geometric identiﬁcation of an elastokinematic model in a...

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Geometric identification of an elastokinematic model ina car suspensionJ Meissonnier1, J-C Fauroux1*, G Gogu1, and C Montezin2

1Laboratoire Mecanique et Ingenierie, Universite Blaise Pascal, Institut Francais de Mecanique Avancee, Aubiere Cedex,France2Manufacture Francaise des Pneumatiques Michelin, Centre de Technologies de Ladoux, Clermont-Ferrand, France

The manuscript was received on 15 December 2005 and was accepted after revision for publication on 31 March 2006.

DOI: 10.1243/09544070JAUTO239

Abstract: This paper deals with the modelling of a car suspension including rubber bushings.Significant differences are found between data from the numerical elastokinematic model andexperimental results. One reason for these differences comes from geometrical shifts of bushingcentres in the real suspension with respect to the numerical model. The aim of the presentwork is to reduce the differences by identifying the geometric parameters of the elastokinematicmodel. To achieve this goal, a method is proposed for computing the location and orientationof each part and joint in the assembled suspension. The first stage of the method uses measure-ments on separate parts to define joint location in the coordinate system local to the part. Inthe second stage, measurements are performed on the assembled suspension mechanism usinga portable coordinate measuring machine for locating parts and joints in a global coordinatesystem. Based on these data and elastic joint stiffnesses, bushing deflection is computed atthe static equilibrium of the vehicle and used to identify the real joint locations on parts. Thisidentification method is tested on a pseudo-McPherson suspension and improves modelbehaviour during vertical wheel movement.

Keywords: car suspension, elastokinematic, geometric identification, bushing, multi-bodysimulation, ADAMS

1 INTRODUCTION Obtaining a full correlation between a model anda given suspension is a complex task. Several causes

The automotive industry has been an early user of of discrepancy between model and reality can bemulti-body simulation software for vehicle behaviour given.analysis and suspension mechanism design. The useof these tools makes it possible to reduce the time 1. The complex behaviour of rubber bushings usedneeded to develop new suspensions. However, before for vibration filtering is often not fully modelledusing a model for design studies, it is important [1–3].to achieve a good correlation between the model 2. The real behaviour of spherical joints, includingbehaviour and the real suspension behaviour in order elasticity, dry friction, and functional clearance, isto ensure that modelling assumptions and para- not modelled.meters are valid. That is why model identification is 3. Some parts, such as the car chassis or McPhersona recurrent problem. strut, considered as perfectly rigid in the model,

may show significant flexibility.4. Part dimensions and location in the mechanism* Corresponding author: Laboratoire Mecanique et Ingenierie,

are not perfectly established as a consequence ofUniversite Blaise Pascal, Institut Francais de Mecanique Avancee,

manufacturing or measurement dispersion andCampus de Clermont-Ferrand – Les Cezeaux, BP 265, Aubiere

Cedex, 63175, France. email: [email protected] assembly clearance.

JAUTO239 © IMechE 2006 Proc. IMechE Vol. 220 Part D: J. Automobile Engineering

1210 J Meissonnier, J-C Fauroux, G Gogu, and C Montezin

These difficulties increase when no parametric computed using multi-body simulation software. Thisis needed finally to identify the geometric parametersdata from the computer aided design (CAD) modelfor the elastokinematic model. Results from experi-of the suspension are known and when each para-mental validation obtained on a pseudo-McPhersonmeter has to be determined in a limited number ofsuspension are given and discussed.measurement operations.

The formalism used to represent positions andPrevious methods proposed for elastokinematicorientations of all the parts and joints is based onparameter identification in order to achieve corre-homogeneous coordinates and operators [8]. Thelation are based on the analysis of wheel motionsimulation software used in this work for suspensionunder quasi-static load cases. The behaviour ofmodelling and behaviour analysis is ADAMS andthe real suspension for this type of load case isits add-on specialized for the automotive industry,generally obtained using a kinematic and complianceADAMS/Car. A preliminary report on the work pre-(K&C) test rig. Geometric and stiffness parameterssented in this paper was read at the 17th Frenchcan be obtained using optimization routines on aCongress of Mechanics [9].mathematical model of the suspension [4]. Test data

are used to define an objective function for themodel, and the identification is then similar to aproblem of optimal design [5]. Another solution is

2 PART GEOMETRIC MODELLINGto perform a sensitivity analysis using techniquesof statistical design of experiments (DOE) that are

In general-purpose multi-body simulation softwareimplemented in commercial simulation software

such as ADAMS, geometric parameters are deter-[6, 7]. Based on the results of this analysis, a set of mined by relative positions and orientations of eachparameters is found to achieve the correlation. joint on each part. In order to define these positions

These methods have two main disadvantages. as precisely as possible, parts comprising the suspen-Firstly, they involve a high number of simulations, sion mechanism are measured separately with awhich is time consuming, and advanced knowledge CMM. This measurement operation is done to buildof numerical computation is required. Secondly, a geometric model of the part. This model shouldeven if the model behaviour correlates with the real describe, in a local coordinate system R

p, the position

suspension behaviour, there is no certainty that the and orientation of each joint on the part.identified parameter set will give an accurate repre- A bushing is typically composed of a hollowsentation of the suspension component properties elastomer cylinder contained between inner andas the solution may not be unique. outer cylindrical steel sleeves. The most common

This paper is part of a larger study aiming to way to join two parts with a bushing is to bolt theimprove modelling techniques for any given suspen- inner sleeve to the first part while the outer sleeve ission mechanism. The work presented concerns, in shrunk on the second part.particular, the identification of geometric parameters When the measured part holds the bushing outerunder the assumption of perfectly rigid bodies sleeve, as shown for part 2 in Fig. 1, the bushingand known stiffness parameters for springs and position is defined by point O

b, the centre of the

bushings. The aim of the identification is to achieve outer sleeve. The bushing orientation can be definedan elastokinematic model of the suspension that is using two lines. Line l

zrepresents the sleeve axis

geometrically correct, i.e. each joint position and while line lx, perpendicular to l

z, indicates the main

orientation computed with the identified model at radial direction. As the bushing is shrunk on the part,the static equilibrium of the vehicle are similar to these geometric elements fully define the jointthose observed on the real suspension. position and orientation within the precision limit of

To achieve this goal, the authors present a com- the CMM employed.prehensive method for determining the precise When the measured part is designed to hold theposition and orientation of joints in an assembled inner sleeve, the only functional surfaces that canand loaded suspension using a portable coordinate be measured to define the bushing position are themeasuring machine (CMM). The proposed method is holes for the bolted assembly. However, owing tocomposed of two measuring operations. The first necessary assembly clearance, the joint positionone, on separate parts, defines a geometric model definition based on these functional surfaces mayfor each part. The second one determines each part not correctly represent the bushing position on theposition and orientation in the assembled and loaded part once the suspension is assembled. Figure 1 gives

a typical example of a bushing assembly. An essentialsuspension. Then, bushing elastic deflections are

JAUTO239 © IMechE 2006Proc. IMechE Vol. 220 Part D: J. Automobile Engineering

1211Geometric identification of an elastokinematic model in a car suspension

Fig. 1 Error on geometric parameter LP1

on part 1 generated by assembly clearance

geometric parameter for part 1 is the distance The uncertainty on geometric parameters existsbetween the spherical joint and the bushing. When also for spherical and revolute joints since boltedmeasuring the detached part, the centre of the bush- assemblies are used in a similar way. Measurementing housing is considered as the nominal location of a spherical joint must define a point O

Sat the

and the distance between the spherical joint and joint centre.the bushing is estimated by the value L

P1. To To achieve the geometric identification, it is

make the assembly possible, the hole diameter wout

necessary to compute each part location andand the inner sleeve diameter w

inmust be greater orientation on the completely assembled vehicle.

than the bolt diameter wbolt

. As a consequence, when This will be done using a set of four reference points,the suspension is assembled and the bolt is tightened, also called marks, on each part. Their geometry isthe effective distance between the spherical joint chosen to allow an easy and precise point coordinateand the bushing may be L

P1+DL , with the maximum measurement with a portable CMM (Fig. 2). The use

value of DL given by of a conic hole as a target and a spherical probe onthe CMM is specially indicated to perform a point

DLmax=hout

2−hbolt+

hin2

(1) coordinate measurement. Moreover, this kind ofmark can be directly manufactured on parts using a

However, the centre Oh

of the bushing housing is used spotting drill. The use of a 2 mm diameter probeas an initial estimation for the bushing position on requires marks with a minimum depth h

minof

part 1. Measuring uncertainties and geometric default 0.7 mm. This dimension is small enough to assumeon the bushing (the inner and outer sleeves may not that it will not reduce the mechanical strength of thebe perfectly concentric) adds extra uncertainty to part, and painted tags are required to visualize markthis approximation. The bushing orientation on location easily.part 1 is not fixed until the suspension is assembled. The position of marks on parts must be chosenTo define this orientation, it is necessary to measure following three constraints.the housing axis l

hand to establish the relative

1. Marks have to be easily accessible to CMMorientation of parts 1 and 2 in the final assembly.measurement once the suspension is assembledThis last point will be detailed in the following

section. and should not be hidden by other parts. As the



Fig. 2 Mark geometry and CMM coordinate measurement

CMM will be fixed under the vehicle, marks aretypically set on the lower face of parts.

2. Marks should be set as far apart as possible fromeach other in order to reduce imprecision on com-puted part orientation consequent to coordinatemeasuring uncertainties.

3. To avoid confusion between marks during forth-coming computations, the distance between twomarks should be different from one mark pair toanother. In the distance matrix D

P, where P

iis the

position vector of mark i, each non-zero termFig. 3 Position of marks on a suspension wishboneshould be different from any other, with

extract point position vectors and unitary line vectorsfrom the IGES file and to build the geometric model

DP=CdP1−P

2d dP

1−P3d dP

1−P4d

0 dP2−P3d dP

2−P4d

0 0 dP3−P4dD (2) of the part.

In this model, bushing position and orientationon the part are defined using an homogeneous

For instance, Fig. 3 gives the mark positions on the operator [BR

p

]. This operator is built from thewishbone of a pseudo-McPherson suspension. Marks position vector P

Ob

of point Ob

, the unitary vectorsare set on the lower face of the arm linking the w

lz

of lz, and u

lx

of lx. The homogeneous operator

spindle plate to the subframe. No mark is set on the [BR

p

] represents the position and orientation of acoordinate system R

bassociated with the bushingarm linking the two bushings because this side of

with respect to Rp

(see Fig. 4). Operator [BR

p

] isthe wishbone is hidden by the subframe when thedefined bysuspension is assembled.

Once all measurement operations are done on apart, results have to be analysed in order to define ageometric model in a mathematical form. Measure-

[BRp2

]=C 1 0 0 0

xb xu2

xv2

xw2

yb yu2

yv2

yw2

zb zu2

zv2

zw2D (3)ment data are saved in a file using the IGES (Initial

Graphics Exchange Specification) format. The authorsused Matlab programming language automatically to



Fig. 4 Elements of a part geometric model, with mark and joint location

where the suspension. Measurement performed on thecomplete suspension must be made in a coordinate

POb

= [xb yb zb ]T system similar to that used for suspension modelling,R

m. In general, this coordinate system is defined withw

lz

= [xw2

yw2

zw2

]Tthe Z axis vertical, the X axis in the forward oriented

ulx

= [xu2

yu2

zu2

]Tlongitudinal direction of the vehicle, and the origincentred between the two tyre contact patches.v

2=ulz

×ulx

= [xv2

yv2

zv2

]TCoordinates of each mark are measured and results

(4) are saved in an IGES file. The analysis of this filedefines a set of four position vectors for each partIn the geometric model, the location of the spherical

joint centre is defined by a position vector in homo- Q*i/Rm

= [1 xQi

yQi

zQi

]TRm

, 1∏ i∏4 (7)geneous coordinates based on measured point O

SAs the measuring order of marks may not be

SRp

= [1 xS yS zS ]T (5) respected (for instance, Q*1

may represent the thirdmark instead of the first one), points Q*

1have to beMark positions are also represented by a position

ordered. This can be done automatically by analysingvector in homogeneous coordinatesthe distance between each pair of points. One

Pi/Rp

= [1 xPi

yPi

zPi

]TRp

, 1∏ i∏4 (6) solution is to test for each permutation of Q*i

definedby the ordered quadruple ( j, k, l, m). For each per-Figure 4 represents the various elements of themutation a distance matrix D

Qis built and comparedgeometric model of a typical suspension linking arm

with the reference distance matrix DP

defined inincluding a spherical joint and a bushing. The elasticequation (2), withbehaviour of bushings and springs are measured

separately when the suspension is disassembled. DQ

( j, k, l, m)

3 PART AND JOINT LOCATION ON THE=CdQ*

j−Q*kd dQ*

j−Q*ld dQ*

j−Q*md

0 dQ*k−Q*ld dQ*

k−Q*md

0 0 dQ*l−Q*mdD (8)ASSEMBLED SUSPENSION

In order to complete the data acquired during separateThe permutation is considered valid if D

Pand D

Qarepart measurement and to achieve identification, it

close enough, i.e. DQ

verifiesis necessary firstly to establish part location andorientation in the assembled suspension at the static

max( |DP−DQ

( j, k, l, m) |)∏e (9)equilibrium of the vehicle. To compute part positionsand orientations for this load case, mark coordinates Parameter e is chosen depending on the precision of

coordinate measurement. For example, if a portableare measured on the assembled suspension. Thevehicle is placed on a measuring bench or a K&C test CMM with a volumetric length accuracy of 0.1 mm

is used, e is set to 0.4 mm. The valid permutationrig, and a portable CMM is fixed on the bench under



defines the ordered point set Qi

with [AR

p2/R

p1

] with

[ARp2/Rp1

]= [ARp1/Rm

]−1 [ARp2/Rm

] (13)Q1=Q*j

, Q2=Q*k

, Q3=Q*l

, Q4=Q*m

This operator makes it possible to express the(10)vector v

2defined in equation (4) in the coordinate

If no permutation of points Q*i

can be achieved insystem R

p1equation (9), this means that a point among Pi

or Qi

v*2= [A

Rp2/Rp1

]v2

(14)is erroneous owing to measurement error or abad interpretation of the measurement result file.

Measurement of part 1 gives the position vector PO

hThis point must not be considered, and followingof the bushing housing centre O

h, and the unitary

computations are realized with a set of three points.vector u

lh

of lh

(see Fig. 1). To represent the bushingOnce points are ordered, it is possible to compute

position and orientation on part 1, the operatorthe part position and orientation in the assembled

[BR

p1

] is defined bymechanism represented by an homogeneous operator[A

Rp

/Rm

] (see Fig. 5). It is assumed that the part isperfectly rigid, and ideally [A

Rp

/Rm

] should give riseto [B

Rp1

]=C 1 0 0 0

xh xu1

xv1

xw1

yh yu1

yv1

yw1

zh zu1

zv1

zw1D (15)

Qi/Rm

= [ARp/Rm

]Pi/Rp

, 1∏ i∏4 (11)

Owing to measurement uncertainties, this equationcannot be satisfied simultaneously for the four wherepoints. Therefore, it is necessary to define the best

POh

= [xh yh zh ]trade-off. This problem is known as the absoluteorientation problem and consists in finding the w

lh

= [xw1

yw1

zw1

]Ttransformation matrix [A

Rp

/Rm

] that minimizes C withu1= [xu1

yu1

zu1

]T

C= ∑4

i=1d [ARp/Rm

]Pi/Rp

−Qi/Rm

d2 (12) v1= [xv1

yv1

zv1

]T

(16)Various algorithms exist to solve this problem [10],but the authors chose to use the direct solution According to the assembly process, the bushingdeveloped by Arun et al. [11]. torsion around its axis l

zis null for the static

If a bushing between parts 1 and 2 as represented equilibrium of the vehicle. The bushing torsion anglein Fig. 5 is considered, the coordinates of mark Q

ion is computed using the following equation [12]

part 2 are used to compute the operator [AR

p2/R

m

] andthe same method is used to compute [A

Rp1

/Rm

]. dRz

=arctanA u1Ωv*2u1Ωulx

B=0 (17)The position of part 2, relative to part 1, is given by

Fig. 5 Mark position in global coordinate system and part location



To achieve this condition, unitary vectors u1

and v1

For the static equilibrium of the vehicle, thecontact force between tyre and ground is vertical andare computed withits norm depends on the vehicle weight. Startingfrom the joint position and orientation defined inu

1=v*2×wlhdv

2×wlh

d section 3, it is possible to compute the force andtorque in each joint of the suspension. Multi-bodyv

1=−u

1×wlh simulation software generally uses the Lagrange

(18) formulation of the equations of motion to performthis task. For each bushing, a force F

B/Rb

and torqueTo build a model with multi-body simulationT

B/Rb

are computed and given in the local coordinatesoftware, each joint position has to be given insystem of the bushing, R

bthe global coordinate system Rm

. The coordinatetransformation is realized using operators [A

Rp

/Rm

] FB/Rb

= [Fx

Fy

Fz]T

SRm

= [ARp/Rm

]SRp

TB/Rb

= [Tx

Ty

Tz]T

[BRm

]= [ARp/Rm

] [BRp

] (22)(19)

Measurement of the bushing stiffness parameters ledto the definition of the force/deflection relation. TheIn order to build a model using multi-body simu-force and torque computed above imply a translationlation software, joint orientation in R

mshould be

and rotational deflection of the bushing on theexpressed using Euler rotation angles rather than asuspensionrotation matrix. The most common rotation sequence

used is the z–x–z sequence (yaw–roll–yaw). The three(dTx

, dTy

, dTz

)= f (Fx, Fy, Fz)Euler angles are given by the following relations [8]

(dRx

, dRy

, dRz

)=g(Tx, Ty, Tz)

w=arctana32

a21−a22

a31

a11

a32−a31

a12

(23)

These deflection values correspond to the displace-h=arctan

√12(a213+a223+a232+a231

)

R33

ment of the inner sleeve relative to the outersleeve. This displacement can be represented bythe operator [D]

Q=arctana13

a22−a12

a23

a11

a23−a13

a21

(20)

[D ]=C 1 0 0 0

dTx

a11

a12

a13

dTy

a21

a22

a23

dTz

a13

a23

a33D (24)with the terms of the homogeneous operator [B

Rm

]as follows

Terms aij

of the rotation matrix are deduced from the[BRm

]=C 1 0 0 0

xRb/Rm

a11

a12

a13

yRb/Rm

a21

a22

a23

zRb/Rm

a31

a32

a33D (21) definition of the three projected torsion angles [12]

used to represent the bushing torsion deflection

−a23

a33=tan(d

Rx

)

4 PARAMETER IDENTIFICATION a13

a33=tan(d

Ry

)

As detailed in section 2, the bushing location definedon part 1 is only an approximation owing to assembly a

21a11=tan(d

Rz

)clearance and experimental uncertainty. To improvethe model quality, a new bushing location on part 1 (25)is computed on the basis of the forces in the bushinggenerated by the static load of the wheel and the A new position of the bushing on part 1 is set accord-

ing to the relative position of parts 1 and 2, and thestiffness parameters measured on the detachedbushing. computed bushing deflection (Fig. 6). The operator



Fig. 6 Bushing position identification

[B*R

p1

] representing this position is computed by parts are separately measured on a surface plateusing a measuring arm. This type of CMM has a

[B*Rp1

]= [ARp1/Rm

]−1 [ARp2/Rm

] [BRp2

] [D ] (26)measurement repeatability of 0.1 mm. Bushing andspring stiffnesses are measured separately. The mainAs kinematic joints are also concerned with geometric

uncertainties, their location has to be identified. stiffness parameters are given in Table 1.Based on these data, it is possible to verify theFor these joints without elastic deformation, the

identification is simpler. If a spherical joint linking assumption of perfectly rigid bodies. The stiffness ofthe rear arm (part S3) is computed for traction forcesparts 2 and 3 is considered, the identified position

of the joint centre on part 3 [S*R

p3

] is given by using a finite element model. With a stiffness of40 000 N/mm and a traction force of 1840 N in the

[S*Rp3

]= [ARp3/Rm

]−1 [ARp2/Rm

] [SRp2

] (27)rear arm at the static equilibrium of the vehicle, thearm elongation is about 0.046 mm. This dimensionalwhere [A

Rp3

/Rm

] is the homogeneous operatorrepresenting the position and orientation of the variation is half the CMM precision, and the hypo-

thesis of a perfectly rigid body used to computereference frame Rp3

local to the part in relation tothe global reference frame R

m. part and joint locations in the mechanism can be

considered as valid.

Table 1 Stiffness parameters of the bushings5 EXPERIMENTAL RESULTSand spring

Experimental validation for the proposed method was Translational stiffness Rotational stiffness(N/mm) (N mm/deg)made on a pseudo-McPherson suspension. Figure 7

presents a kinematic diagram of this suspension. J1 300 1600This axle system has the particularity of using a J2 13 000 4000

J4 650 5500virtual ball joint formed by the front and rear armSpring 22

instead of a wishbone. As described in section 2,

Fig. 7 Kinematic diagram and graph of the suspension



Table 2 Joint locations computed from mark positions

Joint X Y Z y h Q

J1 −294.7 −365.1 246.1 129.5 90.22 181.79J2 61.1 −383.5 186.2 79.58 90.24 4.11J3 −73.7 −353.4 228.3J4 73.7 −594.7 762.6 321.5 166.7 53.68J5 63.1 −617.8 635.1 −28.84 168.1 0J6 −135.3 −694.6 207.4J7 5.87 −679.9 164.8J8 −25.5 −691.2 233.0J9 0.0 −787 305 −0.26 90.69 0

Figure 8 reproduces two photographs of the the force and deformation of bushings at the staticequilibrium of the model. Radial and axial displace-experimental set-up. Parts of the suspension (c)

are attached to a mechanically welded frame that ment values are given in Table 3 for the three bush-ings of the suspension. In this particular example,reproduces the original vehicle attachment points (b).

This frame is fixed on the K&C test rig and the spindle observed torsion deflections could be set aside. Thesevalues are used to define the initial deflection onis linked to the bench actuators (e). While applying

controlled forces and torques, the K&C test rig each bushing. It has been verified that, after identifi-cation, each joint at the model static equilibriummeasures the wheel motion with a computer vision

system (d). The load applied on the spindle to has the same location as measured on the realsuspension.represent the vehicle static equilibrium is a 4500 N

vertical force. The portable CMM (a) is fixed on Once all marks are located, the suspensionbehaviour is characterized with the K&C test rig.the test rig in front of the suspension to allow mark

coordinate measurement. Actuators generate forces applied on both wheelsthat continuously vary between 1000 and 7500 NAs point location has been defined to facilitate

measurements on the assembled axle, this operation vertical, 2000 and −2000 N longitudinal, 2000 andis quite rapid. The experiment showed that measuringall points on the mechanism (20 points) takes less

Table 3 Bushing deflection for thethan 15 min. All measurement results are saved inmodel static equilibrium at

IGES files. The analysis of these data as described the standard load caseabove has been implemented in a program written

Radial deflection, Axial deflectionwith Matlab that automatically computes jointd

Tx

(mm) (mm)positions and orientations. Results of this analysisare given in Table 2 and are used to create the elasto- J1 1.23 0

J2 0.43 0kinematic model with the multi-body simulationJ4 0.37 7.01

software Adams/Car. This model is used to compute

Fig. 8 Suspension close-up and global view of the experimental set-up



−2000 N lateral. All wheel motions are recorded 180 mm [Fig. 9(a)] is improved by the identification.The steering amplitude is 0.3° for the real suspension,during the load cases. The simulation software

includes a virtual test rig that can reproduce each 0.6° for the initial model, and 0.45° for the identifiedmodel. If a tyre drift stiffness of 2 kN/deg at a verticaltest made with K&C. The quality of a model is judged

by comparing force versus wheel movement charts load of 6 kN is considered, a variation of 0.1° of thesteer angle implies an extra lateral force of 200 Narising from test data and simulation.

The behaviour of two models is compared to during a cornering event of the vehicle.The identified model still does not perfectlyevaluate the improvement provided by geometric

identification. The first model uses part dimensions correlate with the real suspension behaviour. Tounderstand the origin of this difference, a steerobtained from disassembled part measurement. In

the second model, part dimensions are modified angle variation under lateral load chart is presented[Fig. 9(b)]. This behaviour is mainly based on stiff-according to the identification method. Steer angle

variation during a vertical wheel displacement of ness properties, and geometric identification is not

Fig. 9 Comparison of real suspension behaviour with elastokinematic models



sufficient to improve the model for lateral load, in The measurement of successive positions ofmarks located on each part during a K&C testspite of the fact that all bushing stiffnesses have beenallows bushing deflection tracking on the assembledmeasured separately. The wheel behaviour undermechanism. This information could be used tolongitudinal loading, presented in Fig. 9(c), dependsperform stiffness parameter identification. This willessentially on the stiffness of bushing J1 (frontbe the focus of future work. Another perspectivearm). As a consequence, the geometric identificationopened up by the part location method is to computedoes not provide a significant improvement. Thisjoint location using only the analysis of part motionexample demonstrates the limits of pure geometricin the assembled mechanism during a K&C test.identification. To improve the model behaviour,The solution of this problem could lead to a newstiffness identification should be set up. To identifysuspension modelling method with no need tothe hysteresis that can be seen in the last twodisassemble.charts, dry friction in spherical joints and the visco-

elastic behaviour of rubber bushings should also bemodelled.

ACKNOWLEDGEMENT

The authors would like to acknowledge the Michelin6 CONCLUSIONSGroup for funding the research programme on whichthis paper is based.A method has been proposed in this paper to

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main reason for the lack of correlation between designed experiments on analytical vehicle models.the elastokinematic model and real suspension Adams User Conference, 1997, paper UC970016.behaviour for a lateral or longitudinal load case. The 7 Rao, P. S., Roccaforte, D., and Campbell, R.

Developing an ADAMS model of an automobilegeometric identification is a first step towards a fullyusing test data. Proceedings of SAE Automotiverepresentative model. The next stage in improvingDynamics and Stability Conference and Exhibition,the model behaviour is to achieve a better estimationDetroit, Michigan, May 2002, technical paper 2002-

of bushing stiffnesses and take into account part 01-1567.elasticity. Then, more complex phenomena, such as 8 Gogu, G. and Coiffet, P. Representation dudry friction or rubber viscoelastic behaviour, could mouvement des corps solides, 1996 (Editions Hermes,

Paris).be included in the model.



9 Meissonnier, J., Fauroux, J. C., Montezin, C., and 11 Arun, K. S., Huang, T. S., and Blostein, S. D.Gogu, G. Identification des parametres geometriques Least-squares fitting of two 3-D point sets. IEEEdu mecanisme de liaison au sol d’un vehicule auto- Trans. Pattern Analysis and Mach. Intell., 1987, 9,mobile. Proceedings of 17th French Congress of 698–700.Mechanics, Troyes, France, August 2005, paper 311. 12 MSC Software ADAMS technical support. AX

10 Eggert, D. W., Lorusso, A., and Fisher, R. B. calculated differently for GFORCE and BUSHING.Estimating 3-D rigid body transformations: a com- Solution 1-KB8295. http://support.mscsoftware.com/parison of four major algorithms. Mach. Vision and kb/results_kb.cfm?S_ID=1-KB8295 7Applic., 1997, 9(5–6), 272–290.


geometric identiﬁcation of an elastokinematic model in a...

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