tolerance analysis of planar mechanisms based on a...

Research ArticleTolerance Analysis of Planar Mechanisms Based ona Residual Approach A Complementary Method to DLM

Hector A Tinoco 12 and Sebastiaacuten Durango 3

1Experimental and Computational Mechanics Laboratory Universidad Autonoma de Manizales Antigua Estacion del Ferrocarril170001 Manizales Colombia2Institute of Physics of Materials Czech Academy of Sciences Zizkova 22 61662 Brno Czech Republic3Grupo Diseno Mecanico y Desarrollo Industrial Universidad Autonoma de Manizales Antigua estacion del Ferrocarril170001 Manizales Colombia

Correspondence should be addressed to Hector A Tinoco htinocoautonomaeduco

Received 20 January 2019 Revised 4 April 2019 Accepted 11 April 2019 Published 5 May 2019

Academic Editor Ramon Sancibrian

Copyright copy 2019 Hector A Tinoco and Sebastian DurangoThis is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The kinematic performance of mechanisms is affected by different uncertainty sources involved in the manufacturing andassembling cycle among these are the geometric variations It is known that the effects of these variations produce position errorswhich are not usually included in the design process With this objective a complementary method for the tolerance analysis ofplanar mechanisms that incorporate geometric variations is presented in this paper The approach is based on Direct LinearizationMethod (DLM) that does not consider all the kinematically admissible solutions DLMnaturallyminimizes a residual functional119867however it is possible to maximize the residual by means of a proposed complementary method called H-Based Residual Method(RMH) From the proposed methodology local and global error domains can be defined to predict the maximum and minimumposition errors caused by the input variations DLM and RMH were applied in a four-bar mechanism with dimensional andangular variations to estimate positioning errors The results show intervals where output positions were invariant with respect toangular variations of the crankThese computations were performed through a distance ratio established with the output deviationsdetermined with nominal angular variations Furthermore domain errors were predicted for a set of positions generated by amultivariate normal random algorithm with 1000 combinations of input variations (links lengths) These domains delimited allsolutions created in each position stage It means that by applying the proposedmethodology it is possible to estimate the geometricerrors of any combination of variations

1 Introduction

Positioning errors in mechanisms are generated by differ-ent uncertainty sources related to manufacturing assemblydesign among others [1ndash3] These uncertainties can beexpressed as geometric variations that play a key role in thedesign process since manufacturing costs are function ofthese variations (manufacturing costs with tight tolerancesare greater on the other hand greater tolerances could makean assembly unfeasible with cheaper manufacturing [4])In mechanisms geometric variations can be considered assmall defects in the link lengths or in the angular positionof the joints According to standards ANSI Y145-2018 [5]

and ISO 1101 [6] tolerances and variations are defined asthe allowable limits that are inherent in manufacturing andassembly processes This means that geometrical variationsare partially controllable parameters (tolerance) and in thosecases the effects of small variations influence the kinematicperformance of themechanismsTherefore appropriate anal-ysis tools should be developed to assess the kinematicsof mechanisms including the different types of variationsas mentioned by Chase et al [7] years ago Their workanalyzed different tolerance analysis with applications tothe mechanism and machine design this shows that thedevelopment of new tools to predict the effects of all type ofvariations is not a new challenge

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 9067624 13 pageshttpsdoiorg10115520199067624

2 Mathematical Problems in Engineering

The effects of small variations (link lengths and jointpositions) on the kinematic performance can be assessed inearly stages of the mechanism design process which corre-sponds to the need for improving the tolerance analysis Lara-Molina et al [8] present a study on planar 5R symmetricalmechanisms in which Monte Carlo simulation was usedto determine the effects of small deviations of link lengthsand joint clearances in several design atlases (MaximumInscribedWorkspace Global Conditioning Index and Mini-mumConditioning Index) Jawale et al [9] revise the use of aPartial Derivative Formulation (PDA) to estimate the output(angular) error in four-bar mechanisms (FBM) subjected tolink tolerances Their works validate that in the diversity ofthe planar mechanisms most of the designs include FBM byits simplicity and functionality which shows that studyingFBM is a current topic Further several error definitionswere presented by Jawale et al [9] (maximum error RMSerror and total error) and the results were verified by meansof a geometrical approach The PDA allows isolating theeffect of each link tolerance on each error category showinga significative difference between the total error (which iscorrelated to the geometric parameters) and the overesti-mated maximum error Consequently other analytic studiesof the error are available eg Hofmeister et al [10] showa model that assumes known joint clearances and estimatesthe error by transforming to the kinematic image spaceA FBM cognate mechanisms and multiloop mechanismswere analyzed Flores [4] calculates the relation betweenvariations in the dimensional parameters and the variationin the generalized coordinates (position error) by derivinga sensitivity matrix from kinematic constraints Worst casescenario and statistical considerations were reported TheDirect LinearizationMethod (DLM) is recurrent in toleranceanalysis being [7 11] seminal works DLMhas been advancedby several authors eg Wittwer et al [12] consolidated awork based on the estimation of the individual contributionof small variations of geometric parameters on assembly vari-ables (dependent variables which guarantee the fulfillmentof the kinematic constraints) To assess the contributionswith DLM a first-order Taylorrsquos series expansion is usedto linearize the variations of the kinematic constraints thatrepresents a linkage The method is usually expressed by aformulation of Jacobian matrices which allows deterministicand probabilistic error assessments eg worst case error androot-sum-square error DLM has been applied in differentmechanism types by [13 14] from rigid to flexible-bodyTheir work described how the coupler path of a four-barmechanisms is affected (shifting in variance and in covarianceof positioning errors) and unveiling critical parameters withhighest contributions to the variations of assembly specifica-tions

Tolerance analysis in mechanisms and machines remainscurrent eg Steering and suspension systems of automobileswere analyzed byKim et al [15]Wang et al [16] andMalvezziand Hess-Coelho [17] Kim et al [15] studied the effects ofaccumulated geometric tolerances on wheel alignment bymeans of Monte Carlo simulations (probabilistic method)Modelling of high reliable steering systems according to theAckermanrsquos principle was assessed by Wang et al [16] Given

the complexity of the kinematic structure of the steeringmechanism a probabilisticmodel based on a global surrogatemodel (build from a set of local surrogate models) was usedto predict the maximal kinematic error of the system Theassessment of kinematic errors in the design of an active rearsuspension was based on a three-degrees-of-freedom parallelmechanism as presented by Malvezzi and Hess-Coelho [17]Two error models were evaluated the first one is based ondirect kinematics (an analyticmodel) that allows themappingof links tolerances and actuator inaccuracies to the wheelworkspace and the second one is based on a parametricoptimization in which the objective function is the wheelalignment Common to Kim et al [15] Wang et al [16]and Malvezzi and Hess-Coelho [17] is the development ofanalysis tools for the design of safer automotive systems inthe kinematic sense

Tolerance analysis applied to transmissions has been con-sidered by Armillotta [18] and Lin et al [19] An analogy fromstatic analysis to tolerance analysis (analytic model) havebeen presented by [18] in which external forces correspondto assembly-level errors and calculated forces (reactionsinternal forces which guarantee equilibrium) correspond tothe sensitivity of the gearing part tolerances on the totalerror Kinematic error and tolerance analysis of cycloidaltransmissions were performed byMonte Carlo simulations in[19] by mapping geometric and manufacturing parametersto the transmission performance through gearing theoryResulting kinematic error distributions allows optimizing thetolerance of the geometric parameters to minimize manufac-turing costs with reliable gearing performance Both worksArmillotta [18] and Lin et al [19] are valuable design tools fortransmissions with a required kinematic error performance

Kinematic error and tolerance analysis in mechanismsparallel robots and structures have been developed recentlyby several authors For example Rameau et al [20] assessthe calculation of joint clearances necessary to guarantee themobility in overconstrained planar and spatial mechanismssubjected to errors in the geometric parameters The methodis based on a geometrical model that allows estimatingmanufacturing parameters for mass production of the over-constrained mechanisms The evaluation of the assemblyprecision in planar mechanisms is presented by Zhao etal [21] The error prediction is conducted by a three-stepassembly algorithm and each step has associated a particularerror model with which was assessed the assembly precisionDifferent from Rameau et al [20] the three-step algorithm isintended for the evaluation of static structures (interpretedas overconstrained mechanisms) Fan et al [22] present akinematic calibration of a multiaxis machine tool based ongeometric errors of the guidewaysThe calibration performedmapping measured kinematic errors to the geometric profileof the guideways The fitted error model allows predictingkinematic errors (position and orientation) in the workspaceof the machine tool then it is possible to modify themanufacturing code (eg G-code) tominimize the kinematicerrors Design of a parallel robot with full-circle rotationbased on error modeling and tolerance analysis is developedin Ni et al [23] Tolerances mapping were assessed througha kinematic model to perform a sensitivity analysis of the

Mathematical Problems in Engineering 3

end moving platform with respect to the geometric errorsThe sensitivity analysis was used as a tolerance design toolby means of nonlinear constrained optimization Commonto Rameau et al [20] Zhao et al [21] Fan et al [22] and Niet al [23] is the use of tolerance analysis as a keystone in thedesign process of mechanisms and machines with requiredperformance (to guarantee mobility and assembly and tominimize positioning errors in machine tools and parallelrobots)

The estimation of individual contributions of geometricalvariations to the positioning error of mechanisms by eitherprobabilistic or deterministic methods it is a commonplace in the aforementioned works This work presents amethodology to predict position errors in planarmechanismswith geometric variations (tolerance analysis) Jawale et al[9] and Wang et al [16] recognized the importance ofimproving DLM (when comparing it with respect to MonteCarlo simulations) and find practical limits (possible subes-timation of maximum normal-to-path errors) Therefore acomplementary method to DLM is developed and applied inthis study The methodology is based on the minimizationand maximization of a functional obtained from numericalresiduals that generate the variations in the closed loop vectoranalysis Error domains are designed from the deviationsto determine the positional variation as a deterministic orprobabilistic solution

2 Theoretical Background

21 Kinematics of PlanarMechanisms Let us consider a 119899-barmechanism composed by one loop which is represented by aclosed vector loop as shown in Figure 1(a) For a mechanismof 119898minusloops the same principles can be applied that will beexposed for one loop mechanisms [24] For describing thekinematics of planar mechanisms a vector notation is usedin a closed loop such that for one loop its mathematicalrepresentation is given by

ℎ = 119899minus1sum119895=1

119903119895119890119894120579119895 minus 119903119899119890119894120579119899 = 0 (1)

where 997888rarr119903 119895 = 119903119895119890119894120579119895 forall119895 = 1 2 119899 Applying Eulerrsquos identityon (1) we have two projections on a coordinate system asfollows

ℎ119909 = 119899minus1sum119895=1

119903119895 cos (120579119895) minus 119903119899 cos (120579119899) = 0 (2)

and

ℎ119910 = 119899minus1sum119895=1

119903119895 sin (120579119895) minus 119903119899 sin (120579119899) = 0 (3)

By organizing (2) and (3) a system of equations is establishedin the following way

A0X minus B0U = 0 (4)

Equations (2) and (3) define two sets of variables anindependent set X = 119902119895 | 119896 119899 sub 119895 and a dependentset U = 119909119894 | 119901 119903 sub 119894 The dependent set is afunction of kinematic variables and geometrical parameters(lengths 119902119896 119909119901 and angles 119902119899 119909119903) Equation (4) describes thekinematics of a mechanism without variations where A0and B0 are composed matrices by known parameters Thesolution U determines the nominal values that satisfy thekinematics for given inputs X

22 Direct Linearization Method (DLM) Direct Lineariza-tion Method (DLM) is a method established for solvingthe kinematics of mechanisms with geometric variations(dimensional and angular variations) DLM was initiallyproposed by Marler [11] and it is based on approximatingvariational parameters of closed vector loops by means ofa linearized functional through Taylorrsquos series expansion asexplained in [14 25] The approximations are obtained fromnominal values Y = XU with the aim of determiningthe variations dU = 119889119909119894 | 119901 119903 sub 119894 that depend onknown parameters dX = 119889119902119895 | 119896 119899 sub 119895 Before applyingDLM the solutions 119902119895 should be solved as well as the set ofexpected variations 119889119902119895 Taking into account the followingconsiderations a variational closed vector loop is defined forthe mechanism represented in Figure 1(b) Letrsquos consider afunctional 119867(Y + dY) linearized about ℎ by using Taylorrsquosseries expansion truncated in the second term therefore itis determined that

119867119909 = ℎ119909 + 119896+119899sum119895=1

120597ℎ119909120597119902119895 119889119902119895 +119901+119903sum119894=1

120597ℎ119909120597119909119894 119889119909119894 = 0 (5)

and

119867119910 = ℎ119910 + 119896+119899sum119895=1

120597ℎ119910120597119902119895 119889119902119895 +119901+119903sum119894=1

120597ℎ119910120597119909119894 119889119909119894 = 0 (6)

As established in (2) and (3) it is known that ℎ119909 =ℎ119910 = 0 (obtained from nominal values) since both solutionsrepresent a closed vector loop If (5) and (6) are organized aset of equations is written as a system of equations such as

AdX + BdU = 0 (7)

being

A = [[[[120597ℎ1199091205971199021 120597ℎ1199091205971199022 sdot sdot sdot 120597ℎ119909120597119902119896+119899120597ℎ1199101205971199021

120597ℎ1199101205971199022 sdot sdot sdot 120597ℎ119910120597119902119896+119899]]]]

B = [[[[[[

120597ℎ1199091205971199091 sdot sdot sdot 120597ℎ119909120597119909119901+119903120597ℎ1199101205971199091 sdot sdot sdot 120597ℎ119905120597119909119901+119903]]]]]]

(8)

where dX = [1198891199021 1198891199022 sdot sdot sdot 119889119902119896+119899]119879 and dU =[1198891199091 sdot sdot sdot 119889119909119901+119903]119879A andB are Jacobianmatrices determined


(a) (b)

y

x

y

x

1 + d1rarrr n

rarrr 3rarrr 2

rarrr 1 r1

r2

rn

dr2

dr1

dr3

drn dr4

r3

r4

rarrr 4

rarrr 5rarrr j

1

rjj forallj = 1 2 3 n

rj + dr

j

Figure 1 General vector loop (a) without variations (b) with variations

of (2) and (3) A solution is obtained from (7) on this basis itcan be calculated that

dU = minusBminus1AdX = SdX (9)

where S is a sensitivity matrix as mentioned by Leishmanet al [14] All basic principles about calculations for DLMcan be reviewed in references [12 13 25] For velocity andacceleration calculations Leishman et al [14] showed indetail the procedure which is based on the differentiation ofthe position equations without and with variations

23 119867-Based Residual Method (RMH) In this section 119867-Based Residual Method (RMH) is presented as an alternativemethod to solve the variations dU it considers119867(Y + dY) asa residual functional since this is determined by numericalapproximations DLM The proposition of the method isbased on the propagation of the numerical errors that aregenerated on 119867(Y + dY) by the truncated Taylorrsquos series Itmeans that there are residual errors Basically the solutionthat will be presented is complementary to DLM Thereforethe following aspect is analyzed when 119867(Y + dY) = 0 thefound solutions are those that minimize the residuals119867119909 and

119867119910 such as it is carried out in DLM However there arekinematically admissible solutions in which the errors canmaximize 120597119867(Y+dY)120597Y = 0 It implies that some kinematicsolutions can propagate errors higher than DLM The abovestatement can be verified from the sensitivity of each variationin 119902119895 and 119909119894 with respect to119867Then the statement is appliedon (5) and (6) to determine maximum or minimum valueson119867 in the following way at 119909minusdirection it is calculated as

119899+119896sum119895=1

120597119867119909120597119902119895 +119903+119901sum119894=1

120597119867119909120597119909119894 = 0 (10)

and at 119910minusdirection it is determined that

119899+119896sum119895=1

120597119867119910120597119902119895 +119903+119901sum119895=1

120597119867119910120597119909119894 = 0 (11)

From (10) and (11) the following system of equations isestablished as follows

dU = minusBsminus1AsdX = SsdX (12)

where

As =[[[[[[[

120597ℎ21199091205971199021120597119902119899+1 120597ℎ21199091205971199022120597119902119899+2 sdot sdot sdot 120597ℎ2119909120597119902119899120597119902119899+119896 120597ℎ21199091205972119902119899+1 + 120597ℎ2119909120597119902119899+11205971199021 sdot sdot sdot 120597ℎ21199091205972119902119899+119896 + 120597ℎ2119909120597119902119899+119896120597119902119899120597ℎ211991012059711990211205971199021120597ℎ21199101205971199022120597120579119899+2 sdot sdot sdot

120597ℎ2119910120597119902119899120597119902119899+119896120597ℎ21199101205972119902119899+1 +

120597ℎ2119910120597119902119899+11205971199021 sdot sdot sdot120597ℎ21199101205972119902119899+119896 +

120597ℎ2119910120597119902119899+119896120597119902119899]]]]]]]

Bs = [[[[[[

120597ℎ211990912059721199091 + 120597ℎ21199091205971199091120597119909119899+1 sdot sdot sdot 120597ℎ21199091205972119909119899 + 120597ℎ2119909120597119909119899120597119909119901120597ℎ211991012059721199091 +120597ℎ21199101205971199091120597119909119899+1 sdot sdot sdot

120597ℎ21199101205972119909119899 +120597ℎ2119910120597119909119899120597119909119901

]]]]]]

(13)

and S119904 is a sensitivity matrix of second order Equation (12)is similar to (9) in its structure but matrices A119904 and Bs havedifferent meanings

24 Geometric Variations dU from Eigenvalues In this sec-tion is shown how variations can be determined from eigen-values obtained from Jacobian matrices these solutions help


to define an error domain inwhich the output position shouldexist inside it Given that both methods (Sections 22 and 23)show a similar equation to solve dU it is possible to computea solution from the variations of the output parameters suchthat the following eigenvalue problem can be written using(7) or (12) as follows

minusAdX = BdU = 120582119868dU (14)

and then we know that

|B minus 120582119868| dU = 0 (15)

where 120582 represents the eigenvalues of the sensitivity matrixB after determining 120582119896 119896 = 1 2 we can obtain the followingsolutions applying (14) as

dUeig = minusAdX120582119896 forall119896 = 1 2 (16)

To determine maximum variations dUmax = dUeig theseshould be obtained with 120582119898119894119899 = min1205821 1205822 It was explainedbefore that dU can be determined from any method forexample RMH and DLM and that there exist two solutionsfor each one those determined from the nominal extremesof the input variations 119889119902+119896 and 119889119902minus119896 So dU+max and dU

minusmax are

determined from the eigenvalues solution

3 Position Prediction

31 Residual Approach for ℎ It was discussed in Section 2that by definition ℎ = 0 which satisfies a closed vector loopwithout variations However if the variations are included inthe mechanism the obtained solutions generate a numericalerror or residual on ℎ(119902119895 +119889119902119895 119909119894 +119889119909119894) = 119890 forall119895 = 1 2 119899caused by the approximation determined for dU (DLM andRMH methods) In Figure 2 a scheme of the propagatederror by the approximation dU is shownThe numerical erroris considered as a vector it denotes that ℎ(119902119896 + 119889119902119896 119909119901 +119889119909119901) minus 119890 = 0 should satisfy the equality The value of 119890 canbe estimated with admissible solutions determined from thefollowing cases if 119889119902+119896 and 119889119902minus119896 Let us consider that 119889119902119896 isa known parameter of input (variation) which has a nominalvalue defined before solving dUThen for 119889119902+119896 gt 0 and 119889119902minus119896 lt0 two different solutions are achieved for dU Each solutiongenerates a residual such that for 119889119902+119896 there are determinedℎ+119909 and ℎ+119910 that represent the projections of the directionalresiduals on both real and imaginary axes Analogously for119889119902minus119896 there are computed ℎminus119909 and ℎminus119910

Considering the directional residuals for each case ofvariations (119889119902minus119896 and 119889119902+119896 ) the Euclidean norm of the residualsis calculated as

119890minus = radic(ℎminus119909)2 + (ℎminus119910)2 if exist119889119902minus119896 lt 0and 119890+ = radic(ℎ+119909)2 + (ℎ+119910)2 if exist119889119902+119896 gt 0

(17)

e

y

x

1 + d1

rarrr n

d rarrr 4d rarrr n

d rarrr 3

d rarrr 2

d rarrr 1 rarrr 3

rarrr 2

rarrr 1

rarrr 4

d rarrrj

rarrrj +

Figure 2 Error propagation over the general vector loop withvariations

To maximize the propagated error by the solutions dU anormalization is proposed as follows

119890 = radic(119890minus)2 + (119890+)2 (18)

Equation (18) can be defined as the maximum propagation ofthe error produced by the approximations In the case ofDLMthe solution dUgenerates a residual of 119890 asymp 0 due to the natureof the solution A discussion will be performed in the nextsections The principal purpose of the error propagation isto extend an error domain from the estimated positions withdU which are kinematically admissible as for example thosesolutions obtained with RMH and DLM

32 Position Prediction in a Local ErrorDomain for Input Vari-ations According to the traditional mechanism design theoutput positions are deterministically predicted Howeverwhen the geometry of the mechanisms varies by differentcircumstances the output positions should exist inside anerror domain which can be seen as a tolerance as proposedby different studies [13 26 27] With this purpose a solutionspace is established and delimited by five points that belongto the output positions of a mechanism with fixed variationsdX It means that geometric variations are known Thosevariations are chosen as steady parameters since the variationsource will be taken only from the input variable (119902119896) Thispresents a range of variation given by 119889119902119896 isin (119889119902minus119896 119889119902+119896 ) Eachestimated output position will be specified in the followingway 119901119896 is the position without including variations in theinput 119901+119896 is the position when exist119889119902+119896 and calculated withdU+ 119901minus119896 is the position when exist119889119902minus119896 and calculated with dUminus119901120582+119896 and 119901120582minus119896 are the positions obtained from the eigenvalueswhen exist119889119902+119896 and 119889119902minus119896 with dU+max and dUminusmax The five pointsare depicted in Figure 3(a) Hence an error domain Ω119896 forthe output positions is defined as a place specified by theboundary of a circumference Γ119896 with an established center in

119901119896 = 13 (119901119896 + 119901120575(119896)) (19)

where 119901120575(119896) = (119901+119896 + 119901minus119896 )2 Equation (19) defines aprediction that represents the closest distance However an


e

pk

y

x

Predicted position

Positions with input variations

trajectory

Positions from eigenvalues

Position without variations

local error space

p+k

p-k

pminuskp+

k

pk

rmaxestimated Ωk

dqminusk and dq+k

obtained with ＞５minusＧＲ and ＞５+

ＧＲ

(a)

y

x

trajectoryglobal error space

error space delimitedby all variations

p+k

p-k

pkpminuskp+

k




dqminusk and dq+k


ＧＲ

estimated Ψk

ok

Ψk

ba

(b)

Figure 3 (a) Local domain error (b) Global domain error

error domain is defined by the radius Γ119896 determined by themaximum distance as

119903119898119886119909(119896) = max 10038161003816100381610038161003816119901119896 minus 119901119896 119901+119896 119901minus119896 119901120582+119896 119901120582minus119896 10038161003816100381610038161003816 + 119890119896 (20)

where 119896 indicates any position The circle area generatesa local domain error Ω119896 that includes the residual 119890119896 thatremains after approximating the output position as illustratedin Figure 3(a) The five points mentioned above are depictedalso in the figure The distance between 119901minus119896 and 119901+119896 isdescribed as 120575 According to 120575 it is possible to say thatthe distance is minimum in regions where the errors in thepositions are invariant to 119889119902+119896 and 119889119902minus119896 Further we can pointout that when 119903119898119886119909(119896) is minimum it indicates where themechanism will be more accurate

To complement the theoretical analysis we propose thedistance 120573 established between 119901120582+119896 and 119901120582minus119896 as a sensitiveparameter The following relation is suggested to determinewhen 119901+119896 asymp 119901minus119896 through the expression

120585 = 120573120575 (21)

Equation (21) will permit examining intervals where the out-put positions 119901+119896 and 119901minus119896 are invariant to the input variationsimposed by 119889119902minus119896 and 119889119902+119896 It means that 120575 asymp 0 when themechanism can have any input variation 119889119902119896 isin (119889119902minus119896 119889119902+119896 )33 Position Prediction with a Global Error Domain forAll Kinematic Variations The main purpose of designingmechanisms is to convert a given input motion into a desiredoutput motion The accuracy of the motion is usually mea-sured by the generated output deviationsThese are producedby the geometric variations of its linkages or by angular

variations provided by assembly tolerances For mechanismswith geometric variations there exist a set of solutions thatdefine an error domain determined by all possible outputpositions related to each given combination of variationsIt indicates that each position should be evaluated in allcombinations of input variations with the aim to estimatethe maximum error in each position state However this taskcan be expensive in computational terms since dependingon the number of combinations the number of solutionswill define the computation time in each solution stage Inthis way a methodology to estimate a global error domain ispresented which will represent all possible solutions markedin an error ellipse for each projected positionTherefore let usconsider amechanismwith nominal variations given as inputparameters An approximation can be constructed from thenominal variations to establish a global domain error Ψ119896 itis defined by an ellipse oriented in direction 119900119896 as shown inFigure 3(b) The unitary vector 119900119896 is determined from thefollowing expression

119900119896 = 119901+119896 minus 119901minus1198961003817100381710038171003817119901+119896 minus 119901minus119896 1003817100381710038171003817 (22)

The minor axis of the error ellipse is given by

119887119896 = radic2 (119901120575(119896) minus 119901119896 + 119890119896) forall119896 = 1 119901 (23)

and the major axis is approximated as follows

119886119896 = 3radic2 (119901120582(119896) minus 119901119896 + 119890119896) (24)


where 119901120582(119896) = (119901120582+119896 + 119901120582minus119896 )2 and 119901 means number ofpositions The error ellipse is a domain that defines theprobability of each output position representing all possiblecombinations of variations of the mechanism

4 Case Study

41 Four-Bar Mechanism (FBM) To evaluate the proposedmethod in Section 2 a planar Four-Bar Mechanism (FBM)is considered and shown in Figure 4(a) The fixed link is thenumber 1 as illustrated in the figureThe angular orientationsare labeled as 120579119895 and the lengths of the links are denoted by119903119895 forall119895 = 1 2 3 4Theparameters 120579119896 119903119895 forall119896 = 1 2 are knownand U = 1205795minus119896 are the output dependent parameters

To describe the position of the FBM each link is repre-sented by a vector as shown in Figure 4(b) The set of vectorsis expressed as 997888rarr119903119894 (119903119894 120579119894) = 119903119894119890119895120579119894 then closed vector loop iswritten as

11990311198901198951205791 + 11990321198901198951205792 + 11990331198901198951205793 minus 11990341198901198951205794 = ℎ = 0 (25)

Equation (25) represents a complete description of anygeometric configuration for determining the positioning ofthe FBM To include the variations in the FBM the nominalvectors are extended by means of variational vectors asdepicted in Figure 4(c) Applying the procedures of lineariza-tion exposed in Section 22 it is determined that

119867 = ℎ + 119899minus1sum119895=1

120597ℎ120597120579119895 119889120579119895 +119899minus1sum119895=1

120597ℎ120597119903119895 119889119903119895minus ( 120597ℎ120597119903119899 119889119903119899 + 120597ℎ120597120579119899 119889120579119899)

(26)

where 119899 = 4 Then applying (26) on (25) we obtain thevariational vector loop that is represented by

119903111988912057911198951198901198951205791 + 11988911990311198901198951205791 + 119903211988912057921198951198901198951205792 + 11988911990321198901198951205792+ 119903311988912057931198951198901198951205793 + 11988911990331198901198951205793 minus 119903411988912057941198951198901198951205794 minus 11988911990341198901198951205794= 119867

(27)

It is important to note that a particular case is obtainedfrom (27) and it is given by 119867 = 0 with these solutions119867 is minimized it means that DLM is determined (seeSection 22) To evaluate the FBM shown in Figure 4 inTable 1 the independent parameters are listed correspondingto those proposed by Leishman et al [14] to validate theDLM

42 Solution dU from DLM for the FBM To describe theposition of the FBM

To obtain the solutions with DLM those that determinethe geometric variations Jacobian matrices A and B arecomputed from (8) so it is obtained that

A = [[cos (1205791) cos (1205792) cos (1205793) minuscos (1205794) minus1199031 sin (1205791) minus1199032 sin (1205792)sin (1205791) sin (1205792) sin (1205793) minussin (1205794) 1199031 cos (1205791) 1199032 cos (1205792) ]] (28)

and

B = [[minus1199033 sin (1205793) 1199034 sin (1205794)1199033 cos (1205793) minus1199034 cos (1205794)]] (29)

The solution dU is determined with (9) where dU =[1198891205793 1198891205794]119879 and dX = [1198891199031 1198891199032 1198891199033 1198891199034 1198891205791 1198891205792]119879 It isknown that S = minusBminus1A wherewe can point out that S is calledsensitivity matrix In general terms the solution presented in(9) is a deterministic solution since a value dU is obtainedfor each input variation References [12 14] estimated thekinematic variations dU by means of a statistic model based

on the deterministic model it is considered as worst case andit can be calculated by

du = radic119899=4sum119895=1

(S119894119895dX119895)2 forall119894 = 1 2 (30)

where dX119895 and Sij represent each element of dX and S

43 Solution dU from RMH for the FBM To solve thevariations dU from the sensitivity 119867 (residual functional)(10) and (11) are applied such that the following matrices arecalculated

As = [[minus sin (1205791) minus sin (1205792) minus sin (1205793) sin (1205794) minus1199031 cos (1205791) minus sin (1205791) minus1199032 cos (1205792) minus sin (1205792)cos (1205791) cos (1205792) cos (1205793) minus cos (1205794) minus1199031 sin (1205791) + cos (1205791) minus1199032 sin (1205792) + cos (1205792)

]] (31)


y

x

3

4

1

2

point A

2 1 4

3

(a)

y

x

3

4

1

2

point A

2 1 4

3

(b)

y

x

3

4

1

2

point A

3+d3

1+d12+d2 4+d4

dr3

dr1

dr4

dr2

(c)

Figure 4 (a) FBM (b) Vector loop without variations (c) Vector loop with variations

Table 1 Dimensions and kinematic variations for the FBM [14]

Item X[cm] Variation dX [cm]1199031 5 0021199032 2 0011199033 5 0021199034 45 00151205791 120587 01205792 0 minus 2120587 1198891205792 isin (minus17 17) times 10minus4and the matrix

Bs = [minus1199033 cos (1205793) minus sin (1205793) 1199034 cos (1205794) + sin (1205794)minus1199033 sin (1205793) + cos (1205793) 1199034 sin (1205794) minus cos (1205794)] (32)

knowing that dU = SsdX and Ss = minusBsminus1As

5 Results and Discussion for the FBM

51 Comparison between DLM and RMH In Figure 5 thereare observed the obtained results for 1198891205793 and 1198891205794 consideringthe input variations on 1205792 119889120579plusmn2 = plusmn17 times 10minus3119903119886119889 (119889119902plusmn119896 )as listed in Table 1 The results computed for dU with bothmethodologies DLM and RMH are compared As explainedin previous sections RMH solutions are based on the sensi-tivity of the linear approximations performed for119867 (presentwork see Section 23) and DLM solutions are focused onthe minimization of 119867 (see [12 14]) In general terms itis seen that the results determined for 1198891205793 and 1198891205794 show asimilar trend for both solution cases However it is importantto mention that 1198891205793 and 1198891205794 evidence a different behaviorfor each input variation 119889120579plusmn2 In the example proposed byLeishman et al [14] only 119889120579+2 was taken into account

Figure 5(a) shows that DLM solutions presented highervariations from 3 and 44 rad for 1198891205794 and 1198891205793 On the otherhand for RMH variations were higher in the rest of theintervals approximately This indicates that both methodscan maximize and minimize the variations with differentsolutions that are kinematically admissible Figure 5(b) showsthat DLM presented maximum variations from 0 until 1and 3 rad for 1198891205794 and 1198891205793 Consequently RMH presentedmaximum variations in the remaining part of the domain It

is important to point out that maximum output variations donot mean that position errors are higher for those intervalssince the position depends on the final configuration of themechanism It implies that the all solutions should satisfy thatℎ(119902119896 + 119889119902119896 119909119901 + 119889119909119901) = 119890 asymp 0 (closed vector loop)

Given that all geometric parameters are known includingits variations it is verified that the solutions satisfy ℎ119909 and ℎ119910by means of the propagated error by both methods DLM andRMH as explained in Section 31 In Figure 6 the correlationsbetween ℎ119909 and ℎ119910 are shown for each input variation119889120579+2 and119889120579minus2 It is observed that errors determined by RMHare greaterthan errors calculated with DLM approximately 102 timesThese correlations are explained as the propagated errorsinfluence the geometric configuration of the mechanismTheexistence of the errors indicates that a confidence domaincan be established to predict the output positions with bothmethodologies as it will be explained in the next sections

In Figure 7(a) the propagated errors determined fromℎ(119902119896 + 119889119902119896 119909119901 + 119889119909119901) = 119890 and calculated with (17) and (18)are shown These were obtained with each input variation119889120579minus2 and 119889120579+2 As proposed in Section 31 it is seen that themagnitude of 119890 computed from DLM is much lower than thevalues calculated from RMH as expected by the definitionof the method 119890 is defined as a vector and represents thedeviation of the geometric configuration of the mechanismsit means that 119890 complements the vector chain to close it

52 Accuracy Errors in the Position by Input Angular Varia-tions (119889120579minus2 and 119889120579+2 ) In this section there are presented thecomputations for the output position errors generated by theinput angular variations 119889120579minus2 and 119889120579+2 applying both methods(RMH and DLM) To propose a discussion about the resultstwo concepts are defined output position error and outputaccuracy Output position error refers to the deviation thatpresents the predicted position with respect to the nominaloutput position (119901119896) Error accuracy means the deviationbetween output positions (distance among them 120575) obtainedwith different input variations as for example the extremeangular values 119889120579minus2 and 119889120579+2

In Figure 8(a) the output position error is shown |119901120575(119896) minus119901119896| determined with the mean value 119901120575(119896) = (119901+119896 + 119901minus119896 )2(as described in Figure 3) It is noted that both methods


times10-4

minus16

minus12

minus80

minus40

0

40

80[r

ad]

1 2 3 4 5 60

2 [rad]

present work-RMHd3present work-RMHd4DLM d3DLM d4

(a)

times10-3

minus3

minus2

minus1

0

1

2

[rad

]

1 2 3 4 5 60

2 [rad]


(b)

Figure 5 Solving dU for (a) 119889120579minus2 (b) 119889120579+2 times10-6

times10-5

d+2

d-2

10050 15 20minus50minus10minus15ℎx [cm]

minus25

minus20

minus15

minus10

minus50

00

50

ℎy

[cm

]

(a)

times10-3

times10-3

d+2

d-2

100 05 15 20minus05minus10ℎx [cm]

minus15

minus10

minus05

00

05

10

15

20

ℎy

[cm

]

(b)

Figure 6 Error in ℎ from (a) DLM (b) RMH

develop a similar trend between the deviations Howeverif the prediction is compared with respect to the nominalposition 119901119896 the deviations are maximum and minimum insome regions For instance it is denoted that in the range1205792 isin (2 54) rad the errors in the output positions areminimized It means that the prediction error was reduced8 with respect to the maximum in this zone

To observe the error accuracy Figure 8(b) is computedtaking the distances 120575 and 120573 that are related by (21) Theseparameters are described in Section 3 In practical terms thedistance 120575 shows the sensitivity of the output position with

respect to the input angular variations 119889120579minus2 and 119889120579+2 When 120575takes minimum values it means that in particular positionsthe mechanism is less sensitive to the input variations Forexample in Figure 8(b) we determine the following intervals1205792 isin (069 082) and 1205792 isin (422 466) rad whichwere established between peaks identified for each solutionmethod (DLMandRMH)The extreme values of the intervalsminimize the distance 120575 that corresponds with the peaks itmeans that 120575 = 0 Consequently these positions are favorableto describe the configuration of the mechanism without anyinfluence of 119889120579plusmn2 This information can be useful in the design


times10-5

e-

e+

e

1 2 3 4 5 60

2 [rad]

00

05

10

15

20

25

30

35N

orm

aliz

ed er

ror v

ecto

r [cm

]

(a)

times10-3

e-

e+

e

1 2 3 4 5 60

2 [rad]

04

06

08

10

12

14

16

18

20

Nor

mal

ized

erro

r vec

tor [

cm]

(b)

Figure 7 Absolute vector error (a) DLM (b) RMH

DLM present work-RMH

00148

00150

00152

00154

00156

00158

00160

00162

00164

p(k

)minuspk [c

m]

1 2 3 4 5 60

2 [rad]

(a)

1

10

100

1 2 3 4 5 60

2 [rad]


(b)

Figure 8 (a) Mean deviation for the output positions (b) 119903119898119886119909

since we could predict where the mechanism is invariant tothe input angular variations imposed by 119889120579plusmn2 53 Estimation of a Local Error Domain by Input AngularVariations (119889120579minus2 and 119889120579+2 ) In order to estimate a tolerance forthe position of the point119860 which is shown in Figure 4 a localdomain error is predicted considering the method exposedin Section 32 Normally in a planar mechanism the outputposition (119901119896) is estimated by means of a deterministic valuewhen it does not present any type of variations (angular anddimensional) On the other hand when there are geometricvariations output positions (119901+119896 119901minus119896 and 119901119896) can be delimitedinside a probabilistic error space as mentioned by Luo et al

[3] which considers that the dimensional variations (designvariables) are fixed nominal values and only it will betaken into account input angular variations established insideinterval 1198891205792 isin (119889120579minus2 119889120579+2 ) as uncontrollable parameters Chenet al [28] For this reason all possible solutions will onlydepend on input angular variations 1198891205792 The FBM used asexample (see Section 41) will help to illustrate the exposedapproach

For the point 119860 a local space error was computed basedon the solution obtained by RMH and DLM but one ofthese solutions is shown in Figure 9 As explained beforefor each position 119896 is determined a circular area Ω119896 withcenter in a new predicted position 119901119896 (see (20)) being the


25 255 26 265 27 275 28 285 29 29538

382384386388

39392394396398

4

Section A-A

Exact position without kinematical variations

Error space by kinematical variations

Section A-A

26

28

3

32

34

36

38

4

42

44

46y

[cm

]

15 2 25 3 35 4 45 5 551x [cm]

(a)


1 2 3 4 5 60

2 [rad]

00100

00104

00108

00112

00116

00120

00124

00128

r max

[cm

]

(b)

Figure 9 (a) Prediction of local error space at the point A (RMHmethod) (b) 119903119898119886119909 comparisons

total error domain delimited by the boundary of the set119862 = Ω1 cup Ω2 sdot sdot sdot Ω119899 marked with gray color in thefigure The radius of Ω119896 defines the tolerance of each outputposition Tolerances are represented by 119903119898119886119909 defined in (20)and these are shown in Figure 9(b) It is observed that in someregions the radius is minimum and in others it is maximumThe results show that the tolerances presented comparablevalues to these illustrated in Figure 8(a) It is noted that 119903119898119886119909determined by DLM is greater than 119903119898119886119909 calculated by RMH20 approximately

54 Estimation of a Global Error Domain for All Combinationsof Variations Section 33 deals with the prediction of a globalerror domain that represents the solutions related to allpossible input combination of geometric variations For thispurpose a multivariate normal random algorithm (MNRA)was used to validate the proposed method using randomparameters Chase et al [29] MNRA generates randomnumbers from values defined with its standard deviationsin our case these were represented by the nominal valuesand the standard deviations by the variations This set ofvalues was then used to calculate the output positions withall possible combinations of the input variations For ourexample there were generated 119873 = 1 000 random setsand a SONY VAIO PC (M350 227 GHz i3 CPU 8 GBRAM) was used in the Windows 7 environment for thecomputations Applying both solution methods (DLM andRMH) the generated data by MNRA were used to validatethe predicted error domains as illustrated in Figure 10

In Section 33 the definition of a global error domain waspresented which is sharply demarcated by an ellipse that rep-resents the possible solutions generatedwith the combinationof input variations 1198891199031 isin (minus002 002) 1198891199032 isin (minus001 001)1198891199033 isin (minus002 002) and 1198891199034 isin (minus0015 0015) In Figure 10are presented three cases of solution in which the output

positions were determined byMNRA for 120579 = 20∘ 150∘ 270∘Points blue and green indicate output positions for theinput nominal variations 119889120579+2 = 0017 rad and 119889120579minus2 =minus0017 rad Points cyan and magenta represent the solutionswith the listed values in Table 1 Points red and blackdescribe the positions obtained with the eigenvalue problemexplained in Section 24 The error domains predicted byDLM and RMH are depicted in Figures 10(a) and 10(b) Itis seen that DLM domains are much smaller than RMHdomains if these are compared the ellipse areas but allpredictions delimited a boundary that enclosed the majorityof solutions As a conclusion it can be stated that thepredictions determinedwith the proposedmethod simplifiedthe variation analysis Therefore it could be used as adesign tool of planar mechanisms that include geometricvariations

6 Conclusions

As main conclusion we pointed out that H-Based ResidualMethod (RMH) was successfully applied and was demon-strated that it is a complementary solution of Direct Lin-earization Method (DLM) Both methodologies (RMH andDLM) were compared and the results showed that these aremathematically correlated The reason is given by the natureof each solution since DLM minimizes a residual functionaland RMH maximizes it as explained theoretically in thisstudy Some differences were evidenced in the presentednumerical example for which was proposed a FBM Asrelevant results were determined intervals where outputpositionswere invariant with respect to the angular variationsof the crank DLM and RMH showed small differencesbetween these The computations were performed througha distance ratio established with the output deviations Inorder to make the results more applied error domains were


Nominal variations Combined variations

444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

274

275

276

277

278

279

28

281

282

y [c

m]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]

144 146 148 15142

x [cm]

Angle 2 in grades 20 Angle 2 in grades 150 Angle 2 in grades 270

p+k

p-k

p+k

p-kpminus

k

p+k

Nominal variations Combined variationsp+k

p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

(a)


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

273274275276277278279

28281282283

y [c

m]

142 144 146 148 15 15214

x [cm]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]


(b)

Figure 10 Global space domain for different positions of 1205792 = 20∘ 150∘ 270∘ (a) DLM (b) RMH

calculated for predicting the output positions as geometrictolerances It was proven by means of a variation analysisthat all combinations of input variations generated by amultivariate normal random algorithm were delimited bythe error domains predicted with RMH and DLM Howeverthe error domain determined with RMH presented a highersize than the domain calculated with DLM it indicates thatRMH is a better method to predict tolerances in the outputpositions

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request Howeverthese can be reproduced with the methodology exposed inthe paper

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was funded by a project from UniversidadAutonoma de Manizales with the project code 423-057 andobtained in the announcement of the year 2015

References

[1] S Khodaygan ldquoManufacturing error compensation based oncutting tool location correction in machining processesrdquo Inter-national Journal of Computer Integrated Manufacturing vol 27no 11 pp 969ndash978 2014

[2] W Wu and S S Rao ldquoUncertainty analysis and allocationof joint tolerances in robot manipulators based on intervalanalysisrdquo Reliability Engineering amp System Safety vol 92 no 1pp 54ndash64 2007

[3] K Luo and X Du ldquoProbabilistic mechanism analysis withbounded random dimension variablesrdquo Mechanism andMachine Theory vol 60 pp 112ndash121 2013

[4] P Flores ldquoA methodology for quantifying the kinematic posi-tion errors due to manufacturing and assembly tolerancesrdquo


Strojniski vestnik ndash Journal of Mechanical Engineering vol 57no 06 pp 457ndash467 2011

[5] Y145-2018 ASME ldquoDimensioning and tolerancingrdquo in TheAmerican Society of Mechanical Engineers ASME New YorkNY USA 2018

[6] 11012017 ISO ldquoGeometrical product specifications (gps geo-metrical tolerancing tolerances of form orientation)rdquo 2017

[7] K W Chase and A R Parkinson ldquoA survey of research in theapplication of tolerance analysis to the design of mechanicalassembliesrdquo Research in Engineering Design vol 3 no 1 pp 23ndash37 1991

[8] F A Lara-Molina E H Koroishi V Steffen and L A MartinsldquoKinematic performance of planar 5R symmetrical parallelmechanism subjected to clearances and uncertaintiesrdquo Journalof the Brazilian Society of Mechanical Sciences and Engineeringvol 40 no 4 article 189 2018

[9] H P Jawale andA Jaiswal ldquoInvestigation ofmechanical error infour-barmechanism under the effects of link tolerancerdquo Journalof the Brazilian Society of Mechanical Sciences and Engineeringvol 40 no 8 article 383 2018

[10] A Hofmeister W Sextro and O Roschel ldquoError workspaceanalysis of planar mechanismsrdquo in EUCOMES the first Euro-pean Conference on Mechanism Science Obergurgl Austria2006

[11] J D Marler Nonlinear tolerance analysis using the direct lin-earization method [PhD thesis] Brigham Young UniversityDepartment of Mechanical Engineering 1988

[12] J W Wittwer K W Chase and L L Howell ldquoThe directlinearization method applied to position error in kinematiclinkagesrdquoMechanismandMachineTheory vol 39 no 7 pp 681ndash693 2004

[13] B M Imani and M Pour ldquoTolerance analysis of flexiblekinematic mechanism using DLM methodrdquo Mechanism andMachine Theory vol 44 no 2 pp 445ndash456 2009

[14] R C Leishman and K W Chase ldquoDirect linearization methodkinematic variation analysisrdquo Journal of Mechanical Design vol132 no 7 Article ID 071003 2010

[15] S K Kim S S Kim Y G Cho and H K Jung ldquoAccumulatedtolerance analysis of suspension by geometric tolerances basedon multibody elasto-kinematic analysisrdquo International Journalof Automotive Technology vol 17 no 2 pp 255ndash263 2016

[16] L Wang X Zhang and Y Zhou ldquoAn effective approach forkinematic reliability analysis of steering mechanismsrdquo Reliabil-ity Engineering amp System Safety vol 180 pp 62ndash76 2018

[17] F Malvezzi and T A Coelho ldquoError analysis for an activegeometry control suspension systemrdquo Journal of the BrazilianSociety of Mechanical Sciences and Engineering vol 40 no 12article 558 2018

[18] A Armillotta ldquoTolerance analysis of gear trains by staticanalogyrdquo Mechanism and Machine Theory vol 135 pp 65ndash802019

[19] K-S Lin K-Y Chan and J-J Lee ldquoKinematic error analysisand tolerance allocation of cycloidal gear reducersrdquoMechanismand Machine Theory vol 124 pp 73ndash91 2018

[20] J Rameau P Serre andM Moinet ldquoClearance vs tolerance formobile overconstrainedmechanismsrdquoMechanism andMachineTheory vol 136 pp 284ndash306 2019

[21] Q Zhao J Guo and J Hong ldquoAssembly precision predictionfor planar closed-loop mechanism in view of joint clearanceand redundant constraintrdquo Journal of Mechanical Science andTechnology vol 32 no 7 pp 3395ndash3405 2018

[22] J Fan H Tao C Wu R Pan Y Tang and Z Li ldquoKinematicerrors prediction for multi-axis machine tools guideways basedon tolerancerdquo The International Journal of Advanced Manufac-turing Technology vol 98 no 5-8 pp 1131ndash1144 2018

[23] Y Ni C Shao B Zhang and W Guo ldquoError modelingand tolerance design of a parallel manipulator with full-circlerotationrdquo Advances in Mechanical Engineering vol 8 no 5 pp1ndash16 2016

[24] R L Norton Design of Machinery An Introduction to theSynthesis and Analysis of Mechanisms and Machines McGraw-Hill Boston Mass USA 2nd edition 1999

[25] H A Tinoco andM A Florez ldquoA newmethod for determiningposition errors of planar mechanisms including dimensionalvariations in its linkagesrdquo in Proceedings of the XII Pan-American Congress of Applied Mechanics (PACAM XII) 2012

[26] J Gao K W Chase and S P Magleby ldquoGeneralized 3-d tol-erance analysis of mechanical assemblies with small kinematicadjustmentsrdquo IIE transactions vol 30 no 4 pp 367ndash377 1998

[27] S Rajagopalan and M Cutkosky ldquoError analysis for the in-situfabrication of mechanismsrdquo Journal of Mechanical Design vol125 no 4 pp 809ndash822 2003

[28] W Chen J K Allen K-L Tsui and F Mistree ldquoA procedurefor robust designMinimizing variations caused by noise factorsand control factorsrdquo Journal of Mechanical Design vol 118 no4 pp 478ndash485 1996

[29] K W Chase J Gao and S P Magleby ldquoGeneral 2-d toleranceanalysis of mechanical assemblies with small kinematic adjust-mentsrdquo Journal of Design and Manufacturing vol 5 pp 263ndash274 1995

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The effects of small variations (link lengths and jointpositions) on the kinematic performance can be assessed inearly stages of the mechanism design process which corre-sponds to the need for improving the tolerance analysis Lara-Molina et al [8] present a study on planar 5R symmetricalmechanisms in which Monte Carlo simulation was usedto determine the effects of small deviations of link lengthsand joint clearances in several design atlases (MaximumInscribedWorkspace Global Conditioning Index and Mini-mumConditioning Index) Jawale et al [9] revise the use of aPartial Derivative Formulation (PDA) to estimate the output(angular) error in four-bar mechanisms (FBM) subjected tolink tolerances Their works validate that in the diversity ofthe planar mechanisms most of the designs include FBM byits simplicity and functionality which shows that studyingFBM is a current topic Further several error definitionswere presented by Jawale et al [9] (maximum error RMSerror and total error) and the results were verified by meansof a geometrical approach The PDA allows isolating theeffect of each link tolerance on each error category showinga significative difference between the total error (which iscorrelated to the geometric parameters) and the overesti-mated maximum error Consequently other analytic studiesof the error are available eg Hofmeister et al [10] showa model that assumes known joint clearances and estimatesthe error by transforming to the kinematic image spaceA FBM cognate mechanisms and multiloop mechanismswere analyzed Flores [4] calculates the relation betweenvariations in the dimensional parameters and the variationin the generalized coordinates (position error) by derivinga sensitivity matrix from kinematic constraints Worst casescenario and statistical considerations were reported TheDirect LinearizationMethod (DLM) is recurrent in toleranceanalysis being [7 11] seminal works DLMhas been advancedby several authors eg Wittwer et al [12] consolidated awork based on the estimation of the individual contributionof small variations of geometric parameters on assembly vari-ables (dependent variables which guarantee the fulfillmentof the kinematic constraints) To assess the contributionswith DLM a first-order Taylorrsquos series expansion is usedto linearize the variations of the kinematic constraints thatrepresents a linkage The method is usually expressed by aformulation of Jacobian matrices which allows deterministicand probabilistic error assessments eg worst case error androot-sum-square error DLM has been applied in differentmechanism types by [13 14] from rigid to flexible-bodyTheir work described how the coupler path of a four-barmechanisms is affected (shifting in variance and in covarianceof positioning errors) and unveiling critical parameters withhighest contributions to the variations of assembly specifica-tions

Tolerance analysis in mechanisms and machines remainscurrent eg Steering and suspension systems of automobileswere analyzed byKim et al [15]Wang et al [16] andMalvezziand Hess-Coelho [17] Kim et al [15] studied the effects ofaccumulated geometric tolerances on wheel alignment bymeans of Monte Carlo simulations (probabilistic method)Modelling of high reliable steering systems according to theAckermanrsquos principle was assessed by Wang et al [16] Given

the complexity of the kinematic structure of the steeringmechanism a probabilisticmodel based on a global surrogatemodel (build from a set of local surrogate models) was usedto predict the maximal kinematic error of the system Theassessment of kinematic errors in the design of an active rearsuspension was based on a three-degrees-of-freedom parallelmechanism as presented by Malvezzi and Hess-Coelho [17]Two error models were evaluated the first one is based ondirect kinematics (an analyticmodel) that allows themappingof links tolerances and actuator inaccuracies to the wheelworkspace and the second one is based on a parametricoptimization in which the objective function is the wheelalignment Common to Kim et al [15] Wang et al [16]and Malvezzi and Hess-Coelho [17] is the development ofanalysis tools for the design of safer automotive systems inthe kinematic sense

Tolerance analysis applied to transmissions has been con-sidered by Armillotta [18] and Lin et al [19] An analogy fromstatic analysis to tolerance analysis (analytic model) havebeen presented by [18] in which external forces correspondto assembly-level errors and calculated forces (reactionsinternal forces which guarantee equilibrium) correspond tothe sensitivity of the gearing part tolerances on the totalerror Kinematic error and tolerance analysis of cycloidaltransmissions were performed byMonte Carlo simulations in[19] by mapping geometric and manufacturing parametersto the transmission performance through gearing theoryResulting kinematic error distributions allows optimizing thetolerance of the geometric parameters to minimize manufac-turing costs with reliable gearing performance Both worksArmillotta [18] and Lin et al [19] are valuable design tools fortransmissions with a required kinematic error performance

Kinematic error and tolerance analysis in mechanismsparallel robots and structures have been developed recentlyby several authors For example Rameau et al [20] assessthe calculation of joint clearances necessary to guarantee themobility in overconstrained planar and spatial mechanismssubjected to errors in the geometric parameters The methodis based on a geometrical model that allows estimatingmanufacturing parameters for mass production of the over-constrained mechanisms The evaluation of the assemblyprecision in planar mechanisms is presented by Zhao etal [21] The error prediction is conducted by a three-stepassembly algorithm and each step has associated a particularerror model with which was assessed the assembly precisionDifferent from Rameau et al [20] the three-step algorithm isintended for the evaluation of static structures (interpretedas overconstrained mechanisms) Fan et al [22] present akinematic calibration of a multiaxis machine tool based ongeometric errors of the guidewaysThe calibration performedmapping measured kinematic errors to the geometric profileof the guideways The fitted error model allows predictingkinematic errors (position and orientation) in the workspaceof the machine tool then it is possible to modify themanufacturing code (eg G-code) tominimize the kinematicerrors Design of a parallel robot with full-circle rotationbased on error modeling and tolerance analysis is developedin Ni et al [23] Tolerances mapping were assessed througha kinematic model to perform a sensitivity analysis of the






ℎ = 119899minus1sum119895=1

119903119895119890119894120579119895 minus 119903119899119890119894120579119899 = 0 (1)


ℎ119909 = 119899minus1sum119895=1

119903119895 cos (120579119895) minus 119903119899 cos (120579119899) = 0 (2)

and

ℎ119910 = 119899minus1sum119895=1

119903119895 sin (120579119895) minus 119903119899 sin (120579119899) = 0 (3)





119867119909 = ℎ119909 + 119896+119899sum119895=1

120597ℎ119909120597119902119895 119889119902119895 +119901+119903sum119894=1

120597ℎ119909120597119909119894 119889119909119894 = 0 (5)

and

119867119910 = ℎ119910 + 119896+119899sum119895=1

120597ℎ119910120597119902119895 119889119902119895 +119901+119903sum119894=1

120597ℎ119910120597119909119894 119889119909119894 = 0 (6)


AdX + BdU = 0 (7)

being


120597ℎ1199101205971199022 sdot sdot sdot 120597ℎ119910120597119902119896+119899]]]]

B = [[[[[[


(8)



(a) (b)

y

x

y

x

1 + d1rarrr n

rarrr 3rarrr 2

rarrr 1 r1

r2

rn

dr2

dr1

dr3

drn dr4

r3

r4

rarrr 4

rarrr 5rarrr j

1


rj + dr

j







119899+119896sum119895=1

120597119867119909120597119902119895 +119903+119901sum119894=1

120597119867119909120597119909119894 = 0 (10)


119899+119896sum119895=1

120597119867119910120597119902119895 +119903+119901sum119895=1

120597119867119910120597119909119894 = 0 (11)



where

As =[[[[[[[


120597ℎ2119910120597119902119899120597119902119899+119896120597ℎ21199101205972119902119899+1 +

120597ℎ2119910120597119902119899+11205971199021 sdot sdot sdot120597ℎ21199101205972119902119899+119896 +

120597ℎ2119910120597119902119899+119896120597119902119899]]]]]]]

Bs = [[[[[[


120597ℎ21199101205972119909119899 +120597ℎ2119910120597119909119899120597119909119901

]]]]]]

(13)





minusAdX = BdU = 120582119868dU (14)


|B minus 120582119868| dU = 0 (15)




minusmax are






(17)

e

y

x

1 + d1

rarrr n

d rarrr 4d rarrr n

d rarrr 3

d rarrr 2

d rarrr 1 rarrr 3

rarrr 2

rarrr 1

rarrr 4

d rarrrj

rarrrj +



119890 = radic(119890minus)2 + (119890+)2 (18)



119901119896 = 13 (119901119896 + 119901120575(119896)) (19)



e

pk

y

x

Predicted position


trajectory



local error space

p+k

p-k

pminuskp+

k

pk

rmaxestimated Ωk

dqminusk and dq+k


ＧＲ

(a)

y

x



p+k

p-k

pkpminuskp+

k




dqminusk and dq+k


ＧＲ

estimated Ψk

ok

Ψk

ba

(b)






120585 = 120573120575 (21)







119886119896 = 3radic2 (119901120582(119896) minus 119901119896 + 119890119896) (24)



4 Case Study



11990311198901198951205791 + 11990321198901198951205792 + 11990331198901198951205793 minus 11990341198901198951205794 = ℎ = 0 (25)


119867 = ℎ + 119899minus1sum119895=1

120597ℎ120597120579119895 119889120579119895 +119899minus1sum119895=1

120597ℎ120597119903119895 119889119903119895minus ( 120597ℎ120597119903119899 119889119903119899 + 120597ℎ120597120579119899 119889120579119899)

(26)


119903111988912057911198951198901198951205791 + 11988911990311198901198951205791 + 119903211988912057921198951198901198951205792 + 11988911990321198901198951205792+ 119903311988912057931198951198901198951205793 + 11988911990331198901198951205793 minus 119903411988912057941198951198901198951205794 minus 11988911990341198901198951205794= 119867

(27)





and




du = radic119899=4sum119895=1

(S119894119895dX119895)2 forall119894 = 1 2 (30)




]] (31)


y

x

3

4

1

2

point A

2 1 4

3

(a)

y

x

3

4

1

2

point A

2 1 4

3

(b)

y

x

3

4

1

2

point A

3+d3

1+d12+d2 4+d4

dr3

dr1

dr4

dr2

(c)















times10-4

minus16

minus12

minus80

minus40

0

40

80[r

ad]

1 2 3 4 5 60

2 [rad]


(a)

times10-3

minus3

minus2

minus1

0

1

2

[rad

]

1 2 3 4 5 60

2 [rad]


(b)


times10-5

d+2

d-2


minus25

minus20

minus15

minus10

minus50

00

50

ℎy

[cm

]

(a)

times10-3

times10-3

d+2

d-2


minus15

minus10

minus05

00

05

10

15

20

ℎy

[cm

]

(b)






times10-5

e-

e+

e

1 2 3 4 5 60

2 [rad]

00

05

10

15

20

25

30

35N

orm

aliz

ed er

ror v

ecto

r [cm

]

(a)

times10-3

e-

e+

e

1 2 3 4 5 60

2 [rad]

04

06

08

10

12

14

16

18

20

Nor

mal

ized

erro

r vec

tor [

cm]

(b)



00148

00150

00152

00154

00156

00158

00160

00162

00164

p(k

)minuspk [c

m]

1 2 3 4 5 60

2 [rad]

(a)

1

10

100

1 2 3 4 5 60

2 [rad]


(b)






25 255 26 265 27 275 28 285 29 29538

382384386388

39392394396398

4

Section A-A



Section A-A

26

28

3

32

34

36

38

4

42

44

46y

[cm

]

15 2 25 3 35 4 45 5 551x [cm]

(a)


1 2 3 4 5 60

2 [rad]

00100

00104

00108

00112

00116

00120

00124

00128

r max

[cm

]

(b)






6 Conclusions




444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

274

275

276

277

278

279

28

281

282

y [c

m]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]

144 146 148 15142

x [cm]


p+k

p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

(a)


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

273274275276277278279

28281282283

y [c

m]

142 144 146 148 15 15214

x [cm]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]


(b)



Data Availability




Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018













ℎ = 119899minus1sum119895=1

119903119895119890119894120579119895 minus 119903119899119890119894120579119899 = 0 (1)


ℎ119909 = 119899minus1sum119895=1

119903119895 cos (120579119895) minus 119903119899 cos (120579119899) = 0 (2)

and

ℎ119910 = 119899minus1sum119895=1

119903119895 sin (120579119895) minus 119903119899 sin (120579119899) = 0 (3)





119867119909 = ℎ119909 + 119896+119899sum119895=1

120597ℎ119909120597119902119895 119889119902119895 +119901+119903sum119894=1

120597ℎ119909120597119909119894 119889119909119894 = 0 (5)

and

119867119910 = ℎ119910 + 119896+119899sum119895=1

120597ℎ119910120597119902119895 119889119902119895 +119901+119903sum119894=1

120597ℎ119910120597119909119894 119889119909119894 = 0 (6)


AdX + BdU = 0 (7)

being


120597ℎ1199101205971199022 sdot sdot sdot 120597ℎ119910120597119902119896+119899]]]]

B = [[[[[[


(8)



(a) (b)

y

x

y

x

1 + d1rarrr n

rarrr 3rarrr 2

rarrr 1 r1

r2

rn

dr2

dr1

dr3

drn dr4

r3

r4

rarrr 4

rarrr 5rarrr j

1


rj + dr

j







119899+119896sum119895=1

120597119867119909120597119902119895 +119903+119901sum119894=1

120597119867119909120597119909119894 = 0 (10)


119899+119896sum119895=1

120597119867119910120597119902119895 +119903+119901sum119895=1

120597119867119910120597119909119894 = 0 (11)



where

As =[[[[[[[


120597ℎ2119910120597119902119899120597119902119899+119896120597ℎ21199101205972119902119899+1 +

120597ℎ2119910120597119902119899+11205971199021 sdot sdot sdot120597ℎ21199101205972119902119899+119896 +

120597ℎ2119910120597119902119899+119896120597119902119899]]]]]]]

Bs = [[[[[[


120597ℎ21199101205972119909119899 +120597ℎ2119910120597119909119899120597119909119901

]]]]]]

(13)





minusAdX = BdU = 120582119868dU (14)


|B minus 120582119868| dU = 0 (15)




minusmax are






(17)

e

y

x

1 + d1

rarrr n

d rarrr 4d rarrr n

d rarrr 3

d rarrr 2

d rarrr 1 rarrr 3

rarrr 2

rarrr 1

rarrr 4

d rarrrj

rarrrj +



119890 = radic(119890minus)2 + (119890+)2 (18)



119901119896 = 13 (119901119896 + 119901120575(119896)) (19)



e

pk

y

x

Predicted position


trajectory



local error space

p+k

p-k

pminuskp+

k

pk

rmaxestimated Ωk

dqminusk and dq+k


ＧＲ

(a)

y

x



p+k

p-k

pkpminuskp+

k




dqminusk and dq+k


ＧＲ

estimated Ψk

ok

Ψk

ba

(b)






120585 = 120573120575 (21)







119886119896 = 3radic2 (119901120582(119896) minus 119901119896 + 119890119896) (24)



4 Case Study



11990311198901198951205791 + 11990321198901198951205792 + 11990331198901198951205793 minus 11990341198901198951205794 = ℎ = 0 (25)


119867 = ℎ + 119899minus1sum119895=1

120597ℎ120597120579119895 119889120579119895 +119899minus1sum119895=1

120597ℎ120597119903119895 119889119903119895minus ( 120597ℎ120597119903119899 119889119903119899 + 120597ℎ120597120579119899 119889120579119899)

(26)


119903111988912057911198951198901198951205791 + 11988911990311198901198951205791 + 119903211988912057921198951198901198951205792 + 11988911990321198901198951205792+ 119903311988912057931198951198901198951205793 + 11988911990331198901198951205793 minus 119903411988912057941198951198901198951205794 minus 11988911990341198901198951205794= 119867

(27)





and




du = radic119899=4sum119895=1

(S119894119895dX119895)2 forall119894 = 1 2 (30)




]] (31)


y

x

3

4

1

2

point A

2 1 4

3

(a)

y

x

3

4

1

2

point A

2 1 4

3

(b)

y

x

3

4

1

2

point A

3+d3

1+d12+d2 4+d4

dr3

dr1

dr4

dr2

(c)















times10-4

minus16

minus12

minus80

minus40

0

40

80[r

ad]

1 2 3 4 5 60

2 [rad]


(a)

times10-3

minus3

minus2

minus1

0

1

2

[rad

]

1 2 3 4 5 60

2 [rad]


(b)


times10-5

d+2

d-2


minus25

minus20

minus15

minus10

minus50

00

50

ℎy

[cm

]

(a)

times10-3

times10-3

d+2

d-2


minus15

minus10

minus05

00

05

10

15

20

ℎy

[cm

]

(b)






times10-5

e-

e+

e

1 2 3 4 5 60

2 [rad]

00

05

10

15

20

25

30

35N

orm

aliz

ed er

ror v

ecto

r [cm

]

(a)

times10-3

e-

e+

e

1 2 3 4 5 60

2 [rad]

04

06

08

10

12

14

16

18

20

Nor

mal

ized

erro

r vec

tor [

cm]

(b)



00148

00150

00152

00154

00156

00158

00160

00162

00164

p(k

)minuspk [c

m]

1 2 3 4 5 60

2 [rad]

(a)

1

10

100

1 2 3 4 5 60

2 [rad]


(b)






25 255 26 265 27 275 28 285 29 29538

382384386388

39392394396398

4

Section A-A



Section A-A

26

28

3

32

34

36

38

4

42

44

46y

[cm

]

15 2 25 3 35 4 45 5 551x [cm]

(a)


1 2 3 4 5 60

2 [rad]

00100

00104

00108

00112

00116

00120

00124

00128

r max

[cm

]

(b)






6 Conclusions




444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

274

275

276

277

278

279

28

281

282

y [c

m]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]

144 146 148 15142

x [cm]


p+k

p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

(a)


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

273274275276277278279

28281282283

y [c

m]

142 144 146 148 15 15214

x [cm]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]


(b)



Data Availability




Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018









(a) (b)

y

x

y

x

1 + d1rarrr n

rarrr 3rarrr 2

rarrr 1 r1

r2

rn

dr2

dr1

dr3

drn dr4

r3

r4

rarrr 4

rarrr 5rarrr j

1


rj + dr

j







119899+119896sum119895=1

120597119867119909120597119902119895 +119903+119901sum119894=1

120597119867119909120597119909119894 = 0 (10)


119899+119896sum119895=1

120597119867119910120597119902119895 +119903+119901sum119895=1

120597119867119910120597119909119894 = 0 (11)



where

As =[[[[[[[


120597ℎ2119910120597119902119899120597119902119899+119896120597ℎ21199101205972119902119899+1 +

120597ℎ2119910120597119902119899+11205971199021 sdot sdot sdot120597ℎ21199101205972119902119899+119896 +

120597ℎ2119910120597119902119899+119896120597119902119899]]]]]]]

Bs = [[[[[[


120597ℎ21199101205972119909119899 +120597ℎ2119910120597119909119899120597119909119901

]]]]]]

(13)





minusAdX = BdU = 120582119868dU (14)


|B minus 120582119868| dU = 0 (15)




minusmax are






(17)

e

y

x

1 + d1

rarrr n

d rarrr 4d rarrr n

d rarrr 3

d rarrr 2

d rarrr 1 rarrr 3

rarrr 2

rarrr 1

rarrr 4

d rarrrj

rarrrj +



119890 = radic(119890minus)2 + (119890+)2 (18)



119901119896 = 13 (119901119896 + 119901120575(119896)) (19)



e

pk

y

x

Predicted position


trajectory



local error space

p+k

p-k

pminuskp+

k

pk

rmaxestimated Ωk

dqminusk and dq+k


ＧＲ

(a)

y

x



p+k

p-k

pkpminuskp+

k




dqminusk and dq+k


ＧＲ

estimated Ψk

ok

Ψk

ba

(b)






120585 = 120573120575 (21)







119886119896 = 3radic2 (119901120582(119896) minus 119901119896 + 119890119896) (24)



4 Case Study



11990311198901198951205791 + 11990321198901198951205792 + 11990331198901198951205793 minus 11990341198901198951205794 = ℎ = 0 (25)


119867 = ℎ + 119899minus1sum119895=1

120597ℎ120597120579119895 119889120579119895 +119899minus1sum119895=1

120597ℎ120597119903119895 119889119903119895minus ( 120597ℎ120597119903119899 119889119903119899 + 120597ℎ120597120579119899 119889120579119899)

(26)


119903111988912057911198951198901198951205791 + 11988911990311198901198951205791 + 119903211988912057921198951198901198951205792 + 11988911990321198901198951205792+ 119903311988912057931198951198901198951205793 + 11988911990331198901198951205793 minus 119903411988912057941198951198901198951205794 minus 11988911990341198901198951205794= 119867

(27)





and




du = radic119899=4sum119895=1

(S119894119895dX119895)2 forall119894 = 1 2 (30)




]] (31)


y

x

3

4

1

2

point A

2 1 4

3

(a)

y

x

3

4

1

2

point A

2 1 4

3

(b)

y

x

3

4

1

2

point A

3+d3

1+d12+d2 4+d4

dr3

dr1

dr4

dr2

(c)















times10-4

minus16

minus12

minus80

minus40

0

40

80[r

ad]

1 2 3 4 5 60

2 [rad]


(a)

times10-3

minus3

minus2

minus1

0

1

2

[rad

]

1 2 3 4 5 60

2 [rad]


(b)


times10-5

d+2

d-2


minus25

minus20

minus15

minus10

minus50

00

50

ℎy

[cm

]

(a)

times10-3

times10-3

d+2

d-2


minus15

minus10

minus05

00

05

10

15

20

ℎy

[cm

]

(b)






times10-5

e-

e+

e

1 2 3 4 5 60

2 [rad]

00

05

10

15

20

25

30

35N

orm

aliz

ed er

ror v

ecto

r [cm

]

(a)

times10-3

e-

e+

e

1 2 3 4 5 60

2 [rad]

04

06

08

10

12

14

16

18

20

Nor

mal

ized

erro

r vec

tor [

cm]

(b)



00148

00150

00152

00154

00156

00158

00160

00162

00164

p(k

)minuspk [c

m]

1 2 3 4 5 60

2 [rad]

(a)

1

10

100

1 2 3 4 5 60

2 [rad]


(b)






25 255 26 265 27 275 28 285 29 29538

382384386388

39392394396398

4

Section A-A



Section A-A

26

28

3

32

34

36

38

4

42

44

46y

[cm

]

15 2 25 3 35 4 45 5 551x [cm]

(a)


1 2 3 4 5 60

2 [rad]

00100

00104

00108

00112

00116

00120

00124

00128

r max

[cm

]

(b)






6 Conclusions




444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

274

275

276

277

278

279

28

281

282

y [c

m]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]

144 146 148 15142

x [cm]


p+k

p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

(a)


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

273274275276277278279

28281282283

y [c

m]

142 144 146 148 15 15214

x [cm]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]


(b)



Data Availability




Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018










minusAdX = BdU = 120582119868dU (14)


|B minus 120582119868| dU = 0 (15)




minusmax are






(17)

e

y

x

1 + d1

rarrr n

d rarrr 4d rarrr n

d rarrr 3

d rarrr 2

d rarrr 1 rarrr 3

rarrr 2

rarrr 1

rarrr 4

d rarrrj

rarrrj +



119890 = radic(119890minus)2 + (119890+)2 (18)



119901119896 = 13 (119901119896 + 119901120575(119896)) (19)



e

pk

y

x

Predicted position


trajectory



local error space

p+k

p-k

pminuskp+

k

pk

rmaxestimated Ωk

dqminusk and dq+k


ＧＲ

(a)

y

x



p+k

p-k

pkpminuskp+

k




dqminusk and dq+k


ＧＲ

estimated Ψk

ok

Ψk

ba

(b)






120585 = 120573120575 (21)







119886119896 = 3radic2 (119901120582(119896) minus 119901119896 + 119890119896) (24)



4 Case Study



11990311198901198951205791 + 11990321198901198951205792 + 11990331198901198951205793 minus 11990341198901198951205794 = ℎ = 0 (25)


119867 = ℎ + 119899minus1sum119895=1

120597ℎ120597120579119895 119889120579119895 +119899minus1sum119895=1

120597ℎ120597119903119895 119889119903119895minus ( 120597ℎ120597119903119899 119889119903119899 + 120597ℎ120597120579119899 119889120579119899)

(26)


119903111988912057911198951198901198951205791 + 11988911990311198901198951205791 + 119903211988912057921198951198901198951205792 + 11988911990321198901198951205792+ 119903311988912057931198951198901198951205793 + 11988911990331198901198951205793 minus 119903411988912057941198951198901198951205794 minus 11988911990341198901198951205794= 119867

(27)





and




du = radic119899=4sum119895=1

(S119894119895dX119895)2 forall119894 = 1 2 (30)




]] (31)


y

x

3

4

1

2

point A

2 1 4

3

(a)

y

x

3

4

1

2

point A

2 1 4

3

(b)

y

x

3

4

1

2

point A

3+d3

1+d12+d2 4+d4

dr3

dr1

dr4

dr2

(c)















times10-4

minus16

minus12

minus80

minus40

0

40

80[r

ad]

1 2 3 4 5 60

2 [rad]


(a)

times10-3

minus3

minus2

minus1

0

1

2

[rad

]

1 2 3 4 5 60

2 [rad]


(b)


times10-5

d+2

d-2


minus25

minus20

minus15

minus10

minus50

00

50

ℎy

[cm

]

(a)

times10-3

times10-3

d+2

d-2


minus15

minus10

minus05

00

05

10

15

20

ℎy

[cm

]

(b)






times10-5

e-

e+

e

1 2 3 4 5 60

2 [rad]

00

05

10

15

20

25

30

35N

orm

aliz

ed er

ror v

ecto

r [cm

]

(a)

times10-3

e-

e+

e

1 2 3 4 5 60

2 [rad]

04

06

08

10

12

14

16

18

20

Nor

mal

ized

erro

r vec

tor [

cm]

(b)



00148

00150

00152

00154

00156

00158

00160

00162

00164

p(k

)minuspk [c

m]

1 2 3 4 5 60

2 [rad]

(a)

1

10

100

1 2 3 4 5 60

2 [rad]


(b)






25 255 26 265 27 275 28 285 29 29538

382384386388

39392394396398

4

Section A-A



Section A-A

26

28

3

32

34

36

38

4

42

44

46y

[cm

]

15 2 25 3 35 4 45 5 551x [cm]

(a)


1 2 3 4 5 60

2 [rad]

00100

00104

00108

00112

00116

00120

00124

00128

r max

[cm

]

(b)






6 Conclusions




444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

274

275

276

277

278

279

28

281

282

y [c

m]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]

144 146 148 15142

x [cm]


p+k

p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

(a)


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

273274275276277278279

28281282283

y [c

m]

142 144 146 148 15 15214

x [cm]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]


(b)



Data Availability




Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018









e

pk

y

x

Predicted position


trajectory



local error space

p+k

p-k

pminuskp+

k

pk

rmaxestimated Ωk

dqminusk and dq+k


ＧＲ

(a)

y

x



p+k

p-k

pkpminuskp+

k




dqminusk and dq+k


ＧＲ

estimated Ψk

ok

Ψk

ba

(b)






120585 = 120573120575 (21)







119886119896 = 3radic2 (119901120582(119896) minus 119901119896 + 119890119896) (24)



4 Case Study



11990311198901198951205791 + 11990321198901198951205792 + 11990331198901198951205793 minus 11990341198901198951205794 = ℎ = 0 (25)


119867 = ℎ + 119899minus1sum119895=1

120597ℎ120597120579119895 119889120579119895 +119899minus1sum119895=1

120597ℎ120597119903119895 119889119903119895minus ( 120597ℎ120597119903119899 119889119903119899 + 120597ℎ120597120579119899 119889120579119899)

(26)


119903111988912057911198951198901198951205791 + 11988911990311198901198951205791 + 119903211988912057921198951198901198951205792 + 11988911990321198901198951205792+ 119903311988912057931198951198901198951205793 + 11988911990331198901198951205793 minus 119903411988912057941198951198901198951205794 minus 11988911990341198901198951205794= 119867

(27)





and




du = radic119899=4sum119895=1

(S119894119895dX119895)2 forall119894 = 1 2 (30)




]] (31)


y

x

3

4

1

2

point A

2 1 4

3

(a)

y

x

3

4

1

2

point A

2 1 4

3

(b)

y

x

3

4

1

2

point A

3+d3

1+d12+d2 4+d4

dr3

dr1

dr4

dr2

(c)















times10-4

minus16

minus12

minus80

minus40

0

40

80[r

ad]

1 2 3 4 5 60

2 [rad]


(a)

times10-3

minus3

minus2

minus1

0

1

2

[rad

]

1 2 3 4 5 60

2 [rad]


(b)


times10-5

d+2

d-2


minus25

minus20

minus15

minus10

minus50

00

50

ℎy

[cm

]

(a)

times10-3

times10-3

d+2

d-2


minus15

minus10

minus05

00

05

10

15

20

ℎy

[cm

]

(b)






times10-5

e-

e+

e

1 2 3 4 5 60

2 [rad]

00

05

10

15

20

25

30

35N

orm

aliz

ed er

ror v

ecto

r [cm

]

(a)

times10-3

e-

e+

e

1 2 3 4 5 60

2 [rad]

04

06

08

10

12

14

16

18

20

Nor

mal

ized

erro

r vec

tor [

cm]

(b)



00148

00150

00152

00154

00156

00158

00160

00162

00164

p(k

)minuspk [c

m]

1 2 3 4 5 60

2 [rad]

(a)

1

10

100

1 2 3 4 5 60

2 [rad]


(b)






25 255 26 265 27 275 28 285 29 29538

382384386388

39392394396398

4

Section A-A



Section A-A

26

28

3

32

34

36

38

4

42

44

46y

[cm

]

15 2 25 3 35 4 45 5 551x [cm]

(a)


1 2 3 4 5 60

2 [rad]

00100

00104

00108

00112

00116

00120

00124

00128

r max

[cm

]

(b)






6 Conclusions




444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

274

275

276

277

278

279

28

281

282

y [c

m]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]

144 146 148 15142

x [cm]


p+k

p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

(a)


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

273274275276277278279

28281282283

y [c

m]

142 144 146 148 15 15214

x [cm]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]


(b)



Data Availability




Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018










4 Case Study



11990311198901198951205791 + 11990321198901198951205792 + 11990331198901198951205793 minus 11990341198901198951205794 = ℎ = 0 (25)


119867 = ℎ + 119899minus1sum119895=1

120597ℎ120597120579119895 119889120579119895 +119899minus1sum119895=1

120597ℎ120597119903119895 119889119903119895minus ( 120597ℎ120597119903119899 119889119903119899 + 120597ℎ120597120579119899 119889120579119899)

(26)


119903111988912057911198951198901198951205791 + 11988911990311198901198951205791 + 119903211988912057921198951198901198951205792 + 11988911990321198901198951205792+ 119903311988912057931198951198901198951205793 + 11988911990331198901198951205793 minus 119903411988912057941198951198901198951205794 minus 11988911990341198901198951205794= 119867

(27)





and




du = radic119899=4sum119895=1

(S119894119895dX119895)2 forall119894 = 1 2 (30)




]] (31)


y

x

3

4

1

2

point A

2 1 4

3

(a)

y

x

3

4

1

2

point A

2 1 4

3

(b)

y

x

3

4

1

2

point A

3+d3

1+d12+d2 4+d4

dr3

dr1

dr4

dr2

(c)















times10-4

minus16

minus12

minus80

minus40

0

40

80[r

ad]

1 2 3 4 5 60

2 [rad]


(a)

times10-3

minus3

minus2

minus1

0

1

2

[rad

]

1 2 3 4 5 60

2 [rad]


(b)


times10-5

d+2

d-2


minus25

minus20

minus15

minus10

minus50

00

50

ℎy

[cm

]

(a)

times10-3

times10-3

d+2

d-2


minus15

minus10

minus05

00

05

10

15

20

ℎy

[cm

]

(b)






times10-5

e-

e+

e

1 2 3 4 5 60

2 [rad]

00

05

10

15

20

25

30

35N

orm

aliz

ed er

ror v

ecto

r [cm

]

(a)

times10-3

e-

e+

e

1 2 3 4 5 60

2 [rad]

04

06

08

10

12

14

16

18

20

Nor

mal

ized

erro

r vec

tor [

cm]

(b)



00148

00150

00152

00154

00156

00158

00160

00162

00164

p(k

)minuspk [c

m]

1 2 3 4 5 60

2 [rad]

(a)

1

10

100

1 2 3 4 5 60

2 [rad]


(b)






25 255 26 265 27 275 28 285 29 29538

382384386388

39392394396398

4

Section A-A



Section A-A

26

28

3

32

34

36

38

4

42

44

46y

[cm

]

15 2 25 3 35 4 45 5 551x [cm]

(a)


1 2 3 4 5 60

2 [rad]

00100

00104

00108

00112

00116

00120

00124

00128

r max

[cm

]

(b)






6 Conclusions




444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

274

275

276

277

278

279

28

281

282

y [c

m]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]

144 146 148 15142

x [cm]


p+k

p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

(a)


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

273274275276277278279

28281282283

y [c

m]

142 144 146 148 15 15214

x [cm]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]


(b)



Data Availability




Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018









y

x

3

4

1

2

point A

2 1 4

3

(a)

y

x

3

4

1

2

point A

2 1 4

3

(b)

y

x

3

4

1

2

point A

3+d3

1+d12+d2 4+d4

dr3

dr1

dr4

dr2

(c)















times10-4

minus16

minus12

minus80

minus40

0

40

80[r

ad]

1 2 3 4 5 60

2 [rad]


(a)

times10-3

minus3

minus2

minus1

0

1

2

[rad

]

1 2 3 4 5 60

2 [rad]


(b)


times10-5

d+2

d-2


minus25

minus20

minus15

minus10

minus50

00

50

ℎy

[cm

]

(a)

times10-3

times10-3

d+2

d-2


minus15

minus10

minus05

00

05

10

15

20

ℎy

[cm

]

(b)






times10-5

e-

e+

e

1 2 3 4 5 60

2 [rad]

00

05

10

15

20

25

30

35N

orm

aliz

ed er

ror v

ecto

r [cm

]

(a)

times10-3

e-

e+

e

1 2 3 4 5 60

2 [rad]

04

06

08

10

12

14

16

18

20

Nor

mal

ized

erro

r vec

tor [

cm]

(b)



00148

00150

00152

00154

00156

00158

00160

00162

00164

p(k

)minuspk [c

m]

1 2 3 4 5 60

2 [rad]

(a)

1

10

100

1 2 3 4 5 60

2 [rad]


(b)






25 255 26 265 27 275 28 285 29 29538

382384386388

39392394396398

4

Section A-A



Section A-A

26

28

3

32

34

36

38

4

42

44

46y

[cm

]

15 2 25 3 35 4 45 5 551x [cm]

(a)


1 2 3 4 5 60

2 [rad]

00100

00104

00108

00112

00116

00120

00124

00128

r max

[cm

]

(b)






6 Conclusions




444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

274

275

276

277

278

279

28

281

282

y [c

m]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]

144 146 148 15142

x [cm]


p+k

p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

(a)


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

273274275276277278279

28281282283

y [c

m]

142 144 146 148 15 15214

x [cm]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]


(b)



Data Availability




Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018









times10-4

minus16

minus12

minus80

minus40

0

40

80[r

ad]

1 2 3 4 5 60

2 [rad]


(a)

times10-3

minus3

minus2

minus1

0

1

2

[rad

]

1 2 3 4 5 60

2 [rad]


(b)


times10-5

d+2

d-2


minus25

minus20

minus15

minus10

minus50

00

50

ℎy

[cm

]

(a)

times10-3

times10-3

d+2

d-2


minus15

minus10

minus05

00

05

10

15

20

ℎy

[cm

]

(b)






times10-5

e-

e+

e

1 2 3 4 5 60

2 [rad]

00

05

10

15

20

25

30

35N

orm

aliz

ed er

ror v

ecto

r [cm

]

(a)

times10-3

e-

e+

e

1 2 3 4 5 60

2 [rad]

04

06

08

10

12

14

16

18

20

Nor

mal

ized

erro

r vec

tor [

cm]

(b)



00148

00150

00152

00154

00156

00158

00160

00162

00164

p(k

)minuspk [c

m]

1 2 3 4 5 60

2 [rad]

(a)

1

10

100

1 2 3 4 5 60

2 [rad]


(b)






25 255 26 265 27 275 28 285 29 29538

382384386388

39392394396398

4

Section A-A



Section A-A

26

28

3

32

34

36

38

4

42

44

46y

[cm

]

15 2 25 3 35 4 45 5 551x [cm]

(a)


1 2 3 4 5 60

2 [rad]

00100

00104

00108

00112

00116

00120

00124

00128

r max

[cm

]

(b)






6 Conclusions




444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

274

275

276

277

278

279

28

281

282

y [c

m]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]

144 146 148 15142

x [cm]


p+k

p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

(a)


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

273274275276277278279

28281282283

y [c

m]

142 144 146 148 15 15214

x [cm]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]


(b)



Data Availability




Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018









times10-5

e-

e+

e

1 2 3 4 5 60

2 [rad]

00

05

10

15

20

25

30

35N

orm

aliz

ed er

ror v

ecto

r [cm

]

(a)

times10-3

e-

e+

e

1 2 3 4 5 60

2 [rad]

04

06

08

10

12

14

16

18

20

Nor

mal

ized

erro

r vec

tor [

cm]

(b)



00148

00150

00152

00154

00156

00158

00160

00162

00164

p(k

)minuspk [c

m]

1 2 3 4 5 60

2 [rad]

(a)

1

10

100

1 2 3 4 5 60

2 [rad]


(b)






25 255 26 265 27 275 28 285 29 29538

382384386388

39392394396398

4

Section A-A



Section A-A

26

28

3

32

34

36

38

4

42

44

46y

[cm

]

15 2 25 3 35 4 45 5 551x [cm]

(a)


1 2 3 4 5 60

2 [rad]

00100

00104

00108

00112

00116

00120

00124

00128

r max

[cm

]

(b)






6 Conclusions




444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

274

275

276

277

278

279

28

281

282

y [c

m]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]

144 146 148 15142

x [cm]


p+k

p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

(a)


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

273274275276277278279

28281282283

y [c

m]

142 144 146 148 15 15214

x [cm]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]


(b)



Data Availability




Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018









25 255 26 265 27 275 28 285 29 29538

382384386388

39392394396398

4

Section A-A



Section A-A

26

28

3

32

34

36

38

4

42

44

46y

[cm

]

15 2 25 3 35 4 45 5 551x [cm]

(a)


1 2 3 4 5 60

2 [rad]

00100

00104

00108

00112

00116

00120

00124

00128

r max

[cm

]

(b)






6 Conclusions




444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

274

275

276

277

278

279

28

281

282

y [c

m]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]

144 146 148 15142

x [cm]


p+k

p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

(a)


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

273274275276277278279

28281282283

y [c

m]

142 144 146 148 15 15214

x [cm]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]


(b)



Data Availability




Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018










444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

274

275

276

277

278

279

28

281

282

y [c

m]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]

144 146 148 15142

x [cm]


p+k

p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

(a)


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k


p-k

p+k

p-kpminus

k

p+k

444

446

448

45

452

454

456

y [c

m]

368369

37371372373374375376377378

y [c

m]

273274275276277278279

28281282283

y [c

m]

142 144 146 148 15 15214

x [cm]

502 504 506 508 51 512 514 5165

x [cm]

244 246 248 25 252 254242

x [cm]


(b)



Data Availability




Acknowledgments


References







































Journal of












Journal of







Volume 2018






Volume 2018










































Journal of












Journal of







Volume 2018






Volume 2018








tolerance analysis of planar mechanisms based on a...

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