kinematic coupling interchangeabilitypergatory.mit.edu/.../aspe_interchangeability_published.pdf ·...

Precision Engineering 28 (2004) 1–15

Kinematic coupling interchangeabilityAnastasios John Hart∗, Alexander Slocum, Patrick Willoughby

Precision Engineering Research Group, Department of Mechanical Engineering, Massachusetts Institute of Technology,77 Massachusetts Avenue, Room 3-470, Cambridge, MA 02139, USA

Received 3 June 2002; accepted 14 November 2002

Abstract

The deterministic nature of kinematic couplings enables closed-form characterization of interchangeability error, parametrized in termsof the magnitudes of manufacturing tolerances in the interface manufacturing and assembly processes. A process is suggested for calibratingkinematic couplings to reduce the interchangeability error, based on measurement of the contact points and calculation of a transforma-tion matrix between the interface halves. A Monte Carlo analysis is developed and validated for predicting interchangeability of canoeball kinematic couplings, and repeatability measurements and interchangeability simulation results are presented for kinematic couplinginterfaces for the base and wrist of an industrial robot. Total mounting error, defined as the sum of the interchangeability and repeatabilityerrors, appears to be dependent to first order only on the interface repeatability and the error of the interface calibration procedure.© 2003 Elsevier Inc. All rights reserved.

Keywords: Kinematic coupling; Repeatability; Interchangeability; Modular; Error

1. Introduction

Traditional studies of kinematic couplings, such as thoseby Slocum and Donmez[1], Muller-Held [2], and Pooveyet al. [3], have focused on the need for high interface re-peatability; however, modular machines and instruments re-quire rapid, accurate interchangeability. Interchangeability ofa kinematic coupling is the tendency of the centroidal frame1

of the top half of the interface to return to the same positionand orientation relative to the centroidal frames of differentfixed bottom halves when switched between them[4,5]. Thecentroidal frame is shown inFig. 1, and interchangeabilityerror is shown schematically by the mismatched centroidalframes inFig. 2.

When a kinematic coupling is used to mount a machineor component, the mounting error arises from irregularitiesin the surface and preload conditions, manufacturing varia-tion in the interface geometry, and environmental influencessuch as temperature changes. The translational and rotationalcomponents of these errors are reflected through the struc-tural path of the machine by geometric transformations, giv-

∗ Corresponding author. Tel.:+1-617-258-8541; fax:+1-617-258-6427.E-mail address: [email protected] (A.J. Hart).

URL: http://pergatory.mit.edu.1 The centroidal frame has its origin at the centroid of the coupling triangle,

x-axis aligned with the segment connecting the lower two balls,y-axis normalto thex-axis and in the plane defined by the three couplings (the couplingplane), andz-axis normal to the coupling plane.

ing the error contribution from the kinematic coupling at apoint of measurement interest, such as the tool tip. The goalof this work is to model interchangeability and to determineif measurement of kinematic coupling contacts before inter-face mating can be used to decrease interchangeability error.

2. Kinematic coupling designs

A typical kinematic coupling mates a triangular configura-tion of three hemispheres on one interface plate to three “vee”grooves on another interface plate, thus enabling essentiallyexact constraint of the six degrees of freedom between thetwo bodies by Hertzian surface contact at six small regions.The main caveat to traditional ball–groove couplings, wherethe sphere diameters are approximately the widths of the veegrooves to which they mount, is that their kinematic naturemeans that their load capacity is limited to that of the smallcontact regions.

To achieve greater load capacity yet maintain repeatability,the “canoe ball” shape (named as such because it looks likethe bottom of a canoe), evolved as shown inFig. 3to includean integrated tooling ball for calibration (discussed later). The“canoe” emulates the contact region of a ball as large as 1 min diameter in an element as small as 25 mm across. However,a drawback of this design is the high cost of custom precisioncontour grinding the “ball” contact surfaces.

0141-6359/$ – see front matter © 2003 Elsevier Inc. All rights reserved.doi:10.1016/S0141-6359(03)00071-0

2 A.J. Hart et al. / Precision Engineering 28 (2004) 1–15

Fig. 1. Coupling triangle, showing coupling centroid and centroidal framedirections[5].

As a cost-accuracy compromise, the three-pin quasi-kinematic coupling, shown inFigs. 4 and 5, was developed[4,6,7]. The three-pin coupling consists of an upper interfaceplate with a triangular arrangement of shouldered or dowelpins, coupled to a plate with a triangular arrangement ofoversized cutouts with flat or curved contact surfaces withwhich the pins make contact. The pins are seated against thecontact surfaces by introducing an in-plane preload force at

x

y

x

y

x

y

GROOVES

BALLS

MATED

Fig. 2. In-plane error motion due to positional and angular perturbations of balls and grooves.

Fig. 3. Canoe ball coupling elements: (a) ball–groove assembly with toolingball measurement feature; (b) exploded assembly.

the first pin using a bolt, compliant pin, or other mechanism.Flat contact surfaces, emphasized inFig. 5, are simple tomachine and ensure low contact stress for a given contact pindiameter. A three-pin coupling is designed by first definingthe pin geometry and in-plane preload force to guaranteethat the interface can be properly statically seated (over-coming friction), then defining the normal-to-plane preloadneeded to guarantee dynamic stability and give the desiredstiffness.

A.J. Hart et al. / Precision Engineering 28 (2004) 1–15 3

Fig. 4. Three-pin interface: (a) model of male half, with preload appliedusing a spring pin; (b) in-plane contact forces, with preloadFp and normalcontact reactionsF1, F2, andF3.

3. Kinematic coupling interchangeability model

Neglecting the small variations in repeatability that maybe caused by relatively larger errors in the coupling geom-etry, a first-order estimate of the total mounting error for akinematic coupling is the sum of the repeatability and theinterchangeability errors. Repeatability of heavily-loadedkinematic couplings is a well-studied effect, measured fortypical ball–groove and canoe ball kinematic couplings to beon the order of 1�m and better under well-controlled mount-ing conditions[1,2]. Friction is one of the biggest detrimentsto repeatability, and cannot be accounted for by calibration.Hale presents a quantitative method for estimating averagefrictional nonrepeatability as a function of ball and groovegeometry and the coefficient of friction between the ball andgroove[8]. Schouten et al. showed that incorporating flexures(e.g. by EDM) into groove surfaces can reduce friction suf-ficiently to increase coupling repeatability by a factor of 2 ormore[9].

Fig. 5. Three-pin interface: (a) model of female half, with shouldered pins above (top plate not shown); (b) close view of engagement area for preloaded pin.Note that the location of contact surfaces speficied inFig. 4bprovides better interface stiffness, but the model shown here was specified by design constraintsfor the robot base.

Interchangeability, on the other hand, is a deterministicgeometric error. The kinematic behavior of a triangular lay-out reduces interchangeability error at the center of stiffness(the coupling centroid) to about one-third of the error of thecoupling placements. The remaining error can be reduced bymapping the geometric errors based on the measured posi-tions and orientations of each of the balls and grooves. Thisallows coupling elements to be measured, and the measureddata to be incorporated in a model that determines a set ofmapping coefficients for the interface.

Additional systematic errors arise from deflection of thecoupling contacts and the interface plates due to applieddisturbance forces and thermal expansion. Considering an in-terface that is designed with proper stiffness and thermal man-agement considerations in mind, these errors are neglectedby this model. Past research has demonstrated that the deflec-tions due to Hertzian contact are not as significant as errorsfrom geometric tolerances when couplings are manufacturedusing traditional machining and assembly processes[10,11].

3.1. Layout

To understand the effects of interchangeability error, firstconsider a general machine design application in which twomodules mate through an interface of ball–groove kine-matic couplings.Fig. 6depicts a cell layout for an industrialrobot with a kinematic coupling base mounting, where thegrooves sit on a plate fixed to the floor and the mating ballsare attached to the foot of the robot. Reference coordinateframes are placed centroidally on the groove set (AGroove)and the ball set (ABall), and the couplings are secured usinga sufficient (e.g. bolted) preload. For the work task, the toolcenter point (TCP) and co-located coordinate frame (ATCP)are offset from the ball coordinate frame by a translationand rotation described by the homogeneous transformationmatrix (HTM) TCPTBall. The measurement system also hasan attached coordinate frame (AMS). When the couplingballs and grooves are placed nominally,ABall and AGroove


Fig. 6. Reference interface, tool, and measurement system coordinate frameswith respect to kinematic couplings on the base of a robot. The ABBIRB6400R industrial manipulator is shown[12].

are coincident. Therefore, the forward kinematics of the ma-chine are represented byTCPTBall. Hence, when the TCP iscommanded to the work location (also neglecting all errorsnot related to the kinematic couplings), the TCP frame andthe work frame coincide, such that:

TCPTBall-nom = WorkTBall-nom. (1)

When error in the kinematic couplings is present,ABall andAGroovebecome offset andEq. (1)is invalid. The translationalerrors are reflected exactly at the TCP, and the rotational er-rors are magnified as sine and cosine errors by the distancefrom the base to the TCP.

3.2. Component errors

The sources of tolerance error from manufacturing andassembly variation of a kinematic interface are:

1. positional tolerances of the mounting holes in the interfaceplate holding the balls and in the interface plate holdingthe grooves;

2. flatness variation of the interface plate that holds the ballsand the interface plate that holds the grooves;

3. feature and form errors in the balls and grooves; and4. errors in press-fitting the ball and groove mounts to the

coupling halves, resulting primarily in translation errornormal to the mounting surfaces and angular error aboutthe insertion axis.

In addition, if the relative placements of the kinematic cou-plings are measured in an attempt to estimate the toleranceerrors, error in the measurement system is important. For

Fig. 7. Error motion of a single mounting hole with respect to the centroidalframe.

a present-day industrial laser tracker, this is a maximum of0.01 mm/m of dead path.2

For example, errors in placement of the interface plateholes for mounting the kinematic couplings can be estimatedby placing normal distributions within specified 3σ diamet-rical tolerance zones of the nominal positions of the holes.Using this method, the perturbed coordinates (xhb1, yhb1) ofa mounting hole, shown inFig. 7, are:

xhb1 = xhb1nom + δposRandN() cos(θrand) (2)

yhb1 = yhb1nom + δposRandN() sin(θrand) (3)

θrand = 2πRand(). (4)

In these equations:

1. (xhb1nom, yhb1nom) is the nominal position of the mountinghole, in the coupling plane.

2. δpos is the position tolerance of the mounting hole, ex-pressed as the 3σ radius of a tolerance zone centered atthe hole’s nominal location.

3. θrandis the random angular direction along which the errormotion is applied, measured counter-clockwise from thex-axis of the centroidal frame.

4. RandN() is anormally distributed random number be-tween−1 and 1, which scales the radial distance of theperturbation from the nominal hole center. The normaldistribution guarantees a heavier weight to smaller radialdistances.

5. Rand() is auniformly distributed random number be-tween−1 and 1, used to calculate the orientation of theperturbation with respect to the coordinate frame of theinterface plate. The uniform distribution guarantees anarbitrary orientation of the error.

2 This excludes systematic temperature dependence, which is reasonablyeliminated by built-in software correction from temperature readings[13].


Fig. 8. Definitions of localz-direction dimensions of a canoe ball.

Similar calculations are made for variations of the interfaceplate thickness, coupling mounting orientation, and dead patherror of the measurement system.

Furthermore, when the coupling is measured at a point off-set from its contact location, form error of the coupling affectsthe interchangeability. To model this effect, a local coordi-nate system is placed at the base of the ball or groove mount,along the axis of its mounting hole, as shown inFig. 8. Theexpected form error stackup between the offset measurementsphere and the ball contact point is calculated as the aver-age of root-sum-square (RSS) and worst-case stackups of theform error components[5]. For example, the localz-directionerrorδz,l in the measurement estimate of the location of thecanoe sphere contact point is:3

δz,l = 1

2

(

2

(δRsph√

2

)+ δhR + δhprot + δhmeas

)

+√

2

(δRsph√

2

)2

+ δ2hR + δ2

hprot + δ2hmeas

, (5)

whereδRsphis the radius tolerance of the contact sphere,δhRis the tolerance of the contact point relative to the bottomof the bulk protrusion, along thez-axis, δhprot is the heighttolerance of the canoe ball, andδhmeasis the height toleranceof the measurement feature relative to the canoe ball.

3.3. Combination of errors

The propagation of errors in the interface components to atotal error at the TCP is shown by the block diagram inFig. 9.First, the nominal geometries of the interface plates, the kine-matic couplings, and the measurement feature are specified.

3 The effective radius tolerance is doubled inEq. (4)because there are twospherical surfaces per coupling mount.

The perturbations in mounting hole placement, machinedform of the kinematic couplings, and of the inserted toolingball measurement feature (discussed in the next section), areintroduced. Insertion errors occur between the measurementfeatures and the kinematic couplings, and between the kine-matic couplings and the interface plates. The total error isderived from the sum4 of the component errors at each of thecontact points, and is expressed as a transformation matrixbetween the nominal and true centroidal frames of each in-terface half. These matrices, calculated using measurementsof the contact points before the interface halves are mated,are denotedBall-trueTBall-nom andGroove-trueTGroove-nom, andare specified to a model of the kinematic constraints betweenthe balls and grooves. This model calculates the mating errorbetween the centroidal frames as a third error transformation,Ball-trueTGroove-true.

The interface transformation (Ball-nomTGroove-nom) ac-counts for the total interchangeability error between thekinematic coupling balls and grooves; hence, it expressesthe relationship between the nominal centroidal frame ofthe grooves (referenced to other objects in the cell) and thenominal centroidal frame of the balls (referenced to the ma-chine structure), expressed in the coordinate frame of themeasurement system:

Ball-nomTGroove-nom = (Ball-trueTBall-nom)−1 Ball-true

TGroove-trueGroove-trueTGroove-nom. (6)

This transformation5 can be added to the forward kinematicsof the machine to reduce the interchangeability error at theTCP.

To relate the interchangeability error at the interface to theerror at the TCP, first recall that if the tolerance errors at theinterface are zero, this model will predict that the stationarywork frame (AWork) and the frame at the TCP (ATCP) are coin-cident. When tolerance errors are introduced without appro-priate correction of the machine’s work location, there willbe a mismatch between the work and TCP frames, given bythe transformation:

WorkTTCP= (TCPTBall-nomBall-nomTGroove-nom)−1Work

× TBall-nom. (7)

Then, the vector giving the interchangeability error at theTCP is:

ETCP = TCPTBall-nomBall-nomTGroove-nomVTCP, (8)

whereVTCP is the vector from the origin ofABall to the (nom-inal) origin ofATCP. This vector goes from the desired work

4 As in Eq. (5)the average of the RSS and worst-case sums is taken.5 Note thatFig. 9 distinguishes between the perfect interface transfor-

mation (True,Ball-nomTTrue,Groove-nom) which would give zero residual errorat the TCP and the interface transformation calculated from measurements(Ball-nomTGroove-nom). Hence, the residual error at the TCP comes from er-ror in measuring the kinematic couplings, and in practice also from errorsexcluded from this model.


Nominal

Geometry

Interface Plate:

Balls:

Nominal

GeometryForm error

Form error

Measurement Feature:

Nominal

GeometryForm error

Insertion

error

(ball assembly)

+

True interface

transformationTrue-Groove-nomT

True-Ball-nom

Measurement

Error

Interface Kinematic Model

(In controller)

Σ

-

=

Error at

TCP

without

calibration

=

(<<)

+

Insertion

error

Calculated

interface

transformationGroove-nomTBall-nom

Error at

TCP after

calibration

ERRORS IN BALL SET:

ERRORS IN GROOVE SET:

INTERFACE MATING

(True behavior)

(Measurements)

TCP-nomTGroove-nom

VTCP

TBALL-TCPVTCP

+

+

+ +

+

+

+

+

Nominal

Geometry

Interface Plate:

Balls:

Nominal

GeometryForm error

Form error

Measurement Feature:

Nominal

GeometryForm error

+

+

Insertion

error

+

+

+ +

+

+

+

Insertion

error

(groove

assembly)

+

TCP-nomTGroove-nomVTCP

Fig. 9. Interchangeability error stackup for a kinematic coupling.

point to the true TCP location found by considering the in-terchangeability error. The next section describes the proce-dure for deriving the interface transformation directly fromthe measured locations of the coupling balls and grooves.

4. Solution method for interface calibration

When mounting an interface in practice, the errors betweenthe ideal (nominal) and real centroidal frames of the inter-face halves can be estimated by measuring appropriate fea-tures of the balls and grooves. Knowing the ideal positionsof the contact surfaces, the measured positions are inputs toa kinematic model of the interface geometry which predictsBall-nomTGroove-nom. This section presents and validates of aninterchangeability model for canoe ball couplings. A modelof the three-pin interface was also built, with simplificationof the contact constraints to give a deterministic seating po-

sition. A brief discussion of this model is inAppendix Aandthe reader is referred to Hart[4] for more details.

4.1. Canoe ball interface model

When contact surfaces or offset features such as toolingballs of kinematic couplings are measured, the geometricmating relationship between the centroidal frames of the in-terface halves is found by solving a system of 24 linear equa-tions. Specifically, measurement of the canoe ball interfacegives location estimates6 for the following features of the

6 When an offset feature such as a tooling ball is measured, these valuesare predicted based on the nominal geometry of the coupling.Eq. (5)giveserror of this prediction in addition to the error of the measurement system.An example of direct measurement is taking multi-point measurements ofthe spherical surfaces of the canoe balls and the vee flats of the grooves. Inthis case, the error of the predictions is solely the error of the measurementsystem, including the fitting algorithms.


Fig. 10. Measured parameters of canoe balls and vee grooves.

balls and grooves:

1. [R1, . . . , R6]: The radii of the six spherical contact sur-faces.

2. [⇀q S1, . . . ,⇀q S6]: Position vectors7 directed from each

sphere center to the centroid of the ball interface. Forexample,⇀q S1= 〈uSn1, vSn1, wSn1〉.

3. [b1, . . . , b6]: The base points of the six groove flats, rela-tive to the measurement frame (AMS). For example,b1 =(xb1, yb1, zb1).

4. [⇀N1, . . . ,

⇀N6]: Normal vectors to the six groove flats, in

AMS. For example,⇀N1= 〈xN1, yN1, zN1〉.

These features are shown inFig. 10. The six unknown restpositions (18 unknown coordinates) of the sphere centers aredenoted [pS1, . . . , pS6] = [(xS1, yS1, zS1), . . . , (xS6, yS6,

zS6)]. The remaining six unknowns are the six error off-sets between the centroidal frames of the interface plates,[εx, εy, εz, θx, θy, θz].

Separating variable and constant coefficients, the systemhas the matrix formAX = B, whereX is the 24-element vec-tor containing the unknown final coordinates of the sphereswith respect to the measurement system and the six error mo-tions of the interface. The components of the 24× 24 matrixA and the 24-element column vectorB will be apparent fromthe equations discussed shortly. After invertingA and mul-tiplying the result byB, the six error motions between thecentroidal frames are elements ofX. With small angle ap-proximations, the transformation between centroidal frames

7 These can be expressed in an arbitrary coordinate frame; only the dis-tances between the ball centers are important. The simulation model ex-presses the position vectors with respect to the centroidal ball frame (FBall).

is then:

Ball-trueTGroove-ture =

1 −θzc θyc δxc

−θzc 1 −θxc δyc

−θyc θxc 1 δzc

0 0 0 1

. (9)

Ball-nomTGroove-nom is calculated by combining this resultwith Ball-trueTBall-nom and Groove-trueTGroove-nom (known di-rectly from the measurements) according toEq. (6). Then,given a kinematic model of the machine, the error vector isfound usingEq. (8).

To construct the system of equations, first consider thatwhen the interface is seated, the projected center of eachspherical surface will be as close as possible to its matinggroove. Hence, the line passing through the projected centerof the each sphere and the contact point between the sphereand its mating groove flat will be normal to the flat. Then, thedistance between the projected sphere center and the grooveflat is equal to the measured radius of the spherical surface.For example, the mathematical constraint between the firstsphere and mating flat for the first canoe ball to groove pair,with unknownp1, is:

(p1 − b1)· ⇀N1

|| ⇀N1 || = R1. (10)

A group of six similar equations, one for each sphere/flatpair, contains the 18 final coordinates of the sphere centersas unknowns.

Second, the measured distances between the sphere cen-ters, represented by [⇀q S1, . . . ,

⇀q S6], must not change. Themotions [εx, εy, εz, θx, θy, θz] of the centroidal frame of theball interface (ABall) with respect to the centroidal frame of


Fig. 11. Relationship between error motions of centroidal frame and offset sphere center.

errortransform.m :

Calculates errortransformation from

perturbed inter facedimensions

I N P U T :

Number o f runs

Cal ibrat ion level

Measure error (y/n)?Feature error (y/n)?

kincalibrate.m

- Speci f ies geometry

- Specif ies feature tolerances

Generates random var iates and

perturbs dimensions, serial lyincorporating placement,

a l ignment, form, and

measurement errors.

Compares measured and t rueinterface transformations

Returns error transformation withdistance error.

O U T P U T :

Error transformation

Distance error

Fig. 12. Structure of MATLABTM model for kinematic coupling interchangeability analysis.

the groove interface (AGroove), can be expressed in terms ofthe final positions of the sphere centers. For example, the fi-nal position of the first sphere center is:8

xS1= δxc + uS1[c(θzc)c(θyc)]

+ vS1[c(θzc)s(θyc)s(θxc) − s(θzc)c(θxc)]

+ wS1[c(θzc)s(θyc)c(θxc) − s(θzc)s(θxc)] (11)

yS1= δyc + uS1[s(θzc)c(θyc)]

+ vS1[s(θzc)s(θyc)s(θxc) − c(θzc)c(θxc)]

+ wS1[s(θzc)s(θyc)c(θxc) − c(θzc)s(θxc)] (12)

zS1= δzc + uS1[−s(θyc)]

+ vS1[c(θzc)s(θxc)]

+ wS1[c(θyc)c(θxc)]. (13)

8 The operations(·) denotes sin(·) andc(·) denotes cos(·).

In order to calculate the matricesA andB, small angle ap-proximations must be made such that:

xS1 = δxc + uS1 − vS1θzc + wS1θyc (14)

yS1 = δyc + uS1θzc + vS1 − wS1θxc (15)

zS1 = δyc − uS1θyc + vS1θxc + wS1. (16)

These relationships are diagrammed inFig. 11. Taken for theposition of each sphere, they are the final 18 equations of thesystem.

Therefore, the matrix of coefficientsA contains the com-ponents of the six groove normal vectors (as inEq. (11))in its upper six rows, and the centroidal position vectors[⇀q S1, . . . ,

⇀q S6] (components as inEqs. (12)–(14)) of the ballcenters in its lower 18 rows. The constant vectorB containsthe projections of the groove normal vectors along the posi-tion vectors [⇀q S1, . . . ,

⇀q S6] as its upper six elements, andthe constant terms in the centroidal error motion equations asits lower 18 elements.


Table 1Calibration options for canoe ball interface when offset features are measured

Complexity Example calibration procedure

0 Measure nothing, assuming nominal geometry1 Measure the position of a single tooling ball on each vee groove2 In addition to (1), measure the center location of the bolt hole in each vee groove. This enables calculation of the vee groove orientations3 In addition to (2), measure the position of a single tooling ball on each canoe ball4 Measure single tooling balls on each canoe ball and vee groove5 In addition to (3), measure the center location of the bolt hole in each canoe ball. This enables calculation of the canoe ball orientations

Table 2Calibration options for canoe ball interface when contact surfaces are mea-sured directly

Complexity Example calibration procedure

0 Measure nothing, assuming fully nominal placement1 Perform a sphere fit to the curved surfaces of each

canoe ball, calculating center positions and radii2 Perform a three-point plane fit to each vee groove

flat, calculating base points and normal vectors3 Combine (2) and (3)

4.2. Simulation model

The interchangeability model was built as a series ofMATLAB TM scripts, structured as shown inFig. 12. Param-eters within the scripts specify the nominal geometry anderror tolerance values. Given this information, the modelpredicts the average interchangeability error at the TCP for avariety of levels of calibration detail. For calibration of canoeball interfaces including tooling balls as offset measurementfeatures, the six levels of calibration listed inTable 1areidentified. When the contact surfaces are measured directly,the three levels of calibration listed inTable 2are tested.Simulation results for each of the calibration complexitylevels are presented for the industrial robot base inSection5. The reader is encouraged to download9 the MATLABscripts to follow the mathematical calibration routine anddemonstrate the interchangeability model.

4.3. Physical experiments

The canoe ball interchangeability model was validated bybuilding a series of small prototype models and measuringthe error in the positions and orientations of their centroidalframes over all possible combinations of ball sets and groovesets. A large baseplate had two arrangements of six groovesspaced at equal 60◦ angles around a center point. Ten smallertop pallets were also manufactured, each with an equilateralarrangement of canoe balls. To ensure statistical confidence inthe calibration-interchangeability relationship, the locationsof the coupling mounting and alignment hole pairs on each

9 The scripts can be downloaded from the tools section ofhttp://pergatory.mit.edu/kinematiccouplings.

plate were intentionally perturbed within circular toleranceszones of 0.64 mm (3σ diameter) from their nominal positions.Reference measurement spheres were placed at identical po-sitions with respect to the coupling centroids on the palletsand baseplate.

Fig. 13shows the setup while it was being measured ona Brown & Sharpe MicroVal PFX CMM. The measurementspheres of each pallet were measured in each mounting con-figuration, and after applying the known offsets between thesphere locations and the nominal coupling locations, the in-terface transformation was calculated by directly specifyingthe measured positions of the contact points to the algorithmdiscussed in the previous section.

Fig. 14plots the in-plane angular error of each interfacecombination (choice of a pallet, groove set on the baseplate,and relative orientation), as measured between the centroidalframes, and after the transformation correction was appliedto the measurements. The combinations are grouped for eachof the five pallets. The fifth pallet, for which the interchange-ability correction actually increases the error for some trials,was machined with no more than 0.01 mm deviation from thenominal mounting hole locations. Over all trials, applying theinterface transformation reduced the placement error by anaverage of 92%, specifically from 1.5×10−3 to 1.4×10−4 radfor in-plane rotation. The average total error (positional errorplus sine and cosine errors) reported at a 100 mm circle fromthe coupling centroid was then 0.015 mm, which was within

Fig. 13. Interchangeability measurement setup.

http://pergatory.mit.edu/kinematiccouplings



-0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

1 6 11 16 21 26 31 36 41 46 51

Combination

An

gu

lar

Err

or

[rad

]

After Interchangeability Correction

Measured

1 2

3 54

Fig. 14. In-plane angular error of prototype pallets (numbered groups) before and after transformation correction.

the accuracy limits imposed by the CMM and the tooling ballplacements by CNC machining10 and a light press-fit.

5. Application to industrial robots

5.1. Coupling designs

Kinematic couplings were designed for the base and wristinterfaces of an ABB IRB6400R six-axis industrial robot ma-nipulator, shown inFig. 15. The base interface sits betweenthe robot foot and the factory floor, and is normally restrainedwith eight 20 mm diameter bolts. The new three-bolt alterna-tives are a canoe ball interface (shown inFig. 16), a three-pininterface, and a groove–cylinder interface (see[4] for details).The interface between the robot wrist (the module providingthe fifth and sixth rotational motions) and the robot arm is nor-mally restrained with eight bolts clamping friction-holdingplates between planar contact surfaces. New wrist mountingswere designed using canoe ball couplings, and a three-pincoupling (shown inFig. 17).

5.2. Repeatability performance

The repeatability of the base and wrist interfaces was mea-sured using a Leica LTD500 Laser Tracker[13], which is tra-ditionally used for calibration of the IRB6400R. A “cat’s eye”retroreflector was mounted at the robot TCP, and static mea-surements were taken at five points in the robot’s workspace.In each case, the interface was fully dismounted and re-

10Error of the CNC machine used to machine the plates was approximately0.05 mm/m of travel, more than an order of magnitude below the prescribedperturbations for the mounting holes.

mounted between measurement trials, giving the average re-peatability values shown inFigs. 18 and 19.

The repeatability of the canoe ball base and wrist mount-ings in the shop environment is noticeably much higher thanwould be expected based on documented laboratory mea-surements of kinematic couplings, notwithstanding the largeamplification in angular errors seen by taking measurementsat the robot TCP. This emphasizes several implications for

Fig. 15. ABB IRB6400R. Base of manipulator shown is conventionallymounted using two-pin locators and eight bolts. The pallet shown is boltedto anchors in the floor.


Fig. 16. Prototype canoe ball kinematic coupling interface plates for indus-trial robot base mounting: (a) plates after coupling insertion; (b) close viewof single coupling with tooling ball for calibration. Production design wouldmachine grooves directly to robot foot, and place balls in floor-mountedbaseplate.

mounting kinematic couplings in high-load industrial situa-tions, such as:

1. Bolt preload should be applied incrementally using atorque gauge wrench, while bolts should be greased andcontacts should be cleaned between mountings. A factorof at least 2:1 improvement in repeatability of the robotbase was observed when the mounting procedure wasrefined in this fashion.

2. Interfaces should be engaged as gently as possible, withinitial contact as close to the final seating position as pos-sible. It was extremely difficult to seat the robot manipula-tor on its base without initial offset, increasing the slidingdistance needed to reach the seating position. Dithering

Fig. 17. Prototype three-pin coupling for mounting robot wrist to robot arm. In-plane preload is applied to the bolt indicated; four normal-to-planebolts providedynamic stiffness.

the interface with low-frequency vibration before tighten-ing the bolts is recommended.

3. As shown by the wrist results, installation orientation isalso important. For an equal-angle, in-plane kinematiccoupling as was tested, installation with the couplingsin the horizontal plane is best. Other groove configura-tions, some discussed in Slocum[5], should be investi-gated when horizontal mounting is not possible.

5.3. Interchangeability simulations

Interchangeability simulations were conducted for the baseand wrist interfaces interface, specifying the geometry of themanufactured prototypes and manufacturing and assemblytolerances representative of high volume production of thecomponents. Ten thousand iterations were conducted for eachlevel of calibration complexity listed inSection 4.2.

Fig. 20 shows the simulation results for the base inter-face. The model predicts that the interface transformation ac-counts for approximately 50% or 0.11 mm of the 0.22 mmaverage total interchangeability error when full calibration isperformed relative to offset measurement features. When thecontact surfaces are measured directly, the interchangeabilityanalysis reduces the tool point error by 88% to 0.02 mm. Inthe latter case the remaining error is solely due to measure-ment error; in the former case, variation in the dimension andplacement of the measurement feature is also a factor. A neg-ligible advantage in accuracy is gained by knowledge of therelative orientations of the balls and grooves. Hence, unlessthe process of mounting the couplings to the plates is poorlycontrolled, only measurement of a single feature is neededfor very good calibration performance when offset measure-ment is performed.

5.4. Estimates of total mounting error

Best-case measured repeatability values for the base andwrist canoe ball designs are added to simulated interchange-ability values with full calibration to give the estimates of to-tal error reported inTable 3. Overall, the accuracy benefit of


0.00

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refined mountingDesign

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eata

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Fig. 18. Repeatability measurements of kinematic coupling robot base mountings. Basic mounting procedure was to tighten bolts sequentially using awrench,applying full preload torque to each at once. Refined mounting procedure was to incrementally (10, 50, 100% of 300 N m limit) tighten the bolts in sequenceusing a torque wrench, and clean the coupling contacts and grease the bolts between trials.

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Fig. 19. Repeatability measurements of kinematic coupling robot wrist mountings. Willoughby[7] presents data for several additional combinations of preloadand mounting configuration. Mounting angle is measured between the robot arm and the floor. Three-pin measurements at 90◦ are not presented because theinterface was damaged.


Table 3Estimated total accuracy of kinematic coupling designs for robot base and wrist

Interface Average repeatability (Figs. 18 and 19) Average interchangeability (simulated) Total accuracy (mm)

Wrist, canoe balls—offset 0.06 0.03 0.09Wrist, canoe balls—direct 0.06 0.01 0.07Wrist, three-pin—direct 0.07 0.01 0.08Base, canoe balls—offset 0.06 0.12 0.18Base, canoe balls—direct 0.06 0.03 0.09Base, three-pin—direct 0.07 0.03 0.10

“Offset” designation refers to interchangeability simulation conducted for an offset measurement feature; “direct” simulates measurement of thecouplingcontact surfaces.

using a precision-machined canoe ball setup is negligible overthe simple three-pin interface based on the measurement re-sults here. Cost of the custom-manufactured canoe balls (US$1000–3000 per three balls and grooves, depending on size,in a quantity of 100) would make them prohibitive for mostindustrial applications based on cost-performance consider-ations. A promising industrial alternative to the designs pre-sented may be found in conical line contact quasi-kinematiccouplings designed by Culpepper[14], which were applied

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Fig. 20. Predicted interchangeability error at TCP vs. interface calibrationdetail for robot base: (a) calibration by measurement of offset measurementsphere and bolt hole; (b) calibration by measurement of spherical surfacesand groove flats.

to the mating halves of an automotive engine block for re-peatable location during successive machining operations.

6. Conclusions

Direct measurement of the contact points on the halves of akinematic interface can greatly reduce the effect of toleranceerrors on mounting accuracy, with the residual interchange-ability error based only on the error of the measurementprocedure. By estimating the total mounting accuracy of akinematic coupling as the sum of the measured repeatabilityand the simulated interchangeability, interface manufactur-ing tolerances and the complexity of the calibration processcan be chosen to satisfy an accuracy requirement at mini-mum cost. While past laboratory measurements of kinematiccouplings have shown micron-level repeatability at rela-tively small scales, a test application to industrial robot baseand wrist mountings shows that measured repeatability erroris approximately equal to interchangeability error derivedfrom simulation. These results show that when interfacesexperience heavy loads, and when variability in bolt preload,cleanliness of the contacts, and mating procedure is present,a quasi-kinematic coupling such as the three-pin interfacemay offer equal performance to a ball–groove coupling, atmuch lower cost.

In both cases, the interface transformation has the potentialto become a universal kinematic handshake between kinemat-ically coupled objects, and could enable a conceptually newinterface-centric calibration process for modular machines,whereby:

1. Interface halves are pre-assembled and encoded with theircoupling calibration information, relative to their cen-troidal coordinate frames.

2. These calibrated interface halves are attached to machinemodules (e.g. robot foot), and the modules are calibratedby mounting the assembly to a reference mating inter-face half. The coupling parameters of the reference inter-face are known; hence a calibrationBall-nomTGroove-nom isknown.

3. When the machine modules are brought to the produc-tion installation site, the production interface transfor-mation is calculated from the coupling parameters ofboth production interfaces. A correction is applied to the


machine module calibration for the difference betweenthe calibrationBall-nomTGroove-nom and the productionBall-nomTGroove-nom. This would allow the machine to bemore accurately programmed off-line.

In production, by making the contact surface measure-ments ahead of time, calculation ofBall-nomTGroove-nomwouldbe a step of the machine calibration routine. Ideally, the soft-ware would take the measurement values for the components,calculate the interface HTM, and apply it to the global se-rial chain of transformations for the machine kinematics. Thepre-measured placements of the contacts could be written toan identification tag on the interface, or the interface serialnumber could serve as a database key to the calibration data.

To this end, a design tool synthesizing measured repeata-bility trends from large body of published measurements andincluding interchangeability models, would be useful to en-gineers in application of kinematic couplings to high volumemachinery products such as industrial robots. In beginningsuch an effort, a comprehensive archive of literature and de-sign tools for kinematic couplings is kept athttp://pergatory.mit.edu/kinematiccouplings. The reader is encouraged tocontribute to this repository.

Acknowledgments

This work was funded by research grants from ABB Cor-porate Research and the Ford-MIT Alliance. John Hart wassupported in part by a National Science Foundation GraduateResearch Fellowship. The authors thank Alec Robertson ofABB Robotics for his assistance in completing the repeata-bility measurements for the robot base and wrist, and GerryWentworth of the MIT LMP Machine Shop for his assistancein manufacturing the prototype wrist interface plates.

Appendix A. Three-pin interchangeability model

The three-pin interface model considers in-plane locationby forcing three vertical pins against three vertical contactsurfaces, and vertical seating by engagement of preloadedhorizontal contact surfaces. Measurement of a three-pin in-terface gives estimates of:

1. the radii of the three-pins;2. the in-plane positions of the pin centers, relative to the

centroidal frame for the three-pins;3. the heights of the normal contact surfaces (e.g. pin shoul-

ders) around the pins;4. the positions (base points) of the flat contact surfaces to

which the pins mate; and5. the normal vectors to the contact structures.

A system of nine linear equations gives the centroidal errormotions of a three-pin interface when parameters of its pinsand contact surfaces are perturbed:

1. Three in-plane constraints are established between themeasured pin centers and the contact surfaces in thebottom plates. Similar to the method for the canoeballs, but here in two dimensions, the lines connectingcontact points with the respective pin centers are paral-lel to the measured normal vectors of the contact sur-faces.

2. Six in-plane individual coordinate constraints are estab-lished between the pin centers and the error motions of thenominal centroidal frame of the pin arrangement. Theseequations are identical toEqs. (14)–(16)for the canoe ballinterface.

3. After the system is solved, out-of-plane error is incorpo-rated by adding and averaging the vertical offsets of thenormal contact surfaces around the pins and on the matingplate.

However, because the three vertical line contacts and threehorizontal plane contacts make the three-pin arrangementquasi-kinematic, the following assumptions are made to es-timate the interface transformation:

1. The vertical contacting surfaces of the pins are perfectlyparallel to the mating vertical cuts in the bottom interfaceplate.

2. The horizontal contact surfaces surrounding the pins inthe top interface assembly are all parallel to the horizontalcontact surfaces on the top of the bottom interface plate.While vertical perturbations of the locations of the hori-zontal contact surfaces are modeled, resulting angular er-rors between the contact pairs imposed by mating of allthree randomly offset pairs at once are ignored.

3. Sufficient preload is always applied to perfectly seatthe interface, and manufacturing variation in the loca-tion of the preload has no effect on the interface matingbehavior.

A further discussion and a presentation of simulation resultsis found in Hart[4].

References

[1] Slocum AH, Donmez A. Kinematic couplings for precision fixturing—Part 2: Experimental determination of repeatability and stiffness. PrecEng 1988;10:115–22.

[2] Muller-Held B. Prinzips der kinematischen Kopplung als Schnittstellezwischen Spindel und Schleifscheibe mit praktischer Erprobung imVergleich zum Kegel-Hohlschaft (Translation: Application of kine-matic couplings to a grinding wheel interface). S.M. thesis, Universityof Aachen, Germany; 1999.

[3] Poovey T,Holmes M, Trumper DL. A kinematically-coupled magneticbearing test fixture. Prec Eng 1994;16:99–108.

[4] Hart AJ. Design and analysis of kinematic couplings for modu-lar machine and instrumentation structures. S.M. thesis, Departmentof Mechanical Engineering, Massachusetts Institute of Technology;2001.

[5] Slocum AH. Precision machine design. Soc. Manufact. Eng. 1992.




[6] Slocum AH, Braunstein D. Kinematic coupling for thin plates andsheets and the like. U.S. Patent No. 5,915,768 (1999).

[7] Willoughby PJ. Kinematic alignment of precision robotic elements infactory environments. S.M. thesis, Department of Mechanical Engi-neering, Massachusetts Institute of Technology; 2002.

[8] Hale LC, Slocum AH. Optimal design techniques for kinematic cou-plings. Prec Eng 2000;25:114–27.

[9] Schouten CH, Rosielle JN, Schellekens PH. Design of a kine-matic coupling for precision applications. Prec Eng 1997;20:46–52.

[10] Schmiechen P, Slocum AH. Analysis of kinematic systems: a general-ized approach. Prec Eng 1996;19:11–8.

[11] Slocum AH. Design of three-groove kinematic couplings. Prec Eng1992;14:67–76.

[12] ABB IRB6400R Product Manual. Vasteras, Sweden: ABB Robotics.[13] Leica LTD500 Product Specifications.http://www.leica-geosystems.

com/ims/product/ltd500.htm.[14] Culpepper ML. Design and application of quasi-kinematic couplings.

Ph.D. thesis, Department of Mechanical Engineering, MassachusettsInstitute of Technology; 2000.

http://www.leica-geosystems.com/ims/product/ltd500.htm

http://www.leica-geosystems.com/ims/product/ltd500.htm

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