An approach to stiffness analysis methodology forhaptic devices

Aftab AhmadMachine Design Dept

Royal institute of Technology-KTHStockholm, Sweden

Email: aftaba@kth.se

Kjell AndersonMachine Design Dept

Royal institute of Technology-KTHStockholm, SwedenEmail: kan@kth.se

Ulf SellgrenMachine Design Dept

Royal institute of Technology-KTHStockholm, SwedenEmail: ulfs@kth.se

Abstract—In this work a new methodology is proposed tomodel the static stiffness of a haptic device. This methodologycan be used for other parallel, serial and hybrid manipulators.The stiffness model considers the stiffness of; actuation system;flexible links and passive joints. For the modeling of the passivejoints a Hertzian contact model is introduced for both sphericaland universal joints and a simply supported beam model foruniversal joints. For validation of the stiffness model a modifiedJP Merlet kinematic structure has been used as a test case. Aparametric Ansys FEM model was developed for this test caseand used to validate the resulting stiffness model. The findingsin this paper can provide an additional index to use for multi-objective structural optimization to find an optimum compromisebetween a lightweight design and the stiffness performance forhigh precision motion within a larger workspace.

Keywords: Haptics; Parallel/Serial kinematic structures;stiffness methodology;

I. INTRODUCTION

A haptic device is a computer-controlled actuated mechan-ical device that provides a physical interface between humansense of touch and computer generated virtual or remoteenvironment. Based on manipulation and interaction withobjects within the virtual or remote environment, these devicesprovide feedback forces and torques to the user. Applicationsof haptic devices are increasing in many fields particularly inmedicine, tele-robotics, engineering design, and entertainment[1]. The intended application of the device presented in thiswork is a medical/dental simulator [1] which will be used toachieve manipulation capabilities and force/torque feedback insix degrees of freedom (6-DoF) during simulation of surgicalor dental procedures in hard tissues such as bone structures[1]. Use of a haptic device in the above described applicationleads to three important design requirements on the design ofthe device [2], [3]:

1) Stiffness2) Transparency and stability3) Haptic feedback of force and torque in 6-DoF.In precise positioning accuracy and high payload capability

stiffness is an essential performance measure criteria. Stiffnesscan be defined as the capacity of a mechanical system tosustain loads without excessive changes of its geometry [4],or in other words these produced changes on geometry, dueto the applied forces, are known as deformations or compliant

displacements. Compliant displacements in a robotic systemproduces negative effects on static and fatigue strength, wearresistance, efficiency (friction losses), accuracy, and dynamicstability (vibration) [4].

By transparency it is meant that motion in free space shouldfeel free while motion in contact with a virtual or remoteobject should result in feedback forces and torques as closeas possible to those appearing in the remote or virtual world.

Haptic feedback is required to reflect both forces andtorques in the case of surgical procedures of hard tissues likebones,which involves removing bone by drilling or milling toprocess channel and cavities, hence required 6 DoF capability.

Current trends in mechanical design of robotic manipulatorshave targeted essential reduction of moving masses, in orderto achieve high dynamic performances with relatively smallactuators and low energy consumption [5]. This motivatesusing advanced kinematical architectures, high strength andlight-weight materials, as well as minimization of cross-sections of all critical elements. The primary constraint forsuch minimization is the mechanical stiffness of the manipu-lator, which is directly related with the robot accuracy definedby the design specifications.

There are still open problems related with stiffness. It hasnot yet been completely solved, for example, the problemof improving the stiffness analysis in order to have a bettermatch between theoretical and experimental results. This as-pect would require the development of more accurate stiffnessmodels by taking into account also phenomena that cannot beeasily modeled such as friction and backlash. Manufacturingtolerances and other geometrical indeterminacies in the kine-matic model of a multi-body robotic system should be alsoproperly considered.

The aim of this paper is to present a methodology forgenerating a compact and accurate generalized model forstiffness analysis of an any-DoF parallel/serial/hybrid manip-ulator. The generated model e.g. could be used for structureoptimization within the workspace. The paper is organized asfollows: section II summarizes a literature review on stiffnessanalysis of 6-DoF manipulators, and section III presents thestiffness analysis methodology. Section IV validates the pre-sented methodology with an illustrative example comparingthe proposed generalized stiffness model and a fairly detailed

978-963-8111-77-7

FEM model. In addition the calculated stiffness variation inworkspace, based of the stiffness model, is presented. Finallysection V concludes the presented work.

II. RELATED WORK

Several methods exist for computation of the stiffnessmatrix: the virtual joint method (VJM), that is often called thelumped modeling [6]–[10] , Finite Element Analysis (FEA)[10]–[12] and Matrix Structural Analysis (MSA) [13]–[16].

The first of them, the virtual joint method is based onthe calculation of the Jacobian matrix that relate the jointdisplacement in joint space to the Tool Center Point (TCP)deflection in Cartesian space. In this method, only activejoint stiffness is considered and links of mechanism areassumed strictly rigid. FEA is reliable for calculating thestiffness of components with arbitrary shape and complexcontact interactions between components in a system. Forexample, the FEA model is adopted to characterize robotstatic rigidity and natural frequencies in [17] and it is usedto validate an analytical model in [18]. However, FEA doesnot provide the analytical relationship between stiffness andstructure dimensions of the mechanism. It does not fit theplanned multi-objective optimization procedure in which theperformance index such as stiffness index, most likely is to berepresented as a function of design parameters. However thismethod is usually applied at the detail design stage becauseof the high computational expenses required for the repeatedre-meshing of a complex structure.

The third of them, the MSA incorporates the main ideasof FEA, but operates with rather large elements and flexiblebeams to describe the manipulator structure [13]–[16].

Uchiyama [19] has derived an analytical model for the stiff-ness of a compact 6-DoF Haptic device, which he utilizes thedesign of a stiff platform for translational motion. The elasticelements were modeled as beam elements and considering theradial stiffness of the bearings. The model shows that thecompliance matrix is a function of kinematic parameters aswell as elastic parameters of each mechanical element. Alsoit was assumed that the base and traveling plates were rigid.

In literature, the compliance of joints is mostly modeled asa constant stiffness. Bonnemains et al. [20] has considered thestiffness of a spherical joints in the stiffness computation andidentification of a kinematic machine tools.

III. STIFFNESS ANALYSIS METHODOLOGY

In this section we will describe a general methodology andthe different steps required to develop a stiffness model ofa robotic structure. This methodology can be applied on anyrobotic structure, but the discussions in this paper is focusedon haptic devices based on parallel kinematic structures. Thebasic idea and assumption in this methodology is that thestiffness of the end-effector of a robotic structure can beobtained from the individual stiffness properties of its linksusing basic principles of the mechanical relation between forceand displacement. The underlying idea is to resolve the forceinto individual link forces and then to compute the individual

link deflection from the link and joint stiffness properties andfinally to transform and add all these displacements to obtainthe final stiffness matrix.The contact stiffness of passive jointslike spherical and universal joints have been calculated asHertzian springs [21]. The nominal point contact model wasused for spherical joints and nominal line contact model wasused for universal joints, while the bending stiffness of anuniversal joint is calculated based on the assumptions that thetwo axes act like a simply supported beam.

An example structure that we will use for formulating thedifferent steps of the methodology is the Delta structure with3 DoF [22], actuated by three rotational motors placed at thefixed base as shown in Fig.1.

Fig. 1. ABB Flexible Automation’s IRB 340 FlexPicker .

In a parallel kinematic structure we divide the total systeminto different kinematic chains. Each of these chains containa number of components that need to be described in order tofind the stiffness of one chain. The systems and componentsthat need to be handled are;

A. Stiffness of actuation systemB. Stiffness of flexible linksC. Stiffness of passive joint

And as a final step the stiffness for the whole structure withN number of kinematic chains is formulated as

KTotal =

N∑i

Kchaini(1)

Each kinematic chain can be considered as a serial manipu-lator, so the same methodology is also applicable to serial andhybrid manipulators. The stiffness of each kinematic chain isgiven by equation (2):

K−1chain = K−1act +K−1C

+K−1sp

+K−1U

(2)

Where Kact,KC , Ksp and KU are stiffness of actuation sys-tem, flexible links, spherical and universal joints, respectively.

A single kinematic chain can be modeled as shown in Fig.2,where the stiffness model of each compliant element is givenin the coming sections.

A. Stiffness of actuation systemThe stiffness of actuation system depends on stiffness of ac-

tuator and transmission system. The total stiffness of actuationsystem is given in equation (3):

K−1act = K−1r +K−1Trans (3)

y

{A}x

z

{U} xA

yA

zA

xB

yB

zB

{B}

xC

yCzC

{C}

2β

{P}xP

yP

zP

2α

Proximal link

Base

Linear Guideway

UC

UC

F

M

UB

UB

F

M

Fig. 2. Simple kinematic chain.

where Kr and KTrans is the stiffness of actuator andtransmission system.

The stiffness of the actuator was calculated from the tor-sional stiffness. The torsional stiffness of the actuator can bedefined by simple expression given in equation (4):

Krotor =JactG

Lr(4)

where Jact is the polar moment of inertia, G modulus ofrigidity and Lr is the length of motor shaft.

The principle of virtual work can be used to find thestiffness of the actuation system for the case of parallel/serialmanipulators [23] , Ks,Kp , as given in equation (5) and (6).

Kr = Kp = JT diag[ki]J (5)

Kr = Ks = J−T dia [ki] J−1 (6)

Here J jacobian matrix derived by Khan et. al. [3], andki = Krotor is the rotor stiffness of motor shaft.

The stiffness of transmission system depends on the in-tended application for the manipulator. Here we have chosento discuss three different types of transmission systems;

1) Cable based linear transmission2) Cable based rotational transmission3) Timing belt transmission

1) Cable based linear transmission: The cable basedtransmission consists of cable based linear actuator driven by aDC motor shown in Fig.3. Both the cable and shaft of the DCmotor can be considered as compliant. A cable with stiffnesscoefficient Kcable, length li and pretension τ , that staticallybalance the force Fe , then stiffness coefficient Kcable of thecable can be found by approximation as given in Equation (7):EQ13

KTrans = Kcable =AE

li[I]i×i (7)

Where A is the cross sectional area of the cable, E is theyoung modulus of the cable and i is the number of cables inthe system.

Fig. 3. Actuation system for 6 DoF haptic device.

2) Cable based rotational transmission: In the case of rota-tional transmission both the cables are acting in parallel, whichis actuated by DC motor, so the effective stiffness coefficientof the rotational transmission system can be expressed inequation (8):

KTrans = Kcable1 +Kcable2 (8)

3) Timing belt transmission: The stiffness coefficient fortiming belt can be calculated based on the assumption that,stress is proportional to strain. Defining the stiffness of a unitlength and a unit wide belt as specific stiffness csp, the stiffnesscoefficients of the belt on the tight and slack side, k1 and k2are proposed by [24], and are expressed by equation (9):

k1 = cspb

L1, k2 = csp

b

L2(9)

Where L1and L2 are the un-stretched lengths of the tightand slack sides of the cable respectively, and b is the beltwidth.

Since the tight and slack sides can be considered as springsacting in parallel, their stiffness add linearly to form a resultantstiffness constant KTrans given in equation (10):

KTrans = k1 + k2 = bcspL1 + L2

L1L2(10)

B. Stiffness of flexible linksTo calculate the stiffness of flexible links i.e the base link

and proximal link we have used the following methodology:we have transformed the force acting on the TCP of theplatform into individual link forces, then we compute theindividual link deflections from the link stiffness properties,and finally we transform and add all these displacements toobtain the final stiffness matrix. The stiffness of the base andproximal links can be calculated by transformation of forcesand moments from frame C to B. The following steps will beused to calculate the stiffness of flexible links.

1) Force/moment transformation: Let consider aforce/torque vector

(CF,CM

)Texpressed in frame C.

To express this in frame B we need to know how C isoriented with respect to B. This information is typicallyspecified by a rotation matrix BRC ∈ R3×3 where the first,second and third columns of BRC are unit vectors describingthe orientation of x, y and z axes of frame C expressedin frame B. The required force transformations is given inequation (11):[

BFBM

]=

[BRC 0

0 BRC

][CFCM

](11)

We take the straightforward approach to compute(UFB ,

U MB

)Tby taking a force and moment balance of the

proximal link, as shown in Fig.4

{C}

{B}

UC

UC

F

M

UB

UB

F

M

Fig. 4. Force/Moment Transformation from C to B.

The force balance equation results in equation (12):UFB =U FC (12)

While taking moments about point C yields the momentbalance equation (13):

UMB =U MC + (URBBPC)×U FC (13)

Where BPC is the position vector from {B} to {C}expressed in frame {B}. Equations (12) and (13) can becombined to a more compact matrix form of equation (14):[

UFBUMB

]=

[I 0⌊

(URBBPC)×⌋

I

]︸ ︷︷ ︸

UBJC,force

[UFCUMC

](14)

we use the notation bp×c ∈ <3×3 to represent the lineartransformation of a cross product with a vector p ∈ <3

bp×c y = p× y, y ∈ <3

where

bp×c =

0 −pz py

pz 0 −px−py −px 0

(15)

UBJC,force ∈ <3×3 as the force transformation Jacobian

matrix which transforms the frame of application of theforce vector from frame C to frame B. The forces are allexpressed in the same frame U. UBJC,force involves the relativeorientation between frame B and frame U as well as theposition vector describing the origin of frame C in frame B.Similarly, using Fig.2 to derive the transformation involvingdisplacements. With infinitesimal displacements, equationsrelating velocity transformations are equivalent to those fordisplacement transformations. Imagine frames B and C to bemoving instantaneously with respect to frame U.

2) Displacement transformation: The linear and angularvelocities

(UvB ,

U ωB ∈ <3), respectively) of frame B may be

computed from the velocities of frame C using:UvB =U vC +U ωC ×

(URUCPB

)(16)

UωB =U ωC (17)

The velocities in (16) and (17) may be replaced with theinfinitesimal displacements UdB and UθB , and combined tothe following compact matrix form in equation (18):[

UdBUθB

]=

[I⌊(URBBPC)×

⌋0 I

]︸ ︷︷ ︸

UBJC,disp

[UdCUθC

](18)

We refer to UBJC,disp ∈ <3×3 as the displacement (or

velocity) transformation Jacobian which transforms the framethat is in motion from fame B into frame C. Frame B andFrame C can be viewed as instantaneously attached to thesame rigid body, and the transformation involves determiningthe motion of frame B due to the motion of frame C. Allmotions are expressed in the same frame U. UBJC,disp involvesthe relative orientation between frames B and A as well as theposition vector describing the origin of frame C in frame B.The force experience at frame B can be expressed in frame Uusing equation (19).[

UFBUMB

]=UB JC,force

[UFCUMC

](19)

Where UBJC,force =

[I 0⌊

(URBBPC)×⌋

I

]3) Transformation of stiffness and compliance matrices:

The stiffness matrix of a rotated beam in a task-based coordi-nate frame U can be obtained from the local link stiffness ma-trix LK from the knowledge of rotating matrix URL ∈ R3×3

that describes the orientation of the limb with respect to thetask frame. Since forces/moments and translational/rotationaldisplacements are related by:[

UFUM

]=U JL

[LFLM

]

and

[UdLUθL

]=U JL

[LdLθ

]and from the definition of the stiffness matrices:[UFUM

]=U K

[UdUθ

]

and

[LFLM

]=L K

[UdUθ

]UK is obtained from LK using the same transformation

matrix UJL and its transpose UJTL =L JU =U J−1LUK =U JLLK

UJTL (20)

Similarly, from the definition of the compliance matrices USand LS, i.e., being the inverse of the stiffness matrices:[

UdUθ

]=U S

[UFUM

]

and

[LdLθ

]=L S

[LFLM

]So we have

US =U JLLSUJTL (21)

C. Stiffness of passive joints

The contact stiffness of spherical and universal joint iscalculated based on Hertzian contact model. The nominalpoint contact model was used for spherical joint and nominalline contact model was used for universal joint, while thebending stiffness of universal joint is calculated based on theassumptions that two axis act like a simply supported beam.

1) Nominal point contact model: The contact stiffnessof the spherical joint can be find out by Hertzian theory,considering the nominal point contact between a sphere withradius R and a plane loaded by force F as shown in Fig.5,the radius r of the circular contact area thus formed is givenby the equation proposed by Hertz (Timoshenko and Goodier[25], Johnson [26]),

r =

(3F

R′

E′

) 13

(22)

Where E′ is the effective modulus of elasticity defined by:

1

E′=

1− v212E1

+1− v222E2

(23)

And R′ the effective radius related to the individual compo-nents1

R′=

1

R′x+

1

R′y,1

R′x=

1

r1,x+

1

r2,x,1

R′y=

1

r1,y+

1

r2,y(24)

The relation between contact radius and the indentation depthδis given by Johnson [26]

r = (2R′δ)12 (25)

Substituting equation (22) into equation (25), results in contactstiffness Ksp of spherical joint

F = (8

9R′)

12E′δ

32 (26)

Ksp =dF

dδ= E′(2R′δ)

12 (27)

2) Nominal line contact model: Stiffness of a universaljoint is calculated from the Hertzian contact model for linecontact, the line contact is show in Fig.5, while the bendingstiffness of universal joint is calculated based on the assump-tions that two axes act as a simply supported beam. The semi-axis b of a nominal line contact is derived by Hertz theory

b = 2

√2FR′

πlE′(28)

b = (2R′δ)12 (29)

Where E′and R′ are defined in equation (23) and (24),comparing both the equations (28) and (29), we get contactstiffness KUC of the universal joint

F =πlE′δ

4, KUC =

dF

dδ=πlE′

4(30)

The general expression for the deflection of simply sup-ported beam with length Luis given as

δujoint =PL3

u

48EI(31)

so the bending stiffness KUB of the universal joint is

KUB =48EI

L3u

(32)

so the total stiffness of the universal joint is

KU = KUC +KUB (33)

r

R

2b

l

2R

Fig. 5. Contact model of spherical and universal joint.

IV. METHODOLOGY VALIDATION

In the following section, the methodology described abovewill be applied to a haptic device based on a parallel mech-anism in the form of a modified version of the JP Merletkinematic structure [3], [27]. The structure consists of a fixedbase, a moving platform, and six identical legs connecting theplatform to the base as shown in Fig.6. Each kinematic chainconsists of an active actuator fixed to the base, a sphericaljoint, a constant length proximal link, and a universal joint.The joint attachment point pairs are symmetrically separated120o and lie on a circle, both on the base and platform. Theplatform attachment points are rotated 60o clockwise from thebase attachment points. The 6-PSU (active prismatic, passivespherical and universal) joints configuration is used to get 6-DoF motion at TCP. Six independent DC motors are connectedvia cable transmissions to provide linear motion to each linearguideways. A parallel mechanism is a close loop mechanismconsisting of a base plat, a traveling plate and elementarychains that connect the two plates. Its stiffness is determinedby the stiffness of each kinematic chain. We assume the baseand traveling plates are rigid, we begin with the analysis ofelementary chains and drive a compliance matrix of the targetchain.

y

{A}x

z

{U} xA

yA

zA

xB

yB

zB

{B}

xC

yC

zC

{C}

2β

{P} xP

yP

zP

2α

Spherical jo int

Base

1L

Platform

Universal joint

2L

Linear guideway

Pr oximal link

Fig. 6. A schematic of 6 DoF haptic device.

A. Stiffness modeling

The kinematic chains consist of actuation system, linearguideways, spherical joint, constant length proximal link anduniversal joint. The following method has been used to calcu-late the stiffness of each kinematic chain.

B. Displacement due to linear guideway

From the local compliance matrix of the linear guideways,we transform the compliance matrix to the reference frame Uusing (21):

USB =U JAASUBJ

TA (34)

ASB =

EiSi

Li0 0 0 0 0

0 12EiIziL3

i

0 0 0 −6EiIziL3

i

0 012EiIyi

L3i

06EiIyi

L2i

0

0 0 0 GiJiLi

0 0

0 06EiIyi

L2i

06EiIyi

Li0

0 −6EiIziL2

i

0 0 0 4EiIziLi

−1

(35)ASB is the stiffness matrix of the linear guideways. The

displacement cause by the force acting on B is then:

dB =

[UdBUθB

]=A SB

[UFBUMB

]=U JUAS

UBJ

TA

[UFBUMB

](36)

The displacement dB is the contribution of the flexibility ofthe linear guideways. The effect of this displacement on C iscomputed using (18), and transforming the local compliancematrix of the beam to the universal frame U using

[UdCUθC

]link

= UCJB,disp

[UdBUθB

]= UCJB,disp

USB

[UFBUMB

]

= UCJB,disp

USBUBJC,force

[UFCUMC

]

= UCJB,disp(

UJAASB

UJTA )UBJC,force

[UFCUMC

](37)

where

UCJB,disp =

−1[I⌊(URBBPC)×

⌋0 I

]︸ ︷︷ ︸

UBJC,disp

UBJC,force =

[I 0⌊

(URBBPC)×⌋

I

]Next we express the force and moment vector acting on C

in frame B using (11) to yield:[BFCBMC

]=B JU

[UFCUMC

](38)

Where BJU is the rotation matrix from link from B to FrameU given by

BJU =

[BRU 0

0 BRU

](39)

BRU is the unit vector describing the orientation of x, yand z axes of frame B expressed in frame U. From the localframe {B} of proximal link, the equations for rotation matrixBRUcan be derived as

Zpli =[

L2ix

L2

L2iy

L2

L2iz

L2

]T

Xpli =

[− L2iy√

L22ix

+L22iy

L2ix√L2

2ix+L2

2iy

0]T

and Ypli = Zpli ×Xpli

Finally, the rotation matrix for proximal link is given byequation (40)

BRU =[Xpli Ypli Zpli

](40)

C. Displacement due to proximal link

From the local flexibility matrix of the proximal link BSC ,we compute the displacement caused by UFC due to theflexibility of proximal link only:

[BdCBθC

]=B SC

[BFCBMC

]=B SBC JU

[UFCUMC

](41)

Where Kprox =B SC

=

12EiIyi

L3i

0 0 0−6EiIyL2

i

0

0 12EiIxi

L3i

0 6EiIxi

L2i

0 0

0 0 πr2

Li0 0 0

0 6EiIxi

L2i

0 4Ixi

Li0 0

−6EiIyi

L2i

0 0 04EiIyi

Li0

0 0 0 0 0 GiJizLi

−1

The displacement solely due to the flexibility of the link 2is expressed in the reference frame U using (39):[

UdCUθC

]link2

=U JB

[BdCBθC

](42)

And substituting the (41) yields:

[UdCUθC

]prox

=U JBBSBC JU

[UFCUMC

](43)

D. Total compliance due to linear guideways and proximallinks

The total displacement of C due to the flexibility oflinear guideways and proximal link is the sum of theindividual displacement expressed in the reference frame U:[

UdCUθC

]=

[UdCUθC

]link1

+

[UdCUθC

]prox

=[UCJB,disp(

UJAASUBJ

TA )

UBJC,force +

U JBBSBC JU

] [ UFCUMC

]From the definition of the compliance matrix we therefore

have:USC =

[UCJB,disp(

UJAASUBJ

TA )

UBJC,force +

U JBBSBC JU

]and the overall stiffness matrix of linear guideways andproximal link is then

KC =U KC

=(UCJB,disp(

UJAASUBJ

TA )

UBJC,force +

U JBBSBC JU

)−1(44)

Overall stiffness of the kinematic chain can be calculatedby considering stiffness of the transmission system , flexiblelinks and passive joints given in equations (3), (27), (33) and(44) by considering all the compliant elements in series andusing (2) as given in equation (45).

K−1chain = K−1acts +K−1C

+K−1sp

+K−1U

(45)

so the overall stiffness of stewart platform is

KTotal =

6∑i

Kchaini (46)

In order to validate the developed model experimentallyin the future work, we have considered static analysis, sothe stiffness of the actuation system is not considered in thesimulation.

E. FEM-based validation

In order to validate the stiffness model developed in sectionB, a parametric model was developed with the Ansys Para-metric Design Language (APDL). The aim was to validate thestiffness model at different configurations. In this parametricFEM model, the inverse kinematics developed in [3] wasused to calculate the length of linear guideways at differentconfigurations of the TCP. Beam, shell, and solid elements,as well as multi-point constraint (MPC) equations were usedto model the linear guideways, the proximal, spherical anduniversal joints, and the platform. Table I shows the designparameters and material properties of the FEM model. Fig.7shows prototype of a 6-DoF haptic device. Fig.8 shows theFEM mesh at the nominal configuration.

F. Comparison of generalized stiffness and FEM Models

To compare FEM analysis and stiffness modeling, thedeflections are calculated at 9 points in the workspace of which8 are shown in Fig.9. The first point (0) is located in the centerof the cube. In this simulation a force of 50N was applied ineach x,y and z-direction at the tool center point(TCP)of the

platform, and then corresponding deflection in each directionwas calculated. The resulting deflections and comparison be-tween Analytical and FEM model is shown in Fig.10. Thecomparison shows that validity of the proposed methodology.The deflection in x, y directions at each configuration showsthat when the TCP are at points 1,2, 3 and 4, the deflection ishigh due to bending of the linear guideways.

Table I:Design Parameters.

Length of linear guide way, L1 Inverse kinematic [mm]Length of proximal link, L2 125.5 [mm]Radius of platform, RP 55 [mm]Radius of Base, RB 118 [mm]Angle between the base joints, β 25Angle between the platformjoints, α

16

Modulus of Elasticity, E 210 [GPa]Radius of sp-joint, R′ 4 [mm]Length of U-Joint, Lu 10 [mm]Radius of U-Joint, R′ 1.5 [mm]

Fig. 7. Prototype of 6 DoF haptic device [3].

Fig. 8. FEM Model.

0

Fig. 9. TCP positions in workspace.G. Stiffness Variations in the workspace

To check the validity of the mathematical model developed,the TCP of the platform was traversed in the workspace andthe deflection and stiffness was calculated in a number of gridpoints The stiffness variation is shown in Fig.11 and indicatethat stiffness is isotropic in the workspace; the stiffness ishigh when the linear guideways are at the minimum actuationlength, when the length of linear guideways is increasing dueto bending phenomena the stiffness of the system decreaseswhich is clear from the blue color.

0 1 2 3 4 5 6 7 8−0.2

0

0.2

0.4

0.6

0.8

Points in workspace

Line

ar D

efle

ctio

ns in

X,Y

and

Z [m

m]

Linear Deflections by applied force 50 N in X,Y,Z−direction

X−AnalyticalX−AnsysY−AnalyticalY−AnsysZ−AnalyticalZ−Ansys

Fig. 10. Linear deflections by applied force of 50 N in X,Y,Z-direction.

Fig. 11. Stiffness variation of TCP of platform.

V. CONCLUSIONS

This paper proposes a new methodology to compute a quasi-static stiffness model of parallel, serial, and hybrid manipu-lators. The proposed methodology is applied to an existingarchitecture of a 6-DoF haptic device based on a modifiedJP Merlet kinematic structure. The proposed model takes intoaccount the stiffness of the actuation system, linear guideways,proximal links, and passive joints. In this methodology wehave decomposed the force acing at the TCP of the platforminto individual link forces, then computed the individual linkdeflections from the link stiffness properties, and finally trans-formed and added all these displacements to obtain the finalglobal stiffness matrix. Another systems modeling approachcovered by the proposed methodology is the introduction ofthe Hertzian contact model for both spherical and universalpassive joints and a simply supported beam model for theuniversal joint.

The proposed model was validated with quasi-static simu-lations with a parametric Ansys FEM model. The comparativeanalysis between the proposed stiffness method and the fairlydetailed FEA model shows that the stiffness results at the TCPare very close to each other.

This methodology also creates possibilities to develop akinetostatic control algorithm, which will allow us to improveaccuracy of the classical kinematic control and to compensateposition errors caused by elastic deformation in links and jointsdue to external loading.

REFERENCES

[1] E. M. G, “Haptic and visual simulation of a material cutting process,”KTH, Stockholm Sweden, Licentiate Thesis., 2006.

[2] K. Suleman, K. Andersson, and J. Wikander, “A design approach fora new 6-dof haptic device based on parallel kinematics,” in ICM 2009.IEEE International Conference on Mechatronics, 2009, april 2009, pp.1 –6.

[3] S. Khan, K. Andersson, and J. Wikander, “Optimal design of a 6-dof haptic device,” in Mechatronics (ICM), 2011 IEEE InternationalConference on, april 2011, pp. 713 –718.

[4] Rivin, E.I, Stiffness and Damping in Mechanical Design. New York:Marcel Dekker Inc, 1999.

[5] K. D. C. A, Pashkevich. A, “Nonlinear effect in stiffness modeling ofrobotic manipulators,” in Proceedings of International Conference onComputer and Automation Technology (ICCAT 2009), vol. 58, october2009, pp. 581 –586.

[6] C. Gosselin, “Stiffness mapping for parallel manipulators,” IEEE Trans-actions on Robotics and Automation,, vol. 6, pp. 321–342, jun 1990.

[7] B. El-Khasawneh and P. Ferreira, “Computation of stiffness and stiffnessbounds for parallel link manipulator,” Int. J. Machine Tools & manufac-ture,, vol. 39, no. 3, pp. 377 –382, February 1999.

[8] C. M. D. Zhang, F. Xi and S. Lang, “Analysis of parallel kinematicmachine with kinetostatic modeling method,” Robotics and Computer-Integrated Manufacturing,, vol. 20, pp. 151– 165, April 2004.

[9] D. Zhang, “Haptic and visual simulation of a material cutting process,”Laval University, Quebec, Canada, Ph.D Thesis., April 2000.

[10] C. Gosselin and D. Zhang, “Stiffness analysis of parallel mechanismsusing a lumped model,” Int. J. of Robotics and Automation,, vol. 17, pp.17–27, April 2002.

[11] Y. Wang, T. Huang, X. Zhao, J. Mei, D. Chetwynd, and S. Hu,“Finite element analysis and comparison of two hybrid robots-the triceptand the trivariant,” in Intelligent Robots and Systems, 2006 IEEE/RSJInternational Conference on, oct. 2006, pp. 490 –495.

[12] Y. Yun and Y. Li, “Comparison of two kinds of large displacementprecision parallel mechanisms for micro/nano positioning applications,”in Robotics, Automation and Mechatronics, 2008 IEEE Conference on,sept. 2008, pp. 284 –289.

[13] R. S. Gonalves and J. C. M. Carvalho, “Stiffness analysis of parallel ma-nipulator using matrix structural analysis,” in Proceedings of EUCOMES08, M. Ceccarelli, Ed. Springer Netherlands, pp. 255–262.

[14] D. Deblaise, X. Hernot, and P. Maurine, “A systematic analytical methodfor pkm stiffness matrix calculation,” in Robotics and Automation, 2006.ICRA 2006. Proceedings 2006 IEEE International Conference on, may2006, pp. 4213 –4219.

[15] Przemieniecki, J.S, Theory of Matrix Structural Analysis. New York:Dover Publications, Inc, 1985.

[16] W. Dong, Z. Du, and L. Sun, “Stiffness influence atlases of a novel flex-ure hinge-based parallel mechanism with large workspace,” in IntelligentRobots and Systems, 2005. (IROS 2005). 2005 IEEE/RSJ InternationalConference on, aug. 2005, pp. 856 – 861.

[17] G. G. Y. H. B.C. Bouzgarrou, J.C. Fauroux, in 35th InternationalSymposium on Robotics (ISR), March. 2004.

[18] Y.-W. Li, J.-S. Wang, and L.-P. Wang, “Stiffness analysis of a stewartplatform-based parallel kinematic machine,” in Robotics and Automa-tion, 2002. Proceedings. ICRA ’02. IEEE International Conference on,vol. 4, 2002, pp. 3672 – 3677 vol.4.

[19] M. Uchiyama, Y. Tsumaki, and W.-K. Yoon, “Design of a compact 6-dof haptic device to use parallel mechanisms,” in Robotics Research,ser. Springer Tracts in Advanced Robotics, S. Thrun, R. Brooks, andH. Durrant-Whyte, Eds. Springer Berlin / Heidelberg, vol. 28, pp.145–162.

[20] T. Bonnemains, H. Chanal, B.-C. Bouzgarrou, and P. Ray,“Stiffness computation and identification of parallel kinematicmachine tools,” Journal of Manufacturing Science and Engineering,vol. 131, no. 4, p. 041013, 2009. [Online]. Available:http://link.aip.org/link/?MAE/131/041013/1

[21] Anton van Beek, Advanced Engineering Design Lifetime Performanceand Reliability. Delft: TU Delft, 2009.

[22] Parallemic - the parallel mechanisms information center. [Online].Available: http://www.parallemic.org

[23] Lung-Wen Tsai, Robot Analysis: The Mechanics of Serial and ParallelManipulators. Delft: Wiley-Interscience, 1999.

[24] Gates mectrol, inc. [Online]. Available: http://www.gatesmectrol.com[25] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity. Cambridge, UK:

Cambridge Univ. Press, 1970.[26] Johnson, K. L, Contact Mechanics. Cambridge, UK: Cambridge Univ.

Press, 1970.[27] F. Hao and J.-P. Merlet, “Multi-criteria optimal design of parallel

manipulators based on interval analysis,” Mechanism and MachineTheory, vol. 40, no. 2, pp. 157 – 171, 2005. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0094114X04001211.