∘ 003∘

120579119887= 4 arcmin

ℎ119901= 32mm

119889screw = 32mm

119870119888= 39543N sdotmrad

119896119892= 752871198904N sdotmrad

119866 = 2

119871 = 16m(21)










Tm = M (q) q + f (q q) (22)




+ 120578119904119886119905 (Φminus1s)

(23)


e = qd minus q (24)


s = e + Λe (25)


s = qr minus q (26)


qr = qd + Λe (27)




Maximum error







RMS error









[

p1sdotsdotsdot6pbp7

]]

]

(28)



[

119903111990311199021

0d d d

0 119903611990361199026

]]]

]


(29)

where 1198981 1198881 1198961 119898



Ψb =[[[

[

sign (1199031) 0

d

0 sign (1199036)

]]]

]

pb =[[[[

[

1198871

1198876

]]]]

]

(30)

where 1198871 119887




p7 = [1198987 1198987119903119909 1198987119903119910 1198987119903119911 119868119909119909 119868119910119910 119868119911119911]119879

(31)


119909119909 119868119910119910 and 119868



119909119910 119868119909119911 and 119868


by

Ψ7 = J[a7

RΩ7 03times3

01times3

minusa7R R1205951205967

] (32)




a7=[[

[

119889

119889

119889+ 119892

]]

]

a7 =[[

[


119889

119889+ 119892 0 minus

119889

minus119889

119889

0

]]

]

(33)


Ω7 =[[[

[

minus1205962119910minus 1205962119911

minus119911+ 120596119909120596119910119910+ 120596119909120596119911

119911+ 120596119909120596119910

minus1205962119909minus 1205962119911

minus119909+ 120596119911120596119910

minus119910+ 120596119911120596119909119909+ 120596119911120596119910

minus1205962119909minus 1205962119910

]]]

]

1205951205967=[[[

[

119909

minus120596119911120596119910120596119910120596119911

120596119909120596119911

119910

minus120596119911120596119909

minus120596119910120596119909120596119909120596119910

119911

]]]

]

(34)



x y z0

01

02

03

04

05

06

07M

axim

um T

CP p

ositi

on er

ror (

mm

)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0002004006008

01012014016018

Max

imum

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574


x y z0

005

01

015

02

025

03

035

04

RMS

TCP

posit

ion

erro

r (m

m)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0001002003004005006007008009

RMS

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574







Inversekinematics

Adaptivecontrol

Parameterestimation

Lineartransmission

driveHexaglide

robot

HIL

Aerodynamicforces

qd

Xd

qTm

p





119881 (119905) =1


119879

KI int s 119889119905) (35)




p = ΓΨ119879s (37)

and (23) changes to


+ 120578119904119886119905 (Φminus1s)

(38)





6 Conclusions





119909

[mm]119910

[mm]119911

[mm]120572

[degree]120573

[degree]120574



RMS error 018 016 011 002 002 002


0010203040506

t (s)

q er

ror (

mm

)

Slider 1 error

(a) Slider 1

t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

Slider 2 error

q er

ror (

mm

)

(b) Slider 2

t (s)0 05 1 15 2 25 3 35 4 45 5


002040608

1Slider 3 error

q er

ror (

mm

)

(c) Slider 3

t (s)0 05 1 15 2 25 3 35 4 45 5


minus0050

00501

01502

02503

Slider 4 error

q er

ror (

mm

)

(d) Slider 4

t (s)0 05 1 15 2 25 3 35 4 45 5

minus04

minus03

minus02

minus01

0

01

02

03Slider 5 error

q er

ror (

mm

)

(e) Slider 5

t (s)0 05 1 15 2 25 3 35 4 45 5

minus03

minus02

minus01

0

01

02

03

04Slider 6 error

q er

ror (

mm

)

(f) Slider 6



minus08

minus06

minus04

minus02

0

02

04


t (s)0 05 1 15 2 25 3 35 4 45 5

xer

ror (

mm

)


t (s)

minus08

minus06

minus04

minus02

0

02

04


0 05 1 15 2 25 3 35 4 45 5

yer

ror (

mm

)


t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

zer

ror (

mm

)



minus006

minus004

minus002

0

002

004

006


t (s)0 05 1 15 2 25 3 35 4 45 5

120572er

ror (

deg

)



0002004006008


t (s)0 05 1 15 2 25 3 35 4 45 5

120573er

ror (

deg

)



0002004006008

01

t (s)


0 05 1 15 2 25 3 35 4 45 5

120574er

ror (

deg

)



x y z02468

10121416


120572 120573 120574

Erro

r (

) (er

ror lowast

100

am

plitu

de)




Competing Interests


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Maximum error

119909 068mm 070mm119910 068mm 070mm119911 04mm 036mm120572 017

∘016∘

120573 010∘ 010∘

120574 005∘ 005∘

RMS error

119909 035mm 037mm119910 036mm 038mm119911 028mm 026mm120572 009∘ 008∘

120573 006∘

006∘

120574 003∘ 003∘

120579119887= 4 arcmin

ℎ119901= 32mm

119889screw = 32mm

119870119888= 39543N sdotmrad

119896119892= 752871198904N sdotmrad

119866 = 2

119871 = 16m(21)










Tm = M (q) q + f (q q) (22)




+ 120578119904119886119905 (Φminus1s)

(23)


e = qd minus q (24)


s = e + Λe (25)


s = qr minus q (26)


qr = qd + Λe (27)




Maximum error







RMS error









[

p1sdotsdotsdot6pbp7

]]

]

(28)



[

119903111990311199021

0d d d

0 119903611990361199026

]]]

]


(29)

where 1198981 1198881 1198961 119898



Ψb =[[[

[

sign (1199031) 0

d

0 sign (1199036)

]]]

]

pb =[[[[

[

1198871

1198876

]]]]

]

(30)

where 1198871 119887




p7 = [1198987 1198987119903119909 1198987119903119910 1198987119903119911 119868119909119909 119868119910119910 119868119911119911]119879

(31)


119909119909 119868119910119910 and 119868



119909119910 119868119909119911 and 119868


by

Ψ7 = J[a7

RΩ7 03times3

01times3

minusa7R R1205951205967

] (32)




a7=[[

[

119889

119889

119889+ 119892

]]

]

a7 =[[

[


119889

119889+ 119892 0 minus

119889

minus119889

119889

0

]]

]

(33)


Ω7 =[[[

[

minus1205962119910minus 1205962119911

minus119911+ 120596119909120596119910119910+ 120596119909120596119911

119911+ 120596119909120596119910

minus1205962119909minus 1205962119911

minus119909+ 120596119911120596119910

minus119910+ 120596119911120596119909119909+ 120596119911120596119910

minus1205962119909minus 1205962119910

]]]

]

1205951205967=[[[

[

119909

minus120596119911120596119910120596119910120596119911

120596119909120596119911

119910

minus120596119911120596119909

minus120596119910120596119909120596119909120596119910

119911

]]]

]

(34)



x y z0

01

02

03

04

05

06

07M

axim

um T

CP p

ositi

on er

ror (

mm

)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0002004006008

01012014016018

Max

imum

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574


x y z0

005

01

015

02

025

03

035

04

RMS

TCP

posit

ion

erro

r (m

m)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0001002003004005006007008009

RMS

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574







Inversekinematics

Adaptivecontrol

Parameterestimation

Lineartransmission

driveHexaglide

robot

HIL

Aerodynamicforces

qd

Xd

qTm

p





119881 (119905) =1


119879

KI int s 119889119905) (35)




p = ΓΨ119879s (37)

and (23) changes to


+ 120578119904119886119905 (Φminus1s)

(38)





6 Conclusions





119909

[mm]119910

[mm]119911

[mm]120572

[degree]120573

[degree]120574



RMS error 018 016 011 002 002 002


0010203040506

t (s)

q er

ror (

mm

)

Slider 1 error

(a) Slider 1

t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

Slider 2 error

q er

ror (

mm

)

(b) Slider 2

t (s)0 05 1 15 2 25 3 35 4 45 5


002040608

1Slider 3 error

q er

ror (

mm

)

(c) Slider 3

t (s)0 05 1 15 2 25 3 35 4 45 5


minus0050

00501

01502

02503

Slider 4 error

q er

ror (

mm

)

(d) Slider 4

t (s)0 05 1 15 2 25 3 35 4 45 5

minus04

minus03

minus02

minus01

0

01

02

03Slider 5 error

q er

ror (

mm

)

(e) Slider 5

t (s)0 05 1 15 2 25 3 35 4 45 5

minus03

minus02

minus01

0

01

02

03

04Slider 6 error

q er

ror (

mm

)

(f) Slider 6



minus08

minus06

minus04

minus02

0

02

04


t (s)0 05 1 15 2 25 3 35 4 45 5

xer

ror (

mm

)


t (s)

minus08

minus06

minus04

minus02

0

02

04


0 05 1 15 2 25 3 35 4 45 5

yer

ror (

mm

)


t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

zer

ror (

mm

)



minus006

minus004

minus002

0

002

004

006


t (s)0 05 1 15 2 25 3 35 4 45 5

120572er

ror (

deg

)



0002004006008


t (s)0 05 1 15 2 25 3 35 4 45 5

120573er

ror (

deg

)



0002004006008

01

t (s)


0 05 1 15 2 25 3 35 4 45 5

120574er

ror (

deg

)



x y z02468

10121416


120572 120573 120574

Erro

r (

) (er

ror lowast

100

am

plitu

de)




Competing Interests


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Maximum error







RMS error









[

p1sdotsdotsdot6pbp7

]]

]

(28)



[

119903111990311199021

0d d d

0 119903611990361199026

]]]

]


(29)

where 1198981 1198881 1198961 119898



Ψb =[[[

[

sign (1199031) 0

d

0 sign (1199036)

]]]

]

pb =[[[[

[

1198871

1198876

]]]]

]

(30)

where 1198871 119887




p7 = [1198987 1198987119903119909 1198987119903119910 1198987119903119911 119868119909119909 119868119910119910 119868119911119911]119879

(31)


119909119909 119868119910119910 and 119868



119909119910 119868119909119911 and 119868


by

Ψ7 = J[a7

RΩ7 03times3

01times3

minusa7R R1205951205967

] (32)




a7=[[

[

119889

119889

119889+ 119892

]]

]

a7 =[[

[


119889

119889+ 119892 0 minus

119889

minus119889

119889

0

]]

]

(33)


Ω7 =[[[

[

minus1205962119910minus 1205962119911

minus119911+ 120596119909120596119910119910+ 120596119909120596119911

119911+ 120596119909120596119910

minus1205962119909minus 1205962119911

minus119909+ 120596119911120596119910

minus119910+ 120596119911120596119909119909+ 120596119911120596119910

minus1205962119909minus 1205962119910

]]]

]

1205951205967=[[[

[

119909

minus120596119911120596119910120596119910120596119911

120596119909120596119911

119910

minus120596119911120596119909

minus120596119910120596119909120596119909120596119910

119911

]]]

]

(34)



x y z0

01

02

03

04

05

06

07M

axim

um T

CP p

ositi

on er

ror (

mm

)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0002004006008

01012014016018

Max

imum

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574


x y z0

005

01

015

02

025

03

035

04

RMS

TCP

posit

ion

erro

r (m

m)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0001002003004005006007008009

RMS

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574







Inversekinematics

Adaptivecontrol

Parameterestimation

Lineartransmission

driveHexaglide

robot

HIL

Aerodynamicforces

qd

Xd

qTm

p





119881 (119905) =1


119879

KI int s 119889119905) (35)




p = ΓΨ119879s (37)

and (23) changes to


+ 120578119904119886119905 (Φminus1s)

(38)





6 Conclusions





119909

[mm]119910

[mm]119911

[mm]120572

[degree]120573

[degree]120574



RMS error 018 016 011 002 002 002


0010203040506

t (s)

q er

ror (

mm

)

Slider 1 error

(a) Slider 1

t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

Slider 2 error

q er

ror (

mm

)

(b) Slider 2

t (s)0 05 1 15 2 25 3 35 4 45 5


002040608

1Slider 3 error

q er

ror (

mm

)

(c) Slider 3

t (s)0 05 1 15 2 25 3 35 4 45 5


minus0050

00501

01502

02503

Slider 4 error

q er

ror (

mm

)

(d) Slider 4

t (s)0 05 1 15 2 25 3 35 4 45 5

minus04

minus03

minus02

minus01

0

01

02

03Slider 5 error

q er

ror (

mm

)

(e) Slider 5

t (s)0 05 1 15 2 25 3 35 4 45 5

minus03

minus02

minus01

0

01

02

03

04Slider 6 error

q er

ror (

mm

)

(f) Slider 6



minus08

minus06

minus04

minus02

0

02

04


t (s)0 05 1 15 2 25 3 35 4 45 5

xer

ror (

mm

)


t (s)

minus08

minus06

minus04

minus02

0

02

04


0 05 1 15 2 25 3 35 4 45 5

yer

ror (

mm

)


t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

zer

ror (

mm

)



minus006

minus004

minus002

0

002

004

006


t (s)0 05 1 15 2 25 3 35 4 45 5

120572er

ror (

deg

)



0002004006008


t (s)0 05 1 15 2 25 3 35 4 45 5

120573er

ror (

deg

)



0002004006008

01

t (s)


0 05 1 15 2 25 3 35 4 45 5

120574er

ror (

deg

)



x y z02468

10121416


120572 120573 120574

Erro

r (

) (er

ror lowast

100

am

plitu

de)




Competing Interests


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x y z0

01

02

03

04

05

06

07M

axim

um T

CP p

ositi

on er

ror (

mm

)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0002004006008

01012014016018

Max

imum

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574


x y z0

005

01

015

02

025

03

035

04

RMS

TCP

posit

ion

erro

r (m

m)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10




0001002003004005006007008009

RMS

TCP

orie

ntat

ion

erro

r (de

g)

120574 movement 120573 = 10 120572 = 10

120574 movement 120573 = minus10 120572 = 10

120574 movement 120573 = 10 120572 = minus10


120573 movement 120572 = 10 120574 = 10

120573 movement 120572 = 10 120574 = minus10

120573 movement 120572 = minus10 120574 = 10


120572 movement 120573 = 10 120574 = 10

120572 movement 120573 = 10 120574 = minus10



120572 120573 120574







Inversekinematics

Adaptivecontrol

Parameterestimation

Lineartransmission

driveHexaglide

robot

HIL

Aerodynamicforces

qd

Xd

qTm

p





119881 (119905) =1


119879

KI int s 119889119905) (35)




p = ΓΨ119879s (37)

and (23) changes to


+ 120578119904119886119905 (Φminus1s)

(38)





6 Conclusions





119909

[mm]119910

[mm]119911

[mm]120572

[degree]120573

[degree]120574



RMS error 018 016 011 002 002 002


0010203040506

t (s)

q er

ror (

mm

)

Slider 1 error

(a) Slider 1

t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

Slider 2 error

q er

ror (

mm

)

(b) Slider 2

t (s)0 05 1 15 2 25 3 35 4 45 5


002040608

1Slider 3 error

q er

ror (

mm

)

(c) Slider 3

t (s)0 05 1 15 2 25 3 35 4 45 5


minus0050

00501

01502

02503

Slider 4 error

q er

ror (

mm

)

(d) Slider 4

t (s)0 05 1 15 2 25 3 35 4 45 5

minus04

minus03

minus02

minus01

0

01

02

03Slider 5 error

q er

ror (

mm

)

(e) Slider 5

t (s)0 05 1 15 2 25 3 35 4 45 5

minus03

minus02

minus01

0

01

02

03

04Slider 6 error

q er

ror (

mm

)

(f) Slider 6



minus08

minus06

minus04

minus02

0

02

04


t (s)0 05 1 15 2 25 3 35 4 45 5

xer

ror (

mm

)


t (s)

minus08

minus06

minus04

minus02

0

02

04


0 05 1 15 2 25 3 35 4 45 5

yer

ror (

mm

)


t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

zer

ror (

mm

)



minus006

minus004

minus002

0

002

004

006


t (s)0 05 1 15 2 25 3 35 4 45 5

120572er

ror (

deg

)



0002004006008


t (s)0 05 1 15 2 25 3 35 4 45 5

120573er

ror (

deg

)



0002004006008

01

t (s)


0 05 1 15 2 25 3 35 4 45 5

120574er

ror (

deg

)



x y z02468

10121416


120572 120573 120574

Erro

r (

) (er

ror lowast

100

am

plitu

de)




Competing Interests


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Inversekinematics

Adaptivecontrol

Parameterestimation

Lineartransmission

driveHexaglide

robot

HIL

Aerodynamicforces

qd

Xd

qTm

p





119881 (119905) =1


119879

KI int s 119889119905) (35)




p = ΓΨ119879s (37)

and (23) changes to


+ 120578119904119886119905 (Φminus1s)

(38)





6 Conclusions





119909

[mm]119910

[mm]119911

[mm]120572

[degree]120573

[degree]120574



RMS error 018 016 011 002 002 002


0010203040506

t (s)

q er

ror (

mm

)

Slider 1 error

(a) Slider 1

t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

Slider 2 error

q er

ror (

mm

)

(b) Slider 2

t (s)0 05 1 15 2 25 3 35 4 45 5


002040608

1Slider 3 error

q er

ror (

mm

)

(c) Slider 3

t (s)0 05 1 15 2 25 3 35 4 45 5


minus0050

00501

01502

02503

Slider 4 error

q er

ror (

mm

)

(d) Slider 4

t (s)0 05 1 15 2 25 3 35 4 45 5

minus04

minus03

minus02

minus01

0

01

02

03Slider 5 error

q er

ror (

mm

)

(e) Slider 5

t (s)0 05 1 15 2 25 3 35 4 45 5

minus03

minus02

minus01

0

01

02

03

04Slider 6 error

q er

ror (

mm

)

(f) Slider 6



minus08

minus06

minus04

minus02

0

02

04


t (s)0 05 1 15 2 25 3 35 4 45 5

xer

ror (

mm

)


t (s)

minus08

minus06

minus04

minus02

0

02

04


0 05 1 15 2 25 3 35 4 45 5

yer

ror (

mm

)


t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

zer

ror (

mm

)



minus006

minus004

minus002

0

002

004

006


t (s)0 05 1 15 2 25 3 35 4 45 5

120572er

ror (

deg

)



0002004006008


t (s)0 05 1 15 2 25 3 35 4 45 5

120573er

ror (

deg

)



0002004006008

01

t (s)


0 05 1 15 2 25 3 35 4 45 5

120574er

ror (

deg

)



x y z02468

10121416


120572 120573 120574

Erro

r (

) (er

ror lowast

100

am

plitu

de)




Competing Interests


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119909

[mm]119910

[mm]119911

[mm]120572

[degree]120573

[degree]120574



RMS error 018 016 011 002 002 002


0010203040506

t (s)

q er

ror (

mm

)

Slider 1 error

(a) Slider 1

t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

Slider 2 error

q er

ror (

mm

)

(b) Slider 2

t (s)0 05 1 15 2 25 3 35 4 45 5


002040608

1Slider 3 error

q er

ror (

mm

)

(c) Slider 3

t (s)0 05 1 15 2 25 3 35 4 45 5


minus0050

00501

01502

02503

Slider 4 error

q er

ror (

mm

)

(d) Slider 4

t (s)0 05 1 15 2 25 3 35 4 45 5

minus04

minus03

minus02

minus01

0

01

02

03Slider 5 error

q er

ror (

mm

)

(e) Slider 5

t (s)0 05 1 15 2 25 3 35 4 45 5

minus03

minus02

minus01

0

01

02

03

04Slider 6 error

q er

ror (

mm

)

(f) Slider 6



minus08

minus06

minus04

minus02

0

02

04


t (s)0 05 1 15 2 25 3 35 4 45 5

xer

ror (

mm

)


t (s)

minus08

minus06

minus04

minus02

0

02

04


0 05 1 15 2 25 3 35 4 45 5

yer

ror (

mm

)


t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

zer

ror (

mm

)



minus006

minus004

minus002

0

002

004

006


t (s)0 05 1 15 2 25 3 35 4 45 5

120572er

ror (

deg

)



0002004006008


t (s)0 05 1 15 2 25 3 35 4 45 5

120573er

ror (

deg

)



0002004006008

01

t (s)


0 05 1 15 2 25 3 35 4 45 5

120574er

ror (

deg

)



x y z02468

10121416


120572 120573 120574

Erro

r (

) (er

ror lowast

100

am

plitu

de)




Competing Interests


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus08

minus06

minus04

minus02

0

02

04


t (s)0 05 1 15 2 25 3 35 4 45 5

xer

ror (

mm

)


t (s)

minus08

minus06

minus04

minus02

0

02

04


0 05 1 15 2 25 3 35 4 45 5

yer

ror (

mm

)


t (s)0 05 1 15 2 25 3 35 4 45 5


001020304

zer

ror (

mm

)



minus006

minus004

minus002

0

002

004

006


t (s)0 05 1 15 2 25 3 35 4 45 5

120572er

ror (

deg

)



0002004006008


t (s)0 05 1 15 2 25 3 35 4 45 5

120573er

ror (

deg

)



0002004006008

01

t (s)


0 05 1 15 2 25 3 35 4 45 5

120574er

ror (

deg

)



x y z02468

10121416


120572 120573 120574

Erro

r (

) (er

ror lowast

100

am

plitu

de)




Competing Interests


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Competing Interests


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article error analysis and adaptive-robust control...

Documents