Mechanism and Machine Theory 46 (2011) 577–592

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Mechanism and Machine Theory

j ourna l homepage: www.e lsev ie r.com/ locate /mechmt

Workspace analysis of positioning discontinuities due to clearances inparallel manipulators

Oscar Altuzarra a, Jokin Aginaga b,⁎, Alfonso Hernández a, Isidro Zabalza b

a University of the Basque Country UPV/EHU, Faculty of Engineering in Bilbao, Department of Mechanical Engineering, Alameda de Urquijo s/n, 48013, Bilbao, Spainb Public University of Navarre, Campus Arrosada s/n, Pamplona 31006, Spain

a r t i c l e i n f o

⁎ Corresponding author. Tel.: +34 948 16 92 86; faE-mail address: jokin.aginaga@unavarra.es (J. Agin

0094-114X/$ – see front matter © 2011 Elsevier Ltd.doi:10.1016/j.mechmachtheory.2011.01.005

a b s t r a c t

Article history:Received 30 September 2009Received in revised form 13 September 2010Accepted 11 January 2011Available online 3 February 2011

Clearances at joints produce a loss of accuracy when positioning amechanism. The end-effectorpose error due to clearances depends on the mechanism configuration, the magnitude of theclearance itself and applied external wrenches. Sudden changes which occur in the actualposture of the mechanism owing to a change of contact mode at joints can be detected in theneighbourhood of some configuration. These sudden changes lead to positioning disconti-nuities on certain trajectories, or on the workspace. In this paper, a methodology for analysingthe location of the discontinuities by means of a dynamic or kinetostatic analysis is presented.The advantages of choosing either the dynamic or the kinetostatic approach are analysed,making use of the 5R planar mechanism.

© 2011 Elsevier Ltd. All rights reserved.

Keywords:ClearanceWorkspaceDynamicsKinetostaticsPositioning discontinuities

1. Introduction

With exact design dimensions, mechanisms with n degrees of freedom have what can be called a nominal configuration space.However, manufacturing and assembly errors cause changes in this space. Despite this, if the mechanisms are not over-constrained—we will restrict ourselves to these for the time being—the configuration space maintains its dimension when idealjoints are considered. In the configuration space of a mechanism with ideal joints, these changes can be handled with anappropriate calibration.

However, when considering clearance joints, there are additional changes in the configuration space and calibration is notapplicable. Clearances in joints increase the dimension of the configuration space, but within a limited boundary. So for given fixedinputs, there is an uncertainty about exactly where the end effector is, it being located somewhere within a clearance-configuration space limited by the clearance values.

Mechanisms take up clearances in different ways depending on the posture, applied loads and their dynamics. This determinesthe actual pose that the end-effector can take due to clearances. In addition, from one pose to another, there may be suddenchanges in the way mechanisms take up clearances producing discontinuities in the trajectory. The dependency on applied loadsand dynamics as well as the discontinuities make a corrective procedure analogous to calibration difficult.

Accordingly, it is necessary to analyse the effect of clearance and identify discontinuities and their cause. A deterministicaccuracy analysis has to be established so that the error in positioning due to the clearance can be found and we can identify thesituations where discontinuities occur. In these postures we will need an analysis of the transmission of forces through themechanism to work out what is the cause of the discontinuity. This can be achieved using a kinetostatic analysis as most authorsdo, or we could include dynamic analysis as well. We will show that this type of discontinuity is usually due to sudden changes in

x: +34 948 16 90 99.aga).

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578 O. Altuzarra et al. / Mechanism and Machine Theory 46 (2011) 577–592

the relative position between the two parts of a joint. These can be described as changes in the contactmode of a joint, which occurwhen contact is lost. Such changes lead to a sudden change in the direction of joint forces and moments.

Several research groups have worked on the accuracy analysis of mechanisms to predict the actual end-effector positionconsidering clearances. Most published studies propose methods that can be included in the following groups:

• Geometrical methods. Some authors have evaluated the end-effector position influenced by clearances using the positionproblem or relying solely on the geometric characteristics derived from a simple modelling of the clearances. For example, theworkspace of the end-effector due to clearances in revolute pairs for given inputs is obtained in [1]. In [2] and [3], the authors usean equivalent link for the revolute pair and evaluate position deviation using position equations and introducing probabilisticdistributions. The Denavit–Hartenberg closed-loop equation is used in [4] to introduce probabilistic variations in the D–Hparameters due to clearance and manufacturing errors, and an optimisation problem to produce a robust design is proposed.Common drawbacks of these geometrical methods are that: they are based on the position problem which is nonlinear, theyevaluatemaximal or most-probable final positions but not the actual one, and their result is the same irrespective of the load andmotion of the mechanism.

• Kinematic methods. Other authors assess the problem of finding the maximum error resulting from a set of nominal jointclearances on the basis that, for given inputs, the slop of the mechanism due to clearances is so small that it can be evaluatedapproximately with velocity analysis. An objective function is stated with the clearance freedoms defined for several models ofthe joint clearances, and maximised or minimised as necessary. In [5], the authors produce a seminal work on this type ofmethod: it is based solely on geometrical and velocity analysis using screw coordinates, and it is applied to a spatial parallelmanipulator. More recently, in [6], an improved version of the aforementioned technique has been published, in which themaximal pose error is found allowing also passive joint parameters to vary due to clearances, and overconstrained mechanismscan be analysed. In [7], the influence of each joint clearance—including several contact modes—on the pose error is assessedusing a set of six kinetostatic analyses with independent virtual loads applied to the end-effector, and subsequently making useof the principle of virtual work to obtain the total error. This way the maximisation can be very much simplified by consideringthe effects of each joint and at every contact mode separately. Later, in [8], a first step toward a deterministic method was madeby including actual loads through a new kinetostatic analysis. On the one hand, the use of kinematicmodels in thesemethods hasmade the accuracy analysis linear, and more complex contact models of the joints have been added. Also, maximisation analysishas been simplified and overconstrained mechanisms can be handled. On the other hand, the actual load and motioncharacteristics have not yet been included.

• Deterministic methods. These try to calculate the actual error in the end-effector pose under the action of the load on themanipulator. In order to do that, the first step is a kinetostatic analysis to obtain actual reactions on the joints. Severalauthors apply the principle of virtual work to assess the contribution of the clearances of each joint to the pose error, andthen sum these contributions. For example, in [9], the author uses a kinetostatic model but only considers mechanisms withrevolute pairs and the load on one link. This approach is completed, in [8], with the adding of the possibility of additionalloads and opening of the procedure to other pair types. In [10], the authors model the clearance of a revolute pair in theplane with an equivalent link and use the principle of virtual work on a four-bar mechanism identifying sudden changes ofpose at dead centre positions. More recently, the same authors have extended their technique to multi-loop linkages [11].In [12], kinetostatic analysis is used in such a way that an analytical model is obtained for the pose error of the end-effectoranywhere within the workspace. And finally, some authors add the effect of deformations of links to the analysis of theeffect of clearances (e.g., [13]).

• Multibody methods. This kind of method is focused on describing contact forces inside imperfect joints. The response of amechanical system with clearance joints becomes chaotic for high speeds and low contact friction, whereas it remains periodicfor lower speeds and a low coefficient of restitution [14]. For revolute joints, three kinds of relative motion between the journaland the bearing have been distinguished: free movement, impact and continuous contact with or without friction. Some authorshave modelled the impact with finite element models and others use analytical impact force models. The calculated jointreactions have been introduced into the dynamicmodel as external generalised forces [15–17]. Impact forces cause high peaks inaccelerations and staircase-shaped velocities. These phenomena are deadened by lubricated joints [17] or flexible links [18].Multibody methods are appropriate for detailed analysis of advanced design of mechanisms with high dynamic requirements.However, in the early stages of design they have a reduced usefulness because of their high computational cost and demand fordetail.

In this paper, error pose due to clearances is calculated by a deterministic method, which can be applied in the early designstage. In such an analysis, it is commonly assumed that there is always a frictionless contact inside the imperfect joints.Nevertheless, contact loss is detected in the analysis and corresponding conditions where this happens are analysed. Analysingsuch conditions can be considered more realistic, since the actual error is calculated based on clearance magnitude, which isalways uncertain. Bearing this fact in mind, this paper presents an analytical methodology for determining where the contact lossoccurs in applications in which inertial loads can be neglected.

Clearance magnitude is much smaller than other dimensions and consequently it is assumed to be infinitesimal, which leads toa linear analysis [5–9,12]. The use of virtual links proposed in [10,11] can overcome this simplification, but resultant hyper-degree-of-freedom linkages cannot be solved analytically so numerical methods are required. Another weakness of virtual links is thecomplexity when applied to spatial mechanisms with any kind of joints.

579O. Altuzarra et al. / Mechanism and Machine Theory 46 (2011) 577–592

The method developed uses loop closure equations expressed in screw coordinates as in [5,10,11]. Clearance take-up isgiven from the results of a kinetostatic or dynamic analysis of the nominal mechanisms as in [8,9,12]. Then, loop closureequations are solved with clearance freedoms in order to determine variations on passive freedoms, i.e., in order to know howclearances affect them. We note that such an analysis could be improved by an iterative process such as the one in [13].Further, the effect of contact modes described in [7,12] could be taken into account by establishing relations between differentclearances of a certain imperfect joint, as in [5]. The error calculation method proposed has characteristics taken from variousdifferent published methods and exploits the strengths of these methods in an attempt to overcome the limitations that eachof them presents.

When the error calculation procedure is carried out on a path or on the workspace, it provides the locations wherediscontinuities on the end-effector actual position occur due to sudden changes in contact modes. Merlet [19] mentionsflight simulators, pick-and-place, vibration simulation and even high-speed machining tools as applications where dynamicsplay an important role. In such applications, locations where discontinuities occur have a strong dependence on inertialloads. On the other hand, in applications where inertial effects can be neglected, such as machine tools, micro-positioningand surgical robots, these locations only depend on the applied external loads and in the mass distribution of themanipulator itself. This paper shows how this dependence on inertial loads vanishes when slowing down the velocity ofthe manoeuvres.

In order to identify the locations where discontinuities occur in kinetostatic applications and to calculate them efficiently, ananalytical methodology for the so-called discontinuity loci calculation is developed. Discontinuity loci and their neighbourhoodmust be avoided because they correspond to configurations of the mechanism with a high level of uncertainty and calibration isnot efficient. The main contribution of this work is to provide a method to locate regions of the workspace where discontinuitiesdue to changes in the contact mode of the imperfect joints do not appear.

2. Clearance error analysis

The first tool that we need to develop is a powerful procedure for the analysis of pose errors due to clearances. This procedureincludes the possibility of applying any external load to any link of the mechanism, evaluates reactions at joints to decide the wayclearances are actually taken up, and finds the end-effector pose error considering the joint clearances and allowing for posecorrections in passive joints.

The procedure includes the following modules:

• The inverse position problem.Wewill analyse the ideal mechanism to find the nominal pose at one configuration, over a desiredpath, or throughout the workspace.

• An evaluation of joint reactions. By means of a dynamic analysis, the wrenches at the joints under the external loads of themechanism will be calculated.

• An accuracy analysis. Our approach is to employ a deterministic method using the kinematic velocity relations. For everyposture, we will define with the joint reactions the infinitesimal displacements at joints due to clearances. With fixed inputs andthe velocity equations at the posture analysed, we will obtain changes in the passive joint rates due to clearances, and then theend-effector pose error.

• Numerical analysis of discontinuities. The pose error will show if clearance take-up changes suddenly at some postures, whichclearance causes this, and how wrenches are transmitted through the mechanism before and after that. These sudden changesappear at certain poses depending on the configuration of the mechanism and the applied external wrenches. Said poses arefound and determined so-called discontinuity loci.

2.1. Clearance definition

Before starting with the analysis of pose errors, it is necessary to define how clearances are taken up. From now on, we will useby way of example a planar parallel manipulator with two degrees of freedom (DOF) defined by five links joined using revolutepairs, commonly known as a five-bar mechanism. Two revolute pairs are actuated and the other three are passive. It is supposedthat both actuated and passive pairs have clearances.

A passive revolute pair Ji with a clearance is shown in Fig. 1a; the pin canmove freely inside the hub, but it is assumed that theyalways remain in contact. As shown in Fig. 1b, in the model of an actuated joint Ji there are two kinds of clearances: radialdifference between the pin and the hub, and width difference between the cotter and its case. For simplicity, those differentclearances are separated and treated independently. We will call them radial clearance and angular clearance. The first oneproduces two clearance-freedoms with magnitudes ΔxJi and ΔyJi, and angular clearance due to width differences at the cotter hasone clearance-freedomΔθJi. Note that angular clearance can be also produced by backlash inside the motor or reduction gear.

2.2. Inverse position problem

The first module solves the inverse position problem in the nominal mechanism. It consists in finding the values of the actuatedjoints corresponding to a known end-effector pose. A usual approach is the use of geometrical loop equations. The morphology ofthe geometrical equations depends on the mechanism architecture.

Fig. 1. Clearance joints of the 5R planar mechanism. (a) Passive revolute pair with clearance. (b) Actuated pair with clearance.

580 O. Altuzarra et al. / Mechanism and Machine Theory 46 (2011) 577–592

In Fig. 2 we have depicted the kinematic model of the 5R planar mechanism with the parameters used in the analysis. In thepick-and-place application, which is analysed in Section 3, the 5R planar manipulator works upside down, that is, its fixed pointsA1 and A2 are always over the reference point P. Actuated joints are at A1 and A2, respective angles are θ1 and θ2. Passive joints are atB1 and B2, i.e., angles ϕ1 and ϕ2. The end-point is at P, and corresponding output variables are its coordinates x and y. Angle ψ is apassive parameter.

For given values of the output [xPyP], the solution of the inverse position problem consists in finding the intersection of twopairs of circles, i.e., one centred at Pwith radius PBi and the other centred at Ai with radius AiBi. Each one produces two solutions tothe position of Bi, and hence, two different postures of each limb. There are 22 different combinations, and hence four workingmodes for the mechanism (Fig. 3). It is common practice to keep the mechanism in one working mode, so we will always use theworking mode shown in Fig. 3a.

Using this approach on themechanism shown in Fig. 2, where link lengths are all equal to L=0.5 m and actuators A1 and A2 areplaced d=0.5 m apart, the workspace of the mechanism is calculated (Fig. 4).

Further to this analysis, Jacobians can be evaluated to find the singularity locus. The workspace is divided into two partsdepending on the sign of the determinant of the direct Jacobian, i.e., the assembly mode, as shown in Fig. 4. Limits between twocolours are direct singularities. The boundary of the workspace corresponds to inverse singularities found with the inverseJacobian.

The use of screw coordinates allows a compact matrix formulation of the velocity equations. Let P be the reference point of theend-effector, and choose for convenience the link PB2 as the mobile platform. Then, the end-effector twist is defined by:

where

$P = ωp

� �ð1Þ

ω is the angular velocity of the mobile platform, and p is the velocity of end-point P.

Fig. 2. Pose parameters of a 2DOF 5R planar parallel manipulator.

Fig. 3. Working Modes of the 5R planar mechanism.

581O. Altuzarra et al. / Mechanism and Machine Theory 46 (2011) 577–592

The screws for the revolute pairs A1, B1 and P of the 5R planar mechanism are:

where

$A1=

krA1

× k

� �; $B1

=k

rB1 × k

� �; $P = k

0

� �ð2Þ

k is the direction of the revolute pairs in planar mechanisms, rA1and rB1

are the position vectors of A1 and B1 from point P

whererespectively.

Applying Eq. (1), the end-effector twist at point P can be stated via links A1B1 and PB1 as:

$P = θ1 $A1+ ϕ1 $B1

+ ψ$P ð3Þ

θ1, ϕ1 and ψ are joint rates.

whereThe same twist found through links A2B2 and PB2 can be expressed as:

$P = θ2 $A2+ ϕ2 $B2

ð4Þ

$A2and $B2 are:

$A2=

krA2

× k

� �; $B2

=k

rB2 × k

� �ð5Þ

An independent loop can be drawn for this mechanism, and equating Eqs. (3) and (4) produces the following equation:

θ1 $A1+ ϕ1 $B1

+ ψ$P−ϕ2 $B2−θ2 $A2

= 0 ð6Þ

Fig. 4. Workspace of a 2DOF 5R planar parallel manipulator.

582 O. Altuzarra et al. / Mechanism and Machine Theory 46 (2011) 577–592

The nominal velocity analysis is solved by obtaining ϕ1, ψ and ϕ2 for given inputs θ1 and θ2 in:

where

$B1$P−$B2

h i ϕ1ψϕ2

264

375 = −$A1

$A2

h i θ1θ2

� �ð7Þ

ck-substituting in Eq. (3) or (4) to give the twist of the end-effector.

and baAcceleration analysis is performed by differentiating Eq. (6) with respect to time. The accelerations calculated can be used

in the next sections to determine inertial loads.

2.3. Joint reaction evaluation

Once the inverse position problem is solved, dynamics of the mechanism can be analysed. Using the free-body diagramapproach and setting out Newton–Euler equations, it is possible to obtain a linear system of equations. Under a given load,reactions in the joints and actuation forces or torques will be the unknowns of the system.

First, without loss of generality, a link of a spatial mechanism will be analysed. Let us consider the i-th link of a parallelmechanism. At a point Qi of this link, a known external wrench, wei, is applied:

wei=

feimei

+ rQ i× fei

� �ð8Þ

fei is the applied force and mei is the applied external couple; rQ iis the position vector of Qi with respect to a certain point,

where

and then rQ i× fei is the moment of fei with respect to the same point.

Reactions in i-th and (i+1)-th joints are represented by a set of forces and couples (fJi,mJi)T and (fJi+ 1

,mJi+ 1)T. Let P be the

chosen reference point, from which the position vectors of the i-th and (i+1)-th joints are rJi and rJi+ 1respectively.

Writing Newton–Euler equations for the i-th link in the general 3D case, Eq. (9) is obtained:

fJimJi

+ rJi × fJi

� �+

fJi + 1

mJi + 1+ rJi + 1

× fJi + 1

� �+

feimei

+ rQi× fei

� �=

mic::i

Iiωi + ωi × Iiωi + rCi × mic::i

� �ð9Þ

mi is themass of link i, c::i the acceleration of its centre of mass Ci, Ii its inertia tensor at Ci, and finallyωi and ωi are its angular

where

velocity and acceleration respectively. Note that depending on the joint type, some components of vectors fJi andmJi related to thepermitted relative movements may be null. For example, in a passive spherical joint, mJi is equal to zero.

Since the inverse kinematics have been solved previously, the right side of Eq. (9) is known and actuation loads and jointreactions are the only unknowns. Hence, Newton–Euler equations can be expressed in a matrix form and unknowns can be simplycalculated:

A⋅wJ = wi−we ð10Þ

wJ is the vector of joint reactions and loads withstood by actuators, vectorwi contains the inertial loads andmatrix A is built

wherefrom the Newton–Euler equations.we is a column vector which groups the external wrencheswei at any link of the mechanism, asshown in Eq. (11).

we = wTe1;…;wT

el

� �T ð11Þ

l is the number of links of the mechanism.

whereIt is remarkable that matrix A is a geometrical matrix, i.e., it depends only on the mechanism configuration. Note also that its

singularities are the same of those of the Direct Kinematic Problem. In fact, this matrix can be used for detecting the directsingularities of a mechanism. By virtue of the duality of kinematics and statics, Hubert and Merlet [20] used the kinematic Jacobianfor detecting closeness to singularities, considering high values in joint reactions as the measure of closeness. Matrix A of Eq. (10)could be used for the same purpose.

Regarding Eq. (10), for the 5R planar parallel manipulator, the twelve unknowns of vectorwJ are two reactions on each passiverevolute joint and two reactions and one actuation torque on each actuated joint.

wJ = fTA1 fTB1 fTP fTB2 fTA2 MA1 MA2

� �T ð12Þ

fJi is an x–y component column vector and MJi is a scalar.

583O. Altuzarra et al. / Mechanism and Machine Theory 46 (2011) 577–592

Assembly of A for the 5R planar mechanism leads to the next matrix:

where

A =

1 0 −1 0 0 0 0 0 0 0 0 00 1 0 −1 0 0 0 0 0 0 0 0

− yA1−yPð Þ xA1−xPð Þ yB1−yPð Þ − xB1−xPð Þ 0 0 0 0 0 0 1 00 0 1 0 −1 0 0 0 0 0 0 00 0 0 1 0 −1 0 0 0 0 0 00 0 0 − yB1−yPð Þ xB1−xPð Þ 0 0 0 0 0 0 00 0 0 0 1 0 1 0 0 0 0 00 0 0 0 0 1 0 1 0 0 0 00 0 0 0 0 0 − yB2−yPð Þ xB2−xPð Þ 0 0 0 00 0 0 0 0 0 −1 0 1 0 0 00 0 0 0 0 0 0 −1 0 1 0 00 0 0 0 0 0 yB2−yPð Þ − xB2−xPð Þ − yA2−yPð Þ xA2−xPð Þ 0 1

26666666666666666664

37777777777777777775

ð13Þ

Solving Eq. (10), joint reactions f Ji at every joint Ji are obtained, mJi being null for passive revolute joints. In the absence offriction, as reactions in joints are produced by contact forces between the pin and hub, their directions will indicate how theknown radial clearance, ρi, is distributed in x and y, i.e., the actual values of ΔxJi, ΔyJi, namely:

ΔxJi = −ρi

fJifJi

��� ��� ⋅i264

375 ð14Þ

ΔyJi = −ρi

f Jif Ji

��� ��� ⋅j264

375 ð15Þ

i and j are unitary vectors in Cartesian axes.

whereIn Fig. 5 we have depicted how the direction of the joint reactions indicates the contact mode in the revolute joint. The joint

reaction defines the direction that the radial clearance has taken, and therefore the components in the fixed frame XY. The angleindicating the direction of clearance take-up βJi is a very important parameter because it will present sudden changes when thetake-up mode changes. βJi can be found from:

βJi= arctan

fJi ⋅ifJi ⋅j

" #ð16Þ

2.4. Kinematic accuracy analysis

After having solved the Newton–Euler equations of the preceding section, an accuracy analysis is carried out. In this, as inSection 2.2, velocity equations are expressed in terms of screw coordinates in order to achieve a compact formulation. Thesevelocity equations are used to approximate displacements, which are assumed to be infinitesimal.

First, let us consider the end-effector twist $P in Eq. (1). The $P of the nominal mechanism can be calculated by adding upcontributions from each kinematic chain, going from the fixed frame to the end-effector:

$P = ∑Nj

i=1q ji $

jJi

ð17Þ

Nj is the number of joints of kinematic chain in limb j, and q ji and $ j

Ji are, respectively, the rate and the screw at each joint i of

wherethe said chain.

For a fully parallel manipulator (manipulators whose number of chains is strictly equal to the number of DOFs of the end-effector) with n kinematic chains, the number of independent loops will be equal to (n-1). $P can be calculated by means of thevector loop equation. Thus, for every closed loop of a mechanism Eq. (18) can be written:

∑Nj + 1

i=1q j+1i $ j+1

Ji− ∑

Nj

i=1q ji $Ji

j= 0 ð18Þ

j=1..(n−1).

Fig. 5. Model of clearance take up in a planar revolute joint.

584 O. Altuzarra et al. / Mechanism and Machine Theory 46 (2011) 577–592

Terms of Eq. (18) can be separated into those of the passive joints and those of actuated joints. Assembling equations of eachclosed loop, denoting the passive joints with indexP and the actuated or input joints with I , and reordering the terms, an equationsystem can be built in order to solve the nominal velocity analysis:

JP⋅qP = JI ⋅q

I ð19Þ

JI is a matrix with the screws of the input joints and JP groups the screws of the passive joints, being the latter a

wheresquare matrix for fully parallel manipulators. Note that this equation is equivalent to Eq. (7).

From Eq. (19) values at passive joints can be obtained. Matrix JP is configuration dependent and it becomes singular for directsingularities of the mechanism, where Eq. (19) cannot be solved.

Additional freedoms due to clearances can be added to the mechanism and then to Eq. 19, yielding Eq. (20):

JP⋅qP = JI JC½ �⋅ qI

qC

� �ð20Þ

JC is a matrix with the screws from freedoms due to clearances and qCare their velocities.

where

Assuming that displacements are infinitesimal, they can be approximated by velocities. If the actuated joints are fixed, onlyfreedoms due to clearance remain at the mechanism. As mentioned before, relative positions between the two parts of each jointare determined by the reactions calculated in the dynamic analysis (Eqs. (14) and (15). Then the difference between nominalposition and actual position is considered as an infinitesimal displacement of each freedom due to clearance. Eq. (21) is then usedinstead of Eq. (19), where actuated joints have been eliminated and velocities of freedom due to clearance have been replaced byinfinitesimal displacements ΔqC:

JP⋅ΔqP = JC⋅Δq

C ð21Þ

Solving Eq. (21) gives the displacements of the passive jointsΔqP due to clearance freedoms. Let us define vectorΔΓ as the poseerror of the end-effector. By adding it to the nominal pose of a mechanism, the actual pose is obtained.With the calculated ΔqP ,ΔΓcan be computed with Eq. (17) as follows:

ΔΓ = ΔαΔp

� �= $j

J $ jC

h i Δq jJ

Δq jC

24

35 ð22Þ

Δp is the error in position of point P due to clearance take up, Δα is the change in the absolute orientation of the end-

whereeffector due to clearances, and Δq j

J and Δq jC are infinitesimal displacements of nominal joints (only passive joints if actuated

joints are fixed) and clearance freedoms respectively. Superindex j means that ΔΓ can be calculated using any limb j of themanipulator.

585O. Altuzarra et al. / Mechanism and Machine Theory 46 (2011) 577–592

Let us apply the described error pose calculating procedure to the 5R planar mechanism studied, the nominal kinematicshaving been analysed in Section 2.2. As infinitesimal displacements can be approximated by velocities, the relation of Eq. (7) canbe applied to the analysis of the transmission of infinitesimal changes of the inputs:

$B1$P−$B2

h i Δϕ1ΔψΔϕ2

24

35 = −$A1

$A2

h i Δθ1Δθ2

� �ð23Þ

Note that screws $A1and $A2

which represent actuated pairs can be replaced by $a

A1and $

a

A2in order to characterise the angular

clearance described in Section 2.1.If radial clearances at revolute joints are modelled as infinitesimal displacements of the pin in the hub, these can be

decomposed in a fixed frame; for example, at joint A1 into ΔxA1and ΔyA1

. Corresponding screws of imperfect joints are shown inthe next equation:

$xA1

= $xA2

= $xB1

= $xB2

= $xP = 0

i

� �; $y

A1= $y

A2= $y

B1= $y

B2= $y

P = 0j

� �ð24Þ

Adding all clearance freedoms to the loop closure equation, Eq. (6), yields:

Δθ1 $aA1

+ ΔxA1$xA1

+ ΔyA1$yA1

+

+ Δϕ1 $B1+ ΔxB1 $

xB1

+ ΔyB1 $yB1

+

+ Δψ$P + ΔxP $xP + ΔyP $

yP−

−Δϕ2 $B2−ΔxB2

$xB2−ΔyB2 $

yB2−

−Δθ2 $aA2−ΔxA2

$xA2−ΔyA2

$yA2

= 0

ð25Þ

Δθ1 and Δθ2 could be used to model an angular clearance of the rotary actuator, and Δϕ1, Δϕ2 and Δψ are infinitesimal

wherechanges in the passive angles.

Taking the results of the Newton–Euler equations to Eqs. (14) and (15), values of infinitesimal displacements ΔxAi, ΔyAi

, ΔxBi,

ΔyBi, ΔxP and ΔyP were obtained. Further, the sense of moments at links AiBi indicate how the angular clearances of actuators, Δθ1

andΔθ2, have to be defined. Then, using Eq. (25) we can solve the infinitesimal changes in the passive angles, i.e.,Δϕ1,Δϕ2 and Δψ,due to the taking up of clearances:

$B1$P−$B2

h i Δϕ1ΔψΔϕ2

24

35 = −$a

A1$aA2

−$xA1

::: $yB2

h iΔθ1Δθ2ΔxA1

::

ΔyB2

26666664

37777775

ð26Þ

Once these are obtained, the end-effector pose errorΔΓ can be found using an expression analogous to Eq. (4) where clearancesare added:

ΔΓ = ΔαΔp

� �= $a

A2$B2

$xA2$yA2$xB2$yB2

h iΔθ2Δϕ2ΔxA2

ΔyA2

ΔxB2ΔyB2

26666664

37777775

ð27Þ

The procedure described calculates the pose error of the mechanism taking into account the influence of clearance take-up onpassive freedoms. These passive freedoms have been determined from local errors at imperfect joints, which have been obtainedfrom a dynamic analysis of the nominal mechanism, not the actual one. Consequently, dynamics of the actual mechanism are notanalysed, but since clearance magnitudes are infinitesimal, the procedure can be considered an appropriate approach.Nevertheless, it could be made more complete by means of an iterative procedure. In such an iterative procedure, local errors atimperfect joints and passive freedom errors could be used in order to update geometrical parameters of the nominal mechanism,namely, lengths of the links, passive angles and angular variations of actuators. Then, the dynamics of the new nominalmechanismcould be calculated and, with them, local pose errors at imperfect joints and variations on passive freedoms, returning to thestarting point. However, implementation of this type of iterative procedure is not considered in this paper.

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2.5. Discontinuity analysis

As well as the error calculation procedure, it is possible to think of configurations of the manipulator—with applied externalloads—in which some joint reactions become zero. In such configurations the contact between the two parts of certain joint (orjoints) get lost, leading to a free relativemovement between two links. The consequence is an uncertainty on the actual pose of theend-effector or a discontinuity along a path. In this paper, a deep analysis of such discontinuities is carried out and a way tocalculate locations where they occur is presented in Section 4.

3. Error analysis on a pick-and-place trajectory

When the performance of the manipulator requires high accelerations, inertial loads play an important role. This is the case inapplications such as pick-and-place, flight manipulators or high-speed machining tools [19]. In the present section, a pick-and-place trajectory of the aforementioned 5R planar mechanism is analysed. Such a trajectory is considered for the moving of a heavymetal object in a blast furnace. In the numerical example, the mass of the links is 1 kg.and of the metal object 10 kg, while linklengths and distance d between points A1 and A2 are 0.5 m.

In this pick-and-place process, the manipulator picks the object from a certain position, raises it for 0.025 m, displaces ithorizontally for 0.5 m, lowers it for 0.025 m and returns to its starting position. Said path is made smooth by introducing roundedcorners between vertical and horizontal displacements. The object to be moved is picked up by a gripper located at point P. As theend-point P belongs to links PB1 and PB2, it is assumed that the gripper is located on PB2, which has been previously chosen as themobile platform. The trajectory is a round trip with a first part of the path, denoted as stage 1, in which the load is brought to theend point; and a return path of the unloaded manipulator to the starting position identified as stage 2.

Such a trajectory is depicted on the XY plane in Fig. 6a and velocities and accelerations produced at point P through the wholetrajectory (complete trip) can be visualised in 6b.

Results of the error calculation procedure described in Section 2 applied to the pick-and-place trajectory are shown in Fig. 7,where stage 1 is represented in blue and stage 2 in black. Clearance magnitude is exaggerated (10 mm for radial clearance and0.005 rad for angular clearance) in order to show clearly the effect it produces. Apart from the calculated position error, suddenchanges appear in the actual position. These are due to sign changes in joint reactions, which lead to loss of contact at joints.

As mentioned in the literature review, one kind of relative motion between the two parts of a joint is the continuous contact.The other two stages, free flight and impact, produce staircase-shaped variations of the mechanism velocity, according to themultibody approach [16]. In a kinetostatic approach, it is assumed that there is always a contact between the two parts of a joint.Under this assumption, some authors replace joints with clearance by ideal links, which can only carry a tension load. In [10],authors showed that sudden changes in ideal link load lead to discontinuities in a four-bar linkage trajectory. Other authorsdescribe different contact modes between the two parts of a joint. In [21], the switch from tension to compression in the legs of theGeneralised Stewart Platform leads to discontinuities in position and orientation error. In a similar way, changes in the contactmodes have been shown to produce discontinuities in the position error in a 3-UPU robot in [12].

In our work, these staircase-shaped values or sudden changes in contact modes are represented as discontinuities in the actualtrajectory (Fig. 7). A discontinuity represents a loss of contact inside imperfect joints along the trajectory. This causes additionalundesirable effects such as noise and vibrations. In order to avoid these situations, discontinuities and their causes are analysed.

Fig. 6. Pick-and-place trajectory for a 5R mechanism. (a) XY trajectory. (b) Velocity and Acceleration plots.

Fig. 7. Nominal and actual trajectories.

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Taking into account the importance of inertial loads on fast tasks like pick-and-place processes, the influence of the cycle periodis analysed below. As a starting point, we choose the one in Fig. 7, which is the fastest with a period of 0.76 s. Subsequently, theperiod was increased in order to analyse slower cycles and show the influence of inertial loads on discontinuities.

Fig. 8a shows the positioning error as a function of distance covered in the pick-and-place trajectory for different cycleperiods. The distance covered in the cycle is represented on an abscissa axis and the cycle period on an ordinate axis, whilecolour on the vertical axis represents positioning error with respect to the nominal path in the Y direction as shown in Fig. 7.On Fig. 8b a planar view of the same error is depicted, and it can be seen that the coloured curves become closely packed atdiscontinuities. For example, for a cycle with a period of one second, the first discontinuity appears at a distance of 55 mmfrom the starting point and the second one at 145 mm. The figures have two clearly different halves: the left corresponds tostage 1 (loaded) and the right to stage 2 (unloaded). Notice that a discontinuity is produced between these two stages, whenthe transported object is placed.

A deeper analysis of Fig. 8b shows how cycle period affects the location of the discontinuities. For fast cycles with cycle periodslower than 1.2 s, locations of discontinuities remain almost constant. In stage 1 discontinuities disappear for cycle periods greaterthan 1.5 s, while in stage 2 discontinuities remain until cycle periods of 2.2 s. Then, regardless of the unavoidable discontinuityproduced when placing the object, cycles which last more than 2.2 s avoid discontinuities. We conclude that discontinuities havestrong dependence on the inertial loads in high-speed tasks.

In the present section a pick-and-place trajectory has been analysed. Results show that inertial loads have a strong influence onthe location of the discontinuities of the positioning error. For such an application, the location of the discontinuities in theworkspace depends on the specific trajectory1 performed by the mechanism. For cycle periods longer than a certain threshold,inertial loads do not have influence on the location of the discontinuities. Therefore, in applications where dynamics do not play animportant role, inertial loads can be neglected. In the next section, the so-called kinetostatic applications are analysed.

4. Workspace analysis in kinetostatic applications

Kinetostatic applications are those in which inertial loads can be neglected. Among them, the most common are machine tools,micro-positioning or surgical robots. In this approach, the right side term of Newton–Euler equations (Eq. 9) is equal to zero.Therefore, there is no dependence on velocities and accelerations, and the entire workspace of a certain working mode can beanalysed.

Applying the approach of Section 2 to the 5R planar mechanism loaded at P with a vertical and negative load of 100 N, theposition error is calculated within the workspace (Fig. 9) for given values of the clearances. Position errors depend on both thedirection of the applied external load and the magnitude of clearances, but the location of the discontinuities is not dependent onthe latter. This allows the discontinuity locus in the workspace to be assessed in spite of the usual uncertainty in the magnitude ofclearances.

As mentioned above, matrix A defined in Eq. (10) detects the direct singularities of the mechanism. Furthermore, the matrix ispoorly conditionedwhen it is close to a direct singularity. As was the case in thework of Hubert andMerlet [20], that means that inorder to balance external forces, reactions require high values, and then the mechanism could break down. This fact is clearlyshown in Fig. 9. Maximum positive and negative values are located in the same positions as the direct singularities, as can be seenin Fig. 9a. Error in the x coordinate is depicted also in Fig. 9b, in which, apart from the direct singularities, discontinuities are alsonoticeable. The latter form curves that divide the workspace into different areas where the position error is stable. As the externalforce is applied in the y direction, reactions in this direction will always have the same sign, so discontinuities appear only in the xcoordinate error.

As far as discontinuities are concerned, they are due to sign changes in joint reactions. With only an external load applied at P,these changes correspond to positions where links PB1 or PB2 are aligned with the external load. In these situations, reactions areequal to zero on at least one joint, so the relative position between the pin and the hub is undefined. In Fig. 10, one of these

1 The term trajectory takes into account not only the path (positions) but also the task velocities and accelerations.

Fig. 8. Cycle time influence on positioning error. (a) Error in Y. (b) Error in Y.

588 O. Altuzarra et al. / Mechanism and Machine Theory 46 (2011) 577–592

configurations is shown. A vertical position of link PB2 implies that this link supports the external vertical load fe. Links whoseposition is undefined are plotted with dashed lines. They do not transmit any load and reactions in joints B1 and A1 are null.Consequently, these links are not fixed and accordingly the mechanism is unstable in the configuration shown in Fig. 10 and inevery configuration where PB1 or PB2 are vertical. The discontinuity is produced when the mechanism crosses theseconfigurations.

Discontinuities on the workspace divide the error plot into four different surfaces. These surfaces represent a region of theworkspace where there are no discontinuities. Designers may take these regions into account in order to create discontinuity-freezones, where vibrations produced by clearance joint impacts are avoided. In the so-called discontinuity-free regions, errors due toclearance can be calculated and corrected by means of a calibration process.

It is worth noting that the pick-and-place trajectory of Section 3 is located in a discontinuity-free region. That is the reasonwhydiscontinuities disappear when inertial loads are negligible. Notice that the vertical load applied at P of the present section isequivalent to the travelling load of Section 3.

The error plot of Fig. 9 changes if the direction of the external load is modified. In Fig. 11, the position error in P is shownwhenthe mechanism carries a horizontal load. It can be seen how the location of the discontinuities is different depending on theapplied load. Note also that, for this horizontal load, discontinuities now appear only in the y coordinate error.

Near direct singularities, the poor conditioning of matrix A leads to the methodology of error calculation performinginadequately, since infinitesimal displacements cannot be approximated by velocities. In order to overcome this limitation, ananalytical method for discontinuity detection is developed.

Fig. 9. Errors in the workspace. Load Vertical and Negative. (a) Error in X: plane representation. (b) Error in X: 3-D representation.

Fig. 10. 5R mechanism with clearance at a discontinuity.

589O. Altuzarra et al. / Mechanism and Machine Theory 46 (2011) 577–592

So far, detecting discontinuities has been done by solving the Newton–Euler equations of the mechanism at discrete postureson a grid of the workspace. Now, an analytical method to find the discontinuity loci is presented. Since this method deals withkinetostatic applications, inertial loads are neglected and these loci only depend on the applied external loads, the joints withclearance and the configuration of the mechanism.

At this point, it should be emphasised that clearance magnitude is usually uncertain. Furthermore, wear phenomena produceunmeasurable changes in this parameter. Hence, there is always an uncertainty when calculating the position error due toclearance. Since discontinuity loci depend on the joints with clearance themselves and not on their magnitude, this uncertaintydisappears. Then, calculating the discontinuity loci of a mechanism can be more useful than calculating the positioning error.

From Eq. (10) reactions in the joints of a mechanism can be calculated. Using Cramer's rule to solve this equation system, it ispossible to calculate the component of the reaction in a joint Ji (Eq. 28):

wkJi=

Aki

��� ���Aj j ð28Þ

index i is referred to the i-th joint and k to the k-th component, with k=1,…,c; and c the number of restrictions imposed by

wherethe joint Ji. Matrix Ak

i results from replacing the (6·(i−1)+k)-th column ofmatrixA by the known column vectorwe, since vectorwi is neglected.

As mentioned before, A will be singular at the direct singularities, that is, Aj j will be equal to zero. For other points, wkJiwill be

equal to zero when the determinant jAki j is null. Solving this determinant symbolically, an analytical expression can be obtained.

This expression is used to describe the geometrical locus where reaction wkJiis equal to zero. When wk

Jirepresents the reaction

Fig. 11. Errors in the workspace. Load Horizontal and Positive. (a) Error in Y: Plane representation. (b) Error in Y: 3-D representation.

Fig. 12. Locus of poses where Cartesian reactions in joint A1 are equal to zero. Load Vertical and Negative. (a) fA1

x .(b) fA1

y .(c) fA1.

590 O. Altuzarra et al. / Mechanism and Machine Theory 46 (2011) 577–592

moment at A1, MA1, equating Aki

��� ��� to zero gives the corresponding discontinuity loci of MA1. Specifically, the determinant Aki

��� ��� ofreaction MA1 is written in Eq. (29) supposing the mechanism is loaded at P with a vertical load.

AMA1

��� ��� =

1 0 −1 0 0 0 0 0 0 0 0 00 1 0 −1 0 0 0 0 0 0 0 0

− yA1−yPð Þ xA1−xPð Þ yB1−yPð Þ − xB1−xPð Þ 0 0 0 0 0 0 0 00 0 1 0 −1 0 0 0 0 0 0 00 0 0 1 0 −1 0 0 0 0 0 00 0 0 − yB1−yPð Þ xB1−xPð Þ 0 0 0 0 0 0 00 0 0 0 1 0 1 0 0 0 0 00 0 0 0 0 1 0 1 0 0 fPy 00 0 0 0 0 0 − yB2−yPð Þ xB2−xPð Þ 0 0 0 00 0 0 0 0 0 −1 0 1 0 0 00 0 0 0 0 0 0 −1 0 1 0 00 0 0 0 0 0 yB2−yPð Þ − xB2−xPð Þ − yA2−yPð Þ xA2−xPð Þ 0 1

������������������������

������������������������ð29Þ

coordinates xB1, yB1, xB2 and yB2 have to be written as functions of coordinates xP and yP. As discontinuity loci are curves� �

wherecalculated equating AMA1

� � to zero, the obtained geometrical locus does not depend on the magnitude of the vertical load fPy.On the other hand, supposing that f xJi and f yJi are reactions in a revolute joint Ji in Cartesian axes, Eq. (30) describes the

geometrical locus where the reaction in Ji becomes null, that is, where the sudden change in the relative position between the pinand hub occurs.

Axi

�� ��2 + Ayi

�� ��2 = 0 ð30Þ

Notice that a sudden change in a joint with clearance leads to a sudden change in the end-effector, i.e., the discontinuity locus ofjoint Ji is also the discontinuity locus of the end-effector. In order to determine all end-effector discontinuity loci, it is necessary tocalculate Eq. (30) for each clearance joint in the mechanism. This way, the joint which causes the end-effector's discontinuity isalso identified.

For the mechanism studied with a vertical load applied at point P, results of the locus defined by Eq. (30) are shown in Fig. 12cfor sudden changes in the Cartesian position of the pin of joint A1. Fig. 12a and b depict loci where components of reaction f xA1

andf yA1

respectively are equal to zero while Fig. 12c shows the condition for the reactionmodule. As discontinuity loci are calculated bymeans of analytical expressions, the curves of Fig. 12 exceed the workspace limits (red colour area). In any case, points outside theworkspace must not be taken into consideration.

In Fig. 13 the locus of sudden changes of Δθ1 and Δθ2 due to the angular clearance at A1 and A2 can be seen. Discontinuities ofboth Δθ1 and Δθ2 define all the end-effector positions where the mechanism will undergo a sudden change. Discontinuity loci ofjoints A1, B1 and P are the same as that of θ1, while loci of A2 and B2 are the same as that of θ2. This is due to the null reactions on thewhole kinematic chain and can be understood by looking at Fig. 10. When the link PB2 is in a vertical posture, the relative positionin joint P is undefined because its reaction becomes null. As link PB1 does not transmit any load, the reaction at joint B1 is null andin the sameway the reaction at A1 is also null. Similarly, when link PB1 is vertical, reactions in joints B2 and A2 are equal to zero. As aconsequence, discontinuity loci for a vertical load are all the positions of the workspace where link PB2 or PB1 are vertical.

Fig. 13. Locus of poses where the reaction moment at joints A1 and A2 is equal to zero. Load Vertical and Negative. (a) MA1. (b) MA2

. (c) MA1and MA2

.

591O. Altuzarra et al. / Mechanism and Machine Theory 46 (2011) 577–592

Finally, the combination of discontinuity loci of Fig. 13a and b leads to Fig. 13c. It should be emphasised that this showsdiscontinuities appearing in the same positions on the workspace as in Fig. 9a.

In the present section, geometrical loci of discontinuities are calculated, both numerically and analytically. These loci leavediscontinuity-free zones where stages of free flight and impact inside imperfect joints are avoided. In the literature, methods toavoid sudden changes in actual position, such as additional counterweights, springs [22], redundant actuation [23], modification oflink shapes [24] and trajectory planning [25], have been proposed. Designing the mechanism in order to have a desireddiscontinuity-free region could be another effective option.

5. Conclusions

In this paper, a numerical procedure for accuracy analysis of parallel manipulators has been presented. Errors in the pose of theend-effector due to clearance are calculated by means of an approximation of the velocity analysis, while the relative position ofthe two parts of the joints are determined by means of a dynamic analysis. It is possible to calculate the dynamics with theapplication of external loads to every link. Adding the error pose to the nominal pose, the actual position has been obtained.

The error calculation analysis has been applied to a pick-and-place trajectory of the 5R planar parallel manipulator.Discontinuities appear on the actual trajectory due to the loss of contact between the pin and the hub of a joint with clearance.Analysing the trajectory at different cycle periods, it is possible to see how inertial loads affect the location of the discontinuities. Itis also shown that for slow cycles, inertial effects do not have an influence and kinetostatic analysis is applicable.

A way to find out the location of discontinuities has been presented, which is valid for applications where dynamics do not playan important role. In these cases, the error analysis can be widened to the entire workspace for the given mechanism. Now, thelocation of discontinuities depends on the mechanism configuration, the imperfect joints and applied external loads. With thisdata, a computationally efficient analytical procedure has been developed in order to locate the discontinuities, which determinesthe discontinuity loci depending on the applied external loads. Discontinuity loci divide the workspace into discontinuity-freezones, within which errors can be corrected by calibration.

Acknowledgment

The authors wish to acknowledge the financial support received from the Spanish Government via the Ministerio de Ciencia eInnovación (Project DPI2008-00159), the ERDF of the European Union and the University of the Basque Country (Project GIC07/78).

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Oscar Altuzarra received his M. Sc. Mechanical Engineering degree from the Faculty of Engineering of Bilbao, Universidad del PaísVasco (UPV/EHU), Spain, in 1995 and the Ph.D. degree inMechanical Engineering from the same University in 1999. He also attended acourse in Mechanical Engineering at Coventry University, Coventry, UK in 1992. Since 1996, he has been a lecturer and then anAssociate Professor in the Department of Mechanical Engineering at the Faculty of Engineering of Bilbao, Universidad del País Vasco(UPV/EHU). His research interests are kinematics and singularities in parallel kinematic machines, robotics, and computationalapplications to complex mechanical problems.

Jokin Aginaga received the M.E. degree in mechanical engineering, in 2004, from the Public University of Navarra, Navarra, Spain. Heis currently in the Ph.D. program of Mechanical Engineering in the Public University of Navarra. Since 2006 he has been an AssistantProfessor in the Public University of Navarra, in the Department of Mechanical Engineering. His research interests include kinematics,dynamics and accuracy analysis of parallel manipulators.

Alfonso Hernández received a degree in Mechanical Engineering from the Faculty of Engineering of Bilbao, University of the BasqueCountry, Spain, in 1985, and a Ph.D. in Mechanical Engineering from the same University in 1988. He has developed lecture andresearch activities at the University of the Basque Country, where he has been a Professor of Mechanical Engineering since 1995. Hehas had educational materials published in the fields of Machine Theory, Machine Design and Theory of Vibrations. In addition, forlecture (and research) purposes, he has designed various computer programmes for the analysis and design of mechanisms. He hashad numerous papers published in refereed journals in the fields of Reliability of the Finite Element Method, Non-linear Structural

Problems, and Mechanisms. Currently he is working on Advanced Mechanical Design and Analysis, and Parallel Robots for industrialproduction. He is also a member of several technical and scientific associations and committees. Further details can be found at theresearch group webpage http://www.ehu.es/compmech/.

Isidro Zabalza received the B.E. degree in Mechanical Engineering in 1968 and the M.E. degree in Electrical Engineering en 1985 in thePolytechnic University of Catalunya and the PhD degree in Mechanical Engineering in 1999 in the Public University of Navarra. From1968 to 1990 worked in mechanical enterprises and since 1990 has been a lecturer and then an Associate Professor in the Departmentof Mechanical Engineering in the Public University of Navarra. His research activities include the kinematic and dynamic analysis ofparallel manipulators.