APS/123-QED

An interval-based method for workspace analysis of planar flexure-jointed mechanism

Denny OetomoDepartment of Mechanical Engineering,

The University of Melbourne, VIC 3010 Australia

David Daney and Jean-Pierre MerletINRIA, Sophia-Antipolis, 06902 France

Bijan ShirinzadehDepartment of Mechanical Engineering,Monash University, VIC 3800 Australia

(Dated:)

This paper addresses the problem of certifying the performance of a precision flexure-base mech-anism design with respect to the given constraints. Due to the stringent requirements associatedwith flexure-based precision mechanisms, it is necessary to be able to evaluate and certify the per-formance at the design stage, taking into account the possible sources of errors: such as fabricationtolerances and the modeling inaccuracies in flexure joints. An interval-based method is proposedto certify whether various constraints are satisfied for all points within a required workspace. Un-like the finite-element methods that are commonly used today to evaluate a design, where materialproperties are used for evaluation on point-to-point sampling basis, the proposed technique offers awide range of versatility in the design criteria to be evaluated and the results are true for all contin-uous values within the certified range of the workspace. This paper takes a pedagogical approachin presenting the interval-based methodologies and the implementation on a planar 3RRR parallelflexure-based manipulator.

PACS numbers:

I. INTRODUCTION

Flexure jointed mechanisms [1] have been widelyutilised in precision positioning and manipulation devices[2], such as for pattern alignment in semiconductor fabri-cation, micro-assembly, micro-surgery, and various scan-ning microscopy techniques [3, 4], as well as for the designof micro-scaled mechanisms, such as micro grippers andmicro transducers, especially as the scale involved doesnot allow the use of conventional roller-bearing type ofjoints.

Due to the high precision nature of the applications,there is a stringent demand on the performance of themanipulator. Currently, finite element methods are of-ten employed to simulate the performance of the mech-anism design, however this is limited in the types of cri-teria and does not allow a guaranteed solution. It alsotends to be computationally intensive. Topological in-formation of the mechanism is often not considered andevaluation of the performance is on point sampling basis.It is therefore difficult to guarantee that the manipulatorsatisfies the required constraints for all poses of its re-quired workspace. Furthermore, there is also the issue ofmodeling inaccuracies in flexure mechanisms. Generally,when constructing the kinematic model of a flexure-jointmechanism, an ideal model is used for the flexure joint,for example, a revolute flexure joint is modeled as an idealrevolute joint. The deformation of the joint during de-

flection, however, produces residual translational motion.An attempt to take into account this residual motion wasgiven in [5], where a revolute flexure joint was representedas a pair of revolute and prismatic joints. However, anaccurate model for such parasitic motion is complex toobtain.

In this paper, an interval-based method is proposed toevaluate the various constraints and to certify whetheror not these constraints are satisfied within the desiredworkspace of the flexure-based manipulator. This wouldallow a designer to verify that various constraints, such asthe reachable range of motion and the required motionresolutions, are achieved with the specific design. Fur-thermore, various uncertainties, including the inaccura-cies of joint modeling and fabrication tolerances, can beaccommodated as a bounded variations in the kinematicparameters. When a given workspace is certified as hav-ing satisfied all the constraints, the certification is validfor all the continuous values within the bounds, even inthe face of the above-mentioned uncertainties.

The work on obtaining the workspace of a manipu-lator has been presented in the past through geometri-cal approach [6, 7] and screw theory [8]. These meth-ods define the boundaries of the manipulators geometri-cally and provides algebraic expressions to the boundarycurves. The method presented in this paper obtains thesame results as the analysis presented in [6] - where theworkspace boundary of a specific manipulator was de-

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scribed through a geometric approach. The interval anal-ysis method proposed in this paper has the advantagesof being more general, easily adapted to other kinematictopologies, capable of handling any types of constraintsexpressed as mathematical equalities / inequalities, aswell as uncertainties in parameters and rounding error.The geometric approaches require complex mathematicalderivation prior to numerical computation to obtain theworkspace boundaries, specific to the mechanism, how-ever it is computationally more efficient. It also calcu-lates only the workspace boundary given the topology,but it is not able to take into account joint limits or anyother types of constraints, such as motion resolution andsingularities.

This paper presents a complete workspace evaluationtechnique for a flexure mechanism. A brief introductionof flexure mechanisms and the overall constraint satis-faction method through interval analysis are presentedin Section 2 and 3, respectively. The implementation ofthe algorithm, through constraints commonly requiredfor precision flexure mechanism, is presented in Section4, illustrated through the example of a 3RRR planarflexure-mechanism. The paper discusses the results ofthe implementation through the direct implementationof the described algorithm, as well as through the moreadvanced interval techniques available to achieve a bet-ter (sharper) result. Concepts of the advanced techniquesare elaborated, with proper references given to cover thedetails of these techniques. A note on the numerical effi-ciency of the algorithm is presented and conclusions aredrawn in Section 5.

II. FLEXURE MECHANISMS

Flexure mechanism [1] is formed by significantly reduc-ing the cross sectional area of a member at a particularpoint so that deflection through elastic deformation canbe induced about that point while treating the rest of themember as an ideal rigid body. As such, flexure jointsdo not suffer from any nonlinearities commonly associ-ated with conventional joints such as friction, stiction,and backlash. They do, however, provide a much smallerrange of displacement compared to conventional joints.Hence, they are suitable for precision manipulation.

The range of deflection that a flexure joint can undergodepends on the shear modulus of material and the designof the joint. This range of deflection provides a naturalbound to the joint displacement variables which then actsas a constraint in determining the achievable workspaceof the end-effector through interval analysis method. Dueto the high precision nature of the applications, flexure-based mechanisms have a strict fabrication tolerance andtherefore high cost of fabrication. This is true for thelarger flexure mechanisms for micro/nano precision ma-nipulation as well as for the micro-scaled mechanisms. Inthis paper, we focus on the notch type joints [1], which(in ideal case) produce a one DOF revolute joint motion,

without loss of generality in the algorithm presented forperformance evaluation and the guarantee of constraintsatisfaction. This type of flexure joints, as demonstratedin [5], exhibits a residual translational motion in additionto the ideal revolute displacement. This motion is com-plex to model accurately and it affects the accuracy of theforward and inverse kinematics of the mechanism. Othertypes of potential errors in the fabrication and assemblyof a flexure jointed mechanism are outlined in [9].

III. INTERVAL-BASED KINEMATICS

The goal of the algorithm is to solve the kinematicsof a given mechanism to obtain the range of end-effectorworkspace such that the required constraints are satis-fied. In this paper, x is defined as a vector containingthe task space variables (x, y, θ)T , h is a vector contain-ing the design parameters of the mechanism (such as linklengths), and C(x,h) ≤ 0 is the mathematical inequalityrepresenting the constraints to be satisfied. The problemcan therefore be formulated such that:

{∀x ∈ [x,x], ∀h ∈ [h,h]; C(x,h) ≤ 0} (1)

where x,x and h,h are the lower and upper bounds ofthe range of values in x and h, respectively.

As an overview, the interval analysis method involvesseveral main components (1) interval extension orevaluation of functions, (2) constraint satisfaction:testing against the constraints and obtaining the inner,outer, or boundary boxes, (3) filtering, to enforce theconsistency of various variables in the constraints, and(4) branch-and-bound loop within which the othercomponents are carried out.

The goal of the strategy is to certify whether a particu-lar range of workspace [x,x] and design parameters [h,h]yield either inner or outer boxes to constraint C(x,h). Ifthe range of the solution is too wide, often it is not pos-sible to obtain a decision, in which case, filtering andbranch-and-bound process are utilised. Filtering pro-cess sharpens the result of constraint evaluation whilethe branch-and-bound process splits the variables intosmaller ranges and evaluates the constraints as a functionof each subset of variable individually. The details of thecomponents are presented in the following sub-sections.

A. Interval Extension

The interval extension of variable x is defined as X,bounded within its lower and upper bounds [x, x], wherex ≤ x ≤ x. The width of the range is defined as x −x. The interval extension of a function is the evaluationof a function with interval variables. Types of functioninterval extension are natural extension [10] and Taylor-form extension [11, 12]. Natural extension is where realvariables in a function are substituted by the equivalent

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interval variables. Hence, in this case:

∀x ∈ X, f(x) ∈ F (X); (2)

is the natural extension of f(x). Taylor form extensionutilises the partial derivative of the function f(x). In-terval methods can be used conveniently to bound theremainder of truncated Taylor series. In this paper, thenatural extension was utilised.

During the interval evaluation of a function, as numer-ical values are substituted into the function, the relation-ship between various variables are lost. Overestimationoccurs when the same variables appear more than oncewithin the function and they are regarded as independentvariables. The evaluation of a function where all variablesinvolved appear only once is sharp (within rounding er-rors), meaning it is bounded within the smallest possible“box”. For example, let X = [1, 2] and Y = [3, 6]. Evalu-ating F (X,Y ) = X+Y = [4, 8] would therefore be sharp.However, evaluating G(X,Y ) = X − X = [1, 2] − [1, 2]results in [−1, 1] and is not equal to zero, although weknow it should. This is because the two variables X aretaken as being independent and not as the same variable.The effect of overestimation can be seen further in latersubsections.

B. Types of Solutions

After the evaluation, it is necessary to test whethera required constraint is satisfied in the system. In ourproblem, it is desired to verify whether the required per-formance constraint C(X,H) of the mechanism is truefor the set of given workspace pose (variables) of interest(x ∈ X) and mechanism parameters (h ∈ H), as pre-sented in (1). There are two types of constraints to beevaluated:

• Inequality Constraint. When this is the case, theevaluated function C(x,h) is said to satisfy the re-quired constraints when:

∀x ∈ X, ∀h ∈ H; CR ≤ C(X,H) ≤ CR; (3)

where CR and CR are the lower and upper boundsof the requirements for the constraints. This isalso termed the inner solution or the inner box ofthe constraint. The outer box is obtained when∀x ∈ X,∀h ∈ H; (C(X,H) ≤ CR) or (CR ≤C(X,H)). Boundary solution or a boundary box isfound when it cannot be decided whether (x,h) isan inner or outer box of the constraint.

• Equality Constraint. When it is desired to obtainthe solution of

∀x ∈ X,∀h ∈ H; C(X,H) = 0, (4)

then it is comparatively easier to obtain the outerbox of the constraint. The outer box is obtained by

solving for: ∀x ∈ X, ∀h ∈ H; 0 /∈ C(X,H). When0 ∈ C(X,H), then it is only possible to deducethat a solution exist within (X,H), however it isnot enough to define inner boxes.

C. Filtering

Filtering process enforces the consistency in the vari-ables involved in the evaluation of an interval function/ constraint. These techniques originated from the con-straint programming field of study. It involves removingthe segments in the interval variables involved that donot hold within the constraints. In this paper, the filter-ing process is used to reduce the effect of overestimationon the interval extensions of functions.

Overestimation of an interval function makes it dif-ficult to decide whether or not a set of interval vari-ables satisfies the given constraints. Consistency filter-ing is therefore required to sharpen the resulting boxes.The process utilises the additional information containedwithin the mathematical equations or the physical con-straints. In the case of mathematical equations, differentways of expressing the same equations yields differentbounds to the interval evaluation, due to varying degreesof overestimation. This is utilised in the filtering processby ensuring the consistency of the solutions throughoutthe various ways of expressing the same equation. In thecase of physical constraints, additional information ob-tained from the physical or mechanical properties of thesystem are utilised to obtain a consistent solution. In thispaper, for example, a parallel mechanism is constructedout of several articulation chains that connect the baseplatform to a common moving platform. Forward kine-matics of each chain to the common end-effector, for ex-ample, provides additional constraints that can be usedto reduce the effect of overestimation.

In solving for the consistency of an equation, the con-cept is to ensure that the interval extension of a functionproduces a solution that is consistent with the given con-straint. For example, let f(x, y) = x2 − xy + 2y = 0 bethe specified constraint and that the initial estimates ofthe interval extension of variables x and y be X ∈ [3, 9]and Y ∈ [1, 4], respectively. If the interval extensionof function f(x, y) is evaluated, we obtain F (X, Y ) =[9, 81] − [3, 36] + [2, 8] = [−25, 86]. However, to ensurethe consistency in the constraint, it is possible to rewritethe equality such that:

X2 = Y (X − 2) = [1, 28]X = [−5.3, 5.3]. (5)

Taking the intersection of (5) and the initial estimate ofX to obtain X ∈ [−5.3, 5.3)] ∩ [3, 9] = [3, 5.3]. Similarly,this can be performed on variable Y with the improvedestimate of X. This can be iterated until such time thatthe improvement in the sharpness of the bounds is nolonger worth the computational effort.

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The procedure described above is termed 2B consis-tency [13, 14]. Other filtering techniques are availablesuch as 3B and interval Newton [10, 11, 15, 16].

D. Branch-and-Bound

It is often difficult to conclude whether a given boxconstitute an inner or outer box when it is evaluated as afunction of interval variables with large width. While fil-tering process contracts the box and attempts to obtaina sharp solution, it can only return the sharpest box thatwould contain the solution. Within the box, the solutionoften occupies only a portion of the bounded space. Thebranch-and-bound strategy is therefore utilised to auto-mate the solution search of the algorithm such that abetter definition of the solution may be found.

The branch-and-bound algorithm searches for the so-lution by evaluating a box and deciding whether it yieldsan inner, outer, or boundary box. If it yields a bound-ary box, then the box is split (bisected) and each of thebisected boxes is iteratively evaluated following the sameprocess. The bisection process is iteratively performeduntil an inner or outer box is found, or until a thresholddimension of the variable boxes ε is reached. Boxes thatremains as boundary solution at this point will form theboundary solutions of the system. In this paper, to eval-uate the performance of a mechanism within the speci-fied workspace X with a specific mechanism parametersH, bisection is performed on the M dimensional inter-val box X where M is the number of the task spacepose (workspace) variables. In the case of our planarparallel mechanism, M = 3 and is made up of 2 trans-lation and one orientation degrees of freedom. Bisec-tion process is performed across all the pose variables indifferent order, depending on the bisection algorithms.Several possible bisection algorithms are available [17].The simpler approaches are the round robin (bisection ofvariables by turn) and the largest first (bisection of thevariables with the largest width first). Both the strate-gies above suffer from the simplistic approach. A moreintelligent bisection algorithm was proposed [18] by util-ising the derivative of the system. This algorithm definesthe smear function of the system and aims to identifythe dominant variable that affects the system the mostat that iteration. The most dominant variable is thenselected for bisection. This approach is effective as it at-tacks the problem where bisection would yield the mosteffect. However, it has a drawback that it may continu-ously bisect specific dominant variables until ε is reached.This is amended by defining a minimum allowable thresh-old in the ratio of the width of a variable to its originalwidth, beyond which bisection turn is handed over to thenext dominant variable.

E. Summary of Algorithm

The proposed algorithm is summarised in Table I.Constraint function C(X,H) can contain a single designconstraint to be evaluated, expressed as mathematicalequalities or inequalities, or a list of constraints. In thecase where there are multiple constraints to be evaluated,an inner box is obtained only if Xi satisfy all the con-straints (forms an inner box to all constraints) and anouter box is obtained if Xi fails to satisfy any of the con-straint. The resulting solutions are contained in lists (seeTable I): list LIN contains the inner solutions that sat-isfy all the constraints, while LOUT contains outer boxes,i.e. when a box fails to satisfy any one of the constraintsin C. The boundary boxes are given in list LB .

The variable X contains the end-effector workspace de-scription, i.e. position (x, y) and orientation (θ) for theplanar case considered in this paper (3 DOF planar mech-anism). However, a 6DOF mechanism will have the bi-section process performed across the six-dimensional boxdescribing the workspace (as can be deduced from thealgorithm in Table I). This would increase the compu-tational cost of the algorithm. It should be noted, how-ever, that the complexity in formulating and using thealgorithm does not change a great deal. This is one ofthe advantages of the proposed algorithm, i.e. its versa-tility - often as a trade-off to computational cost, whencompared to a more problem-specific algorithm. As longas constraints are expressed mathematically, most of thework will be carried out by the numerical computation.In contrast to other methodologies, such as the geometricapproach, as mentioned in the introduction, where math-ematical derivation is very much mechanism specific anda 6DOF mechanism will be much harder to solve than a3DOF.

IV. INTERVAL ANALYSIS: 3RRR PARALLELFLEXURE MECHANISM

In this section, the workspace verification problem of a3RRR planar flexure mechanism with respect to the vari-ous constraints relevant to the functionality of a precisionmanipulator is solved using the interval based techniquespresented in Section III. The problem is to evaluate theperformance of the manipulator defined by the design pa-rameters H, with respect to design constraints C(X,H),at the required workspace pose X and to certify whetherX is a solution to the performance constraints.

The performance criteria of the planar manipulatorthat are presented to help illustrate the algorithm in thispaper are (1) the amount of workspace reachable throughthe allowable deflection of the flexure joints, (2) singu-larity free workspace, and (3) the workspace that yieldsthe required motion resolution given the resolution of thejoint space. The 3RRR planar parallel mechanism, withthe definitions of the variables and frame assignments, isgiven in Fig. 1.

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TABLE I: Summary of algorithm for workspace constraintanalysis with interval analysis.

1 Initialise empty lists LIN , LOUT , and LB .2 Initialise list L containing initial task

space intervals (boxes) to be analysed.3 While (L not empty)

(a) Extract manipulator pose Xi from list L.(b) Evaluate constraints C(X, H)(c) Test C(X, H) against the required

performance [CR, CR].(d) Filtering process is carried out if necessary.(e) Return whether X constitutes an inner, outer,

or boundary box.(f) Case result is inner, outer, or boundary box:

(a) Case 1: The solution lies within the ALLconstraints C(X, H)Remove Xi from L and add to list LIN

(b) Case 2: The solution lies outside ANY ofthe constraints in C(X, H)Remove Xi from L and add to list LOUT

(c) Case 3: If Xi is a boundary solutionIf (dimension of box Xi) > ε)

Bisect Xi into Xi1 and Xi2Remove Xi from L and add Xi1 and Xi2

into the list L.Else If (threshold dimension ε has

been reached)Remove Xi from L and add to list LB

End If(g) End Case

4 End While

FIG. 1: A 3RRR planar parallel mechanism

The planar workspace of the manipulator is defined asX =O pe = (xe, ye, θ)T , which comprise the positionand the orientation of the end-effector, respectively (seeFig. 1). For simplicity, the task space variable is alwaysexpressed with respect to the base frame O, therefore ref-erence to the frame is omitted in this paper, i.e. the taskspace variable will be written simply as pe = [xe, ye, θe].Joint space variables are the (αi, βi, γi), where i = 1, 2, 3represents each of the three serial chains connecting the

TABLE II: Parameters of the case study 3RRR planar parallelflexure-based mechanism.

Parameters Values

Origin of O1 (0, 0)T mmOrigin of O2 (167.27, 0)T mmOrigin of O3 (83.64, 144.86)T mm

|di| 10 mmri 66 mmli 46 mm

(xem, yem, θem)T (83.64, 48.29,−10.3o)T

base and moving platform. It is assumed that only jointsα1, α2, α3 are actuated (RRR chains) and displacementsensor feedbacks are only available on these joints. Thisis used as the case study to illustrate the algorithm, al-though it is possible in precision mechanisms to installdisplacement sensors to measure motion directly at theend-effector. Several planar parallel micro-positioningmechanisms in the literature are designed in this topol-ogy [5, 19].

The position vectors of points B1, B2, B3 with respectto base points O1,O2,O3 are defined as p1,p2,p3, re-spectively (where pi = (xi, yi)T ). These points movewith the end-effector, hence their positions are given bythe position of the end-effector (xe, ye), plus the dis-placement caused by the orientation of the end-effectorθ due to the offset distance d = [d1,d2,d3]T . With re-spect to the global origin O, these vectors are defined asOp1,

O p2,O p3. The position vectors p1,p2,p3 are ob-

tained through:

pi = Opi −Oi;

= pe + RZ(θ + ∠di).

00d

−Oi;

(6)

where d is the length of vectors d1,d2,d3, which areassumed to be of equal length, (∠d1, ∠d2, ∠d3) =(7π

6 ,−π6 , π

2 ) are the headings of vectors d1,d2,d3 whenthe flexure joints are at rest.

The manipulator design is used as an example in thefollowing sub-sections to better illustrate the algorithmproposed. The various link lengths of the selected 3RRRplanar parallel mechanism are summarised in Table II.The pose of the end-effector when the 3RRR mechanismis symmetrical is defined as (pem) = (xem, yem, θem)T ,which is also the pose when the flexure joints are at rest,i.e. when they undergo zero deflections. The algorithmwas implemented in C++ with ALIAS library developedupon BIAS/Profil platform within the COPRIN project[20].

A. Workspace by Joint Limits

It is imperative that the joint displacement in a flexurejoint takes place only within the linear deformation re-gion of the member. This depends primarily on the shear

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modulus of the material and the design of the joint. Aftertaking into account the safety margin, the selected valuesof maximum allowable joint deflection used in the inter-val analysis of the flexure mechanism workspace shouldconstitute a bound for which the joints behave linearlywithin the elastic region.

To obtain the joint displacement of the mechanism fora given end-effector pose, inverse kinematics is carriedout on the desired end-effector workspace (X), with thegiven interval link length parameters (H). The inversekinematics of planar parallel mechanisms is often dis-cussed in the literature [21] [22]. Generally, inverse kine-matic solution can be obtained by first calculating theangle βi, which has two possible solutions within [0, 2π].Choosing one of the two possible solutions for each βi,the joint displacement of angle αi can be obtained. Ob-taining the inverse cosine or inverse sine of an intervalvariable does not uniquely define the solution angle asthese trigonometric functions are periodic. To overcomethis problem, the constraints on the allowable joint de-flection are expressed as the limits on the sine and cosineof the joint limit angles. For a flexure jointed mechanism,however, the unique solution to βi can be pre-determined(whether it is the elbow up or elbow down solution), dueto the limited motion of the mechanism. The closed-form solution of the inverse kinematics is therefore givenas follows:

cos(βi) = x2i +y2

i−(r2i +l2i )

2liri,

cos(αi) = xi(ri+licos(βi))+yilisin(βi)x2

i +y2i

,

sin(αi) = −xilisin(βi)+yi(ri+licos(βi))x2

i +y2i

.

(7)

Although angles γi are not usually considered theoret-ically in the inverse kinematics of a 3RRR planar parallelmechanism, it is important in practical cases to take thelimits of these joints into account - such as to avoid col-lisions among the links of the mechanism. In the case offlexure jointed mechanisms, it is also necessary to imposea joint limit constraint on these joints, as they are alsoflexure joints. From Fig. 1, it can be observed that

αi + βi + γi = ∠(−di). (8)

1. Constraint Definition

The interval extension of end-effector workspace vari-ables pe = (xe, ye, θe)T were utilised to describe the de-sired range of the workspace X. The interval variablesfor the link lengths (H), however, were used to expressthe fabrication tolerances and other un-modeled sourcesof errors. With these variables defined, the constraintfor the workspace as defined by the allowable joint de-flections. As trigonometric functions are periodic, thejoint limits are expressed as the upper and lower boundsof their trigonometric functions (cosine and sines). Thejoint displacement limit for a joint qi can therefore be

expressed as:

Cos(qi) ∈ [cos(q), cos(q)] AND Sin(qi) ∈ [sin(q), sin(q)].

(9)

A box of solution Xi is an inner solution when all of thejoint limit constraints are satisfied. The workspace de-scribed by the inner box is reachable by the end-effectorof the mechanism, given the allowable flexure joint de-flection. A box Xi is an outer box if any of the intervalinverse kinematic solutions falls within the complementof (9).

2. Results and Discussion: Workspace by Flexure JointLimit

In this example, the allowable joint deflection was setat ±3o. The workspace to be evaluated is selected as amotion range of ±2.5 mm in x and y direction (xr = yr =2.5mm) and ±17.5 milli radian (θr = 1o) for orientation,about (xem, yem, θem)T , respectively.

(a)81.5 82 82.5 83 83.5 84 84.5 85 85.5 86

46

46.5

47

47.5

48

48.5

49

49.5

50

50.5

x (mm)

y (

mm

)

(b)81.5 82 82.5 83 83.5 84 84.5 85 85.5 86

46

46.5

47

47.5

48

48.5

49

49.5

50

50.5

x (mm)

y (

mm

)

FIG. 2: Workspace of the 3RRR planar parallel mechanism,under joint limit constraint (a) without considering fabrica-tion tolerances and (b) assuming ±50µm tolerance on linklength r and l; at constant θ = θm = −10.3o.

Due to the overestimation, direct interval evaluationof the constraints yielded no solutions in the inner boxes.Consistency filtering is required to sharpen the evaluationof the inverse kinematic solution. The multiple chainsi which connect to the common moving platform wasutilised as additional constraint to reduce the amount of

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uncertainty involved in the interval calculation of the in-verse kinematic problem. The following set of physicalconstraints, essentially forming the direct kinematics ofvector pi, were utilised as additional constraints in theenforcing the consistency of the inverse kinematic solu-tions:

C1(X, H) = RiCos(αi) + LiCos(αi + βi)−Xi = 0;C2(X, H) = RiSin(αi) + LiSin(αi + βi)− Yi = 0;C3(X, H) = (Cos(αi))

2 + (Sin(αi))2 − 1 = 0;

(10)

A 2B/3B consistency filtering was implemented andthe improvement on the inner solution is shown in Fig.2(a). This two dimensional figure display the innerand outer boxes of the manipulator workspace when theflexure-joint limits are imposed, after consistency filter-ing. It is taken at constant θ = θm = −10.3o. The effec-tiveness of the filtering technique can be clearly seen inthe ability of the algorithm in admitting inner solutions.

The result in Fig. 2(a) was produced by assumingzero uncertainties in the fabrication tolerances and nouncertainties in the flexure joint modeling. In this case,interval variable Ri and Li were defined as degenerate in-tervals, such that they have the same values of the lowerand upper bounds (zero interval width). As explained inSection II and presented in [5], the un-modelled kinemat-ics of a notch type revolute flexure joint manifest itselfin an additional amount of translational motion. Thiscan be modelled as additional uncertainties in the linklengths Li and Ri. This un-modelled degree of freedomis complex to model and comparatively small in magni-tude. In our technique, the bounds of the error estimateare used to account for the additional translation in theabsence of a complex and accurate model. Additionaluncertainties due to fabrication tolerances are also addedto the interval variables Li and Ri. To include such un-certainties into the evaluation process, these bounds ofuncertainties are added to the link lengths Li and Ri.

In our example, the additional uncertainties due to un-modelled kinematics and the flexure fabrication toleranceare defined as being bounded within ±50µm for eachlink length ri and li. The resulting workspace withinthe limits of allowable flexure joint deflection, is shownin Figure 2(b) as comparison. As expected, there is alarger area of boundary solutions compared to when fab-rication tolerances and modelling errors were not consid-ered. However, the inner solutions exist such that therepresented workspace range is certified to be within therequired constraints, with the fabrication tolerances andthe uncertainties in kinematic model taken into account.It is therefore demonstrated that various uncertainties,including fabrication limitations, can be included in thecalculation during the design process to guarantee thatthe performance of the resulting mechanism is within thespecified requirements.

Figure 3 demonstrates the workspace of the mechanismbounded by the allowable joint deflection for a range oforientation Θ, represented by the vertical-axis of the plot.The range of interval Θ is ±17.5mrad.

FIG. 3: The inner solution admitted into the workspace ofthe 3RRR planar parallel flexure mechanism. The orientationrange is θI = θm ± 17.5mrad.

B. Singularity

It is also desired to evaluate the workspace of the mech-anism to certify that the operational region is free ofsingularity. The constraint, in this case, is the functiondefining the loci of singularity. The singularity free regionis the end-effector workspace that can be guaranteed tocontain no solution to the constraint. For simple mecha-nisms, it is possible to obtain the symbolic expression ofthe determinant of the Jacobian matrices. However, ob-taining symbolic expression of singularity for more com-plex mechanisms with higher degrees of freedom may notbe practical. An efficient method was proposed in [23] toevaluate the regularity of the interval Jacobian matrixnumerically. This method is used in this paper to obtainthe non-singular workspace of the mechanism.

The differential kinematic relationships can be ob-tained from [21] as

[fTi , fT

i d⊥i ][

pe

ω

]= rifT

i

[ −sin(αi)cos(αi)

]αi, (11)

where fTi is the unit vector in the direction of the recip-

rocal screws passing through the revolute joints at pointsA and B:

fTi =

1li

[xi − ricos(αi)yi − risin(αi)

], (12)

d⊥i is the vector perpendicular to di, or d⊥i =(−diy, dix)T , and α is the vector containing the rate ofthe actuated joints (α1, α2, α3), as defined in Fig. 1. Theoverall differential kinematics of the mechanism can bedescribed by:

J1.xe = J2.α (13)

where xe = (pTe , ω)T = (xe, ye, θ)T and J1, J2 are the

3× 3 Jacobian matrices, with each row representing the

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relationship (11) for individual leg i. Note that J2 is adiagonal matrix.

1. Constraint Definition

The constraint for singular workspace of the mecha-nism is defined as:

C1(X,H) = det(J)1 = 0;C2(X,H) = det(J)2 = 0; (14)

It is possible to guarantee that a specific box in theworkspace does not contain singularity by solving for theworkspace such that:

{∀x ∈ X,∀h ∈ H, (0 /∈ C1(X,H) ∧ (0 /∈ C2(X,H)} (15)

2. Results and Discussion: Singularity-free workspace

Singularities associated with a rank deficient J1 andJ2 are the internal and the boundary singularities of themechanism, respectively. To demonstrate the interestingfeatures in this evaluation, the singularity analysis is per-formed considering the entire workspace the 3RRR mech-anism, independent of the joint displacement limits. Theresulting loci of singular-free configurations, obtained bythe direct evaluation of constraint (15).

For the clarity of further analysis, the singularity-freeworkspace for constant orientation is presented as 2 di-mensional plots in Fig.4(a), taken at θ = 40o. It canbe seen that in this example, direct evaluation does notproduce a very sharp result even after the consistencyfiltering process, leaving a large region of boundary so-lutions. As highlighted earlier, it is difficult to decideon the solutions of equality constraint. To improve thesharpness of the solution, an advanced numerical regular-ity test, that has been implemented on the ALIAS library[20] was then utilised to enhance the performance of thealgorithm. This regularity test utilises

algorithm component 1: the Rohn consistency test todetermine the interval matrices that are regular.The Rohn consistency test states that for an inter-val matrix I, if a well defined set of scalar matricesderived from I have determinants of the same sign,then there is no singular matrix in I. This is apowerful test to certify the outer box of the equal-ity, i.e. to certify that a box of workspace does notcontain any singularity, i.e. 0 /∈ C(X,H).

algorithm component 2: a matrix regularity testthrough the sign of the matrix determinant (as pro-posed in [23]) to certify that a matrix contains sin-gularity. It is carried out by sampling points withina given box and comparing the signs of the deter-minants for these points. If there exists any point

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FIG. 4: Singularity free workspace of the 3RRR planar par-allel mechanism, taken at θ = 40o obtained (a) by directevaluation of constraint (15) and (b) by the matrix regularitytest algorithm, as provided by the ALIAS library. The loci ofsingular workspace is marked in red (solid colour) for clarity.

in the interval box that displays a different sign ofdeterminant from other points, then singularity ex-ists within the interval box. This provides a strongtool to certify the existence of a singularity withinan interval matrix.

Combined, the two techniques provide an effective toolto evaluate the singularity of mechanisms.

Another point to note is if this test concludes that asolution to the equality exists in the interval box (X,H),it does not mean the entire box is singular (as singularityis a point). Therefore, the interval box should not be im-mediately assigned as singular but as a boundary solutionto be bisected further to localise the singularity loci. It istherefore possible to narrow down the boundary boxes tothe size of ε (the threshold of the smallest dimension ofworkspace region where bisection process is terminated).This method identifies singularity numerically down tothe threshold dimension ε. The large improvement onthe sharpness of the solution, provided by this approach,over the direct evaluation method is shown in Fig. 4(b),where the loci of singular configurations are marked inred solid line for a clearer view.

It should be noted that evaluating the singularity-freeconstraint over the end-effector workspace within the al-lowable joint displacement limits (Section IVA) yields nosingularity. Referring to Fig.4, the centre portion of the

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workspace is where the determinant of the Jacobian ma-trix is negative, while the three portion along the edgesare of positive determinant. It is important that oper-ational workspace and the motion path planning of theend-effector do not cross between a positive and nega-tive determinant regions. This demonstrates the easewith which interval analysis techniques can be adaptedto verifying a singularity-free path planning problem.

It might also be of interest to note that an optionaltechnique exists to trace the loci of singularity once onesingular point is found in the workspace. Once a singu-larity point is found, the continuation method [24, 25]can be employed to trace and identify the loci of sin-gular configurations. This may provide a faster solvingalgorithm.

C. Task Space Motion Resolution

Another important characteristic in flexure mechanismis the task space motion resolution. As these mechanismsare generally employed for precision manipulation, theresolution of the smallest step possible in the motion ofthe end-effector is often an important performance cri-terion. Generally, it is possible to directly establish thebounds of joint space motion resolutions from the speci-fications of the sensors and actuators used. Incrementalstep size in the task space can therefore be calculated bythe differential kinematic relationship with the incremen-tal step size in joint space displacement.

J1.∆xe = J2.∆q, (16)

where ∆q is the incremental step in joint space. Since a3RRR mechanism is considered in our case, then only thethree base joints are actuated. It can be assumed thatonly these active joints are equipped with displacementsensors. Hence, ∆q = (δα1, δα2, δα3)T . It is then re-quired to solve for ∆xe in the linear equation (16). Thisis done in this paper using Gaussian method [16]. Severalinterval arithmetics packages have Gaussian solving func-tions ready, hence users do not need to code this functionfrom scratch.

1. Constraint Definition

The constraint to be satisfied is therefore defined as:

|∆Xe| ≤ ∆Xmax; (17)

where ∆Xe is the interval extension of vector xe as de-fined in (16) and ∆Xmax is the largest end-effector mo-tion resolution required for the task. Inner boxes areobtained when the specified workspace satisfies (17) andouter boxes when |∆Xe| > ∆Xmax;

For this example, it is given that joint space displace-ment resolution is bounded within 0.06 mrad. The de-sired resolution (∆Xmax) for the task space translational

motion and orientation are set at 0.5µm and 0.5mrad,respectively.

2. Results and Discussion: Task Space Resolution

When a Gaussian elimination technique was utilised,the algorithm yielded no inner solution to the constraints.Preconditioning process [11] was performed to improvethe sharpness of the solution. Preconditioning was ap-plied by pre-multiplying both sides of the equation withmatrix M, where M = (mid(J1))−1, and where mid(J1)is the matrix containing the mid values of the elements ofthe interval matrix J1. When (16)-(17) is evaluated forthe workspace of our 3RRR planar parallel manipulator,the workspace that satisfies the required motion resolu-tion is given in Fig. 5(a). The solution was obtainedthrough a variation of the Gaussian technique, namedHansen-Bliek solving algorithm [26–28], which is numer-ically pre-conditioned.

Further improvement can be obtained through sym-bolic preconditioning, as proposed in [23]. It has beenshown to be effective in complex systems to improvethe sharpness of the solutions. This approach is pos-sible when symbolic expressions of the linear system ofequations is given. The idea is to minimise symbolicallythe number of multiple occurrences of the variables in aninterval function. This method is performed by evaluat-ing matrix M but keeping J1 symbolic. Pre-multiplyingthe system with M and keeping J1 symbolic allows theelements of the resulting matrix to be rearranged sym-bolically to minimise the multiple occurrences of vari-ous interval variables, hence reducing the effect of de-pendency. Consistency filtering is also included in thealgorithm to further sharpen the results. The improve-ment in the algorithm ability to admit inner solution isdemonstrated in the amount of workspace that can becertified as the inner solution of the constraint dictatedby the desired end-effector motion resolutions. This isshown in Fig. 5(b). The results in Fig. 5 were obtainedwith identical conditions, taken at constant θ = θm, withthe only differences being the algorithms for solving thelinear equations: (a) the preconditioned Hansen-Bliek al-gorithm, and (b) the symbolically preconditioned Gaus-sian elimination method.

D. Overall Available Workspace

The certified available workspace of the planar parallelmechanism can be evaluated by imposing all of the con-straints that have been presented and discussed above. Adesired interval of end-effector workspace can be testedagainst the set of constraints. For the desired workspaceto satisfy all the performance criteria required of the ma-nipulator, it is necessary that the procedure results ininner boxes to all the given constraints for the entire de-sired workspace. It is therefore desirable to be able to

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FIG. 5: Two dimensional workspace at constant θ = θm =−10.3o that satisfies the required motion requirement, giventhe joint space motion resolution. Solving algorithms were(a) preconditioned Hansen-Bliek and (b) symbolic precondi-tioning with Gaussian elimination.

obtain a sharp solution to decide whether or not an in-terval in the workspace satisfies the design requirements.If a bounded solution cannot be obtained, then it cannotbe guaranteed that all the design constraints are satis-fied. It is then necessary to alter the design or to relaxsome of the requirements.

In the usage of interval analysis, it is important tonote that various constraints require different levels ofcomputational resources. Computational load increasesexponentially with every additional bisection in the algo-rithm. In implementing our algorithm, computationallycheap constraints were calculated first within each itera-tion and, whenever possible, used to eliminate boxes thatdo not satisfy the constraint before calculating the morecomputationally expensive constraints. These can thenbe housed in a nested heuristic structure where the com-putationally most expensive constraints are calculatedonly when every other (computationally cheaper) con-straint has failed to produce an inner or outer box. Ifan inner or outer box is not obtained even after filtering,then bisection is performed.

In the case of flexure mechanism, the joint limit is gen-erally the largest contributor in reducing the availableworkspace, as flexure joint workspace constitutes only avery small portion of the overall mechanism workspace.In our example, it can be seen that the workspace al-lowed by the joint deflection limit is situated well within

the singularity free region with negative determinant. Anexample of the usable workspace that satisfies all thecriteria (joint limit, singularity-free, and required mo-tion resolution) is therefore given by Fig. 6. The fig-ure shows the zoomed-in view of the workspace, locatedwell within the singularity-free region at the centre ofthe manipulator workspace, with nominal orientation atθm = −10.3o. The two most stringent constraints in thisexample were the joint limit and the end-effector motionresolution. The workspace allowed by the joint limit isreproduced in Fig. 6 (a), the workspace that producesan end-effector motion resolution of 0.5µm (translation)and 0.5mrad (orientation), given joint motion resolution(δαi) of 0.05mrad, is given in Fig. 6(b) and the resultingintersection, which is the workspace that satisfies all thegiven criteria, is given in Fig. 6(c).

To test a desired workspace X, with a particular ma-nipulator design and link lengths H, the interval vari-ables of the workspace are substituted directly into thealgorithm and certified whether or not they form innerboxes of the constraints. This is the case when the entiredesired workspace consists of only inner boxes to all thegiven constraints.

Implemented on Dual Core Intel 2.4GHz processors(4MB cache) with 1GB RAM, the algorithm took 15s toverify that the flexure mechanism workspace with a rangeof ±1mm in translation and ±1o in orientation from itszero deflection point (xm, ym, θm) is entirely containedwithin the inner solution of all the constraints (repre-sented by 8728 inner boxes). This demonstrates the effi-ciency of the algorithm.

V. CONCLUSIONS

A technique to address workspace verification prob-lem of a precision flexure-based mechanism is presentedin this paper. The technique certifies whether or notthe required workspace satisfies certain a set of perfor-mance criteria, taking into account the modelling andfabrication uncertainties. Performance features relevantto a flexure-based mechanism are presented and the effi-cient interval-based methods in evaluating and resolvingthe features were proposed, implemented, and discussed.Future work is aimed at extending the algorithm to aproduce an efficient synthesis algorithm that would en-able the determination of design parameters of a mecha-nism that satisfy a set of given constraints. This methodwould require not only the verification of constraint sat-isfaction for a set of nominal design parameters but alsofor a continuous range of possible design parameters.

Acknowledgements

This study is conducted under Project ARES (Assem-bly of Reconfigurable Endoluminal System), a NEST-Adventure European Union funded project (EU contractno. 015653).

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FIG. 6: Overall workspace due to multiple constraints. (a) The workspace allowable by limits of joint deflection. (b) Theworkspace that satisfies the required motion resolution. (c) The intersection of all given constraints.

[1] J. M. Paros and L. Weisbord, Machine Design 37, 151(1965).

[2] F. E. Scire and E. C. Teague, Rev. Sci. Inst. 48, 1735(1978).

[3] D. Choi and C. Riviere, Procs. IEEE Conf. Engineeringin Medicine and Biology Society pp. 2325–2328 (2005).

[4] D. Kim, D. Kang, J. Shim, I. Song, and D. Gweon, Rev.Sci. Inst. 76, 073706.1 (2005).

[5] B.-J. Yi, G. B. Chung, H. Y. Na, W. K. Kim, and I. H.Suh, IEEE Trans. Robs. & Autom. 19, 604 (2003).

[6] J.-P. Merlet, C. Gosselin, and N. Mouly, Mech. Mach.Theory 33, 7 (1998).

[7] G. Pennock and D. Kassner, ASME J. Mechanical Design115, 269 (1993).

[8] V. Kumar, ASME J. Mechanical Design 114, 368 (1992).[9] T.-F. Niaritsiry, N. Fazenda, and R. Clavel, in Procs.

IEEE Intl. Conf. Robs. & Autom. (2004), vol. 4, pp.4091–4096.

[10] R. Moore, Interval Analysis (Prentice-Hall, NJ., 1966).[11] E. Hansen and G. Walster, Global Optimization using

Interval Analysis (Marcel Dekker, NY., 2004), 2nd ed.[12] M. Berz and G. Hoffstatter, Reliable Computing 4, 83

(1998).[13] F. Benhamou, F. Goualard, and L. Granvilliers, in Procs.

Intl. Conf. on Logic Programming (Las Cruces, USA,1999), pp. 230–244.

[14] M. Collavizza, F. Delobe, and M. Rueher, Reliable Com-

puting 5, 1 (1999).[15] O. Lhomme, in Procs. IJCAI 93 (Chambery, France, Au-

gust 1993), pp. 232–238.[16] A. Neumaier, Interval Methods for Systems of Equations

(Cambridge University Press, London, 1990).[17] R. B. Kearfott, ACM Transactions on Mathematical

Software 14, 399 (1988), ISSN 0098-3500.[18] R. B. Kearfott and M. Novoa III, ACM Trans. Mathe-

matical Software 16, 152 (1990).[19] T.-F. Lu, D. C. Handley, Y. K. Yong, and C. Eales, In-

dustrial Robot 31, 355 (2004).[20] J.-P. Merlet, in SEA (Toulouse, 14-16 June 2000).[21] I. Bonev, Ph.D. thesis, Universite Laval, Quebec, QC,

Canada (2002).[22] J.-P. Merlet, Parallel Robots (Kluwer, Dordrecht, 2000).[23] J.-P. Merlet and P. Donelan, in Proceedings of the In-

ternational Symposium of Advances in Robot Kinematics(ARK) (Ljubljana, Jun 2006), pp. 41–48.

[24] J.-P. Merlet, International Journal of Robotics Research23, 221 (2004).

[25] M. Raghavan, ASME Design and Automation Confer-ence 32, 397 (1991).

[26] E. Hansen, SIAM J. Numerical Analysis 29, 1493 (1992).[27] J. Rohn, Interval Computations 4, 13 (1993).[28] A. Neumaier, Reliable Computing 5, 131 (1999).