Multibody Syst DynDOI 10.1007/s11044-014-9408-9

The decoupling of the Cartesian stiffness matrixin the design of microaccelerometers

Ting Zou · Jorge Angeles

Received: 5 January 2013 / Accepted: 8 January 2014© Springer Science+Business Media Dordrecht 2014

Abstract The 6 × 6 Cartesian stiffness matrix obtained through finite element analysis forcompliant mechanical structures may lead to spurious coupling that stems from discretiza-tion error. The coupling may lead, in turn, to inaccurate results of the translational androtational displacement analysis of the structure, for which reason a reliable decouplingtechnique becomes essential. In this paper, the authors resort to a decoupling technique ofthe Cartesian stiffness matrix, reported elsewhere, which is applied to the stiffness matrix ofa class of accelerometers. In doing this, the generalized eigenvalue problem is first recalledas a powerful tool that is pertinent to the design task at hand (Ding and Selig in Int. J. Mech.Sci. 46(5): 703–727, 2004). The decoupled submatrices are then investigated by means ofeigenvalue analysis. As a consequence, the translational and rotational stiffness matrices canbe analyzed independently. Meanwhile, the decoupled stiffness matrices reveal compliancealong the sensitive axes and high off-axis stiffness, thereby satisfying the ultimate designobjectives for microaccelerometers with isotropic, monolithic structure.

Keywords Cartesian stiffness matrix · Decoupling · Accelerometer · Finite elementanalysis · Screw theory

1 Introduction

A growing type of accelerometers is currently designed as compliant mechanisms, namely,flexure-based monolithic structures. Flexure hinges are commonly employed in this context,rather than lower kinematic pairs, to produce a desired motion [8]. According to Lobon-tiu [15], a compliant mechanism comprises at least one component that is highly deformable(compliant) as compared to other links. In comparison with multi-articulated rigid-body

T. Zou (B) · J. AngelesCentre for Intelligent Machines & Department of Mechanical Engineering, McGill University,Montreal H3A 0C3, Canadae-mail: ting@cim.mcgill.ca

J. Angelese-mail: angeles@cim.mcgill.ca

T. Zou, J. Angeles

mechanisms, compliant mechanisms offer some inherent advantages, such as low-cost, com-pactness, backlash-free and no need of assembly. However, due to the complex and nonlineardeformation of compliant components that are designed to offer compliant motions, the cou-pling of translational and rotational stiffness components appears in the Cartesian stiffnessmatrix [27].

The 6×6 Cartesian stiffness matrix of a class of multibody systems contains the 3×3 ro-tational, translational, and coupled stiffness blocks of the system [16]. For compliant mech-anism design, the Cartesian stiffness matrix can be obtained in terms of finite element anal-ysis (FEA) [30]. A means to predict the stiffness of the mounting of a rigid body on anelastic suspension relies on the entries of the stiffness matrix [12]. Another means relies onthe eigenvalues and eigenvectors of the same matrix, which convey more information thanthe individual entries, as the former are frame-invariant [6]. However, spurious coupling oftranslational and rotational stiffnesses may occur, as first pointed out by the authors [29].The coupling prevents an independent and straightforward analysis of the translational androtational stiffness of the system. Moreover, the Cartesian stiffness matrix has entries withdifferent physical units, thereby calling for a generalized eigenvalue analysis [5, 24]. Thedecoupling of the Cartesian stiffness matrix was discussed by Angeles [2]. In some typicalapplications, such as accelerometer design, the structure is desired to exhibit compliancealong the sensitive axes and high stiffness along the remaining axes of the rigid-body mo-tion space. Due to this feature, screw theory can be employed to help investigate the motionsof compliant mechanisms [2].

Two fundamental concepts arise in screw theory: twist—velocity and angular velocity—of a rigid body about an instantaneous axis in space and wrench, the force and momentacting on that body. At the early stage within its development history, screw theory wasused to address rigid-body motion problems. Ball [3] established screw theory to studythe motion of rigid bodies, while Hunt [10] developed a mathematical representation forscrew theory. Phillips [17] studied further the application of screw analysis to mechanismsynthesis. Recently, screw theory has become the subject of intensive research in connectionwith the design of compliant mechanisms [9, 28]. Lipkin [14] applied screw theory to thecompliant analysis of robotic manipulators; Kim [11] investigated the characterization ofcompliant blocks on the basis of screw theory for compliant mechanism design.

The design objectives of both simplicial1 uniaxial accelerometers (SUA) and their biaxialcounterparts (SBA), discussed in this paper, call for an extremely high torsional stiffness, asthe accelerometers are expected to undergo pure translations. Based on their structures, theCartesian stiffness matrix is obtained by means of FEA. Afterwards, the decoupling is per-formed in terms of a similarity transformation defined over the space of “small-amplitude”screw displacements. Decoupling allows for the independent characterization of the rota-tional and translational stiffness properties of the accelerometers under design, which alsobrings about a much more precise dynamic analysis.

2 The stiffness matrix of the structure of monolithic accelerometers

Illustrated in Fig. 1 is a Π -joint—a parallelogram linkage—that is proposed here to real-ize the compliant motions of the proof-mass in accelerometer design. In this realization

1The qualifier, derived from the mathematical-programming concept of simplex, stems from the number oflimbs, n + 1, assigned to couple the proof-mass with the accelerometer frame, where n is the number ofsensitive axes.

Decoupling of Cartesian stiffness matrix

Fig. 1 Compliant realization ofthe Π -joint

the flexure hinges are designed to provide the compliance along the sensitive directions.In structures with flexure hinges, the motion in one direction is expected to have a muchlarger amplitude than in the other directions. Under applied loading, the compliant mecha-nisms can undergo six motions: three translations and three rotations. Meanwhile, couplingbetween translations and rotations may be observed [7, 13]. Assuming that K denotes theaccelerometer 6 × 6 stiffness matrix, the matrix admits a natural partitioning into four 3 × 3blocks:

K =[

Krr Krt

KTrt Kt t

], (1)

where Krr is the rotational stiffness submatrix with torsional stiffness in N m, Kt t is thetranslational stiffness submatrix with units of N/m, and Krt is the coupling stiffness subma-trix with units of N. The first two blocks are symmetric and at least positive-semidefinite.

2.1 Stiffness along the sensitive axis

Accelerometers can be uni-, bi-, and triaxial, depending on the number of their sensitiveaxes. The main objective of accelerometer design is to provide high compliance along thesensitive axes and high stiffness along all other directions [20]. With the aid of screw the-ory, we analyze the stiffness matrix via the associated generalized eigenvalue problem. Thebackground on eigenscrew theory is included in Appendix B. Assuming that a unit screwki , i = 1, . . . ,6, is the ith eigenvector of K and κi is the corresponding eigenvalue, theeigenvalue problem is formulated as

Kki = κiΓ ki , (2)

where κi can be proven to have units of force (N) from the expansion of Eq. (2), while Γ

denotes the 6 × 6 permutation matrix defined as

Γ =[

O3×3 13×3

13×3 O3×3

], Γ = Γ −1, (3)

in which O3×3 denotes the 3 × 3 zero matrix, and 13×3 is the 3 × 3 identity matrix, Γ thusconverting the screw axis into radial coordinates [10]. Henceforth κi is referred to as the itheigenforce. In screw theory, the reciprocal product of two screws is defined as the power Π

developed by a wrench w acting on a rigid body that moves with a twist t, namely,

Π = tT Γ w. (4)

T. Zou, J. Angeles

Screws constitute vector spaces that do not admit an inner product [23].A twist is said to be reciprocal to a wrench when their reciprocal product is zero, i.e.,

when Π = 0. The idea of reciprocal product will be employed for obtaining the eigenscrewsin Appendix C.

The physical significance of screws on accelerometer stiffness is worth investigating. Inthe case of uniaxial accelerometers, for example, suppose that the accelerometer is mountedon a rigid body that undergoes translational and rotational motions simultaneously. Accord-ing to screw theory, we can assume that the accelerometer is subjected to a small-amplitudedisplacement—one involving a small angle of rotation—determined by a screw s. Then,the screw will induce a wrench. In general, the screw axis does not coincide with thewrench axis. For this to happen, an elastically isotropic structure is needed. In multiaxial-accelerometer design, the structure need not be fully isotropic, but only isotropic in thesubspace of motions associated with the sensitive axes. In the subspace associated with theother axes, isotropy need not be imposed, as long as the stiffness along these axes is or-ders of magnitude higher than otherwise.2 This criterion is known in the literature as theforce-compliant feature-based condition for compliant mechanisms [22].

3 Decoupling of the stiffness matrix

3.1 Benefits to accelerometer design

Despite the intensive research work on screw formulations of compliant mechanisms, ascrew-based stiffness analysis is seldom found. Accelerometer design requires meeting theforce-compliant or torque-compliant criterion under arbitrary loading conditions. Neverthe-less, coupling of translations and rotations always occurs, which reduces the precision andsensitivity of accelerometers. Another criterion in evaluating accelerometer design lies in itsfrequency-ratio readouts. The frequency ratio denotes the ratios of the frequencies of dis-placements along the nonsensitive directions of the full six-dimensional displacement spaceover those along the sensitive directions. The higher the frequency ratio, the more sensitivethe accelerometer is to accelerations in the direction(s) of interest—sensitive direction(s).Therefore, an analysis of the translational and rotational stiffnesses is significant in the de-sign of accelerometers.

Because of the approximative nature of the finite element model of accelerometers, spu-rious coupling may occur within the numerical calculation of the Cartesian stiffness matrix.Moreover, the different units of the entries of the Cartesian stiffness matrix also prevent astraightforward analysis of the translational and rotational stiffnesses separately. Throughdecoupling, the coupling blocks in the Cartesian stiffness matrix are filtered out, leavingonly the translational and rotational stiffness blocks. By doing so, the accelerometer fre-quency ratios can be directly studied, and the accelerometers optimally designed for a bettersensitivity and precision.

3.2 Coupling in Cartesian stiffness matrix of compliant mechanisms

For the purpose of demonstrating the coupling behavior of compliant mechanisms and show-casing the importance of decoupling of the Cartesian stiffness matrix, the flexure hinge inaccelerometer design is first studied. The flexure hinge based on the Lamé curve3 is intro-

2This statement is rather loose and should be taken with a grain of salt, as the 6 × 6 stiffness matrix involvesentries with units of N/m, N, and m. The issue is made clearer in terms of the natural frequencies.3The definition of a Lamé curve is provided in Appendix A.

Decoupling of Cartesian stiffness matrix

Fig. 2 CAD model ofLamé-notched flexure hinge

duced and used in the numerical examples of both SUA and SBA, as illustrated in Fig. 2.Compared to other types of flexure hinges, the Lamé-notched hinge distinguishes itself inits low stress-concentration values. In our example, the 4th-order Lamé-notched hinge isfixed at one end and subjected to three-dimensional loading at the free end. The six degreesof freedom (dof) of the free end comprise three translations, ux,uy, uz, and three rotationsθx, θy, θz, as measured in the reference coordinate frame Oxyz of Fig. 2.

The 6 × 6 compliance matrix of the Lamé-notched hinge takes the form [15]:

Ch =

⎡⎢⎢⎢⎢⎢⎢⎣

CθxMx 0 0 0 0 00 CθyMy 0 0 0 CθyFz

0 0 CθzMz 0 CθzFy 00 0 0 CuxFx 0 00 0 CuyMz 0 CuyFy 00 CuzMy 0 0 0 CuzFz

⎤⎥⎥⎥⎥⎥⎥⎦

, (5)

where CuyMz = CθzFy and CuzMy = CθyFz .The 6 × 6 stiffness matrix of the Lamé-notched hinge is obtained as the inverse of Ch,

namely,

Kh = C−1h , (6)

where Kh has the same block structure as that of Ch.Figure 3 illustrates the translational stiffness components KuxFx , the rotational compo-

nent KθxMx , and the coupled term KuyMz of the Cartesian stiffness matrix. As shown inthe figure, all translational, rotational, and coupled components in the Cartesian stiffnessmatrix behave nonlinearly and exhibit sensitivity to the design parameters, as illustrated inFig. 4. Hence, any slight change of design parameters will have an effect on the stiffness.For this reason, stiffness analysis plays an important role in the whole process of accelerom-eter design. However, the compliant nature of flexure hinges accompanies a much higherpossibility of coupling between the translational and rotational stiffness blocks, in compar-ison with conventional mechanisms. In accelerometer design, numerous flexure hinges areemployed to provide compliant motions, whence the coupling terms may not be negligible.Therefore, under such situations, the decoupling of the Cartesian stiffness matrix becomessignificant in providing a precise stiffness analysis of the whole system.

3.3 Decoupling process

Decoupling of the Cartesian stiffness matrix is possible if and only if the 3 × 3 couplingblock Krt is singular, of rank 2 or 1 [2]. Moreover, decoupling is achieved by means of asimilarity transformation that involves only a shift of the origin.

T. Zou, J. Angeles

Fig. 3 Geometry of a Lamé-notched flexure hinge

Let the stiffness matrix under study be denoted by [K]A when represented in a coordinateframe labeled A. With this matrix representation known, the representation [K]B of the samematrix is required in a second frame B, under the assumption that the orientation of the axesof the two frames are different, and so are their origins.

The matrix S that transforms the components of a unit screw s from B-coordinates intoA-coordinates is given by [18]

S =[

Q O3×3

DQ Q

], (7)

Decoupling of Cartesian stiffness matrix

Fig. 4 Some stiffnesscomponents vs. designparameters

where Q denotes the rotation matrix that orients the frame B into the orientation of A, whileD is the cross-product matrix of the displacement d that carries the origin of frame B intothe origin of A. Moreover, D is defined as [1]

D = CPM(d) = ∂(d × r)∂r

∀r ∈R3. (8)

The matrices [K]A and [K]B are displayed below:

[K]A =[

Krr Krt

KTrt Kt t

], [K]B =

[K′

rr K′rt

K′Trt K′

t t

]. (9)

The similarity transformation that relates [K]A with [K]B is known to be [21]

[K]B = Γ SΓ −1[K]AS−1, (10)

where, in light of Eq. (3),

Γ SΓ −1 = Γ SΓ =[

Q DQO Q

]. (11)

Substituting Eq. (9) into Eq. (10) yields

K′rr = Q(Krr − KrtD)QT + DQ

(KT

rt − Kt tD)QT ,

K′rt = (QKrt + DQKt t )QT ,

K′t t = QKt tQT .

(12)

Under the decoupling condition K′rt = O, with no condition imposed on Q, which can

thus be freely chosen, the simplest choice is the identity matrix 1. If Q = 1, Eqs. (12)yield,

K′rr = Krr − KrtD + D

(KT

rt − Kt tD), (13a)

DKt t = −Krt , (13b)

K′t t = Kt t , (13c)

whence D can be determined from Eq. (13b). However, D being a 3 × 3 skew-symmetricmatrix, the product DKt t is at most of rank 2, according to Sylvester’s theorem [26].

T. Zou, J. Angeles

Fig. 5 Accelerometer designflowchart, where DO = DesignObjectives

Therefore, the right-hand side of Eq. (13b) is bound to be of rank 2 or less. If Krt is offull rank, then decoupling is not possible. Under the assumption that Krt is singular, D isfound upon taking the axial vector of both sides of Eq. (13b), which yields

Md = krt , M = 1

2

[1tr(Kt t ) − Kt t

], (14)

where krt is the axial vector of Krt , which is defined, for every vector r ∈ R3, as Krtr =

krt × r. Moreover, M has units of force, and tr(Kt t ) is the trace of Kt t .If M is invertible, then d = M−1krt ; otherwise, one of two cases occur: (a) tr(Kt t ) = 0,

and (b) Kt t is a rank-one matrix. In either case, M fails to be invertible, but d can still becalculated, as explained in [2].

4 Numerical examples

In this section, the decoupling of the stiffness matrix is implemented on a uniaxial and abiaxial accelerometer. For each case, two approaches based on screw theory are employedto find the eigenscrews of the Cartesian stiffness matrix. First, following the decouplingprocess outlined in Sect. 3, the stiffness matrix is decoupled, and its eigenvalue analysis isconducted upon the decoupled stiffness submatrices. Then, for comparison purposes, thegeneralized eigenscrew problem described in Appendix C is solved for each example.

The general process of accelerometer design is illustrated in Fig. 5. Apparently, throughthe whole process, stiffness analysis plays a significant role.

4.1 Simplicial Uniaxial Accelerometer

A compliant-mechanism realization of the Simplicial Uniaxial Accelerometer (SUA) is il-lustrated in Fig. 6, in which the proof-mass, a homogeneous cube of side 46.71 mm, islocated at the center, suspended by means of two compliant limbs to the moving object

Decoupling of Cartesian stiffness matrix

Fig. 6 Eigenscrew illustration ofthe SUA architecture

whose acceleration is to be measured. The proof-mass is assumed to be rigid, each of thetwo identical limbs being composed of two parallelogram mechanisms of rigid links, withan intermediate link of a triangular-prismatic shape. All rigid links are coupled by means oflinearly elastic joints. The dimensions of the SUA, including parameter optimization of theflexure hinges, can be found in [30]. In ANSYS, the stiffness matrix of the structure assumedto be fabricated of Titanium alloy TiAl6V4, with a Young modulus E = 1.17 × 105 MPa, aPoisson ratio ν = 0.33, and a density ρ = 4453 kg/m3, is conducted in four steps:

1. Apply, successively, a unit force at the center of the proof-mass and a unit moment inthe corresponding translational and rotational directions x, y, z. The loading forces andmoments are stored as w = [Fx Fy Fz Mx My Mz]T . Six loading cases {wi}6

1 are definedwith wi having all but its ith entry equal to 0, this entry being set equal to 1 N or,correspondingly, 1 N m.

2. Conduct a finite element analysis for each loading case wi and obtain the position andthe orientation of the proof-mass, the position in question being that of the center of thecube; the respective “small” translational and angular displacements are stored in thesix-dimensional vector �si , which is the ith column of C, the compliance matrix.

3. For a general load w, �s is obtained as

�s = Cw. (15)

4. The stiffness matrix is simply the inverse of C:

K = C−1. (16)

As a result, the rotational, translational, and coupling stiffness matrices are obtained as

Krr =⎡⎣8196.18 0 0

0 32125.74 58.990 58.99 31762.12

⎤⎦ N m,

Kt t =⎡⎣795630 0 775.85

0 12086590 0775.85 0 12217446

⎤⎦ N/m,

Krt =⎡⎣24.04 0 0

0 0 00 0 0

⎤⎦ N.

(17)

T. Zou, J. Angeles

Notice that, by virtue of the geometric and structural symmetry of the accelerometer layoutof Fig. 6, the second and third diagonal entries of Krr and Kt t should be identical, while the(2,3) and (3,2) entries of the former and the (1,3) and (3,1) entries of the latter shouldvanish since the structure is elastostatically isotropic in the y–z plane. These entries fail tobe as expected because of the discretization error introduced by the FEA. As a matter offact, the nonzero Krt is also attributable to a result of discretization error, as made apparentbelow.

4.1.1 Decoupling of the stiffness matrix

Apparently, the rank of Krt is 1, and hence, K is decouplable.Matrix M of Eq. (14), in N/m, is calculated as

MSUA =⎡⎣1.22 0 0

0 0.65 00 0 0.64

⎤⎦ × 107. (18)

It is apparent that MSUA is invertible, d following from Eq. (14) by simple inversion ofMSUA:

dSUA =⎡⎣ 0

0.960

⎤⎦ × 10−9 m, (19)

which represents a displacement of magnitude in the nanometric scale, i.e., six orders ofmagnitude below the dimensions of the accelerometer.

Following the relation D = CPM(d), we obtain:

DSUA =⎡⎣ 0 0 0.96

0 0 0−0.96 0 0

⎤⎦ × 10−9 m. (20)

Then, with reference to Eqs. (13a), (13b), (13c), K′rr and K′

t t are calculated as

K′rr =

⎡⎣8196.18 0 0

0 32125.74 58.990 58.99 31762.12

⎤⎦ N m,

K′t t =

⎡⎣795630 0 775.85

0 12086590 0775.85 0 12217446

⎤⎦ N/m.

(21)

Apparently, the numerical values of these matrices are identical to those of their counterpartsbefore decoupling, a result of the extremely small value of dSUA.

Since K is decoupled, the translational and rotational stiffness submatrices can be ana-lyzed by means of a simple eigenvalue problem. Based on this, the corresponding eigenval-ues and eigenvectors of K′

rr , stored in arrays frr and Λrr , respectively, are displayed below:

frr =⎡⎣0.82

3.173.21

⎤⎦ × 104 N m, Λrr =

⎡⎣1.00 0 0

0 0.16 −0.990 −0.99 −0.16

⎤⎦ (22)

Decoupling of Cartesian stiffness matrix

those of K′t t , stored likewise, being

ft t =⎡⎣0.08

1.211.22

⎤⎦ × 107 N/m, Λt t =

⎡⎣1.00 0 0

0 1.00 00 0 1.00

⎤⎦ . (23)

It is noteworthy that the second and third eigenvalue entries of ft t , which are the eigen-values in the y and z directions, are almost identical. Moreover, they are more than 15 timeslarger than the first eigenvalue, which means that the structure is stiff in the y and z di-rections and compliant in the x direction. In addition, notice that the eigenvector array Λt t

is diagonal, which indicates that the coordinate axes of the frame included in Fig. 6 areprincipal directions of translational stiffness.

4.1.2 Solution of the generalized eigenvalue problem

In order to solve the generalized eigenvalue problem, the procedure of Appendix C is appliedto the stiffness matrix K, whose partitioning is given in Eq. (17).

With reference to the generalized eigenvalue problem of Eq. (42), as solved with Maple,the six eigenvalues stored in λ are displayed below with three significant digits:

λ = [0.810 −0.810 6.22 6.24 −6.24 −6.22

]T × 105, (24)

the six eigenvectors stored in Λ being

Λ =

⎡⎢⎢⎢⎢⎢⎢⎣

0.99 −0.99 0 0 0 00 0 −0.64 0.76 0.76 −0.640 0 0.76 0.65 0.65 0.76

0.10 0.10 0 0 0 00 0 −0.03 0.04 −0.04 0.030 0 0.04 0.03 −0.03 −0.04

⎤⎥⎥⎥⎥⎥⎥⎦

. (25)

The six eigenforces are obtained from the above data by means of the procedure of Ap-pendix C:

κ1 = −κ2 = 8.04 × 104 N,

κ3 = −κ6 = 6.22 × 105 N,

κ4 = −κ5 = 6.23 × 105 N,

(26)

the corresponding eigenpitches being

p1 = −p2 = 0.10 m/rad,

p3 = −p6 = 0.05 m/rad,

p4 = −p5 = 0.05 m/rad.

(27)

T. Zou, J. Angeles

The foregoing eigenscrews are now stored in the matrix S:

S =

⎡⎢⎢⎢⎢⎢⎢⎣

1.00 −1.00 0 0 0 00 0 −0.64 0.76 0.76 −0.640 0 0.76 0.65 0.65 0.76

0.10 0.10 0 0 0 00 0 −0.03 0.04 −0.04 0.030 0 0.04 0.03 −0.03 −0.04

⎤⎥⎥⎥⎥⎥⎥⎦

, (28)

whose elements are dimensionless.Upon application of the procedure of Appendix C, all screw axes are found to pass

through the center of mass O of the proof-mass. As displayed in Eq. (28), the four screw axes3 to 6 lie in the yz plane, and the other two in the x axis. A drawing of the six eigenscrewsis illustrated in Fig. 6, where s1, s2, s3, s4, s5, and s6 correspond to the respective column ofS in Eq. (28). It is shown in Fig. 6 that s1, s3, s4 point in the positive directions of the cor-responding axes, while s2, s5, s6 in the negative directions. The lines representing the axesof the eigenscrews in Fig. 6 carry two sets of diagonal dashes in opposite directions. Thesedashes indicate that each axis is associated with two pitches of opposite hands. Since theSUA features a symmetric stiffness layout, the eigenscrews are in good agreement with theresults obtained from the decoupling approach. If we look at the eigenpitches in Eq. (27),it goes without saying that there is neither pure translational nor pure rotational stiffnesssince the eigenpitches are neither infinite nor zero. However, the finite and nonzero eigen-pitches denote spurious screw motions, which tallies with the results displayed in Eqs. (22)and (23). It is also noteworthy that the eigenpitches p1 and p2 are larger, in absolute value,than the other four, which means that the dominant motion is along the x-axis, as per thedesign objectives.

It is noteworthy that the rotational and translational eigenstiffnesses and their correspond-ing eigenvectors could be obtained from two simple eigenvalue problems, as applied tomatrices K′

rr and K′t t of Eqs. (21). However, this was possible only after decoupling the

6 × 6 stiffness matrix. The generalized eigenvalue problem obviates the need of decoupling,which is not always possible. Indeed, Selig and Ding [24] found that decoupling—block-diagonalizable in the cited paper—is possible if and only if the 6 × 6 stiffness matrix hasan inversion symmetry but admitted that this characterization is not “very easy to use com-putationally.” Angeles [2] proved, in turn, that decoupling is possible if and only if the off-diagonal block is singular. The foregoing discussion should shed light on the relevance of thegeneralized eigenvalue problem in the design of accelerometers using MEMS technology.

4.2 Simplicial Biaxial Accelerometer

The layout of the Simplicial Biaxial Accelerometer (SBA) is illustrated in Fig. 7. In theSBA, the center triangular proof-mass plate is attached by three flexible legs to the ac-celerometer frame. The accelerometer is designed for fabrication with MEMS using sil-icon, with a Young modulus E = 1.618 × 105 MPa, Poisson ratio ν = 0.222 and den-sity ρ = 2.33 × 10−15 kg/µm3. The equilateral triangular proof-mass has a side length of3333.3 µm and the regular hexagon a side length of 3466.6 µm. Moreover, the thickness w

of the device is 300 µm.This structure is designed to allow for a two-degree-of-freedom pure translation of the

triangular proof-mass in the plane of the figure. At the same time, a complex deformation isdetected on the flexure hinges connecting the proof-mass to the frame. However, spurious

Decoupling of Cartesian stiffness matrix

Fig. 7 Eigenscrew illustrationfor SBA

motions of the proof-mass in the other four directions also occur due to the flexibility ofthe hinges. In addition, the stiffnesses associated with these motions are much higher thanthose of the translational motions. The foregoing features will become apparent with theeigenvalue and eigenscrew analyses of the stiffness matrix.

A similar procedure to that described for the SUA is employed here to obtain the stiffnessmatrix of the SBA structure under ANSYS. As a result, the blocks of the stiffness matrix areobtained as displayed below:

Krr =⎡⎣1.20 0.06 0

0.06 1.21 00 0 2.77

⎤⎦ × 10−2 N mm,

Kt t =⎡⎣5.58 0 0

0 5.58 00 0 25.07

⎤⎦ N/mm,

Krt =⎡⎣ 0 0 0.05

0 0 4.060.44 0.01 0

⎤⎦ × 10−4 N.

(29)

4.2.1 Decoupling of the stiffness matrix

For the SBA model, it is not difficult to prove that Krt is of rank 2, decoupling thus beingpossible, with matrix M of Eq. (14) obtained as

MSBA =⎡⎣15.32 0 0

0 15.32 00 0 5.58

⎤⎦ N/mm, (30)

T. Zou, J. Angeles

which is diagonal, with the first two diagonal entries coinciding to the first four digits. Sim-ilar to the SUA case, M is invertible, whence d and D are obtained as

dSBA =⎡⎣0.13

0.010

⎤⎦ × 10−4 mm,

DSBA =⎡⎣ 0 0 0.01

0 0 −0.13−0.01 0.13 0

⎤⎦ × 10−4 mm,

(31)

whose entries are five orders of magnitude smaller than the dimensions of the device.Further, K′

rr and K′t t are calculated in turn as

K′rr =

⎡⎣1.20 0.06 0

0.06 1.21 00 0 2.77

⎤⎦ × 10−2 N mm,

K′t t =

⎡⎣5.58 0 0

0 5.58 00 0 25.07

⎤⎦ N/mm,

(32)

which show no difference with their counterpart matrices of Eq. (29), up to the digits dis-played.

The corresponding eigenvalues and eigenvectors are arrayed as in the case of the SUA:

frr =⎡⎣1.14

1.262.77

⎤⎦ N mm, Λrr =

⎡⎣−0.74 0.68 0

0.68 0.74 00 0 1.00

⎤⎦ ,

ft t =⎡⎣ 5.58

5.5825.06

⎤⎦ × 103 N/mm, Λt t =

⎡⎣1.00 0 0

0 1.00 00 0 1.00

⎤⎦ .

(33)

It is thus found that the SBA exhibits a quasi-isotropic structure in the xy-plane. As out-of-plane translational stiffness is about three times higher than its in-plane counterpart, thereis still room for improvement by means of a dimensional fine-tuning of the current model.Furthermore, tilt motions occur along axes L of Fig. 7, which are more likely to happen thanthe in-plane rotation.

4.2.2 Solution of the generalized eigenvalue problem

The same procedure as in the first example is adopted to implement the generalized eigen-value analysis for K in Eq. (29). The eigenvalue and the eigenvector arrays λ and Λ, respec-tively, are displayed below:

λ = [0.83 0.27 0.25 −0.25 −0.27 −0.83

]T(34)

Decoupling of Cartesian stiffness matrix

and

Λ =

⎡⎢⎢⎢⎢⎢⎢⎣

0.61 −0.69 −0.72 −0.72 0.69 0.610.12 0.72 −0.69 −0.69 −0.72 0.120.99 −0.19 0.55 −0.55 −0.19 −0.990.12 −0.33 −0.32 0.32 −0.33 −0.120.18 0.34 −0.31 0.31 0.34 −0.180.33 −0.14 0.17 0.12 0.14 0.33

⎤⎥⎥⎥⎥⎥⎥⎦

. (35)

The eigenforces were obtained as

κ1 = −κ6 = 0.83 N,

κ2 = −κ5 = 0.26 N,

κ3 = −κ4 = 0.25 N,

(36)

the corresponding eigenpitches being

p1 = −p6 = 0.03 mm/rad,

p2 = −p5 = 0.05 mm/rad,

p3 = −p4 = 0.04 mm/rad.

(37)

The six eigenscrews are given below:

S =

⎡⎢⎢⎢⎢⎢⎢⎣

0 −0.69 −0.72 −0.72 0.69 00 0.72 −0.69 −0.69 −0.72 0

0.99 0 0 0 0 −0.990 −0.03 −0.03 0.03 −0.03 00 0.03 −0.03 0.03 0.03 0

0.03 0 0 0 0 0.03

⎤⎥⎥⎥⎥⎥⎥⎦

, (38)

whose elements are dimensionless. Notice that the three eigenscrew axes of Fig. 7 also beardashes indicating pitches of opposite hands.

All screw axes are found to pass through the center of mass of the proof-mass O . Asper the numerical results in Eq. (38), four screw axes—the second, third, fourth and fifthcolumns in S—lie in the x–y plane, the remaining two on the z axis, which is in goodagreement with the symmetric layout of the structure. However, none of the eigenpitchesdisplayed in Eq. (37) vanishes, and none is unbounded, thereby ruling out both pure rotationsand pure translations. Therefore, the finite and nonzero eigenpitches denote spurious screwmotions, which tallies with the eigenvectors listed in Eq. (33). The six eigenscrews, namely,the columns of S of Eq. (38), are displayed in the 3D layout of the SBA in Fig. 7. Likewise,s1, s2, s3 are pointing to the positive directions of the corresponding axes, while s4, s5, s6 arepointing to the negative directions.

It is worth noting that slight errors are detected from the comparison of the decoupledstiffness submatrices with their coupled counterparts. Considering that the stiffness matrixis obtained by means of FEA, discretization errors are deemed to be the source of the spu-rious results, which is reflected in the coupling submatrix. The FEA error thus needs to befiltered for design purposes. Hence, when the submatrix Krt is rank-deficient, decouplingis achievable by means of a “small” shift of the origin. Thus, the discretization error is fil-tered by means of decoupling. After that, the translational and rotational stiffness properties

T. Zou, J. Angeles

can be obtained independently via a simple eigenvalue problem, associated with two 3 × 3symmetric, positive-definite matrices.

5 Conclusions

This paper focuses on the decoupling of the Cartesian stiffness matrix and its applicationsin the design of accelerometers designed with a compliant-mechanism structure. Based oneigenscrew theory, the Cartesian stiffness matrix is decoupled, and the stiffness submatricesK′

t t ,K′rr are compared with their counterparts Kt t ,Krr before decoupling. Consequently, for

the two cases discussed, SUA and SBA, K′t t and K′

rr are almost numerically identical to Kt t

and Krr , respectively. The discretization error inherent to FEA is deemed to be the source ofthe spurious results, which is reflected in the nonzero coupling submatrix. Upon zeroing ofthis submatrix, the independent translational and rotational stiffness analyses are possible.Afterwards, the eigenvalue analysis is conducted on the decoupled stiffness submatrices. Forvalidation purposes, the eigenscrews are also computed, based on a generalized eigenvalueanalysis. The real, symmetric pairs of eigenvalues yield the corresponding eigenpitches forboth cases, while the corresponding eigenvectors help explain the motions of the proof-mass. It is apparent that the paper design objective of independent analysis of translationaland rotational displacement of accelerometers, by means of decoupling the system Cartesianstiffness matrix, is met.

Acknowledgements The support of Quebec’s Fonds de recherche sur la nature et les technologies throughgrants FQRNT PR-112531 and FQRNT PR-112531-Equip is highly acknowledged. Further support fromCanada’s NSERC in the form of a Discovery Grant to the second author is dutifully acknowledged.

Appendix A: Lamé curve

Within the framework of [19], Lamé curves are defined by the equation

∣∣∣∣ x

ax

∣∣∣∣η

+∣∣∣∣ y

ay

∣∣∣∣η

= 1, η = 1,2, . . . , (39)

where ax and ay are the side lengths of the circumscribing rectangle, as illustrated in Fig. 8.Even-order Lamé curves are analytic everywhere. For η = 2, the curve becomes a circle;

as η → ∞, the curve approaches a rectangle. An important shortcoming intrinsic in com-pliant mechanisms is that the high stress concentration at the notched hinges may lead tofatigue failure. Curvature discontinuities of the structure profile generate stress concentra-tion. We recall the concept of G2-continuity, which denotes position, tangent, and curvaturecontinuity over a given geometric curve [19]. G2-continuity is a significant criterion in as-signing the level of stress concentration that a flexure hinge can take.

Appendix B: A summary of stiffness decoupling

In screw theory, the general spatial motion of the rigid body is represented by a rotation ofthe body about a line, called the screw axis, and a concomitant translation parallel to thataxis [4, 25], as illustrated in Fig. 9(a). Furthermore, a scalar quantity p denotes the pitch,which couples the rotation with the translation.

Decoupling of Cartesian stiffness matrix

Fig. 8 Lamé curves withvariable η

A unit screw s is represented as a six-dimensional array, namely,

s =[

eμ

]=

[e

r × e + pe

], (40)

where e is the unit vector parallel to the direction of the screw axis L, while μ denotes trans-lation of the screw nut under a unit rotation of the latter. Moreover, r is the position vectorpointing from a point R on the screw axis L to the origin O . Without loss of generality, R

can be assumed to be the point of L closest to O , as depicted in Fig. 9(a). It is worth notingthat a pure rotation is characterized by a zero pitch, while an infinite pitch characterizes apure translation.

As illustrated in Fig. 9(a), a rigid body is displaced from Pose I to Pose II around thescrew axis L with a pitch p.

Based on the “small-amplitude” displacement screw assumption, the screw displace-ment s, the rigid-body twist t, and the wrench w are obtained by multiplying the unit screws in Eq. (40) by a “small” amplitude Θ , an arbitrary amplitude Ω with units of angularvelocity, and an amplitude F with units of force, respectively:

s = Θ s =[

θ

r × θ + pθ

], t = Ω s =

[ω

r × ω + pω

],

w = F s =[

fr × f + pf

].

(41)

Figure 9(b) depicts the screw elements of a twist t for a rigid body, i.e., the angularvelocity ω and the velocity vB of a point B of the rigid body, the screw nut, which coincidesinstantaneously with the origin O .

Appendix C: Procedure to obtain the eigenscrews

At the outset, it is recalled that scientific code assumes that all the entries of the matrixunder eigen-analysis bear the same units, and hence, its eigenvectors can be normalized torender them of unit norm. In the context of eigenscrew analysis, the said norm cannot be

T. Zou, J. Angeles

Fig. 9 Screw of a rigid body

defined, but the putative unit eigenvectors are still useful to extract the desired informationfrom them.

With reference to Angeles [2], based on K, the procedure to obtain the eigenscrews isoutlined below:

Let λi and λi denote the ith generalized unit eigenvector and its corresponding eigen-value, as returned from an eigenvalue solver when applied to K:

Kλi = λiΓ λi , (42)

whence,

κiΓ ki = λiΓ λi . (43)

Decoupling of Cartesian stiffness matrix

Since Γ is nonsingular, it is apparent that

κiki = λiλi . (44)

According to Eq. (40), it is convenient to expand Eq. (44) into block-form to calculateall the factors on the left-hand side, as

κi

[ei

pi × ei + piei

]= λi

[ηi

ζ i

], (45)

where ηi and ζ i are three-dimensional blocks of λi for simplicity.Expressions for ei and κi are readily obtained upon taking the norm of the upper blocks

of the foregoing equation and recalling that ‖ei‖ = 1, whence,

ei = ηi

‖ηi‖, κi = λi‖ηi‖. (46)

The ith eigenpitch pi is obtained upon equating the lower blocks of Eq. (45) and dot-multiplying the equation, thus resulting by ei , which yields

pi = λi

κi

eTi ζ i . (47)

Similarly, pi is expressed in terms of the lower blocks in Eq. (45). For convenience, theequation is rewritten in a simple form as

Eipi = −λi

κi

ζ i + piei , (48)

where4 Ei = CPM(ei ). An additional condition is imposed upon pi , by defining it to be theposition vector of the point of the ith screw axis Li of minimum magnitude:

eTi pi = 0. (49)

Then, by adjoining Eq. (49) to Eq. (48), an augmented system of four linear equations in thethree unknown components of pi is obtained:

Aipi = bi , (50)

in which

Ai ≡[

Ei

eTi

], bi ≡

[− λi

κiζ i + piei

0

]. (51)

It is worth noting that Eq. (50) is an “overdetermined” linear system, but overdeterminacy isonly formal, as the four equations are consistent, hence the quotation marks. Finally, basedon Eq. (50), the unique solution for vector pi of minimum Euclidean norm yields

pi = ηi × ζ i

‖ηi‖2, (52)

thereby computing all the eigenscrew parameters in the generalized eigenvalue problemassociated with the stiffness matrix.

4Ei is defined as the 3 × 3 skew-symmetric matrix with the property Eipi ≡ ei × pi .

T. Zou, J. Angeles

References

1. Angeles, J.: Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms,3rd edn. Springer, New York (2007). http://www.springer.com/engineering/robotics/book/978-0-387-29412-4

2. Angeles, J.: On the nature of the Cartesian stiffness matrix. Ing. Mec. 3, 163–170 (2010). Available, withits Corrigenda at http://revistasomim.net/revistas/3_5/art1.pdf

3. Ball, R.S.: Theory of Screws. Cambridge University Press, Cambridge (1998)4. Davidason, J.K., Hunt, K.H.: Robots and Screw Theory: Applications of Kinematics and Stat-

ics to Robotics. Oxford University Press, London (2004). http://www.oupcanada.com/catalog/9780198562450.html

5. Ding, X., Selig, J.M.: On the compliance of coiled springs. Int. J. Mech. Sci. 46(5), 703–727 (2004).doi:10.1016/j.ijmecsci.2004.05.009

6. Gosselin, C.: Stiffness mapping for parallel manipulators. IEEE Trans. Robot. Autom. 6(3), 377–382(1990). doi:10.1109/70.56657

7. Gosselin, C., Zhang, D.: Stiffness analysis of parallel mechanisms using a lumped model. Int. J. Robot.Autom. 17(1), 17–27 (2002)

8. Hartenberg, R.S., Denavit, J.: Kinematic Synthesis of Linkages. McGraw-Hill, New York (1964)9. Hopkins, J.B., Culpepper, M.L.: A screw theory basis for quantitative and graphical design tools that

define layout of actuators to minimize parasitic errors in parallel flexure systems. Precis. Eng. 34(4),767–776 (2010). doi:10.1016/j.precisioneng.2010.05.004

10. Hunt, K.H.: Kinematic Geometry of Mechanisms. Oxford University Press, New York (1978)11. Kim, C.: Functional characterization of compliant building blocks utilizing eigentwists and eigen-

wrenches. In: Proc. of the 32nd Mechanisms and Robotics Conference at IDETC2008, New York, NY,pp. 3–6 (2008). No. DETC2008-49267

12. Kövecses, J., Angeles, J.: The stiffness matrix in elastically articulated rigid-body systems. MultibodySyst. Dyn. 18, 169–184 (2007). doi:10.1007/s11044-007-9082-2

13. Li, Y., Xu, Q.: Design and analysis of a totally decoupled flexure-based XY parallel micromanipulator.IEEE Trans. Robot. 25(3), 645–657 (2008). doi:10.1109/TRO.2009.2014130

14. Lipkin, H., Patterson, T.: Geometric properties of modelled robot elasticity: Part II—decomposition. In:ASME Design Technical Conferences, Scottsdale, 13–16 September, pp. 186–193 (1992)

15. Lobontiu, N.: Compliant Mechanisms: Design of Flexure Hinges. CRC Press, Boca Raton (2003). http://www.crcpress.com/product/isbn/9780849313677

16. Melodie, F.M., Nur, A.F.S., Oliver, M.O.: On Cartesian stiffness matrices in rigid body dynamics:an energetic perspective. Multibody Syst. Dyn. 24, 441–472 (2010). doi:10.1007/s11044-010-9205-z

17. Phillips, J.: Freedom in Machinery. Cambridge University Press, Cambridge (1984)18. Pradeep, A.K., Yoder, P.J., Mukundan, R.: On the use of dual-matrix exponentials in robotic kinematics.

Int. J. Robot. Res. 8(5), 57–66 (1989). doi:10.1177/02783649890080050519. Risler, J.J.: Mathematical Methods for CAD. Cambridge University Press, Cambridge (1992). http://

www.cambridge.org/ca/knowledge/isbn/item1143247/?site_locale=en_CA20. Robert, H.B.: Mechatronic Systems, Sensors, and Actuators: Fundamentals and Modeling. CRC Press,

Boca Raton (2007). http://www.crcpress.com/product/isbn/978084939258021. Roberts, R.G.: On the normal form of a spatial stiffness matrix. In: Proc. 2002 IEEE International Con-

ference on Robotics & Automation, Washington, DC, 11–15 May, pp. 556–561 (2002). doi:10.1109/ROBOT.2002.1013417

22. Saxena, A., Ananthasuresh, G.K.: Topology synthesis of compliant mechanisms for nonlinearforce-deflection and curved path specifications. J. Mech. Des. 123(1), 33–42 (2004). doi:10.1115/1.1333096

23. Selig, J.: Geometric Fundamentals of Robotics. Springer, London (2005). http://www.springer.com/computer/ai/book/978-0-387-20874-9

24. Selig, J.M., Ding, X.: Diagonal spatial stiffness matrices. Int. J. Robot. Autom. 17, 100–106 (2002)25. Smith, D.K.: Constraint analysis of assembles using screw theory and tolerance sensitivities. Master of

Science thesis, Department of Mechanical Engineering, Brigham Young University, UT, USA (2001)26. Strang, G.: Linear Algebra and Its Applications, 3rd edn. Harcourt Brace Jovanovich, San Diego (1988)27. Su, H.J.: A pseudorigid-body 3R model for determining large deflection of cantilever beams subject to

tip loads. J. Mech. Robot. 1(2), 021008 (2009). doi:10.1115/1.304614828. Zhu, D., Feng, Y.: A spatial 3-dof translational compliant parallel manipulator. Appl. Mech. Mater. 34–

35, 143–147 (2010). doi:10.4028/www.scientific.net/AMM.34-35.143

Decoupling of Cartesian stiffness matrix

29. Zou, T., Angeles, J.: Decoupling of the Cartesian stiffness matrix: a case study on accelerometer design.In: Proc. ASME 2011 International Design Engineering Technical Conferences & Computers and In-formation in Engineering Conference IDETC/CIE 2011, Washington, DC, 28–31 August, 2011. PaperDETC2011-47914. doi:10.1115/DETC2011-47914

30. Zou, T., Angeles, J., Zsombor-Murry, P.: A comparative study of architectures for uniaxial accelerom-eters. In: Proc. Canadian Society for Mechanical Engineering Forum, Victoria, British Columbia, 7–9June, 2010