Proceedings of DETC’02 ASME 2002 Design Engineering Technical Conferences and

Computers and Information in Engineering Conference Montreal, Canada, September 29-October 2, 2002

DETC2002/MECH-34207

DESIGN OF LARGE-DISPLACEMENT COMPLIANT JOINTS

Yong-Mo MoonResearch Fellow

NSF ERC for Reconfig. Machining Systems University of Michigan

Brian Patrick TreaseGraduate Research Assistant

Dept. of Mechanical Engineering University of Michigan

Sridhar KotaProfessor

Dept. of Mechanical Engineering University of Michigan

ABSTRACT Flexure joints are widely used to approximate the function

of traditional mechanical joints, while offering the benefits of high precision, long life, and ease of manufacture. This paper investigates and catalogs the drawbacks of typical flexure connectors and presents several new designs for highly-effective, kinematically-behaved compliant joints. A revolute and a translational compliant joint are proposed (Figure 1), both of which offer great improvements over existing flexures in the qualities of (1) large range of motion, (2) minimal axis drift, (3) increased off-axis stiffness, and (4) reduced stress-concentrations. Analytic stiffness equations are developed for each joint and parametric computer models are used to verify their superior stiffness properties. A catalog of design charts based on the parametric models is also presented, allowing for rapid sizing of the joints for custom performance. Finally, two multi-degree-of-freedom joints are proposed as modifications to the revolute joint. These include a compliant universal joint and a compliant spherical joint, both designed to provide high degrees of compliance in the desired direction of motion and high stiffness in other directions.

(a) Revolute Joint (b) Translational Joint

Figure 1. Proposed Compliant Joints

INTRODUCTION Rigid mechanical connections, such as hinges, sliders,

universal joints, and ball-and-socket joints, allow different kinematic degrees of freedom between connected parts.

However, the clearance between mating parts of rigid joints causes backlash in mechanical assemblies. Further, there is relative motion in all of the above joints, which causes friction that leads to wear and increased clearances. A kinematic chain of such joints compounds the individual errors from backlash and wear, resulting in poor accuracy and repeatability.

The objective of this research is to provide a flexible

means of connecting parts using large-displacement compliant joints. Flexible joints (a.k.a. flexures) utilize the inherent compliance of a material rather than restrain such deformation. These joints eliminate the presence of friction, backlash, and wear. Further benefits include up to sub-micron accuracy due to their continuous monolithic construction. The monolithic construction also simplifies production, enabling low-cost fabrication.

In the last 50 years, many flexible joints have been

researched and developed, most of which are considered one of two varieties: notch-type joints (Figure 2(a, b)) (also see Table 1(b-d) and Table 2(a)) and leaf springs (Figure 2(c)) (also see Table 1(a) and Table 2(b-i)). Notch-type flexible joints (a.k.a. fillet joints) were first analyzed by Paros and Weisbord (1965) and have since become well understood by many researchers and designers. Today, notch-type joint assemblies are widely used for high-precision, small-displacement mechanisms (Penette et al., 1997). These joints have also been applied by Howell and Midha (1994) to develop the field of psuedo-rigid- body compliant mechanisms.

Leaf springs provide the most generic flexible translational

joint, composed of sets of parallel flexible beams (Figure 2(c)). In addition to high-precision motion stages, leaf spring joints are also widely used in medical instrumentation (Smith, 2000) and MEMS devices (Saggere et al., 1994).

1 Copyright © 2002 by ASME

Thank you for using our work. When referencing this research, please refer to the journal version of this paper, found in the Journal of Mechanical Design, with the same article title. Vol. 127, Number 4, July 2005. [DOI: 10.1115/1.1900149]

x

(a) planar notch joint (b) spherical notch joint (c) leaf-springs

Figure 2. Basic Flexible Joint Components The benefits gained from using conventional flexure joints

come at the cost of several disadvantages that must be taken into account when designing. To overcome these drawbacks and develop better flexures, a set of criterion must be established for benchmarking. The four most important criterion are (1) the range of motion, (2) the amount of axis drift, (3) the ratio of off-axis stiffness to axial stiffness, and (4) stress concentration effects.

Range of Motion All flexures are limited to a finite range of motion, while

their rigid counterparts rotate infinitely or translate long distances. The range of motion of a flexible joint is limited by the permissible stresses and strains in the material. When the yield stress is reached, elastic deformation becomes plastic, after which, joint behavior is uncontrollable and unpredictable. Therefore, the range of motion is determined by both the material and geometry of the joint.

Axis Drift In addition to limited range of motion, most flexure joints

also undergo imprecise motion referred to as axis drift or parasitic motion. For notch joints, the center of rotation does not remain fixed with respect to the links it connects. With translational flexures, there can be considerable deviation from the axis of straight-line motion. For example, a simple four-bar leaf spring experiences curvilinear motion.

Axis drift can be improved by adding symmetry to the

design of a joint. However, this often increases the stiffness of the joint in the desired direction of motion. Further, more space is required to accommodate any symmetric joint components.

Off-Axis Stiffness While most flexure joints deliver some degree of

compliance in the desired direction, they typically suffer from low rotational and translational stiffness in other directions. A high ratio of off-axis to axial stiffness is considered a key characteristic of an effective compliant joint.

Stress Concentration Most notch-type joints have areas of reduced cross-section

through which their primary deflection occurs. Depending on

the shape of these reduced cross-sections, the joints may be prone to high stress concentrations and hence a poor fatigue life. Refer again to Figure 2(a, b) for examples of flexures with stress concentrations.

RELATED RESEARCH As mentioned, primitive joints previously developed

typically fall into one of two categories: notch joints or leaf spring joints. These joints are often combined in assemblies and most commonly used as revolute joints, universal joints, or parallel four-bar translational joints. Most commercially-available flexible joints are such derivatives of the primitive joints, with the addition of any variety of packaging and connections to suit particular engineering needs.

Most of the existing translational joints are based on a

parallel four-bar building block. Their flexibility is derived from leaf springs (Table 1(a)) or notch joints (Table 1(b)). The compound four-bar joints in Table 1(c) and Table 1(d) deliver a larger range of straight-line motion. All four joints have acceptable off-axis stiffness, but the range of motion is very limited, even for the compound joints. The newly proposed Compliant Translational joint is shown in Table 1(e).

The comparisons offered in this paper are primarily

qualitative rather than quantitative. The purpose is to characterize the existing joints based on their inherent form without respect to scale. For example, the range of motion of many flexures can be doubled by simply doubling the size. A comparison that provides actual numbers would require some normalization (e.g. device footprint) and is an interest to be considered in future research.

2 Copyright © 2002 by ASME

Table 1. Benchmarked Flexible Translational Joints (X: poor, ∆: normal, O: good)

Ran

ge o

f M

otio

n

Axi

s Dri

ft

Stre

ss

Con

cent

ratio

n

Off

-Axi

s St

iffne

ss

Com

pact

ness

(a)

X X ∆ ∆ O

(b)

X X X ∆ O

(c)

X X X ∆ O

(d)

X O X ∆ O

(e)

O O O O O

Many revolute joints have also been created using notch

and leaf spring primitives. Table 2(a) shows a spherical joint created by a cylindrical notch cut. Leaf springs can also be used in a variety of ways to create revolute joints, as shown in Table 2(b) (Kyusojin, 1988), Table 2(c) (Bona, 1994), Table 2(d) (Smith, 2000), and Table 2(e) (Smith, 2000). 2(c) is also recognized as the well-known “free-flex” or “cross-spring” pivot, commercially available in many forms.

Figure 3. Commercial “Free-Flex” or “Cross-Spring” Joint

Two leaf spring joints that offer almost no axis drift are depicted in Table 2(f) (Weinstein, 1965) and Table 2(g) (Kyusojin, 1988). However, the triple-leaf spring (f) offers a limited range of motion and both joints are bulky and have only moderate off-axis stiffness.

Universal joints fabricated from circular leaf springs are

shown in Table 2(h) and Table 2(i) (Smith, 2000). Both of these joints also provide axial translation, which is useful in self-alignment applications. However, they both have stress concentrations which limit their ranges of motion.

Goldfarb et al. (2000) suggest a split-tube revolute joint,

shown in Table 2(j). This joint offers the off-axis stiffness of a solid circular tube while having a low torsional stiffness. While the axis drift of a split-tube is small, it is not zero. Perfect rigidity would require infinitely thin line contact between the connecting link and the tube. Further, this joint exhibits a tradeoff between range of motion and off-axis stiffness. Under large displacements, the gap separation increases and the tube warps out of circular shape, greatly reducing the off-axis stiffness.

Our proposed Compliant Revolute joint, shown in Table

2(k), maintains zero axis drift under moment-loading. Of all the flexible revolute joints, it is the only one to have a large range of motion combined with a high ratio of off-axis stiffness to stiffness in the desired direction of motion. In comparison, the rotation axis of a popular cross-spring pivot of comparable size to our joint (diagonal leaf-springs 114mm long) drifts 5.5mm while rotating through 40 degrees. (See Haringx (1949) for design tables.) Even under typical axial loading, the proposed compliant joint’s axis of rotation drifts only nanometers, as described later in this paper.

3 Copyright © 2002 by ASME

Table 2. Benchmarked Flexible Revolute Joints (X: poor, ∆: normal, O: good)

Ran

ge o

f M

otio

n

Axi

s Dri

ft

Stre

ss

Con

cent

ratio

n

Off

-Axi

s St

iffne

ss

Com

pact

ness

(a) x

X X X X O

(b)

∆ X O X ∆

(c)

∆ X O X X

(d)

X X ∆ X O

(e)

X ∆ X ∆ ∆

(f)

X O ∆ X X

(g)

O O O X X

(h)

X O X X X

(i)

X ∆ X X ∆

(j)

O ∆ O ∆ ∆

(k)

O O O O ∆

COMPLIANT TRANSLATIONAL JOINT

Conceptual Design The Compliant Translational (CT) joint, shown in Figure 5,

is proposed as a novel improvement of the leaf spring joint. The problems typical leaf spring joints encounter are poor off-axis stiffness, limited range of motion, and deviation from straight-line motion.

Leaf spring joints are simple plates that have three degrees

of freedom (DoF), depicted schematically in Figure 4. To design a translational joint, the rotational DoF, R2, should be constrained, which is accomplished by the parallelogram arrangement of the plates.

The other DoF’s (R1 and R3) normally allow rotation and

translation at the tip of a single plate. However, the parallelogram constrains the rotation, creating a curvilinear trajectory. Thus, a single parallelogram configuration does not generate straight-line motion. To eliminate the axis drift, a symmetric folded configuration of plates is used (Figure 5(a)). By using a set of four parallelograms in an over-constrained arrangement of parallel beams (leaf springs), the CT joint also ensures parallelism between the two ends of the joint.

Figure 4. One DoF Configuration for Tra

R1 R2 R3

t

The off-axis (lateral) stiffness of the planar j

the axial stiffness of the individual beams andproportional to beam cross-section area, A. Bprovides the axial (translational) stiffness, whicproportional to the area moment of inertia, IB. Iratio of the off-axis stiffness to the axial stiffincreasing the area while decreasing IB. This is in the CT joint by using groups of three parallerather than two. The formulas for the simple tbeam cases are derived in Table 3. The last rowshows that by simply increasing the number ofkeeping the total area constant, the stiffness ratiby 9/4 = 2.25.

Using distributed compliance in long beam

lumped compliance in short and narrow cross-sefor greater displacements before local joint yieldmultiple thin beams further increases the range ofload is distributed among all the beams, and th

4 Copyright © 20

ranslation

fixed ends

nslation

oint is due to is therefore eam bending

h is therefore ncreasing the ness requires accomplished l leaf springs wo and three of the table

beams while o is increased

s, rather than ctions, allows ing. Having motion. The in beams can

02 by ASME

flex farther than thick beams before the maximum bending stress is reached.

Table 3. Stiffness Ratios for Multiple Beam Flexures (both figures have the same out-of-plane beam width)

Off-Axis stiffness L

EwhL

EA=

LEwh

LEA

=

Axial Stiffness 3

3

3 3122L

EwhLEI

=⋅ 3

3

3 34123

LEwh

LEI

=⋅

Stiffness Ratio 2

31

⎟⎠⎞

⎜⎝⎛

hL

2

43

⎟⎠⎞

⎜⎝⎛

hL

When off-axis forces are applied outside of the group of

beams, the 3-beam construction offers the beefit of reduced compressive loads. With only two beams under such loading, both are in compression. For three beams, however, two are in compression while the beam farthest from the applied load is in tension. With both tension and compression present, the load in each beam is reduced.

Straight-line motion is achieved by the symmetry of the configuration. The planar CT joint has two sets of leaf springs symmetric about the longitudinal axis. The spatial configuration is two planar CT joints intersecting at 90 degrees, giving it rotational symmetry about its axis of motion.

(a) Planar ConFigure

Analytical ModeMathematical

provide initial andobtained using linbeing analyzed aare still relativelyremain in the realm

The flexibility of the CT joint comes from simple cantilever elements (leaves) of rectangular cross-section. The remaining elements (connectors) are considered rigid. Calculating the axial translational stiffness is straightforward; the structure of the planar CT joint is 2 sets of 6 parallel cantilever beams, connected in series. The resulting axial stiffness is 3 times that of a single beam. The remaining off-axis stiffnesses cannot be calculated in such a straightforward manner. Analytic expressions are calculated using the parameterized model in Figure 6.

off-axis

axis

Figure 6. Parameters used in Planar Analytic Model

The two-dimensional model is fully constrained at one end

and has a general axial force, lateral force, and moment applied at the free end. The solution of the linear system of Euler beam equations leads to the stiffness matrix:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

3323

2322

11

00

00

kkkk

kk

Table 4. Stiffness Matrix Elements for CT Joint

34

3

11 3L

wEtk = )365(

)2(3 23

232

224

222

22 LtLLLLtLEtwk+++

+=

)365(2)2)(2(323

232

224

2223

23 LtLLLLtLLLEtwk

+++++

=

1024241212241512224

222

42

2223

223

323

232

223

22222

24

33

LLLLLLLLLLtLLtLtLtLtt

Etwk

⎟⎟⎠

⎞⎜⎜⎝

⎛

+++++++++

=

interbeam spacing leaves

sbl

pace between halves

eam ength

figuration (b) Spatial Configuration 5. CT joint Conceptual Designs

ling of Compliant Translational Joint models of both joints were created to rapid design tools. Stiffness matrices were ear beam theory. While the displacements

re large compared to notch-type joints, they small when considered as cantilevers and of linear theory.

)365(4 33224 LtLLLL +++

It is noted that the k11 term in the matrix equals that already

predicted by adding the springs in series and parallel, i.e. three times that of an individual beam. For clarity, the stiffnesses of a typical CT joint are listed in Table 5. The ratio of each component with respect to the axial stiffness is included to demonstrate the effectiveness of the joint. The dimensions used are: width (w) = 10mm, t = 1mm, L4 = 35mm, L = 52.5mm, L2 = 12mm, and E = 200GPa.

connectors

5 Copyright © 2002 by ASME

Table 5. Analytic Stiffness for Sample Planar CT Joint Stiffness Value Ratio to k11

k11 140 N/mm 1 k22 8489 N/mm 61 k23 577264 N/rad 4125 mm/rad k33 43425400 N-mm/rad 310311 mm2/rad

The formulas for the planar joint can be used to

approximate the axial and lateral stiffness of the spatial joint. With respect to axial stiffness, the spatial joint is essentially two planar joints working in parallel, giving the spatial joint twice the stiffness of the planar joint:

34

3

6)(L

wEtspatialk axial =

For the off-axis lateral stiffness, the out-of-plane stiffness (kzz) of the planar joint is first estimated to be 3Etw3/L3. In the spatial joint, k22 of one of the planar joints acts in parallel with kzz of the other. Added together, the joint’s lateral stiffness is obtained:

⎟⎟⎠

⎞⎜⎜⎝

⎛+

++++

= 24

2

23

232

22

222

4 36523)(

Lw

LtLLLtL

LEtwspatialk lateral

Table 6 lists the stiffness values of a spatial joint with the same parameters used in the above example for the planar joint. The spatial joint offers an improved ratio of lateral to axial stiffness of 80.3.

Table 6. Analytic Stiffness for Sample Spatial CT Joint

Stiffness Value kaxial 280 N/mm klateral 22483 N/mm

An initial optimization study of the analytic stiffness

matrix points to some general trends. When attempting to maximize the ratio of lateral stiffness to axial stiffness, beam length and interbeam spacing (L2) tend to be maximized, while the beam cross-section and the gap between the two-halves of the joint (L3) tend to be minimized. However, these trends are limited by the assumption of the rigidity of the connecting members.

Parametric Model of Compliant Translational Joint MDI’s ADAMS software was used for motion analysis of

both the CR and CT joints. Fully parametric, three-dimensional, flexible models were created to be quickly reconfigured to any design specification. The discrete flexible links used by ADAMS can deflect under axial, bending, shear, and torsional loads. As with the analytic study, the connecting members are still considered rigid. For all of the results given in this report, the flexible members have been designated as steel with E = 207 GPa and G = 80.2 GPa.

The ADAMS model of the CT joint (Figure 7) consists of 24 flexible beams connected by 6 rigid connectors. Each flexible beam is divided into 8 flexible elements. ADAMS allows for the use of “follower forces” which cannot easily be applied in most analytic schemes or finite element analysis software. The follower forces match the orientation and displacement of the tip of the beam as it is loaded, which more accurately represents the operation of this joint under lateral loading. Note that the parameters chosen for the computer model differ slightly from those used for the analytic model.

Figure 7. CT Joint Parameters used in ADAMS Model

The parameters studied include the interbeam spacing (L2),

the length of the input/output arms (L1), and the beam dimensions (width, thickness, and length). Variations of these parameters were analyzed for their effect on axial and lateral stiffness.

Parametric Study The catalog of graphs obtained from the parametric studies

serves as a quick and effective design tool for sizing new joints. If a maximum axial stiffness or minimum lateral stiffness is specified, that value can be found on the vertical axis of the corresponding graph. For example, when attempting to meet a given axial stiffness, the given value corresponds to a horizontal plane cut through the graph. Many of these planes are already shown in the following figures (e.g. the 2000 N/mm line in Figure 11). Any point located below this plane indicates the feasible design space left to meet any other design specifications.

A designer may next wish to go to the lateral stiffness

graph and find the greatest lateral stiffness that can be achieved from the subspace determined by the previous graph. This technique requires only a few iterations and can be used to meet stiffness requirements, spatial limitations, and weight limitations. The slope of a graph at any point also gives the designer an idea of what changes may be implemented to improve future designs, and by what degree.

6 Copyright © 2002 by ASME

Compliant Translational Joint Design Charts Four studies were performed with the CT joint model.

Three of these considered parametric effects on the moment-loaded lateral stiffness (N-mm/degree) and are shown in Figures 7, 8, and 9. For the designs in Figure 8, the cross-section is held constant: width = 10mm and thickness = 2mm. It is noted here that the gap between the two halves of the joint (L3) does not effect the moment-loaded lateral stiffness of the joint.

0

1000

2000

3000

4000

5000

6000

7000

8000

0 10 20 30Space between Beams (mm)

Late

ral S

tiffn

ess

(kN

-mm

/deg

)

40

Figure 8. Lateral Stiffness of CT Joint

(thickness = 2mm, width = 10mm)

In Figure 9, the gap is constant at 30mm, the interbeam spacing is 5mm, and the beam width is 10mm. In Figure 10, the gap and interbeam spacing are held at the same values, but the beam thickness is instead fixed at 1mm.

30 40 50 60 70 80 90

12

34

5

0200400600800

1000120014001600

Late

ral S

tiffn

ess

(N

-mm

/deg

)

Beam Length (mm)

Beam Thickness

(mm)

1400-16001200-14001000-1200800-1000600-800400-600200-4000-200

Figure 9. Lateral Stiffness of CT Joint (width = 10mm)

20 30 40 50 60 70 80

515

2535

45

0100200300400500600700800900

1000

Late

ral S

tiffn

ess

(N

-mm

/deg

)

Beam Length (mm)Beam Width

(mm)

900-1000800-900700-800600-700500-600400-500300-400200-300100-2000-100

Figure 10. Lateral Stiffness of CT Joint

(thickness = 1mm) Due to the assumption of rigid connecting members, L2

and L3 have no effect on the axial stiffness of the joint. This is evident by visual inspection of the joint. Further, only one study (Figure 11) is needed to evaluate the parametric effects on axial stiffness, using beam length and moment of inertia about the axis of bending of a single beam, IB, as the variables. The analytic equations predict that the axial stiffness should only be a function of IB, which takes into account both the beam width and thickness. This direct dependence holds only for the axial stiffness; the lateral stiffness has a more complex dependence on beam width and thickness, requiring the two separate studies shown in the previous figures.

Figure 11. Axial Stiffness of CT Joint

Note in Figure 11 that the moment of inertia has a linear

effect on axial stiffness while the beam length has an inverse relationship with axial stiffness, both of which are predicted by the analytic formula kaxial = 72 EIB/L3.

7 Copyright © 2002 by ASME

COMPLIANT REVOLUTE JOINT

Conceptual Design The Compliant Revolute (CR) joint is designed to generate

pure rotational motion. The majority of the existing flexible revolute joints are notch-type joints with considerable axis drift.

As demonstrated for the CT joint, a simple plate has three

degrees of freedom, depicted again in Figure 12(a). For a CR joint, only the torsional DoF (R2) should be present; the two bending DoF’s (R1 and R3) should be constrained or reduced. Therefore, two plates are configured as in Figure 12(b) so that the bending DoF’s are constrained and the total DoF is one.

(a) Kinematic Diagram for One Plat

(b) Kinematic Diagram for Two Plates for Figure 12. Kinematic Plate Configura

Using the configuration in Figure 12(b), the

Figure 13 are constructed. The CR joints rely on connecting plates to provide very large compliancaxes and a large range of motion compared to threvolute joints. The intended axis of motion (rocase) is called the motion axis or the functional axis designed to be very stiff against forces or torqudirections (i.e. high off-axis stiffness).

We are presenting two different CR joints:

13(a)) and segmented-cross (Figure 13(b)). Cjoints have ribs to prevent bending and reduced cto provide torsional compliance. For the segmenconnecting rod acts as a rib. A minimum distance between the ends of the joint and the rib to provcompliance yet resist bending about all three orthThe range of motion can be adjusted by changingthe beams.

(a) End-Moment CR Joint (b) Center-Moment CR Joint Figure 13. Cross-Type Compliant Revolute Joints

Analytical Modeling of Compliant Revolute Joint Most of the stiffness components of the CR joint, except

for the primary rotational stiffness, can be calculated with

Motion Axis

R1 R2

Motion Axis Motion Axis

1st connecting link

2nd connecting link

R3

e

Rotation tions

CR joints in torsion of the e about their e notch-type tation in this is. The joint e in all other

cross (Figure ross-type CR ross sections ted joint, the is maintained ide torsional ogonal axes.

the length of

standard beam formurotational stiffness of (2000). A cruciform hcross-section, depictedconsidered as two cru15), thus having twice suggested by Smith. Dlinearity, the resultingdiagonal. The six diagof Figure 15, are givewidth and thickness, afor torsional stiffness value.

Table 7. Analy

Torsional Stiffness k66 (

Bending / Rotational Stiffness

k44 (Mk55 (

Bending Stiffness

k11 (k33 (

Axial Stiffness k22 (

Note: Ix = Iy = I =

Figure 14. Cross The stiffnesses for

and displayed in Tawidth = 10mm, thicknmoment arm = 55mm

widt

8

ribs

las. An empirical formula for the a cruciform hinge is described by Smith inge is a torsion bar with a cross-shaped in Figure 13(a). The CR joint is ciform hinges used in parallel (Figure the axial, bending, and torsional stiffness

ue to symmetry and the assumption of 6x6 spatial stiffness matrix is purely onal elements, based on the coordinates

n in Table 7. “w” and “t” represent the s labeled in Figure 14. The formula used is accurate to within 4% of the actual

tic CR Joint Stiffness Formulas

Mz/θz) LtG

tw

34

373.04

⎟⎠⎞

⎜⎝⎛ −

x/θx), My/θy)

2 EI/L

Fx/dx), Fz/dz)

24 EI/L3

Fy/dy) 2 AE/L

1/12*(wt3 + tw3 – t4); A = 2wt – t2

-section Parameters for CR Joint

a typically sized CR joint are calculated ble 8. The dimensions used are: ess = 1mm, beam length = 50mm,

. To illustrate the benefits of this joint,

h

Copyright © 2002 by ASME

the ratio of each off-axis stiffness with respect to the torsional stiffness is also included.

Table 8. Analytic Stiffnesses of Example CR Joint

Because translational and rotational stiffness have different

units, their ratio has units of radians/mm2, which indicates dependence on the length of the moment arm (MA). Dimensionless values are attained by multiplying these ratios by MA2. In Table 8 (MA = 55mm), multiplying the direct ratios (k11/k66 = 0.1623 rad/mm2 and k22/k66 = 7.6410 rad/mm2) by (55mm)2 gives the modified ratios of k11/k66 = 491 and k22/k66 = 23,114. With the smallest off-axis ratio at 33.8 and the others much higher, it is evident that this is a very effective flexible joint.

Computer Model of Compliant Revolute Joint The CR joint, shown in Figure 15, consists of two beams

each of 8 flexible elements, connected in the middle by a rigid arm. The desired motion of this joint is rotation about the longitudinal axis of the flexible beams (θz). Again, follower forces are applied to the tip of the momemt arm to represent actual loading of the joint. Not only were all stiffnesses of this joint considered, but also the drift of the center of rotation for various loading conditions. A rigid center of rotation, which is not found in most traditional flexible joints, is critical in emulating the kinematics of true revolute joints. The parameters studied in the CR joint include the cross-section of the beams, defined by the thickness and width (see Figure 14), the length of the beams, and the length of the moment arm.

Figure 15. Computer Model of CR Joint

The typical range of motion in one direction is shown in

Figure 16.

Stiffness Value Ratio to k66

Torsional k66 (Mz/θz) 20,589

N-mm/rad 1

Bending / Rotational

k44 (Mx/θx), k55 (My/θy)

696,210 N-mm/rad 33.8

Bending k11 (Fx/dx), k33 (Fz/dz)

3,342 N/mm 491

Axial k22 (Fy/dy) 157,320 N/mm 23,114

Figure 16. Range of Motion of a Typical CR Joint From analysis of a typical joint, (width = 10mm,

thickness = 1mm, beam length = 50mm, moment arm = 55mm), the Center of Rotation Drift (CRD) is determined to be minimal. During normal moment-loaded operation there is no CRD for any degree of rotation. If x-direction loading is present during operation, it contributes only 215nm to the CRD per newton of applied force, Fx. These values indicate negligible drift for practical applications.

Compliant Revolute Joint Design Charts Nine dual-parameter studies were done with the CR joint,

represented by 3-D surface plots of the output variables. The nine studies consisted of the following three sub-groups:

Table 9. Sub-cases for CR Parametric Study

x-deflection of the tip due to Fx (100N) (Figure 17a) y-axis rotation due to Fz (100N) (Figure 17b)

z-axis rotation due to Fy (10N) In each of these studies, the following three combinations of parameters were inspected:

Width and Length Thickness and Length

Width and the Ratio of Thickness to Width (RTW)

9 Copyright © 2002 by ASME

(a) Fx/dx (b) Fz/θy

Figure 17. Graphical Depiction of CR Joint Stiffness Of the 9 studies, several significant ones are included in

this report. As with the CT joints, these parametric studies serve as design charts to aid in creating new joints. Interested readers may contact the authors to obtain the complete set of design charts.

The first quantity considered is the desired motion of the

joint: torsional compliance when a force is applied in the y-direction. To maximize the desired compliance, the torsional stiffness, illustrated in Figure 18 and Figure 19, must be minimized. The first plot indicates that stiffness decreases nonlinearly with respect to width when the RTW is constant and vice versa.

3

91521

27

1030

5070

90

0800

160024003200400048005600640072008000

Width (mm)Percent Ratio t/d

7200-80006400-72005600-64004800-56004000-48003200-40002400-32001600-2400800-16000-800

Tors

iona

l Stif

fnes

s (k

N-d

eg/m

m)

Figure 18. z-Rotational Stiffness of CR Joint

(MZ = 100N-mm, beam length = 50mm)

Figure 19 shows the combined effects of beam length and width on the torsional stiffness. Beam width has only a linear effect on stiffness for a given beam length. However, beam length nonlinearly decreases the stiffness for a given width.

20 40 60 80

100 5

1525

3545

00.5

11.5

22.5

33.5

44.5

5

Tors

iona

l Stif

fnes

s

(kN

-deg

/mm

)

Length (mm)Width (mm)

4.5-54-4.53.5-43-3.52.5-32-2.51.5-21-1.50.5-10-0.5

Figure 19. z-Rotational Stiffness of CR Joint

(MZ = 100N-mm, thickness = 1mm)

While the first two plots suggest small widths, small thicknesses, and long beams for minimal torsional stiffness, these conflict with the requirements for maximum off-axis stiffness. This requires referring to Figure 20, which shows the rotational bending stiffness for an axially applied load, illustrated in Figure 17(b). From the plot, it is evident that maximum bending stiffness requires shorter beams with thicker flanges. This contradiction verifies the need for a design tool to balance both objectives.

20 30 40 50 60 70 80 90

100 1

59

1317

01.5

34.5

67.5

910.5

1213.5

15

Length (mm)Thickness

(mm)

13.5-1512-13.510.5-129-10.57.5-96-7.54.5-63-4.51.5-30-1.5

y-ax

is R

otat

iona

l Stif

fnes

s

(MN

-mm

/deg

)

Figure 20. y-Rotational Stiffness of CR Joint

(MZ = 10kN-mm, width = 20mm) Figure 21 shows the stiffness of the CR joint when it is

loaded as a fixed-fixed beam with a perpendicular force (i.e. x-direction) applied at its center, as in Figure 17(a). Increased width and reduced length are required to increase the x-axis stiffness. The effect of width is nearly linear for a given length, but the length has a nonlinear effect for a constant width.

10 Copyright © 2002 by ASME

20 30 40 50 60 70 80 9010

0 515

2535

450

80160240320400480560640720

x-A

xis

Stif

fnes

s (k

N/m

m)

Length (mm)Width (mm)

640-720560-640480-560400-480320-400240-320160-24080-1600-80

Figure 21. x-Stiffness of CR Joint

(Fx = 100N, thickness = 1mm)

The length of the moment arm (MA) was not included in the above studies because its effect on the joint can be easily summarized in a single chart, as seen in Table 10. These relationships are almost entirely unaffected by the other parameters.

Table 10. Effect of Moment Arm on CR Joint Stiffness x-axis stiffness (Fx/dx) independent of MA y-axis stiffness (Fy/dy) ~ proportional to MA2

z-axis stiffness (Fz/dz) ~ proportional to MA2

y-axis rotational stiffness (Fz/θy) directly proportional to MAz-axis rotational stiffness (Fy/θz) directly proportional to MA

FUTURE WORK

Other Compliant Joint Concepts The CR joint in Figure 13(b) requires a large space in the

direction of the axis of rotation (motion axis). While this may be acceptable for some applications, others may be limited by different size constraints. An alternate CR joint configuration, shown in Figure 22, allows for the tradeoff of joint footprint in the xy-plane and joint depth in the z-direction.

Figure 22. Alternate CR Joint Conceptual Design

Compliant Universal Joint To further increase the library of compliant joints for the

design of generic mechanisms, two CR joints are concatenated to create a compliant universal (CU) joint.

Figure 23. CU Joint Conceptual Design

The CU joint allows only two rotational degrees of

freedom, as does its traditional mechanical counterpart. However, a Compliant Spherical (CS) joint with 3 degrees of freedom can be built by connecting CU and CR joints as demonstrated in Figure 24.

Figure 24. CS Joint Conceptual Design

CONCLUSIONS This paper presented new types of compliant joints for

rotational and translational motions. The new compliant joint designs allow for a larger range of motion than that of the conventional flexure joints. The joints presented in this paper are modeled and analyzed with both analytical and CAE methods. For the CR joint, the smallest and largest off-axis stiffnesses are 33 and 23,000 times the joint stiffness, respectively. The CT joint has off-axis stiffness 80 times greater than its axial stiffness. Further, the overconstrained CT joint delivers exact straight-line motion and the rotation axis of the CR joint drifts only 140nm for ±90° motions. The design charts presented in this paper, based on extensive parametric analyses, aid in sizing the joints for various applications.

x

z

y

1st connecting link

2nd connecting link

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ACKNOWLEDGEMENT The authors gratefully acknowledge the financial support

of the NSF Engineering Research Center for Reconfigurable Machining Systems (NSF Grant EEC95-92125) at the University of Michigan and the valuable input from the Center's industrial partners.

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Haringx, J.A., 1949, “The Cross Spring Pivot as a Constructional Element,” Applied Scientific Research, Vol. A1, No. 5-6

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