Kineto-static Mechanical Synthesis for Nonlinear Property Designof Passive Stiffness using Closed Kinematic Chain

Masafumi Okada and Jun Takeishi

Abstract— To prevent the human injury or breakage ofthe robot caused by interaction or collision with the robotand its environment, it is an important issue to introducerobot softness. However, because not only softness but alsostiffness is required for precise task execution, the simultaneousrealization of softness and stiffness using time varying stiffnessis required in the real environment. We focus on the robotmotion, and propose a realization method of time varyingstiffness using robot motion and nonlinear passive stiffness. Torealize the purpose-designed property of the time dependentstiffness, the nonlinear property of the passive stiffness has tobe arbitrary designed considering the robot dynamics. In thispaper, we propose the nonlinear property design method ofpassive stiffness based on kineto-statics with closed kinematicchain. The mechanism is synthesized based on the optimizationof the generative force or torque, and purpose-designed stiffnessis realized. The proposed method is evaluated by simulationsand experiments using the prototype of the landing mechanism.

I. INTRODUCTION

Mechanical softness of robot members or joints is animportant issue for safety and adaptation to its environmentspoints of view. As shown in Fig.1, the softness absorbs

Fig. 1. Impact absorption and task execution with elastic joints

the impact of collision with humans and prevents from abreakage by itself. However, for task execution, stiffnessis required for precise control of the robot with a heavyweight. From these considerations, simultaneous realizationof stiffness and softness is required, and time variant stiffness

This research is supported by the “Motion emergence and mutual pro-gression of robot body and intelligence from the dynamical point of view”under Grants-in-Aid for Scientific Research (Category Encouragement ofYoung Scientists (A)) of Japan Society for the Promotion of Science.

Masafumi Okada and Jun Takeishi are with Dept. of Mechanical Sciencesand Engineering, Tokyo TECH, 2 – 12 – 1 Ookayama Meguro-ku Tokyo,JAPAN, okada@mep.titech.ac.jp, takeishi.j.aa@m.titech.ac.jp

or compliance plays an important role. Some researchershave developed variable compliance so far [1].

(1) Active compliance / Impedance control realizes avirtual spring and damping by torque and force controlof actuators [2]–[5]. This method is effective becausean arbitrary characteristic of compliance or viscositywill be realized, however, does not deal with theinstantaneous impact force because of a latency of theclosed loop bandwidth of control.

(2) Programmable passive compliance [6] is realized byadditional actuators that change the stiffness of the ma-terial. For example, a tendon mechanism with nonlin-ear spring [7], [8] consists of two redundant actuatorsthat change the internal force of springs. Morita et al.developed Mechanical Impedance Adjuster (MIA) [9].Wimbock et al. designed a robot hand with variablestiffness [10]. Ham et al. developed MACCEPA for legcompliance [11]. Tonietti and Wolf developed specialmechanism for programmable passive compliance [12],[13]. These mechanisms realize an arbitrary compli-ance however the additional actuator occupies volumeand weight of the robot.

On the other hand, passive compliance is a simple way torealize robot softness. It is realized by mechanical softnessand has high reliability and robustness, however, low change-ability because of its time invariant complaint characteristic.

We focus on that the robot is accompanied by “motion”.The landing motion changes the height of the body and theswing motion moves the robot arm to hit a ball, which meansthe position of the robot p is time variable and assumedto be written by p = p(t). On the other hand, nonlinearstiffness K means the position dependent stiffness whichis represented by K = K(p). By connecting “motion”and “nonlinear stiffness”, we can realize the time varyingstiffness K = K(p(t)). Because p is decided by the robotdynamics, K(p) has to be designable to realize an arbitraryproperty of the time varying stiffness.

(a) (b)

Fig. 2. Nonlinear spring

The 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems October 18-22, 2010, Taipei, Taiwan

978-1-4244-6676-4/10/$25.00 ©2010 IEEE 4213

So far, some nonlinear stiffness mechanisms have beenproposed as shown in Fig.2. Fig.2-(a) is used for a humanoidshoulder mechanism [14] and Fig.2-(b) has high nonlinearityof stiffness based on singularity of the closed kinematic chain[15]. Though these mechanisms realize high nonlinearityof stiffness, they have small design possibility because ofthe inherent nonlinearity of mechanism. On the other hand,a closed kinematic chain has high nonlinearity of input-output relationship and high design possibility by mechanicalsynthesis.

The purposes of this paper are as follows.1) By the combination of “motion” and “nonlinear stiff-

ness”, we propose the realization strategy of timevarying passive stiffness.

2) To realize the arbitrary property of time varying stiff-ness, the design method of nonlinear property of thepassive stiffness is proposed based on kineto-staticmechanical synthesis and closed kinematic chain.

3) The effectiveness of the proposed method is shown bya prototype of the landing mechanism to absorb theimpact force.

II. KINETO-STATIC MECHANICAL SYNTHESIS FORNONLINEAR STIFFNESS

A. Closed kinematic chain and restoring torque

The stiffness of the closed kinematic chain with a linearspring is designable by changing link parameters. Considerthe restoring torque τ of the mechanism (four-bar linksystem) as shown in figure 3-(a), for example. The linkparameters (length of the links and joint angles) are definedin Fig.3-(b). �i and θi mean the length and rotational angle of

θ1

θ2 φ

θ3

θ4

�1

�2�3 �4

r

y

xA

B C

D

T

R

(xT, yT)

(xR, yR)

Linear spring τ

(a) (b)

L

Fig. 3. Closed kinematic chain with four-bar link system and linear spring

link i, r is the length of the one side of the trigonal link andφ is its angle. (xj , yj) (j = T,R) is the position of the edgeof the linear spring with the length L. From the kinematics ofthis mechanism, the following closed loop constraints haveto be satisfied.

4∑i=1

θi = 2π,4∑

i=1

�i cosi∑

j=1

θj = 0,4∑

i=1

�i sini∑

j=1

θj = 0

(1)The natural length of the linear spring is set as L0, and ityields the initial configuration of the system as follows.

L0 =√

(xT0 − xR)2 + (yT0 − yR)2 (2)[xT0

yT0

]=

[�1 cos θ10 + r cos(θ10 + θ20 − φ)�1 sin θ10 + r sin(θ10 + θ20 − φ)

](3)

By defining the potential energy E

E =12K(L− L0)2 (4)

that is accumulated to the linear spring with its springconstant K , τ is obtained by

τ = τ(θ1) = − ∂E∂θ1

= −K(L− L0)∂L

∂θ1(5)

The last term ∂L/∂θ1 in equation (5) represents the nonlin-earity of the mechanism, and it is obtained by

∂L

∂θ1=

∂L

∂xT

∂xT

∂θ1+

∂L

∂yT

∂yT

∂θ1(6)

∂L

∂xT=xT − xR

L,

∂L

∂yT=yT − yR

L(7)

∂xT

∂θ1= −�1 sin θ1 − r sin(θ1 + θ2 − φ)

(1 +

∂θ2∂θ1

)(8)

∂yT

∂θ1= �1 cos θ1 + r cos(θ1 + θ2 − φ)

(1 +

∂θ2∂θ1

)(9)

∂θ2/∂θ1 is obtained from the partial differential of the closedkinematic constrains in equation (1) with respect to θ1.

B. Nonlinear property of the stiffness

Based on equation (5), the torque properties are obtainedas shown in Fig.4. Two patterns of the link parameters

60 70 80 90 100 110 120 130 140-2

0

2

4

6

8

10

Torq

ue[N

m]

Rotation angle θ1 [degree]

-25

-20

-15

-10

-5

0

Torq

ue[N

m]

60 70 80 90 100 110 120 130 140Rotation angle θ1 [degree]

Fig. 4. Torque property of mechanism

are selected. The solid line shows the obtained torque andthe link configurations are shown in each joint angle θ1.These results show that by changing the link parameters, thenonlinear property of the torque or stiffness is changed somuch.

C. Kineto-static mechanical synthesis

In this section, nonlinear property design method of thestiffness based on kineto-static mechanical synthesis is illus-trated.

Firstly, the reference torque τref

τref ={τ1ref (θ11), τ2

ref (θ21), · · · , τNref (θN

1 )}

(10)

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is set with respect to the given joint angle of θ1 = θi1 (i =

1, 2, · · · , N). By setting the objective function J as

J =N∑

i=1

∥∥τ iref − τ(θi

1)∥∥2

(11)

the link parameters qj are optimized by the following gradi-ent method

qj ← qj − ∂J

∂qjδ (12)

where δ is a small constant. The candidates of qj are �i, xR,yR, K , r, φ and initial value of θ1.

III. LANDING MECHANISM WITH NONLINEAR STIFFNESS

A. Designed mechanism

Applying the kineto-static mechanical synthesis, the land-ing mechanism is designed. Figure 5 shows the designedpassive mechanism which consists of body part and leg part.The leg part consists of two parallelograms that are coupledwith 3D kinematic chain (Mechanism g in Fig.5) and yieldsthe vertical motion of foot link (link e – f) as shown inFig.6. Link parameters in the body part are defined same as

- θ0

θ1 = 1+nθ0 φ

θ22 φ

θ3

θ4

r

y

xA

B C

D

T

R

a d

b c

e f

body part

leg part

- θ0g : 3D mechanism

(xT, yT)

(xR, yR)

ψ0

ψ0

Fig. 5. Designed mechanism

Fig. 6. Motion of leg part

in Fig.3. φ1 is the inital value of θ1. θ0 = 0 yields the initialconfiguration so that the spring becomes natural length. y�0

is an initial value of the body height and calculated by;

y�0 = 2�� sinψ0 (13)

The motion of this mechanism is shown in Fig.7. Thedistortion of the body height y� yields the rotation of θ0and it causes the rotation of θ1 by the timing belt with itsgear ratio 1:n. Because of the nonlinearity of the four-barlink, the length of the spring changes with high nonlinearity,which causes the nonlinear ground force of this mechanism.

Fig. 7. Motion of the designed mechanism

B. Ground force calculation

According to the change of y�, the length of the springis changed and this mechanism yields the ground force F .From the kinematics of the mechanism, θ0 is calculated as;

θ0 = ψ0 − sin−1

(y�0 + y�

2��

)(14)

The accumulated energy U of the spring is calculated inequation (4), and the ground force F is obtained by;

F =∂E

∂y�=∂E

∂θ1

∂θ1∂θ0

∂θ0∂y�

= K(L− L0)∂L

∂θ1

∂θ1∂θ0

∂θ0∂y�

(15)

From equation (14),

∂θ0∂y�

=1

2�� cos(ψ0 − θ0) (16)

is obtained. On the other hand, the rotation angles of the legpart and body part are connected by timing pulley with itsgear ration 1 : n and

∂θ1∂θ0

= n (17)

is satisfied. From equations (15), (6) ∼ (9), (16) and (17),the ground force is calculated as F = F (y�).

C. Mechanical synthesis of the body part

0 - 0.04 - 0.08 - 0.12 - 0.160

40

80

120

* low stiffness

high

stif

fnes

s

high

stif

fnes

s

Ref

eren

ce fo

rce Fref

[N]

(yf, Mg)

property (a)

property (b)

property (c)

property (d)

property (e)

Fig. 8. Reference force that satisfies the given property

1) Set of the reference ground force: By assuming thatthis mechanism with weight M falls from the height h andcollides with the ground with its drop velocity vM =

√2gh

with the gravity acceleration g, the properties of the referenceground force Fref are set as follows:

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(a) Fref = 0 at y� = 0 is satisfied because the spring isnatural length at y� = 0.

(b) The stiffness K� = ∂Fref/∂y� at y� = 0 is sufficientlylarge. It causes the brake for the drop velocity. We notethat the ground force Fref is small even if stiffness islarge because of small |y�|.

(c) Fref = Mg at y� = yf is satisfied, where yf is thefinal distortion of the height of the body (t→∞).

(d) The stiffness at y� = yf is sufficiently small. It doesnot only contribute to impact absorption but also causesthe attenuation of the vibration.

(e) The stiffness in large |y�| is sufficiently large. It playsthe role of stopper.

It is not sure that these profile is optimal for impactabsorption but come up with kineto-static point of view.Optimization based on biologically inspired method willbe required in the future study. From these conditions, thereference ground force Fref is set as shown in Fig.8.

2) Mechanical synthesis: The objective function is set as;

J =∑

i

∥∥F iref − F (yi

�)∥∥2

(18)

where yi� (i = 1, 2, · · · , N ) are the representative points of

y�. The link parameters are optimized so that J is minimized.For simplicity, ψ0(=75degree), ��(=20mm), �4(=10mm) andn(=1.75) are fixed. ψ0 defines the initial configuration of

Initial configuration Optimized configuration

Fig. 9. Initial and optimized configuration

Gro

und

forc

e F

[N]

0 -0.04 -0.08 -0.12 -0.160

40

80

120

Optimized force

Initial force

Reference force

Fig. 10. Obtained ground force

the leg part. The change of �� and �4 yields a homologousmechanism. n cannot be taken an arbitrary value becauseit is decided by the number of teeth of the timing pulley.

The design parameters are the following nine parameters:�1, �2, �3, r, φ2 (link parameters of four-bar link), φ1 (initialvalue of θ1), xR, yR (root position of the spring) and K(spring constant). By setting an appropriate initial valueof link parameters, the design parameters are optimized.The initial configuration and the optimized configuration ofthe mechanism are shown in Fig.9. Figure 10 shows theground force F of the initial configuration and the optimizedconfiguration.

C F

M

Fig. 11. Model of the mechanism

final positionh = 0.05m

h = 0.15m

h = 0.30m

initial position

Fig. 12. Landing simulations

IV. LANDING SIMULATION RESULTS

By using the optimized mechanism, the landing simula-tions are executed, where the dynamics of the inertia termsof members are ignored. The mechanism is approximated bya mass-damper-spring system with the concentrated mass asshown in Fig.11. C is a constant viscous coefficient which isdetermined from the experimental system in section V andK(y�) is nonlinear stiffness. This mechanism falls from theheight h = 0.05, 0.15, 0.30m and collides to the groundwith its velocity vM . The simulation results are shown inFig.12. The final positions are same because the weight ofthe robot is same. The transient responses in the simulationsare shown in Figs.13 and 14. For comparison, caliculationresults of the ground force and body height using a linearspring (K(y�) = const.) are shown. The spring constant is setso that it satisfies Kyf = Mg, which means the final positionof the body is same even though linear/nonlinear spring isutilized. Fig.13 shows the ground force. This figure showsthat the nonlinear spring reduces the maximum value of theground force comparing the linear spring (see h = 0.05,0.15m), which means the nonlinear spring works as a soft

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Time [sec]

h = 0.05m

Gro

und

forc

e [N

]

0 0.5 1 1.5

0

40

80

0

40

80

0

40

80

h = 0.15m

h = 0.30m

linear springnonlinear spring

linear springnonlinear spring

linear springnonlinear spring

Fig. 13. Simulation results of the ground forces

spring to reduce the impact force. On the other hand, whenh = 0.30m, even though the maximum ground force ofnonlinear spring is larger than linear spring (because of Fref

property in (e)), the body height is not so much changed asshown in Fig.14, which means the nonlinear spring worksas a hard spring. This is because the nonlinear spring has anadvantage of the stopper structure.

0.2

0.3

0.4

Time [sec]

h = 0.05m

0 0.5 1 1.5

h = 0.15m

h = 0.30m0.2

0.3

0.4

0.2

0.3

0.4

linear springnonlinear spring

linear springnonlinear spring

linear springnonlinear spring

Fig. 14. Simulation results of the height of bodies

V. PROTOTYPE OF THE PROPOSED MECHANISM ANDEXPERIMENTAL EVALUATION

A. Prototype of the mechanism

Fig.15-(a) shows the prototype of the proposed mechanismand (b) shows the 3D closed kinematic constrain of g inFig.5. The optimized link parameters in section III-C.2 areemployed. This mechanism is connected to the linear sliderto move vertically and has an encoder and accelerometer tomeasure the body height and body acceleration.

A

(a) Prototype

(b) 3D constrain

accelerometer

encorder

spring

g : 3D mechanism

Fig. 15. Prototype of the proposed mechanism

B. Measurement of nonlinear stiffness

The generative ground force of the prototype is measured.By using a weight, the ground force is changed and theheight of the body y�0 + y� is measured by the encoder.The experimental results are shown in Fig.16. Because ofthe coulomb friction, y� is not decided uniquely and themaximum and minimum values of y� are indicated. Themaximum value is shown by red © and minimum valueis shown by green � for the same ground force. The blueline shows the nominal force. After F crosses over 50N,the difference of the nominal value and experimental valuetends to be large, this is because the length of the springtransgresses the permissible length of linearity. From theseresults, the prototype realizes the reference ground force.

Rea

lized

forc

e F

[N]

0 -0.04 -0.08 -0.12 -0.160

40

80

120nominal value maximum height minimum height

Fig. 16. Ground force of the prototype

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C. Landing experiment

By using the prototype, the landing experiment is executedwith h = 0.15m. The body height y�0 + y� is shown in

0 0.5 1 1.50.2

0.3

0.4

Time[sec]

experimental resultsimulation result

Fig. 17. Experimental result of the height of body

Fig.17. The solid line shows the simulation result (shownin the middle of Fig.14) and “+” shows the experimentalresults. Because of coulomb friction, the body height ofexperimental result does not coincide to the simulation resultat t → ∞, however, the transient response is similar. Andthe acceleration of the body is shown in Fig.18, which isobtained from the accelerometer. The solid line shows the

0 0.5 1 1.5-5

0

5

10

15

20

25

Time[sec]

Acc

eler

atio

n [m

/s2 ]

experimental resultsimulation result

Fig. 18. Experimental result of the acceleration of body

simulation result and “∗” shows the experimental result. Justafter landing, the acceleration of the body is not small, whichis because the initial posture of the leg part is close toa singular posture (ψ0 = π/2), and the influence of theneglected dynamic term is not small. Some modificationsof the initial posture are required. On the other hand, thetransient response is similar to the simulation results. Fromthese results, we can see that the proposed mechanism andsynthesis method is effective for the impact absorption. Theattached movie shows the experimental result.

VI. CONCLUSIONS

In this paper, we focus on that the robot motion plusnonlinear stiffness yields time valiant stiffness, and proposea nonlinear property design method of passive stiffness usingclosed kinematic chain. The results of this paper are asfollows;

1) We propose the design method of nonlinear stiffnessusing a linear spring and nonlinearity of the closed

kinematic chain because the link mechanism has largedesigned possibility by the optimization of the linkparameters.

2) The optimization method of the link parameters isproposed to realize the reference ground force basedon kineto-statics.

3) To reduce the impact force of landing, the property ofthe reference ground force is considered and the linkparameters are optimized.

4) The effectiveness of the proposed method is evaluatedby the landing simulations and experiments using theprototype.

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