an algorithm to the study of elastodynamics

7/28/2019 an algorithm to the study of elastodynamics

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Alessandro Cammarata1

e-mail: [email protected]

Davide Condorelli

Rosario Sinatra

Department of Industrial Engineering,

University of Catania,

Viale A. Doria 6,

95125 Catania, Italy

An Algorithm to Studythe Elastodynamics of ParallelKinematic Machines With Lower

Kinematic PairsIn this paper, an algorithm to help designers to integrate the elastodynamics analysisalong with the inverse positioning and orienting problems of a parallel kinematicmachine (PKM) into a single package is conceived. The proposed algorithm applies con-cepts from the matrix structural analysis (MSA) and finite element analysis (FEA) todetermine the generalized stiffness matrix and the linearized elastodynamics equations ofa PKM with only lower kinematic pairs. A PKM is modeled as a system of flexible linksand rigid bodies connected by means of joints. Three cases are analyzed to consider thecombinations between flexible and rigid bodies in order to find the local stiffness matri-ces. The latter are combined to obtain the limb matrices and, then, the global stiffnessmatrix of the whole robotic system. The same nodes coming from the links discretizationare considered to include joint masses/inertias into the model. Finally, a case study isproposed to show some feasible applications and to compare results to commercial soft-ware for validation. [DOI: 10.1115/1.4007705]

1 Introduction

The need to combine accurate positioning and high payloadcapability in industrial robots has to face the issue of mechanicalsystems stiffness. Evaluating stiffness is necessary for designersto predict positioning and orienting errors due to flexibility, so asto improve quality in positioning and tracking of trajectories.

In other cases, stiffness is not a side effect or a disturbance,rather, a key to design smart devices that are rapidly spreadingacross market and industry: minicompliant and microcompliantmanipulators, lightweight haptic mechanisms, MEMS, and so on.

Researchers and companies seek to develop reliable analysismethods and optimization techniques to increase the performancesof mechanisms and robots under the elastodynamical point ofview. A first classification existing in the literature includes threedifferent approaches: the FEA, the MSA, and the virtual jointmethod (VJM). FEA is the most-used strategy to analyze thestructural behavior of a mechanical system. This is mainly due tothe reliability and precision of the method and to the feasibility tomodel each part of a robot with great detail [1]. As instance, inRefs. [2] and [3], FEA models were used to evaluate the stiffnessof SCARA-type parallel robots, while in Ref. [4] it is employed todevelop the dynamic model of a six-degree-of-freedom (DOF)parallel structure seismic simulator. The main drawback of FEA isthat it implies very tedious and time-consuming routines as FEAmodels need to be remeshed over and over again when the robotpose is changed. For this reason, FEA models are mostly used toverify components during the design stage, while their application

to optimization techniques is unusual as it would require a verylong computational time.

The MSA is derived from FEA and includes some simplifica-tions that make this method more efficient for optimization, con-siderably reducing the computational time. While FEA operateswith small elements (QUAD, TRIA, TETRA, etc.). MSA modelsoperate with larger elements like beams, arcs, cables, or superele-ments, the latter being macro-assemblies of many micro-elements

[57]. An application of MSA is presented in Ref. [8] where atripod-based PKM has been decomposed into two substructuresassociated with the PKM and the machine frame. In Ref. [9], theauthors followed the same strategy to find the stiffness model of aStewart platform-based PKM. In Ref. [10], the stiffness matrix ofa Stewart-Gough platform has been derived without consideringbending of the links. Other approaches, based on MSA, consideredthe minimization of the potential energy of a PKM to find theglobal stiffness matrix. Al Bassit et al. [11] used the total potentialenergy, including either the strain energy or the potential gravita-tional energy, to derive the elastodynamics model of a SchonfliesMotion Generator. Deblaise et al. [12] proposed a similar method

to analytically determine the stiffness matrix of PKMs. The solu-tions of the structure displacements which satisfy the equilibriumconditions were found studying the conditions of extremum of thetotal potential energy augmented adding the kinematic constraintsby means of the Lagrange multipliers. A similar approach was alsoused in Ref. [13] to describe the stiffness analysis of a 6-RSS (Revo-lute Spherical Spherical) parallel architecture. Resuming, methodsbased on MSA are efficient and can provide analytic results for thestiffness matrix of a PKM. Furthermore, approaches based on strainenergy are suitableeven for overconstrained (hyperstatic) PKMs.

While the FEA and MSA methods are also referred to as dis-tributed stiffness modelings, meaning that the stiffness isdistributed all over the entire geometry and shape of compo-nents as a consequence of the spatial discretization, the VJM isreferred to as lumped stiffness modeling [14]. This approach

comes from the work of Gosselin [15] which expanded a rigid-body model of a PKM by adding lumped springs, i.e., virtualjoints, to take into account the elastic deformations of the actua-tors. In the approach followed by Gosselin, the stiffness matrix ofa PKM is mapped onto its workspace by means of its Jacobianmatrix. All the simplified approaches that recur to the Jacobianmatrix to plot the stiffness of the actuators of a PKM inside itsworkspace do not consider that the bending contribute is manytimes larger than the longitudinal in deformations, as alreadypointed out in Ref. [16].

More refined models, also referred to as pseudorigid models,included link flexibility coupling rigid bodies and linear/torsionalsprings located at the existing joints [17,18]. Although more accu-rate than the simplified models based only on the actuators

1Corresponding author.Contributed by the Mechanisms and Robotics Committee of ASME for

publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript receivedFebruary 10, 2011; final manuscript received September 11, 2012; published onlineOctober 19, 2012. Assoc. Editor: Vijay Kumar.

Journal of Mechanisms and Robotics FEBRUARY 2013, Vol. 5 / 011004-1CopyrightVC 2013 by ASME


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stiffness, the pseudorigid models map the stiffness of a PKM ontoa subspace defined by the mobility of its end-effector. In Refs.[19] and [20], the said problem has been solved by introducing amultidimensional lumped-parameter model to consider the linkflexibility by localized 6-DOF virtual springs. The parameter ofeach virtual spring has been obtained using FEA modeling.

Here, we propose a new approach, based on MSA, which com-bines stiffness matrices along with joint-matrices to take intoaccount connections among flexible and rigid bodies. The algo-rithm may be applied to PKMs with arbitrary number of limbs,

being the latter a kinematic chain composed of links and lower ki-nematic pairs. A generic PKM is modeled as a system of flexiblelinks, rigid bodies, i.e., the base and the moving platform (MP),connected by joints. A discretization of the flexible parts, similarto the meshing process of FEA, is first applied to break up eachlimb into a series of smaller subsystems. Three cases are identi-fied, each involving connections between rigid bodies, flexiblebeams, and flexible joints. These cases, used to obtain the localstiffness matrices expressed in terms of the nodal independentarrays, are then composed to obtain the stiffness matrix of a singlelimb. The ensuing expression of the limb stiffness matrices in termsof a global array of nodal displacements, i.e., the array including allthe independent nodal displacements of the PKM, allows the gener-alized stiffness matrix to be derived by simple summation.

The algorithm can take into account the inertial properties ofrigid, flexible bodies, and joints. The generalized inertia matrix is

obtained upon addiction of all the local mass matrices, opportunelyexpressed in terms of the global array of nodal displacements.

As case study, the algorithm is then applied to the tripteron, alower mobility fully translational PKM. Results from the elastody-namics analysis are compared to commercial software forvalidation.

Finally, conclusions summarize the results of the proposedmethod and point out its applications and extensions.

2 Description of the Algorithm

This paper is aimed to introduce the reader to a general algo-rithm to study vibrations of PKMs with lower kinematic pairs. Ageneric PKM is composed of a MP connected to a fixed base plat-form (BP) by means of, at least, two limbs. Each limb, in turn, iscomposed of a number of links connected by means of joints: rev-olute, prismatic, cylindrical, spherical, universal, and so on. Here,we consider each limb as a serial linkage. Studying elastodynam-ics in such a mechanical system is not a trivial issue as its vibra-tional behavior varies when its pose changes. Under amathematical point of view, it means that the generalized stiffnessand inertia matrices of a PKM depend on the pose that the robot isattaining.

Generally, each body is flexible. The notion of rigid body is anabstraction that becomes a good approximation if strains of thestructure are small when compared to displacements. Here, weassume that each link of a generic PKM can be split into flexible3D Euler-beams, while the MP is modeled as a rigid body. Thischoice is justified for the most part of industrial PKMs as the MPis, usually, one order of magnitude stiffer than the remaining

links. Stiffness and inertial properties of beams are distributed andcan be derived from classical FEA elements [1]. Joints can bearmass and inertia and their stiffness is modeled by means of spe-cific joints matrices. Regarding the number of limbs and theirtype, that is, the sequence of joints going from the BP to the MP,the algorithm has no limitations. The contribution of passive jointsto the elastodynamics is included by means of specific joint-matrices, while assuming the actuated joints to be locked.

Here, we introduce the steps necessary for the algorithmdefinition

(1) Solve the inverse kinematic problem (IKP) in order toknow the starting pose, i.e., the undeformed configuration,of the PKM.

(2) Discretize links in the desired number of flexible bodies.The partitioning phase is similar to the meshing process ofFEA. It is noteworthy that the cutting performed during thepartitioning process is purely virtual and needed for algo-rithmic purposes; the continuity of the real link beingpreserved.

(3) Label all bodies and joints2 from the BP to the MP. Eachbody is included between two consecutive joints, whereasprismatic P, revolute R, universal U, spherical S, and fixedF joints are considered.

(4) Introduce dependent and independent joint-arrays and

nodal-arrays for each node.After partitioning, each flexible body has two nodes at itsend-sections. Thereby, a node can couple two consecutivebodies belonging to the same link or two bodies belongingto two different links. The former case implies that thenode lies onto a fixed virtual joint, while the latter casemeans that the node is positioned onto a real joint of thePKM. Joint-matrices and arrays will be described in Sec. 3.

(5) Assemble the generalized stiffness matrix of the PKM.Once all the independent nodes have been identified, threepossible cases allow to assemble the limb stiffness matrix,as will be explained in detail in Sec. 4.

(6) Assemble the generalized inertia matrix of the PKM.Section 5 will rely on the assembling of the inertia matricesof rigid, flexible bodies, and joints.

(7) Solve the elastodynamics equations.This is the final step of the algorithm that leads to the system of

equations to perform elastostatics and elastodynamics analysis.

3 Joint-Matrices and Arrays

As pointed out in Sec. 2, after partitioning, each flexible bodyhas two nodes at its end-sections.

All joints, but the fixed, comprise two nodal-arrays; thefixed joints only one. The six-dimensional nodal-arrayu

ji u

jix u

jiy u

jizu

jiu u

jih u

jiw

T has six nodal dis-placements, three translational and three rotational, referred to thesection of the ith-body coupled by the jth-joint. By the same to-ken, u

ji1 refers to the nodal-array of the body (i 1) coupled by

the same joint j, as shown in Fig. 1.

Two nodal-arrays coupled by the same joint j are linked bymeans of the following equation, namely

uji u

ji1 H

fhf Hchc (1)

where Hf is a 6 mj joint-matrix, while m(j) is the dimensionof the joint-array hf. The terms Hc and hc are, respectively, termedcomplementary joint-matrix and joint-array because these refer tothe degrees of constraint (DOCs) of the joint. Conversely, thejoint-matrix Hf and the joint-array hf depend on DOFs of a joint.The former contains unit vectors indicating geometric axes, the

Fig. 1 Notation of the algorithm

2Hereafter, the letteri will be referred to bodies, while the letterjto joints.

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latter containing joint displacements, either linear sf, for transla-tions, or angularhf, for rotations. This approach allow us to intro-duce joints contribute to elastodynamics in a direct and simpleway. In Ref. [12], the authors recurred to Lagrangian multipliersto introduce joint constraints. This choice introduces both depend-ent and independent nodes along with the Lagrangian multipliersinto the model, thus incrementing the dimension of the final elas-todynamics equations. Besides, the kinematic relations betweennodal displacements are limited to only ideal revolute joints with-out stiffness.

According to the general Eq. (1), the joint-matrices and joint-arrays for the prismatic and revolute joints have the followingexpressions:

Prismatic joint

Hf eT1 0T

T; hf s1 (2a)

Hc e2 e3 0 0 00 0 e1 e2 e3

!; hc s2 s3 h1 h2 h3

T

(2b)

where e1 is the unit vector parallel to the direction of translationof the prismatic joint P; e2 and e3 are two not-parallel unit vectorslying onto the plane normal to e1; s and h are scalar lengths oftranslation and angles of rotation, respectively, along and aboutthe three unit vectors fe1; e2; e3g and 0 is the three-dimensionalzero vector.

Revolute joint

Hf 0T eT1 T

; hf h1 (3a)

Hc e1 e2 e3 0 0

0 0 0 e2 e3

!; hc s1 s2 s3 h2 h3

T

(3b)

where e1 is the unit vector along the axis of the revolute joint R,e2 and e3 are two not-parallel unit vectors lying onto the planenormal to e1.

Using the above definitions of prismatic and revolute joints,other classes of joints might be introduced. As already pointedout, the partitioning introduces virtual cuttings inside a link. Now,in order to respect the continuity of the real link, a fixed joint is

introduced at the common cut section of the two consecutivebodies. The following equation uji u

ji1 u

j stands where thebody index has been deleted.

The choice of the independent nodes is personal and dependson several aspects, such as the degree of accuracy sought, theavailable computational resources, and so on. The independentnodes influence the size of the system of elastodynamics equa-tions, thus, it should be carefully chosen before starting theanalysis.

4 Generalized Stiffness Matrix

Stiffness represents the capacity from a body to enhance ordecrease its strain energy. A flexible body can be modeled as a

generalized spring that is an extension to the three-dimensionalspace of the monodimensional spring. Mathematically, this notionis expressed by means of matrices whose entries bear units ofstiffness.3

Following the above description, a PKM can be regarded as asystem of generalized inertias connected through generalizedsprings.

A generalized spring is expressed by a 12 12 stiffness matrixreferred to the two end-nodes linking the body to its neighbors.

Let Ki be the 12 12 stiffness matrix of the ith-body betweenthe two consecutive joints 1 and 2, defined as

Ki K

1;1

i K

1;2

iK2

;1i K

2;2i

! (4)Each entry of the generic 6 6 block-matrix Kp;qi can be seen as aforcetorqueat the pth-joint of the ith-body when a unit dis-placementrotationis applied to the qth-joint.

The strain energy Vi of the ith-body, under the linearityassumption, is a positive-definite quadratic form ofKi in 12 varia-bles, i.e., the nodal displacements of the end-nodes arrays u1i andu2i of the said body. Defining eui u1Ti u2Ti T, we can write

Vi 1

2euTi Kieui (5)

Figure 2 explains three possibilities to connect rigid and flexible

bodies: rigidflexible, flexibleflexible, and flexiblerigid. Thecase (a) pertains the coupling between the BP of a PKM and thefirst body of the first link of a limb; the case (b) describes two flex-ible bodies belonging to two different links coupled by means of ajoint j; finally, the case (c) depicts the coupling between the lastbody of the last link of a generic limb and the MP.

4.1 Case (a) Rigid BodyFlexible Body. Here, we refer tothe case (a) of Fig. 2. Recalling Eq. (1), we write

u12 u21 H

fhf Hchc (6)

where u21 is the nodal-displacement array of the attachment pointto the rigid body (1) and depends on the independent displacementarray u11 u

1 of the center of mass of the same rigid body.

Let us assume that the latter can accomplish only small dis-placementrotationsstarting from an initial posture and let pbe the displacement of its center of mass and r be the axial vectorof the small rotation matrix R [11], then the six-dimensional dis-placement array u11 of the center of mass of the rigid body is

defined as u11 pT rT

T. The nodal-array u21 can be obtained

in terms ofu11, namely

u21 Gu11; G

1 DiO 1

!(7)

Fig. 2 Possible combinations of bodies and nodes inside a limb: (a) rigid bodyflexible body;

(b) flexibleflexible; and (c) flexible bodyrigid body

3Notice that stiffness units change according to the corresponding degrees offreedom, as instance: position coordinates, slope coordinates and so on.

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where G is a 6 6 matrix, 1 and O, respectively, are the 3 3identity-matrix and zero-matrix, and Di is the cross-product ma-trix of di [21], which is the vector pointing from the center ofmass of the rigid body to the attachment point Pl of the flexible

body. Let us define the 12-dimensional array eu2 u1T2 u2T2 Tcontaining all the displacements array of the beam (2) of Fig. 2,and let Kh be the 6 6 joints stiffness matrix. Then, the expres-sion of the strain energy Va for the case in exam is sum of twocontributes the strain energy of the beam and the strain energy ofthe joint, i.e

Va 1

2euT2 K2eu2 12 hcT Khhc (8)

The latter is function of independent nodal-arrays u22, dependentnodal-arrays u12, and dependent joint-arrays h

f and hc.The joint-arrays hf and hc depend on the elastic properties of

the system and minimize the strain energy Va. This propertyimplies that the derivatives of Va with respect to h

f and hc vanish,i.e.

dVa=dhf 0T6 ; dVa=dh

c 0T6 (9)

where 06 is the six-dimensional zero array.By means of Eq. (9), and after simplifications, we can obtain

hc C1u11 C2u22; h

f F1u11 F2u22 (10)

where, upon introducing Df and Dc, defined as

Df HfT K

1;12 H

f; Dc Kh HcT K

1;12 H

c

HcT

K1;12 HfD1f H

fT K1;12 Hc (11)

the matrices C1; C2; F1, and F2, become

C1 D1c HcT K

1;12 H

fD1f H

fT K1;12 H

cT K1;12 G (12a)

C2 D1c HcT K1

;12 H

fD1f H

fT K1;2

2 HcT K1

;22 (12b)

F

1

D1

f H

fT

K

1;1

2 G D1

f H

fT

K

1;1

2 H

c

C

1

(12c)

F2 D1f HfT K1;22 D

1f H

fT K1;12 HcC2 (12d)

in which each stiffness matrix is expressed into the inertial refer-ence frame. Then, by substituting Eq. (9) into Eq. (6), we obtain

u12 X1u11 X

2u22 (13)

X1 G HfF1 HcC1; X2 HfF2 HcC2 (14)

Let us define the 12-dimensional array eua u1T1 u2T2 T andXU

X1 X2

O 1

!; Ha

C1T

KhC1 C1

T

KhC2

C2T

KhC1 C2

T

KhC2

!(15)

The expression of the strain energy Va becomes

Va 1

2euTa Kaeua; Ka XTUK2XU Ha (16)

4.2 Case (b) Flexible BodyFlexible Body. The case (b) ofFig. 2 describes two flexible bodies linked by means of a flexiblejoint at the node j. Let us define the following arrayseu1 u1T1 u2T1 T and eu2 u1T2 u2T2 T to be used to write thetotal strain energy Vb of the two bodies:

Vb 1

2euT1 K1eu1 12euT2 K2eu2 12 hcT Khhc (17)

where Kh is the stiffness matrix of the flexible joint. In the previ-ous expression, Vb depends on independent nodal-arrays u

11; u

22;

dependent nodal-arrays u21; u12; and dependent joint-arrays h

f and

hc. Similar to Eq. (1), we can write a kinematic constraint between

u21 and u12, i.e.

u21 u12 H

fhf Hchc (18)

Then, by imposing that the derivative ofVb with respect to all the

dependent terms, h

f

; hf

and u1

2, vanish, we derive

hc C1u11 C2u12; h

f F1u11 F2u12 (19)


Df HfT K2;21 H

f; Dc Kh HcT K2;21 H

c

HcT

K2;21 H

fD1f H

fT K2;21 H

c (20)

we obtain the matrices C1; C2; F1, and F2

C1 D1c HcT K2

;21 H

fD1f H

fT K2;1

1 HcT K2

;11 (21a)

C2 D1c HcT K2;21 H

fD1f H

fT K2;21 HcT K2;21 (21b)

F1 D1f HfT K2;11 D

1f H

fT K2;21 HcC1 (21c)

F2 D1f HfT K2;21 D

1f H

fT K2;21 HcC2 (21d)

Besides, the nodal-displacement array u12 becomes

u12 G1u11 G

2u22 (22a)

G1 D1u WTK2

;11 W

TK2;2

1 HfF1 HcC1 (22b)

G2 D1u K1;22 ; Du W

TK2;2

1 W K1;12

W 16 HfF2 HcC2 (22c)

Then, by substituting Eq. (22a) into Eq. (19), the joint-arrays hf

and hc are obtained in terms of only independent nodal-arrays,

namely

hc Z1u11 Z2u22; h

f Y1u11 Y2u22 (23a)

Z1 C1 C2G1; Z2 C2G2 (23b)

Y1 F1 F2G1; Y2 F2G2 (23c)

The dependent nodal-displacement array u21 is readily derived bysubstituting Eq. (22a) and Eq. (23a) into Eq. (18), therebyobtaining

u21 X1u11 X

2u22 (24a)

X1 G1 HfY1 HcZ1 (24b)

X2

G2

Hf

Y2

Hc

Z2

(24c)

Finally, by substituting Eqs. (22a), (23a), and (24a) into Eq. (17),we derive the expression ofVb in terms of only independent joint,i.e.

Vb 1

2euTb Kbeub; Kb XTLK1XL GTUK2GU Hb (25)

where eub u1T1 u2T2 T andXL

1 O

X1 X2

!; GU

G1 G2

O 1

!(26a)



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Hb Z1

T

KhZ1 Z1

T

KhZ2

Z2T

KhZ1 Z2

T

KhZ2

!(26b)

4.3 Case (c) Flexible BodyRigid Body. From Fig. 2, wewrite the strain energy Vc in terms of the 12-dimensional arrayeu1 u1T1 u2T1 T, i.e.

Vc 1

2

euT1 K1

eu1

1

2hc

T

Khhc (27)

The kinematic constraint between the nodal-displacement arraysu21 and u

12 can be expressed in the form

u21 u12 H

fhf Hchc (28)

Following a similar procedure of the case (a), we easily obtain thefollowing expressions for the joint-arrays hf and hc:

hc C1u11 C2u22; h

f F1u11 F2u22 (29)


Df HfT K2

;21 H

f; Dc Kh HcT K2

;21 H

c

HcT

K2;21 HfD1f H

fT K2;21 Hc (30)

the matrices C1; C2; F1, and F2 have the following expressions:

C1 D1c HcT K2

;21 H

fD1f H

fT K2;1

1 HcT K2

;11 (31a)

C2 D1c HcT K2

;21 H

fD1f H

fT K2;2

1 HcT K2

;21 G (31b)

F1 D1f HfT K2;11 D

1f H

fT K2;21 HcC1 (31c)

F2 D1f HfT K2;21 G D

1f H

fT K2;21 HcC2 (31d)

Then, by substituting Eq. (29) into Eq. (28), we obtain

u21 X1u11 X

2u22 (32)

X1 HfF1 HcC1; X2 G HfF2 HcC2 (33)

Let us define the 12-dimensional array euc u1T1 u2T2 T andXL

1 OX1 X2

!; Hc

C1T

KhC1 C1

T

KhC2

C2T

KhC1 C2

T

KhC2

!(34)

The expression of the strain energy Vc becomes

Vc 1

2euTc Kceuc; Kc XTLK1XL Hc (35)

4.4 Assembling of the Global Stiffness Matrix. Once dis-cretization and independent nodes have been chosen, a PKM limbbecomes a combination of the three cases discussed above.4 In

order to better explain how to apply the algorithm to a genericlimb, let us refer to Fig. 3 in which two rigid bodies and two flexi-ble links are coupled by means of three joints. Each link, in turn,is divided into three flexible beams. The bodies, both rigid andflexible, are labeled with circled numbers from 1 to 8, while inde-pendent nodes are labeled with Arabic numbers from 1 to 6. Thestrain energy Vl can be expressed simply adding the contributes ofits flexible bodies: Vl

P7i2 Vi. Then, recurring to results

obtained for the cases (a)(c) of Secs. 4.14.3 for the couplingsrigid body 1flexible body 2, flexible body 4flexible body 5, andflexible body 7rigid body 8, the strain energy becomes

Vl 1

2euTa12Ka12eua12 12euTb45Kb45eub45

1

2euT3 K3eu3 12euT6 K6eu6 12euTc78Kc78euc78 (36)

Besides

eua12

u11

u22

" #

u1

u2

" #;

eu3

u13

u23

" #

u2

u3

" #;

eub45 u14u25

" #

u3

u4" #

; eu6 u16u26

" #

u4

u5" #

;

euc78 u17u28

" #

u5

u6

" # (37)

Let us introduce the nq-dimensional array q containing all the in-dependent nodal-displacement arrays of the kinematic chain:q u1

T

u2T

u3T

u4T

u5T

u6T

Tand let B; a 12 nq

Boolean matrix mapping each 12-dimensional array eu in terms ofq, i.e.

eu

uiuj !

Bi;jq (38)

where i and j, respectively, are the position indices of ui and ujinside q. Through the matrix B it is possible to express the strainenergy in terms ofq

Vl 1

2qTKLq (39)

KL B1; 2TKa12B1; 2 B2; 3

TK3B2; 3

B3; 4TKb45B3; 4 B4; 5TK6B4; 5

B5; 6TKc78B5; 6 (40)

The nq nq matrix KL is the generalized stiffness matrix of thelinkage.

A PKM is composed of a parallel assembly of two o more link-ages, or limbs, therefore its strain energy VPKM is the sum of thecontributes Vl of its limbs. Then, in order to find the generalizedstiffness matrix KPKM, we need to express all terms of the strainenergy VPKM with respect to a global array qPKM containing all theindependent nodal-displacement arrays of the robot. We remem-ber that KPKM depends on the discretization and on the choice ofthe independent nodes, thus, its expression and dimension is notunique but it changes according to the said features. It is importantto notice that stiffness matrices of separate kinematic chains canbe summed as they refer to the same global array q. It is wellunderstood that the summation is valid for other approaches basedon 6 6 Cartesian stiffness matrices [22]. Here, the summation isguaranteed by the use of the binary-entry matrix B that allows for

Fig. 3 Application to a kinematic chain

4If the base platform is considered fixed, the expressions of case (a) are simplifieddeleting all terms containing the nodal-displacement array of the base.



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any stiffness matrix, referred to a link or a limb as well, to beexpanded to the global set of independent displacements.

5 Generalized Inertia Matrix

In this section, we describe how to include the inertial proper-ties of links, rigid bodies, and joints into the model.

5.1 Rigid Bodies. The case of a rigid body is readilyobtained as its mass and inertia are lumped into a unique inde-pendent node located at the center of mass of the body. Therefore,let mi and Ji, respectively, be the mass and the inertia matrix of arigid body, then the generic 6 6 inertia dyad eMi assumes the fol-lowing expression:

eMi mi1 OO Ji !

(41)

5.2 Flexible Links. Flexible links are modeled by means of adistributed approach that considers the true distribution of massinside a beam. Let Mi be the 12 12 matrix associated to the ki-netic energy Ti of a beam [23], the latter being defined as a quad-ratic forms into the time-derivatives _u1i ; _u

2i of the nodal-

displacement arrays u1i and u2i

Ti 1

2

_u1i

_u2i !T M1;1i M1;2iM2;1i M2;2i ! _u

1

i_u2i

! (42)In Sec. 4, we have found how dependent nodal-displacementarrays are expressed in terms of independent ones. Similar expres-sions can be extended at velocity and acceleration level, asdescribed in Ref. [24]. As already done for stiffness matrices, it ispossible to find three matrices Ma; Mb and Mc, defined as

Ma XTUM2XU (43a)

Mb XTLM1XL G

TUM2GU (43b)

Mc XTLM1XL (43c)

referred to the time-derivatives e_ua; e_ub, and e_uc of the independentnodal-displacement arrays eua;eub, and euc.5.3 Joints With Mass/Inertia. Generally, a joint that con-

nects together two bodies is composed of two parts, respectively,attached to the relative body. Therefore, we can split the jointmass/inertia and distribute the ensuing contributes to the twobodies. Let eML and eMR, where the capital letters stand for leftand right, be the mass dyads of the two half-parts of the joint,defined as

eML mL1 OO JL ! eMR mR1 OO JR

!(44)

The kinetic energy TJ of the joint can be written in terms of the

above matrices eML and eMR and of the time-derivatives of thenodal-arrays _u21; _u12, thusTJ

1

2 _u21

T eML _u21 12 _u12T eMR _u12 (45)Then, upon recalling Eqs. (13), (22a), (24a), and (33), TJ can beexpressed in terms ofe_u u1T1 u2T2 T, i.e.

TJ 1

2e_uTMJxe_u (46)

where MJx is the 12 12 generalized inertia matrix of the jointexpressed in terms of independent nodal displacements, defined as

MJa X1

T eMRX1 GT eMLG X1T eMRX2X2

T eMRX1 X2T eMRX2" #

MJb X1

T eMLX1 X1T eMLX2X2

T eMLX1 X2T eMLX2" #

G1

T eMRG1 G1T eMRG2G2

T eMRG1 G2T eMRG2" #

MJc

X1T

eMLX1 X1

T

eMLX2

X2T eMLX1 X2T eMLX2 GT eMRG" #

(47)

Finally, following a procedure similar to that described in Sec.4.4, the inertia matrices of rigid and flexible bodies of Eqs. (41)and (43) and those of joints of Eq. (47) can be extended, by meansof Eq. (38), to become generalized inertia matrices to be added toobtain the global inertia matrix MPKM of the PKM.

6 Elastodynamics Equations System

The derivation of the linearized elastodynamics equations isstraightforward, namely

MPKMq KPKMq f (48)

Algorithm 1 Preliminary steps

Require: IKP (define the starting pose)Hf; Hc e {Joint-matrices definition}Ti Qi {Ti : 12 12 block rotation matrix}

Require: Partitioning

for all Link dolk Lk=Nk {Lk; Nk: length and number of flexible bodies of the kth-link}nr Nk 1 {nr: ind. nodes inside the kth-link}

end fornq 6

Pk nr

Define: q {nq-dimensional global displ. array}

Algorithm 2 Stiffness matrix determination

Require: Preliminary stepsfor all Flexible body do

Ki TiKilkTTi {Ki: defined in body-frame}

ifFlexible body 2 Case a) thenG dDf; Dc Kh; H

f; Hc; K2C1; C2 Hf; Hc; K2; G; Df; DcF1; F2 Hf; Hc; K2; G; DfX1; X2 Hf; Hc; G; C1; C2; F1; F2

XU X1; X2

Ha Kh; C1; C2

Ka XU; K2;HaKa B;

TKaB; else ifFlexible body 2 Case (b) thenDf; Dc Kh; H

f; Hc; K1C1; C2 Hf; Hc; K1; Df; Dc

F1; F2 Hf; Hc; K1; Df; C1; C2Du H

f; Hc; K1; K2; C1; C2; F1; F2

G1; G2 Hf; Hc; K1; K2; Du; C1; C2; F1; F2

Z1; Z2 C1; C2; G1; G2

Y1; Y2 F1; F2; G1; G2

X1; X2 Hf; Hc; G1; G2; Y1; Y2; Z1; Z2

GU G1; G2

XL X1; X2

Hb Kh; Z1; Z2

Kb K1; K2; GU; XL;HbKb B;

TKbB; else ifFlexible body 2 Case (c) then

G dDf; Dc Kh; H

f; Hc; K2C1; C2 Hf; Hc; K1; G; Df; Dc

http://-/?-http://-/?-


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where the nq-dimensional generalized nodal-wrench array f isintroduced. The elastodynamics model of Eq. (48) can be used todetermine natural frequencies and modes of a PKM, as well as tostudy the statics. We stress that the proposed model deals onlywith stationary vibrations since the elastodynamics equations arelinearized around a starting pose. Literature presents iterativestiffness models to take into account the influence of externalloads [22,20] or methods based on the influence of the nominalmotion to study nonstationary vibrations [25]. As pointed out bythe same authors in Ref. [25], the nominal motion has small effecton the eigenfrequencies of a PKM passing from nonstationary tostationary vibrations because the extra inertial force from thenominal motion is always much smaller than the stiffness force inthe system bodies as long as the bodies are made of hard material .Hence, the proposed method can be a good first approximation forPKMs conceived to experience small deformations due to externalforces or low inertia forces compared to the stiffness forces.

Below, we summarize the contributes that the algorithm canprovide

Modular approach to study the elastodynamics of PKMs. Thestiffness and inertia matrices are obtained upon assemblingelementary blocks. This modular approach is feasible tostudy PKMs with different architectures.

Analytic expressions of the global stiffness and inertia matrices.

The analytic expressions of the global stiffness and inertiamatrices are readily obtained as these matrices are directlyassembled by means of the three cases proposed inside thepaper.

Direct calculations of joint displacements and possibility toinclude joint masses/inertias. Unlike most of the approachesappearing in literature, e.g., Refs. [12] and [19], our methodenables a direct introduction of joints into the model thatlets designers to control joint displacements due to deforma-tions. New classes of joint can be easily implemented andmass and inertia can be further included.

Unique set of nodes to derive both stiffness and inertia matrices.Some authors introduce new nodes to study vibrations of aPKM, therefore, the statics and the elastodynamics analyzes rely

on two different models [16]. Here, we overcome this drawbackintroducing an unique set of nodes for both the analyzes.

Following all steps implemented in Algorithms 13, it is possi-ble to obtain the generalized stiffness and inertia matrices neces-sary to write the elastodynamics equations.

7 Case Study: Tripteron

In this section, we apply our formulation to the tripteron, pro-posed by Gosselin in Ref. [26]. The robot is a PKM with reducedmobility composed of three legs of type cylindrical revolute revo-lute (CRR), where C stands for cylindrical joint, while R for revo-lute joint, as shown in Fig. 4. Each cylindrical joint is actuatedalong its axis of translation and connects each limb of the robot tothe base frame B. The remaining revolute joints are passive. Inorder to perform the elastodynamics analysis, we lock the transla-tions of the actuated cylindrical joints. It means that, in the follow-ing analysis, the cylindrical joints turn into revolute joints. Thegeometric, inertial, and structural parameters used for the analysisare reported in Table 1. The algorithm has been used to study theeigenfrequencies and eigenmodes of the tripteron at the centralpose of its workspace. It is noteworthy that different poses may bechosen without changing the sense of the case study in exam.

In Fig. 5, we plotted the first nine eigenmodes and their associ-ated level of strain energy of the tripteron, in which all joints havea stiffness matrix Kh 105 1.

F1; F2 Hf; Hc; K1; G; DfX1; X2 Hf; Hc; G; C1; C2; F1; F2

XL X1; X2

Hc Kh; C1; C2

Kc XL; K1;HaKc B;

TKcB; else

Ki B; TKiB;

end ifend forKPKM

PKx x a; b; c; i

Algorithm 3 Mass matrix determination

Require: Preliminary steps and stiffness matrixfor all Rigid body doeMi mi; Ji; Ji QiJiQTi {Ji: in body-frame}end forfor all Flexible beam do

Ma; Mb; Mc XU; XL; GUend forfor all Joint 2 Cases (a), (b), or (c) do

MR; ML mR; mL; JR; JLMJa eMR; eML; G; X1; X2MJb

eMR;

eML; X

1; X2; G1; G2

MJc eMR; e

ML; G; X1; X2

end forMx B; TMxB; ; x a; b; c; Ja;Jb;Jc; i

MPKM P eMx

Fig. 4 CAD model of tripteron: undeformed pose

Table 1 Geometric, inertial, and structural parameters of thetripteron

Tripteron

Notation Value Description Unit

L1 0.6 Proximal link length (m)L2 0.4 Distal link length (m)r 0.015 Cross section radius (m)

E 70 109 Youngs modulus (Pa) 0.33 Poissons modulus q 2700 Material density (kg=m3)mp 1 Mass of MP (kg)

Ixx 0.05 Inertia entry of MP (kgm2)

Iyy 0.05 Inertia entry of MP (kgm2)

Izz 0.1 Inertia entry of MP (kgm2)

mR; mL 0.2 Joints mass (kg)JL; JR 10

41 Joints inertia matrix (kgm2)



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Figure 6 compares results from our formulation to a FEAmodel, obtained by a commercial software, in which each link issplit into ten 3D Euler-beams and stiffness in the joints is includedby means of DOF spring elements. Besides, CONM2 elements areused to take into account the lumped 6 6 inertia dyads of jointsand MP. The reason why we used FEA instead of a commercialsoftware specifically designed for multibody flexible dynamics isdue to the nature of our formulation. As a matter of fact, locking

the actuation joints at a predefined robots pose converts mecha-nisms into structures in which the mobility is reduced to zero. Weconsidered two partitions: A first one in which the proximal (L1)and distal (L2) links are divided into five and four beams and asecond one in which are partitioned into seven and six beams,respectively. Besides, two simulations, for each of the partitions,are performed increasing the stiffness in the joints of one order ofmagnitude. Results reveal good accuracy with a maximum rela-tive error under 9% for the first ten frequencies and under 4% forthe first three natural frequencies. In the subfigure (a), the first

partition that with a lower number of beams seems to be moreaccurate than the second partition for the first eigenfrequencies;while it is reversed for the subfigure (b). This behavior is a draw-back that can be explained through the ill-conditioning of the elas-todynamics system when its dimension grows. The stiffness in thejoints seems to worsen the conditioning of the elastodynamics sys-tem of equations when the partition becomes higher. On the con-trary, the method seems to converge quickly to FEM results evenwith few beam elements in the partition. When a new partition ischosen, it does not affect the topology of the system, that is, con-nections among rigid and flexible beams still remain the same. Re-ferring to Fig. 3, it means that the new independent nodes, comingfrom the new partition, are internal to the flexible beams, thustheir contributes of the new beams to the stiffness are similar tothose of bodies 36 and can be included by means of simple sum-mation. As we verified in Ref. [24], the algorithm can be success-

fully extended to constrained optimization techniques so as to findglobal indices, as instance based on natural frequency range,defined all over the workspace. The latter complex issue can onlybe accomplished since the algorithm requires low computationalburden for the calculation of the eigenfrequencies at a singlerobots pose.

8 Conclusions

An algorithm to study the elastodynamics of the parallel kine-matic machines was presented. The proposed formulation is basedon the matrix structural analysis and can be applied to PKMs witharbitrary number of limbs and lower kinematic pairs. Links aremodeled as flexible 3D Euler-beams, while the moving platformand the base platform are modeled as rigid bodies. Joints have

stiffness and can bear mass and inertia. An initial discretization ofthe links of a PKM splits each link into two or more flexiblebodies. Then, independent and dependent nodal-arrays, along withjoint-arrays are introduced into the model to identify three possi-ble cases of connections between flexible, rigid bodies, and jointsand to find generalized stiffness and inertia matrices. The latterare combined as elementary blocks to obtain the generalized stiff-ness and inertia matrices of the PKM necessary to write the elasto-dynamics system of equations linearized around a starting pose ofthe PKM.

In order to validate the algorithm, the case of the Tripteron waspresented and compared to a model implemented by a commercialsoftware. Results showed good accuracy in determining naturalfrequencies and modes.

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Fig. 5 The first nine modes and the associated level of strain

energy V

Fig. 6 Relative error versus mode number of the algorithmcompared to FEA commercial software results

http://dx.doi.org/10.1155/2011/489695http://dx.doi.org/10.1155/2011/489695


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[7] Przemieniecki, J. S., 1985, Theory of Matrix Structural Analysis, Dover Publi-cations, Inc, New York.

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http://dx.doi.org/10.1109/70.56657http://dx.doi.org/10.1109/70.56657http://dx.doi.org/10.1017/S0263574704000050http://dx.doi.org/10.1017/S0263574704000050http://dx.doi.org/10.1016/j.mechmachtheory.2008.05.017http://dx.doi.org/10.1016/j.mechmachtheory.2010.12.008http://dx.doi.org/10.1115/1.2748477http://dx.doi.org/10.1115/1.2748477http://dx.doi.org/10.1016/j.mechmachtheory.2010.12.008http://dx.doi.org/10.1016/j.mechmachtheory.2008.05.017http://dx.doi.org/10.1017/S0263574704000050http://dx.doi.org/10.1109/70.56657http://dx.doi.org/10.1109/70.56657

an algorithm to the study of elastodynamics

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