design of a flexibly-constrained revolute pair with non

Kimura, N., Iwatsuki, N., and Ikeda, I.

Paper:

Design of a Flexibly-Constrained Revolute Pairwith Non-Linear Stiffness for Safe Robot Mechanisms

Naoto Kimura, Nobuyuki Iwatsuki, and Ikuma IkedaDepartment of Mechanical Engineering, School of Engineering, Tokyo Institute of Technology

2-12-1 Ookayama Meguro-ku, Tokyo 152-8552, JapanE-mail: {kimura.n.ac@m, nob@mep}.titech.ac.jp

[Received March 5, 2018; accepted December 4, 2018]

A revolute pair with a flexible translational constrainton a plane is proposed as simple mechanism for saferobots. The mechanism is composed of two pairing el-ements, one with a circular and one with a cam profilethat are connected by a linear spring. Flexible transla-tional constraint is generated by spring forces and thereaction force between the two pairing elements. Twomethods for designing the cam profile are proposed inorder to implement the specified non-linear stiffness inthe flexible constraint. Design examples with variousstiffness characteristics are shown. Some prototypesare fabricated, and it is confirmed that they performas designed. As an application, a flexible, underactu-ated link mechanism with the proposed pairs is syn-thesized, and its flexibility and kinematic performanceare investigated.

Keywords: link mechanism, kinematic pair, cam profiledesign, passive compliance, underactuated mechanism

1. Introduction

Robots working in human daily life, such as servicerobots, home robots, and nursing care robots, need to beflexible so that nobody gets hurt in accidental collisions.However, there is a trade-off between flexibility for safetyand rigidity for power transmission.

In order to achieve a balance between flexibility andrigidity in flexible link mechanisms, several approachesto add flexibility in the revolute pairs have been proposed.One approach is the active compliance approach with mo-tor compliance control [1, 2]. This approach achievesappropriate compliance for a desired task with a sim-ple mechanism, but it has difficulty responding to sud-den changes in external forces, such as dynamic collisionwithout sophisticated control systems. Therefore, this ap-proach tends to be expensive. In order to solve this prob-lem, flexibility should be achieved with passive springs.This approach is called the passive compliance approach.In previous works, joint mechanisms to adopt passivecompliance for the desired task by using springs and re-dundant actuators have been proposed. These mecha-nisms can adjust the rotational stiffness by using two non-

linear antagonizing springs [3–6] with two actuators, bychanging the transmission of spring forces [7–9], or bychanging the preload applied to springs [10, 11] with anadditional actuator. However, since an additional actuatoris used in each joint, the entire link mechanism tends tobe heavy. In order to achieve a balance between passiveflexibility and rigidity without redundant actuators, waysto introduce passive non-linear rotational stiffness into thejoint mechanism have been proposed. Flexible joints withnon-linear stiffness can obtain sufficient braking distancein the low stiffness area, and sufficient payloads in thehigh stiffness area. Typical non-linear stiffness character-istics are hardening and softening spring characteristics.Hardening spring characteristics are suitable for applica-tions requiring low precision and a large payload, whilesoftening spring characteristics are suitable for applica-tions requiring high precision and a small payload. Inprevious works, joint mechanisms with softening springcharacteristics have been achieved with a double-slidermechanism with a linear spring [12] or a cam and springmechanism [13]. Joints with hardening spring character-istics have been achieved with a cam and spring mech-anism [13–17] or elastic wire and non-circular pulleymechanism [18–20]. Others have had a rubber and cir-cular pulley mechanism [21] or a link mechanism withelastic links [22]. However, as these mechanisms entail alarge number of mechanical elements, they are not verysimple.

As a design for a flexible link mechanism that has asimple structure, a revolute pair with a flexible transla-tional constraint on a plane is proposed. The structure ofthis pair is like a human joint, as shown in Fig. 1. Thispair is composed of two pairing elements, each with acam profile and a circular profile, and the two elementsare connected by a linear spring. Since both end pointsof the spring allow free rotation, this pair has zero stiff-ness in the relative rotational direction. However, it hasnon-linear stiffness in the relative translational directionfrom the spring force and the reaction force between thetwo pairing elements. Therefore, these pairing elementsare easy to rotate relatively and hard to move relatively inthe translational direction. In other words, this pair can beregarded as a revolute pair that is flexibly constrained inthe translational direction.

This paper focuses on the following two topics.

156 Journal of Robotics and Mechatronics Vol.31 No.1, 2019

https://doi.org/10.20965/jrm.2019.p0156

© Fuji Technology Press Ltd. Creative Commons CC BY-ND: This is an Open Access article distributed under the terms of the Creative Commons Attribution-NoDerivatives 4.0 International License (http://creativecommons.org/licenses/by-nd/4.0/).

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Flexibly Constrained Revolute Pair with Non-Linear Stiffness

Pairingelement 2(Circularprofile)

Pairing element 1 (Cam profile)

Linearspring

Zero stiffness

Non-linearstiffness

Fig. 1. Structure of the proposed kinematic pair (flexiblyconstrained revolute pair).

• The design methodology of the proposed kinematicpair to implement the specified non-linear stiffnessbetween two pairing elements is established.

• The performance of a flexible link mechanism withthe proposed pairs is investigated in the quasi-staticcondition.

In Section 2, the design methodologies of the pair are de-scribed. In Section 3, several design examples of the camprofile are shown. In Section 4, some prototypes of theproposed pair are fabricated and experimentally examinedto confirm the correctness of the proposed design method-ology. In Section 5, a simple closed-loop link mechanismwith the proposed pairs is synthesized, and its flexibilityand kinematic performance are investigated through anal-yses and experiments.

2. Design Methodology

In this section, the design methodology of the cam pro-file of the pairing element is described in order to imple-ment the specified non-linear stiffness in the relative trans-lational direction between two pairing elements, as shownin Fig. 1. For the non-linear stiffness, one of the twoforce-displacement characteristics shown below is used inthe design process.

(a) Relationship between external force and displace-ment of the center of the circle.

(b) Relationship between external force and displace-ment of the contact point between pairing elements.

By using the characteristics of (a), designers can intu-itively understand the motion of the pair when an externalforce is applied. However, the size of the cam profile maybecome large because the size cannot be controlled in thedesign process. On the other hand, by using the character-istics of (b), designers can control the size of the cam pro-file because the cam profile is the same as the path tracedby the contact point between pairing elements. Since theadvantage and disadvantage of the characteristics (a) and

��

�

�

(Curve: � � ��

Pairing element 2(Circle with the radius of �)

Pairing element 1

O

C

Q

Fig. 2. Schematic diagram of the proposed pair used in thedesign process.

(b) are opposite, one of the two methodologies should beselected by designers according to the application. Thesections below describe the design methodology of thecam profile with each of the force-displacement charac-teristics.

2.1. Design Based on the Center of the CircleThe planar model for the design of the pair is shown in

Fig. 2. O-xy coordinates are fixed on pairing element 1(with the cam profile). The circular profile of pairing el-ement 2 is kept in contact with the cam profile at con-tact point Q: (xq,yq) by the linear spring. The end pointsof the spring are connected to the origin, O: (0,0), andthe center of the circle, C: (xc,yc), allowing free rotation.An external force fe acts on the pairing element 2 in thex-direction, and the following equation holds by principleof virtual work.

fedx = (ks+w0)ds . . . . . . . . . . . (1)

where k is the spring constant, w0 is the initial tension ands is the extension of the spring. Note that the frictionalforce between pairing elements is ignored. Integratingboth sides of Eq. (1), s(xc) can be derived as follows.

s(xc) =1k

[−w0 +

√w2

0 +2k(∫ xc

xc,0

fe(ξ )dξ +ks2

02

+w0s0

) ], (2)

where s0 is the initial extension of the spring and cal-culated by the initial position of the center of the circle(xc,0,yc,0) as the following equation.

s0 =√

x2c,0 + y2

c,0− l0 . . . . . . . . . . (3)

Therefore, the position vector of C is geometrically de-rived as follows.

pppc(xc) = [xc yc(xc)]T

=[

xc

√[s(xc)+ l0]2 − x2

c

]T

, . . . . (4)

Journal of Robotics and Mechatronics Vol.31 No.1, 2019 157


where l0 is the natural length of the spring. Note that y =yc(x) represents the trajectory of the center of the circle.The unit normal vector of this trajectory is calculated asfollows.

nnnc(xc) =1√

1+(

dyc

dx

)2

⎡⎣dyc

dx−1

⎤⎦ . . . . . . (5)

Therefore, the vector equation of the cam profile is repre-sented as follows.

pppq(xc) = pppc(xc)+ rnnnc(xc), . . . . . . . . (6)

where r is the radius of the circle. The cam profile pppq(xc)can be derived if spring parameters k, l0, w0, initial posi-tion of the circle (xc,0,yc,0), radius of the circle r, and thespecified non-linear characteristics (relationship betweenexternal force and displacement of the center of the cir-cle) fe = fd(xc) are substituted into Eqs. (2)–(6) as designparameters.

2.2. Design Based on Contact PointWhen an external force fff e = [ fe 0]T acts on the circle,

the following statics equation holds.

fff e +www+ fff n = 000 . . . . . . . . . . . . (7)

where www = [wx wy]T is the restoring force due to thespring, and fff n is the normal force between pairing ele-ments. Note that the frictional force between pairing ele-ments is ignored. Eq. (7) is projected in the tangent direc-tion of the cam profile y = g(x), and the following simpleequation holds.

wy(xq)dgdx

|x=xq = − [ fe(xq)+wx(xq)] . . . . (8)

The restoring force due to the spring is calculated as fol-lows.

www(xq) =

[wx(xq)

wy(xq)

]

= − [k (‖pppc(xq)‖− l0)+w0]pppc(xq)

‖pppc(xq)‖ . . (9)

pppc(xq) is represented by the following equation.

pppc(xq) = pppq(xq)+ rnnnq(xq)

=

[xq

g(xq)

]+

r√1+

(dgdx

|x=xq

)2

⎡⎣−dg

dx|x=xq

1

⎤⎦ ,

. . . . . . . . . . . . . . . . (10)

where nnnq is the unit normal vector of the cam profiley = g(x). The cam profile y = g(x) (yq = g(xq)) canbe obtained if spring characteristics k, l0,w0, initial po-sition of the contact point (xq,0,yq,0), radius of the cir-cle r, and the specified non-linear stiffness characteristicsfe = fd(xq) are substituted into the non-linear differential

equation represented by Eqs. (8)–(10), and then the equa-tion is solved for g(xq).

Note that the non-linear differential equation rep-resented by Eqs. (8)–(10) includes derivative functiondg/dx|x=xq implicitly, so a numerical method such as theRunge-Kutta method cannot be applied just as it is. There-fore, the Runge-Kutta method is extended in order tosolve the non-linear differential equation, Eqs. (8)–(10).

Derivative function dg/dx|x=xq is represented as s, andEqs. (8)–(10) are represented as F(xq,g(xq),s) = 0. Val-ues of the n-th step, xq = xn and gn = g(xn) are assumed asknown. In the (n + 1)-th step, the following calculationsare carried out.

(1) F(xn,gn,sn) = 0 is solved numerically for sn, and thesolution is defined as sn,1.

(2) F(xn +h/2,gn +h/2s1,sn) = 0 is solved numericallyfor sn, and the solution is defined as sn,2.

(3) F(xn +h/2,gn +h/2s2,sn) = 0 is solved numericallyfor sn, and the solution is defined as sn,3.

(4) F(xn + h,gn + hs3,sn) = 0 is solved numerically forsn, and the solution is defined as sn,4.

Then, values of the next step, xn+1 and gn+1, are calcu-lated as

xn+1 = xn +h, . . . . . . . . . . . . (11)

gn+1 = gn +h6(sn,1 +2sn,2 +2sn,3 + sn,4), . (12)

where h is the step length.

3. Design Example

In this section, several cam profiles of the proposed pairare designed to have the specified non-linear stiffness byusing each design methodology described in Section 2.

3.1. Example Based on the Center of the CircleBoth hardening and softening spring characteristics are

used for design examples as the specified non-linear stiff-ness. Hardening spring characteristics are useful forrobots working around people because precise movementis not required but a large pay load is required to carrythings. On the other hand, softening spring characteris-tics are useful for robots that assemble small parts be-cause precise movement is required but a large payloadis not required. Design parameters are listed in Table 1.The specified stiffness characteristics of pattern A in thetable are hardening spring characteristics, while those ofthe pattern B are softening spring characteristics. Notethat the values of spring constants k and the initial tensionof the linear spring w0 are multiplied by 2 because sametwo springs are used in parallel. The specified stiffnesscharacteristics are shown in Fig. 3. Fig. 3(a) depicts thehardening spring characteristics of pattern A in Table 1,while Fig. 3(b) depicts the softening spring characteristicsof pattern B in the same table.



Table 1. Design parameters for examples with externalforce-displacement of the center of the circle relation.

Pattern A Bfd(xc) [N] 0.8(e0.05xc −1) 1.5ln(xc +1)

(Range [mm]) (0 � xc � 50) (0 � xc � 50)k [N/mm] 0.0494×2l0 [mm] 37.5w0 [N] 0.440×2r [mm] 15

(xc,0,yc,0) [mm] (0,45)

0 10 20 30 40 50 xc [mm]

0

2

4

6

8

10

f d[N

]

(a) Pattern A

0 10 20 30 40 50 xc [mm]

0

2

4

6

8

10

f d[N

]

(b) Pattern B

Fig. 3. Relationships between external force and displace-ment of the center of the circle for design examples.

0 20 40 61.85 x [mm]

30

40

50

60

70

80

y [m

m]

(a) Pattern A

0 20 40 55.63 x [mm]

30

40

50

60

70

80

y [m

m]

(b) Pattern B

Fig. 4. The calculated cam profiles of examples with re-lationships between external force and displacement of thecenter of the circle.

Cam profiles obtained through the design methodologydescribed in Section 2.1 are shown in Fig. 4. Fig. 4(a)shows the cam profile with design parameters A in Ta-ble 1, while Fig. 4(b) shows that with design parame-ters B. The entire cam profile of pairing element 1 asshown in Fig. 1 can be obtained by arranging the calcu-lated cam profile symmetrically with the y-axis.

3.2. Example Based on Displacement of ContactPoint

Design parameters are shown in Table 2. The specifiedstiffness characteristics of pattern A in the table are hard-ening spring characteristics, while those of pattern B aresoftening spring characteristics. Stiffness characteristics

Table 2. Design parameters for examples with externalforce-displacement of the contact point relation.

Pattern A Bfd(xq) [N] 0.8(e0.05xq −1) 0.5ln(xq +1)

(Range [mm]) (0 � xq � 50) (0 � xq � 50)k [N/mm] 0.0494×2 0.103×2l0 [mm] 37.5 33.7w0 [N] 0.440×2 1.01×2r [mm] 15

(xq,0,yq,0) [mm] (0,30) (0,40)

0 10 20 30 40 50xq [mm]

0

2

4

6

8

10

f d[N

]

(a) Pattern A

0 10 20 30 40 50 xq [mm]

00.5

11.5

22.5

3

f d [N

]

(b) Pattern B

Fig. 5. Relationships between external force and displace-ment of the contact point for design examples.

0 10 20 30 40 50 x [mm]

30

40

50

60

70

80

y [m

m]

(a) Pattern A

0 10 20 30 40 50 x [mm]

0

10

20

30

40

50

y [m

m]

(b) Pattern B

Fig. 6. The calculated cam profiles of examples with re-lationships between external force and displacement of thecontact point.

are depicted in Fig. 5. Fig. 5(a) is the hardening springcharacteristics of pattern A in Table 2, while Fig. 5(b) isthe softening spring characteristics of pattern B.

Cam profiles were designed using the method describedin Section 2.2. Note that the Newton method was used tosolve non-linear equations in each step. The designed camprofiles are shown in Fig. 6. Fig. 6(a) is the cam profileobtained using the design parameters of pattern A in Ta-ble 2, while Fig. 6(b) is that obtained using the designparameters of pattern B. Since the calculated cam profilesfall into the range 0 � x � 50, it is shown that the size ofthe cam profiles can be controlled by using the proposeddesign methodology.



Pairingelement 2

Pairingelement 1

Force gauge

Spring

Fig. 7. Experimental setup for measuring force-displacement characteristics of the prototyped pair.

4. Prototyping and Evaluation

In this section, some prototypes of the pairs with thecam profiles designed in Section 3 are fabricated. Then,their force-displacement characteristics in the relativetranslational direction between pairing elements are mea-sured and compared with the specified ones in order toconfirm the correctness of the proposed design method-ologies.

Prototypes of the pair with the hardening spring charac-teristics designed in Section 3 (using design parameters Ain Tables 1 and 2) were manufactured using the fused de-position modeling (FDM) of a 3D printer. Fig. 7 showsthe prototype of the pair designed using design parame-ters A in Table 2. Note that the actual pair has depth,although the models of the pairs are assumed to be planemodels. Therefore, the two pairing elements of the paircontact on a line. In addition, the same linear springs areattached on both the front and reverse sides of the pair soas to balance the moment.

Pairing element 1 was fixed on a workbench as shownin Fig. 7. Depending on the specified force displacementcharacteristics, either the center of the circle of pairingelement 2 or the contact point between pairing elementswas placed on a mark of the scale on pairing element 1.Pairing element 2 was pushed by the force gauge untiltwo pairing elements slipped relatively. Then, the peakvalue measured by the force gauge was recorded. Thisprocess was carried out more than once at each relativeposition between pairing elements, and the mean valuewas calculated.

The measured and calculated force-displacement char-acteristics are shown in Fig. 8. The solid line in (a) rep-resents the specified stiffness characteristics of pattern Ain Table 1, while the solid line in (b) represents the char-acteristics of pattern A in Table 2. In these figures, themeasured values (x-points) were larger than the specifiedvalues (solid line). The cause is considered to be frictionbetween actual pairing elements. Therefore, the theoret-ical value was recalculated by taking the friction into ac-count. Assuming that the maximum static friction forcewith the magnitude μ|| fff n(x)|| is generated between pair-ing elements, the theoretical characteristics are recalcu-

0 10 20 30 40 50 xc [mm]

0

2

4

6

8

10

f d[N

]

Theoretical valuewithout frictionMeasured valueTheoretical valuewith friction

(a) Pattern A in Table 1 (relationship between external force and dis-placement of the center of the circle).

0 10 20 30 40 50 xq [mm]

0

2

4

6

8

10 f d

[N]

Theoretical valuewithout frictionMeasured valueTheoretical valuewith friction

(b) Pattern A in Table 2 (relationship between external force and dis-placement of the contact point between pairing elements).

Fig. 8. Comparison of theoretical and measured force-displacement characteristics.

lated using the following equation with the cam profiley = g(x).

fd(x) = −

⎡⎢⎣wx(x)+wy(x)

dgdx

1−μdgdx

⎤⎥⎦ . . . (13)

The recalculated theoretical characteristics, plotted withthe broken lines shown in Fig. 8, agree well with the mea-sured values. Note that the value of μ is 0.2. Therefore,the proposed design methodologies described in Section 2are correct.



P1(Motor)

P3 (Output)

P2(Flexibly constrained)

P4(Flexibly constrained)

P5(Motor)

Fig. 9. Closed-loop planar 5-bar mechanism with the pro-posed kinematic pairs, where the proposed pairs P2, P4 aremanufactured using FDM of the 3D printer and revolutepairs P1, P5 are actuated by DC motors.

5. Application

In the proposed pair, pairing elements easily rotate inthe relative rotational direction because of zero stiffness,while they do not move easily in the relative translationaldirection because of non-linear stiffness. Thus, this pairis regarded as a revolute pair with 1 degree of freedom(DOF) under small loads, while the pairing elements candisplace in the relative translational direction under largeloads. Therefore, if the proposed pairs are implementedin a fully-actuated link mechanism as revolute pairs with1 DOF, a flexible underactuated link mechanism can eas-ily be synthesized. In this section, the flexibility and kine-matic performance of a flexible link mechanism with theproposed pairs is investigated through both analyses andexperiments under static conditions.

5.1. Synthesized Flexible Link MechanismThe synthesized flexible link mechanism is shown in

Fig. 9. This closed-loop planar 5-bar link mechanism iscomposed of active revolute pairs P1 and P5, the proposedpairs P2 and P4, and a passive revolute pair P3. If the pro-posed pairs are regarded as revolute pairs with 1 DOF,this mechanism is a fully-actuated link mechanism with2 DOF. However, since the proposed pairs have 2 DOF,the synthesized mechanism is actually an underactuatedmechanism. Therefore, it has the flexibility to absorb ex-ternal forces.

5.2. Kinematic Analysis of the Tested MechanismThe posture of the synthesized mechanism should be

determined so as to satisfy static balance because the out-put point P3 displaces when an external force acts on it.The motion of an elastically constrained underactuatedlink mechanism such as this mechanism can be analyzedwith the method proposed by Iwatsuki et al. [23]. Thismechanism was analyzed following the process below.

(1) The external force fff e acting on the output point andthe initial posture of the mechanism are assumed.

Normal force: �

��

Virtual force: �

��

Elastic force: �

�

Circularcurve

Cam curve

Displacement: ��

Fig. 10. Forces acting between pairing elements of eachflexibly-constrained revolute pair in the tested mechanism.

Static analysis

��

�� Newton-Raphson

method

Update the posture�

�〜�

�

��

��

��

��〜�

�

Start

End

��

��

��

��

Yes

No

Assume ��

andinitial posture

Fig. 11. Flowchart of the analysis of the elastically-constrained underactuated mechanism.

(2) It is assumed that virtual forces fff v,i (i = 2,4) are actedon the tangential direction between pairing elementsof P2 and P4, as shown in Fig. 10. Then, static equa-tions are solved and virtual forces are obtained. Thesolution of this calculation is uniquely determined be-cause there are 12 statics equations and 12 unknownvalues.

(3) Relative displacements s2 and s4 are updated with theupdate formula of the Newton-Raphson method so asto satisfy fff v,i = 000 because virtual forces (frictionalforce) fff v,i should be zero.

(4) Positions of P2, P3, and P4 are updated by kinematicanalysis.

Processes (2)–(4) mentioned above are repeated untilvirtual forces fff v,i converge in the vicinity of zero. Theflowchart presented as Fig. 11 represents these processes,where ppp1～ppp5 are position vectors of P1～P5.

Note that active pairs P1 and P5 cannot be assumed tobe rigid in the actual mechanism due to servo stiffness. Inthat case, each passive angular displacement of P1 and P5



Table 3. Mechanical constant of the tested mechanism.

Position offixed pairs

P1 (50,0) mmP5 (−50,0) mm

Length of links

P1–P2 150 mmP2–P3 200 mmP3–P4 200 mmP4–P5 150 mm

Parameters ofproposed pairs

fd e0.1xq −1 N(Range) (0 � xq � 25 mm)

k 0.110×2 N/mmwo f f 0.686×2 N

l0 19.4 mmr 14 mm

(xq,0,yq,0) (0,25) mm

also can be calculated by solving the following equationin process (3).

τ j − km(θ j,0 −θ j) = 0 ( j = 1,5), . . . . (14)

where τ j is the torque of P1 and P5 calculated in pro-cess (2), km is torsional stiffness (servo stiffness) of activerevolute pairs, θ j,0 is the initial angle of P1, P5, and θ j isthe angle of P1, P5 in each step.

5.3. FlexibilityForce-displacement characteristics at the output point

of the mechanism in the vertical direction were calculatedusing the method described in Section 5.2. The mecha-nism constants of this mechanism are listed in Table 3,where the specified force-displacement characteristics ofthe proposed pairs, P2 and P4, are relationships betweenexternal force and displacement of the contact point. Therange of the external force was 0–6 N. Initial output pointwas (0,280). Torsional stiffness (servo stiffness) of activepairs was assumed to be km = 1.8×104 N·mm/rad.

The calculated force-displacement characteristics areplotted as a broken line in Fig. 12. These characteris-tics are small in stiffness in the small displacement area,and the stiffness increases as the displacement increases.It means that it has hardening spring characteristics.

Next, force-displacement characteristics of the outputpoint of the fabricated mechanism shown in Fig. 9 weremeasured by a compression testing machine. The mea-sured characteristics plotted as a solid line in Fig. 12 agreewell with the calculated characteristics.

In these results, it is not clear whether the flexibility ofthe proposed pairs really contributed to the flexibility ofthe output point because the results include the effects ofservo stiffness. Thus, the output force-displacement char-acteristics were recalculated without the effects of servostiffness and compared with the results with servo stiff-ness. The results are shown as a dotted line in Fig. 12. Itis shown that the displacement between the solid line andthe dotted line at the same external force is smaller thanthe displacement between the dotted line and Δy = 0-axis.

Contribution of proposed pairs

Contribution ofservo stiffness

Fig. 12. Stiffness characteristics of the tested mechanism.In calculated result 1, the servo stiffness of active pairs isconsidered. In calculated result 2, servo stiffness is not con-sidered.

6N17.2mm

Fig. 13. Behavior of the mechanism when an external forceis applied in −y-direction.

This means that the contribution of the proposed pairs foroutput flexibility is greater than the servo stiffness.

The calculated behavior of the mechanism when the ex-ternal force was acting in the −y-direction is shown inFig. 13. Each pairing element of P2 and P4 was rela-tively displaced in the translational direction, and the out-put point was displaced considerably in the −y-direction.In addition, behavior when the external force was actingin the +x-direction is shown in Fig. 14. Then, the outputpoint also displaced considerably in +x-direction. Theseresult shows that this mechanism has flexibility in multi-ple directions on the plane due to the proposed pairs.

5.4. Kinematic PerformanceThe output trajectories of the flexible link mechanism

were calculated by the method described in Section 5.2.Then, several downward external forces acted on the out-



6N

23.8mm

Fig. 14. Behavior of the mechanism when an external forceis applied in +x-direction.

-75 -60 -45 -30 -15 0 15 30 45 60 75x [mm]

255

270

285

y [m

m]

Target0N (Calculated)1N (Calculated)2N (Calculated)

0N (Measured)1N (Measured)2N (Measured)

Fig. 15. Calculated and measured output trajectories whenexternal forces are applied at the output point.

put point while the output point was generating the spec-ified trajectory. The specified trajectory was the lineartrajectory from (x,y) = (−60,280) to (60,280). Inputangles were simply calculated by inverse kinematics ofthe mechanism where the proposed pairs P2, P4 were re-garded as normal revolute pairs.

The calculated output trajectories under several magni-tudes of the external load are shown in Fig. 15. The outputtrajectory under no load corresponded to the specified tra-jectory. This means that the proposed pairs under no loadbehave as normal revolute pairs. The output trajectorieswith non-zero loads strayed from the specified trajectoryin the same direction as the load. The difference betweenthe trajectories under 1 N and 2 N was smaller than thedifference between the trajectories under no load and 1 N.It is considered that this effect is from hardening springcharacteristics shown in Fig. 12. Therefore, this mecha-nism can satisfactorily generate the specified trajectory byusing the simple inverse kinematics solution without con-sidering the effects of the non-linear stiffness, although ithas flexibility.

In order to confirm the validity of the calculation re-sults, the actual output trajectories of the fabricated mech-anism shown in Fig. 9 were measured. Weights were hungon the output point of the fabricated mechanism, and the

output trajectories were measured by a motion capturesystem. The measured trajectories are shown in Fig. 15.These results are similar to the calculation results. How-ever, the measured trajectories deviate slightly downwardfrom the calculated trajectories. This is considered to bean effect of gravity. Therefore, the calculated results agreewell with the measured results.

6. Conclusion

In this paper, a revolute pair with flexible translationalconstraint on a plane was proposed. It is composed of onepairing element with a circular profile and one with a camprofile connected by a linear spring. The advantage ofthis joint is that the total number of mechanical elementsis smaller than that of conventional joint mechanisms withpassive non-linear stiffness because the cam mechanism isdirectly used as a kinematic pair in this mechanism. Theresults are as follows.

(1) The cam profile that generates the specified relation-ship between the external force and displacement ofthe center of the circle is derived by the relationshipbetween the potential energy of the spring and thework of the external force. These types of force-displacement characteristics are intuitive for design-ers, although it is difficult to control the size of thecam profile.

(2) The differential equation to derive the cam profilethat generates the specified relationship between theexternal force and displacement of the contact pointbetween pairing elements is derived from the staticequations. These equations can be solved usingthe proposed new method, in which ways to solvenon-linear equation are included in each step of theRunge-Kutta method. These types of the force-displacement characteristics are useful to control thesize of the cam profiles, although this is not intuitivefor designers.

(3) The flexible revolute pair with various non-linearstiffness characteristics, e.g., hardening spring char-acteristics, softening stiffness characteristics and soon, in the relative translational direction can be de-signed by the proposed design methodology.

(4) The proposed design methodologies are correct andeffective because the force-displacement characteris-tics of prototyped pairs were in good agreement withthe given one.

(5) A planar closed-loop link mechanism with proposedpairs has both flexibility and sufficient payload whilegenerating the specified motion.

Finally, some prospects of this research are describedbelow for future development.

(a) In order to investigate the effects of the proposedpair for application to shock absorption, as mentioned



in the introduction, some analyses or experimentsin dynamic environment considering friction (damp-ing) are required. Which non-linear stiffness profilesachieve good performance for shock absorption mustalso be ascertained based on dynamic investigation.

(b) The output stiffness of a flexible link mechanismwith the proposed pairs can be roughly controlled bychanging the magnitude of the stiffness of the pro-posed pairs. However, a concrete output stiffness pro-file cannot be controlled. In addition, the output stiff-ness also changes if the posture of the mechanism ischanged. Therefore, the design and control method-ology of the flexible link mechanism with the pro-posed pairs should be investigated to achieve the de-sired output stiffness.

AcknowledgementsThe authors appreciate Prof. Masafumi Okada of the Tokyo Insti-tute of Technology, as he provided helpful advice related to thisresearch in ROBOMECH 2017 in Fukushima, Japan. In addi-tion, this work was supported by JSPS KAKENHI Grant Number18J21095.

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Name:Naoto Kimura

Affiliation:Graduate Student, Department of MechanicalEngineering, Tokyo Institute of Technology

Address:2-12-1 Ookayama Meguro-ku, Tokyo 152-8552, JapanBrief Biographical History:2016 Graduated from Department of Mechanical Intelligent SystemsEngineering, Tokyo Institute of Technology2018 Graduated from Master Course of Department of MechanicalEngineering, Tokyo Institute of Technology2018- Ph.D. Student, Department of Mechanical Engineering, TokyoInstitute TechnologyMain Works:• “Development of kinematic pair consisting of elastically constrained twocurved surfaces which can generate the specified rolling motion,” Trans. ofJSME, Vol.84, No.863, 2018 (in Japanese).Membership in Academic Societies:• The Japan Society of Mechanical Engineers (JSME)



Name:Nobuyuki Iwatsuki

Affiliation:School of Engineering, Tokyo Institute of Tech-nology

Address:2-12-1 Ookayama Meguro-ku, Tokyo 152-8552, JapanBrief Biographical History:1987 Graduated from Doctoral Course of Tokyo Institute of Technology1987- Research Associate, Precision and Intelligent Laboratory, TokyoInstitute of Technology1995- Associate Professor, Faculty of Engineering, Tokyo Institute ofTechnology2004- Professor, Graduate School of Science and Engineering, TokyoInstitute of Technology2016- Dean, School of Engineering, Tokyo Institute of TechnologyMain Works:• N. Iwatsuki and T. Kosaki, “Large Deformation Analysis and Synthesisof Elastic Closed-Loop Mechanism Made of a Certain Spring WireDescribed by Free Curves,” Chinese J. of Mechanical Engineering, Vol.28,No.4, pp. 756-762, 2015.• N. Iwatsuki, N. Nishizaka, K. Morikawa, and K. Kondoh, “MotionControl of a Hyper Redundant Manipulator Built by Serially ConnectingMany Parallel Mechanisms Units with a Few DOF,” Int. J. AutomationTechnol., Vol.4, No.4, pp. 364-371, 2010.• N. Iwatsuki and K. Morikawa, “Sound Generating Mechanism of FrogShaped Guiros,” JSME J. of System Design and Dynamics, Vol.2, No.2,pp. 596-609, 2008.Membership in Academic Societies:• The Japan Society of Mechanical Engineers (JSME)• The Japan Society for Precision Engineering (JSPE)• The Robotics Society of Japan (RSJ)• Japan Society for Design Engineering (JSDE)• Japanese Council of IFToMM

Name:Ikuma Ikeda

Affiliation:Assistant Professor, Tokyo Institute of Technol-ogy

Address:2-12-1 Ookayama Meguro-ku, Tokyo 152-8552, JapanBrief Biographical History:2010- Postdoctoral Researcher, Kyushu University2012- Assistant Professor, Tokyo Institute of TechnologyMain Works:• “Human tremor model considering randomness of refractory period,”Trans. of the JSME, Vol.84, No.857, 2018 (in Japanese).Membership in Academic Societies:• The Japan Society of Mechanical Engineers (JSME)• The Institute of Noise Control Engineering of Japan (INCE/J)• The Acoustical Society of Japan (AST)


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