Stiffness Distribution Control– Locomotion of Closed Link Robot with Mechanical Softness –

MATSUDA TakeshiInterdisciplinary Graduate School of

Science and EngineeringTokyo Institute of TechnologyYokohama, JAPAN 226–0026

Email: matsuda@mrt.dis.titech.ac.jp

MURATA SatoshiInterdisciplinary Graduate School of

Science and EngineeringTokyo Institute of TechnologyYokohama, JAPAN 226–0026Email: murata@dis.titech.ac.jp

Abstract— This paper proposes a new method of locomotioncontrol, named “Stiffness Distribution Control (SDC)”, to realizevarious locomotion of a closed link robot with mechanicalsoftness. SDC directly defines the reference stiffness coefficientat each hinge by SDC string which represents the distribution ofthe stiffness. This simple method does not need costly calculationat all and thus is suitable for controlling closed link robots.We build “BIYOn” as the prototype of closed link robots, thevariable stiffness hinge (VSH) of which is newly designed. SDCis experimented on BIYOn and then it’s found that SDC iseffective in a controlling locomotion in practice. We also develop asystematic optimization method of stiffness distribution composedof two-stage approach and succeed in optimization to obtain afast and smooth rolling motion and a sudden stopping motion.

I. INTRODUCTION

Applications of robotic system is rapidly expanding tovarious non-industrial fields such as nursing, entertainment,education, and security where robots have to do their taskamong other people and other robots in the same environment.The robots which are designed to coexist with humans arerequired to be absolutely safe; they must not harm peoplein any possible situations. Conventional approaches to realizesuch safety are usually based on software servo technique;namely, compliance at the joint is emulated by gain control.However, this method cannot react to impulsive disturbancesbecause of control delays. It also cannot cope with faultyoperations or breakdown of controller processors.

An alternative approach to secure the absolute safety thatdoes not rely on the control is to embed passive softness inthe robot hardware. We consider that there are two importantrequirements to realize such safety, a) mechanical softness ofthe robot hardware, b) closed link configuration of the robot.The former guarantees compliance against any collisions orimpacts. The latter is advantageous for safety, because of theround surface without any protrusions.

We have proposed a modular robot called “BIYOn” basedon the above idea [1]. It is a crawler robot made of identi-cal segments. Instead of driving relative angles between theadjacent segments, each segment has a motor that controlsthe torsion stiffness between the segments. This design is tosatisfy the requirement a), and the loop configuration satisfiesb).

Fig. 1. BIYO10

Control of closed link system usually requires heavy cal-culation of kinematics/dynamics [4], though there are someexceptions1. Even if its dynamics is obtained explicitly2 ,various frictions and boundary constraints caused by theground contact is very difficult to be taken into account.Moreover, BIYOn is regarded as the hybrid system [2], [3],which is a combination of differential equations (dynamics)and discrete logic (ground contact conditions), requires evenmore computational cost (Figure 2).

In terms of intrinsic safety, servo control based on kinemat-ics/dynamics does not make sense, because the robot merelyfollows a given target trajectory, namely, a robot is deprived ofquick reflection to unpredictable disturbances. The compliancecontrol is advantageous to reduce the calculation cost. Insteadof costly calculation of kinematics/ dynamics, it implicitlydetermines the geometrical shape of the link system owingto its loop configuration.

In order to cope with the difficulties in closed loop robotcontrol, various attempts have been made. One of such ap-

1Kinematics of a closed few-link mechanism, containing less than six links,is easy to solve.

2There are several methods to obtain a systematic computational schemeof the dynamics of closed link mechanism [5], [6], [7]. In the methods, theclosed loop is replaced by open loop system by cutting at some hinge, andrestored by adjusting boundary conditions at the cut point. These methods areapplicable only when there are no singular points in the configuration space.

Fig. 2. Hybrid system. System dynamics and boundary conditions arechanged along the motion.

proaches is to reduce the complexity of kinematics/dynamics.Omata et al. [9] proposed a locomotion control for theirfive-link crawler robot. They made three hinges unactuatedand specified a rolling motion around one axis on the floor.This is a very straightforward way to remove redundancies3

and simplify the kinematics/dynamics, however generalitiesof the method are almost lost. Another approach is to useevolutionary computation. Matsuo et al. [10], [11] appliedthe Genetic Algorithm (GA) to their six-link crawler, allthe hinges of which are directly actuated by motors. Theyoptimized their control algorithm by exhaustive calculationof dynamics simulations. GA gives an optimal solution fora specific robot, however it does not suggest any generaldesign methodology. Yokoi et al. [12], [13] proposed a methodcalled “Direct Compliance Control” for grasping motion of aredundant parallel link manipulator. It realizes force control atthe end effector by introducing compliance at each joint. Thismethod only considers external forces at the end affecter, thuscannot be applied to crawlers that have multiple contacts withthe ground.

Our purpose is to develop a systematic design method ofclosed link robot control, which has low computational costand easy implementation.

In this paper, we propose a new control method to generatelocomotion of closed link robots on the ground called StiffnessDistribution Control (SDC). The principle of the method is asfollows: Assuming that stiffness at each hinge of the closedlink can be controlled, the overall shape of the loop depends onthe distribution of hinge stiffness. Namely, we can control theshape of the loop by giving appropriate stiffness distributionon the loop. If the distribution is continuously changed,dynamic motion of the loop such as locomotion (rotation)will be realized. For the simplicity, we use a unique fixedstiffness distribution. The stiffness coefficient at each hinge isdetermined by this stiffness distribution and relative position ofthe hinge in the loop defined by the ground contact condition.At every moment when the contact condition is changed bythe movement, a new stiffness coefficient is reassigned to eachhinge by using the same stiffness distribution. This methodrequires only a simple shifting operation of the stiffnesscoefficients determined by the binary input (Contact or Non-contact) at each link.

II. HARDWARE OF MODULAR CRAWLER

In this section, we give a brief description of the hardwaresystem of BIYOn (Figure 1). BIYOn is a crawler type robot

3A closed five-link mechanism is 2 DOF by Grubler’s formula[8].

TABLE I

MAJOR SPECIFICATIONS OF BIYON

Num. of Connected Modules 8 . . .10Mass (per module) 162(g) (w/o com. cables)Size (per module) W88×L90×H20(mm)Distance bet. neighbor modules 74(mm)Sensors Ground (touch) switch onlyBatteries External supplyMCU PIC16F876

BCA

l

a bP

δ

Fig. 3. Simply Supported Beam

rack and pinion

slider

hinge axis

neighbormodule

pin(leaf spring holder)

θi;rδi

δi / θi;r

(a) Conversion Mechanism

shift

A BC

( t ' 0:5)

Lowstiffness

( t ' 0:25)

Highstiffness

actuator

leaf spring

rotationby actuator

rack and pinion

(b) Leaf Spring Shift Mechanism

Fig. 4. The Variable Stiffness Hinge (VSH). (a) Rotation around the hingeaxis (blue part in the figure) is converted into translation movement of aspring loader (pink) via pinion-rack gears (yellow). (b) Motor slides a slider(light blue) on which a leaf spring (green) simply supported. By changingthe position where the spring loader pushes/pulls the spring, variable stiffnessaround the rotation axis is realized.

composed of the identical modules. (n in BIYOn denotes thenumber of modules.) Its specification is given in TABLE I.

In BIYOn, torsion stiffness between adjacent modules canbe controlled by a compact mechanism called “Variable Stiff-ness Hinge (VSH)” (Figure 4). Motors do not drive the rotationaxes directly, instead, they control torsion stiffness around theaxes. We use a simply supported leaf spring as a source ofstiffness. In Figure 3 the minimum stiffness kmin is obtainedwhen the loader locates at the mid point of the spring asfollows.

δ =P

3lEIa2b2 (1)

∴ kmin =Pδ

∣∣∣∣a=b=l/2

=48EI

l3 (2)

detached

R¡0

R¡1

R¡2R¡3

F+1

F+2F+3

F+0G G

Rolling Direction

R¡0R¡1

R¡2

R¡3

F+1

F+2

F+3

F+0G

S

Fig. 5. Relative coding system for SDC. The labels of the hinges are relativelydefined on the basis of ground contact. The foremost hinge touching theground is called F+0. F+1, F+2, F+3 follow in counter-clock-wise. Likewise,R–0 , R–1 , R–2 , R–3 are defined from the rearmost hinge on the ground.Whenever a link detaches from or attaches to the ground, this coordinationis renewed. ‘G’ is given to hinges on the ground. When some hinges existbetween R–3 and F+3, these hinges are labeled ‘S’.

Pushing ForcePulling Force

SDC string: LHHL__HHLLHigh StiffnessLow Stiffness

Fig. 6. Simple stiffness distribution

where E is Young’s modulus, I is second moment of area, andl is the length of the leaf spring.

When a = tl(0 ≤ t ≤ 0.5) , the stiffness ratio r becomes asfollows.

r =k

kmin=

116t2(1− t)2 (3)

The maximum stiffness ratio rmax is 1.7 in the VSH.

III. STIFFNESS DISTRIBUTION CONTROL

Stiffness Distribution Control (SDC) is a method to controllocomotion of a closed link system. SDC controls torsionstiffness at each connection axis between the modules.

We define a relative coding system to give labels to hingesand relative angles between adjacent modules (Figure 5).

A. Stiffness Control by SDC string

SDC string represents the distribution of the stiffness. It isa string in form of “ABCD EFGH”, where each charactercorresponds to a discrete level of stiffness ki at the locationR–3, R–2, R–1, . . . F+2, F+3. We also define “standardstiffness” around which the stiffness coefficient is varied. Weuse 0.4(mN/rad) for the standard stiffness coefficient. Thisvalue is chosen to give a convex curvature on the loop. Thestandard stiffness is allocated to hinges with label G and S.From now on, the unit of stiffness coefficients are not written.

By using SDC string, we can calculate torque τi at eachhinge, where i denotes i-th hinge from R–3 to F+3.

τi = ki ·θi,r + d · θi,r (4)

ki =1T

(ki − ki) (5)

The second term of Eq. (4) represents viscous friction in theVSH. We assume that the actual stiffness coefficient of the

Fig. 7. Snapshots from simulation

Fig. 8. Snapshots in experiments

hinge ki converges to the reference coefficient ki with timeconstant T shown in Eq. (5).

B. Simulation & experiments

We have evaluated the SDC by dynamics simulation. Weadopt “Open Dynamics Engine ver0.5” to simulate the robotlocomotion throughout this paper. The real specifications ofBIYOn is used for simulation. Other parameters are, T =0.6(s), d = 0.005(mNs · rad−1) and simulation step = 5.0(ms).

The SDC string used in the simulation is LHHL HHLL(L:0.4, H:0.6). We put two hard segments (HH) near theforemost and rearmost contact hinges, that may push the frontand pull up the rear shown in Figure 6. Snapshots from thesimulation are shown in Figure 7. In the simulation, obtainedlocomotion velocity is 0.29(m/s) after the robot reaches asteady state.

Next, we have conducted experiment on the BIYOn hard-ware using the same SDC string. The snapshot of the experi-ment is shown in Figure 8. Obtained speed is 0.08(m/s). The

TABLE II

CODING SYSTEM OF STIFFNESS COEFFICIENT

0 1 2 3 4 5 6 7 8 9A 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9B 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9C 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9D 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9E 4.0 ————————– no-use ————————–

reason why we get slower speed compared to the simulationis not clear. One of the possible reasons is that we couldnot realize equal characteristics among the modules. Anotherpossible reason is a large friction in the VSH.

IV. OPTIMIZATION OF STIFFNESS DISTRIBUTION

CONTROL

In this section, we consider optimization of SDC based onsimulations. Although the simulation parameters are the sameas Section III-B, we use SDC strings with higher resolution.

The stiffness coefficient varies from 0.0 (free hinge) to 4.0(ten times of the nominal stiffness value), discretized with 41levels (TABLE II). For instance, C6 represents 2.6.

In order to reduce the computational cost of dynamicssimulation, we adopt a two-stage optimization strategy. In thefirst stage, which is called Hinge-Wise (HW) optimization,we get the local (coarse) optimal stiffness coefficient foreach hinge by focusing on one particular hinge and allowingonly two values to other hinges. The second stage is calledCollective Decent (CD) optimization that is a kind steepestdescent method applied to a collection of SDC strings. By thismethod, we are able to fine-tune the result of HW optimizationto get the best solution. (It does not guarantee the globaloptimum in the strict sense.)

Each SDC string is tested by dynamics simulation of 20(physical) seconds, and evaluated by its ability to generatelocomotion from the stationary state. The robot is hold atthe starting position during the first four seconds, then inthe subsequent six seconds, it starts locomotion, but thistransient phase is neglected in the evaluation. The SDC stringis evaluated by the following 10 seconds by the averagelocomotion (horizontal) speed and (vertical) vibration speed(the average of absolute vertical speed (m/s) at every samplingtime (0.1(s)).

In Section IV-A, the first stage of optimization is described.Namely, by the HW optimization, we are able to reveal thequalitative functionalities of each position of the stiffnessdistribution. The local optimal SDC string is obtained by con-sidering several aspects such as locomotion speed, vibrationand reliability (the ability that the string always generateslocomotion to a certain direction).

The detail of CD optimization aiming at improving loco-motion speed is described in Section IV-B. Some simulationresults of locomotion will be shown in Section IV-C. InSection IV-D, optimization result of stopping motion (SDCstring which draw up the robot) will be shown. Hereafter, wecall a robot controlled by some SDC string an “individual”.

Velocity

Vibration

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18 4th mode

1st mode

2nd mode

3rd mode

0.1 0.20.0 0.3 0.4 0.5 0.6 0.7 0.8

Fig. 9. Observed four modes.

A. Hinge-Wise optimization of SDC

We have some intuitive understanding of role of each hingein the loop robot. For instance, we put high stiffness coefficientat F+0 and F+1 and they push the robot to the forwarddirection. Other hinges may have some specific functionalityaccording to the desired motion. We propose HW optimizationis to find qualitative functionality of each hinge. The methodis as follows.

(1) To reduce the computational cost, limiting hinge stiff-ness at two levels (A4 and A6) except for one hinge.

(2) Evaluate all possible binary combinations to get qualita-tive functionality and optimal stiffness coefficient at thehinge.

(3) Apply (1), (2) to all the hinges to get the local optimalSDC string. (This is used for the initial string to thefurther tuning in the CD optimization.)

Figure 10 is a scatter plot of all the individuals evaluated bylocomotion speed against the stiffness. Each strip correspondsto allocated stiffness value for one hinge, and the distributionof speed for all the individuals. For example (Figure 10(b)),when stiffness 1.4 is assigned to R–2 hinge, most of individ-uals make forward locomotion, however some shows reverselocomotion. From these plots, we can understand qualitativerole and importance of each hinge for locomotion throughdifferent performances among all stiffness coefficients.

Figure 11 shows locomotion speed and vibration of all theindividuals for each hinge. We can also read some usefulinformation from this plot. For instance, R–1 has strongpositive correlation between speed and vibration at range of0.3, but it becomes strong negative at range of 0.6. If thevalue is larger than 1.6, high speed is realized without largevibration. Figure 9 summarizes the tendencies we found inthose plots. Observed tendencies can be classified into fourmodes. (There is no rigorous explanation of the reason whywe have four different modes.)

1st: Intermittent rotation with low speed and large vibra-tion.

2nd: Transient from intermittent rotation to continuousrotation. Vibration is decreased along with the speed.

3rd: Smooth rotation with low vibration. (Desired)4th: Unstable bouncing.The qualitative knowledge we have extracted from the

results of HW optimization is summarized as follows.

¡0.4

Stiffness Coefficient

Ve

locity

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

0.4Over

Over

Stiffness Coefficient

Stiffness Coefficient

Stiffness Coefficient

Stiffness Coefficient

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

(a) R¡3 (h) F+3

(c) R¡1

Stiffness Coefficient

(f) F+1

(d) R¡0

Stiffness Coefficient

(e) F+0

0.3

0.2

0.1

0.0

¡0.1

¡0.2

¡0.3

¡0.4

Ve

locity

0.4Over

Over

0.3

0.2

0.1

0.0

¡0.1

¡0.2

¡0.3

Ve

locity

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

¡0.4

0.4Over

Over

0.3

0.2

0.1

0.0

¡0.1

¡0.2

¡0.3

Ve

locity

¡0.4

0.4Over

Over

0.3

0.2

0.1

0.0

¡0.1

¡0.2

¡0.3

Velo

city

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

(b) R¡2

Stiffness Coefficient

(g) F+2

¡0.4

0.4Over

Over

0.3

0.2

0.1

0.0

¡0.1

¡0.2

¡0.3

Velo

city

¡0.4

0.4Over

Over

0.3

0.2

0.1

0.0

¡0.1

¡0.2

¡0.3

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

¡0.4

Velo

city

0.4Over

Over

0.3

0.2

0.1

0.0

¡0.1

¡0.2

¡0.3

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

¡0.4

Ve

locity

0.4Over

Over

0.3

0.2

0.1

0.0

¡0.1

¡0.2

¡0.3

Fig. 10. One dimensional scatter plots on velocity per stiffness value. Becauseof a symmetric robot mechanism, plots for R–i and F+i have the reversedcharacteristics. An area of a square at “Over” represents a number of stringswhich generate a rolling motion at over 0.4(m/s).

R–3: Higher stiffness gives forward direction, and low oneprovides 2nd mode.

R–2: Higher stiffness gives 3rd mode.R–1: Higher stiffness gives 3rd mode, however it does not

affect the direction of rolling.R–0: Stiffness must be low for forward locomotion.F+0: Stiffness must be higher than the standard coefficient

(= 0.4) to get forward direction. However, it is too

Velocity

Vibration

0.0 0.1 0.2 0.3 0.4

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0

0.4

1.9

2.4

4.0

(a) R¡3

Velocity

Vibration

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.2

0.6

1.0

1.5

2.1

2.8

3.4

4.0

(b) R¡2

Velocity

Vibration

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.3

0.6

0.9

1.6

2.2

3.0

3.7

(c) R¡1

Velocity

Vibration

0.0 0.1 0.2 0.3 0.4

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0

0.2

0.5

0.7

(d) R¡0

Velocity

Vibration

0.0 0.1 0.2 0.3 0.4

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0

0.3

0.8

1.0

1.3

2.0

(h) F+3

Velocity

Vibration

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.2

0.6

0.9

1.5

2.2

3.9

(f) F+1

Velocity

Vibration

0.0 0.1 0.2 0.3 0.4

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.3

0.6

1.4

1.8

2.6

(e) F+0

Velocity

Vibration

0.0 0.1 0.2 0.3 0.4

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0

0.2

0.7

1.0

1.1

(g) F+2

Fig. 11. Scatter diagram on velocity vs. vibration. Data of several stiffnessvalue are only cited.

high, locomotion becomes 1st mode.F+1: Higher stiffness often gives high speed locomotion,

however sometimes it brings 4th mode. It doesn’taffect the direction.

F+2: Low stiffness gives high probability of forward di-rection. However it must not be too low to obtain2nd or 3rd mode.

F+3: Low stiffness gives 2nd mode and high probabilityof forward direction.

Sum

1.2088

0.0010

33( (:

mean

variance

top( (

1.2144

0.0011

98( (

0

2

0( (

1

1

2( (

2

0

4( (

1.2165

0.0012

186( (

Ran

k a

nd

conver

t to

poin

ts

Characteristicsvalues

2 4 6

HigherMiddleLower

Neighborhoods

Divide into

3 sets.

6561

218721872187

(2)

(3)

(4)

(5)

(6) Higher is better (type III).

(1)

oper

atio

ns

in e

ach h

inge

Searching center SDC string

Expansion&

Evaluation

Fig. 12. One cycle of CD optimization method

TABLE III

RATING OF STRING SET

criterion best middle worst(a) 2 1 0(b) 2 1 0(c) 4 2 0

We derived the local optimal SDC by using the knowledgein total.

Local optimum:A4D9D9A1 A6B8A3A1 (6)

B. Global optimization by collective descent method

In this subsection, we describe a method to fine-tune thelocal optimal solution obtained (Eq. (6)). CD optimizationis an incremental optimization method based on steepestdescent method. Performance of the whole individuals in aneighborhood of a certain central SDC string is evaluated first,and then the center is shifted to the steepest direction, whichimproves the performance by using certain criteria. We use(Eq. (6)) as the initial central SDC string.

Figure 12 shows the procedure of CD optimization. Weexplain the procedure along with the flow in this figure.

a) Generate individuals in a neighborhood of the centralSDC string.: The set of individual in a neighborhood isdefined as follows. For instance, if the central SDC string isA3D5D9A1 A5C7B0A2, we allow three stiffness levels aroundthe central value at every position. Therefore it is given by acombination such as,A2

A3A4

D4D5

D6

D8D9E0

A0A1A2

A4A5

A6

C6C7

C8

A9B0B1

A1A2A3

(7)

Each neighborhood is divided into three sets; Lower, Middleand Higher. These sets include the same number of individuals.If we focus on the hinge F+1, they are

Type I) Equilibrium. Do nothing.

Type II) Lower is better. Shift down.

Type III) Higher is better. Shift up.

poin

ts mid

dle

hig

her

low

er

Fig. 13. Stiffness Coefficient shifting rules.

Lower Set: ∗ ∗ ∗ ∗ ∗ C6 ∗ ∗ ∗Middle Set: ∗ ∗ ∗ ∗ ∗ C7 ∗ ∗ ∗Higher Set: ∗ ∗ ∗ ∗ ∗ C8 ∗ ∗ ∗

where “∗” denotes some levels in the neighborhood (7).b) Evaluate all the individuals by simulation.: Each set

is evaluated by the following measures.(a) Mean of locomotion speed(b) Variance of locomotion speed(c) Number of individuals within top 2.5%

Measure (a) is to guarantee the average high performance, (b)is to suppress the dispersion of the performance and (c) is topursue the highest performance. From the engineering pointof view, we have to combine these measures to get a realisticsolution. In order to compare different sets, a total point ofeach set is calculated by using rules in TABLE III.

Each set has three points corresponding to the measures (a)–(c). For instance, if the set got the first place in terms of meanlocomotion speed it gets 2 points. Other measures are alsoevaluated in the same fashion, and the total point representsevaluation of the set.

c) Shift the central SDC string.: Based on the totalevaluation of each set, the central stiffness value is shiftedto the next level. Depending on the pattern of total pointsdistribution of the High, Middle, Low sets, to shift up or toshift down or unchanged is determined by the rules in Figure13. By applying this procedure to every hinge, we obtain anew SDC string for the next iteration.

d) Repeat above until it converges.:

C. Results of CD Optimization

In Figure 14, the transition of the central SDC string isshown. Starting from the initial SDC string (iteration=0), itconverges to a small periodic oscillation between two SDCstrings after the sixth generation. Therefore, we judged it isconverged to the global optimum.

The result is

Global optimum : A1D8D9A1 A1B9A4A1 (8)

Iteration

Stiffn

ess c

oeffic

ient

0 2 4 6 8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1 3 5 7 9

F+0

F+1

F+2

F+3

Local optimum

R¡3

R¡2

R¡1

R¡0

F+0

F+1

F+2

F+3

Global optimum

R¡1

R¡2

R¡3

R¡0

A4D9D9A1 A6B8A3A1 A1D8D9A1 A1B9A4A1

Fig. 14. Convergence of SDC string by CD optimization method

Figure 15 is a performance plot of all the individuals for dif-ferent generations. First generation (local optimal generation)shows low vibration, however disparity of locomotion speedis large (a). After the first iteration, it forms relatively smallcluster with small disparity both in speed and vibration (b).Along with the iteration, it becomes faster in speed and showsstrong negative correlation between speed and vibration (c)–(e). This is because our SDC evaluation criteria put emphasison the maximum speed (Section IV-B). However, in theeighth generation, 81% of all 6561 individuals have less than0.05(m/s) vibration, which implies that the cluster convergedto high speed and low vibration area as a whole. Simulationof obtained SDC string is also shown in Figure 16.

Locomotion generated by final SDC string“A1D8D9A1 A1B9A4A1” is shown in Figure 16. Itsaverage locomotion speed was 1.26(m/s), vertical vibrationwas 0.22(m/s).

D. Optimization of Stopping motion

We also optimize a SDC string for stopping motion. In thissection the results are briefly described.

SDC strings for stopping motion are evaluated by time tohalt the motion and magnitude of spurious vibration at thecenter of gravity during the stopping motion. These criteriaare evaluated for 10(s) after 10-second run-up to accelerate to1.0(m/s). Parameters are, T = 0.6(s), d = 0.005(mNs · rad−1)and simulation step = 2.5(ms).

Snapshots from the simulation are shown in Figure 17. Thissequence shows the ability of SDC for stopping motion.

V. DISCUSSION & FUTURE WORK

There are several issues remained for future work. Inhardware, we have to improve our module of mechanicalsoftness in terms of reliability, controllability and low-frictiondesign. Actually we are now working on another type of VSH.

Another issue in hardware is that the prototype BIYOndescribed in this paper is one-dimensional. We need additionaldegree of freedom to realize turning motion on a plane. Inorder to do this, we need to add more axes on the crawler,that require complicated control.

0.8 0.9 1.0 1.1 1.2 1.3

0.000

0.050

0.100

Velocity

Vibration

0.8 0.9 1.0 1.1 1.2 1.3

0.000

0.050

0.100

Velocity

Vibration

0.8 0.9 1.0 1.1 1.2 1.3

0.000

0.050

0.100

Velocity

Vibration

0.8 0.9 1.0 1.1 1.2 1.30.000

0.050

0.100

Velocity

Vibration

0.8 0.9 1.0 1.1 1.2 1.3

0.000

0.050

0.100

Velocity

Vibration

0.8 0.9 1.0 1.1 1.2 1.3

0.000

0.050

0.100

Velocity

Vibration

(a) iteration: 0

(b) iteration: 1

(c) iteration: 2

(d) iteration: 3

(e) iteration: 5

(f) iteration: 8

unification

strong negative

correlation

Fig. 15. Transition of velocity-vibration clusters

As software or algorithm issues, unlike Genetic Algorithm,we do not scan over the entire space SDC string, thus wecannot guarantee that the obtained string gives the globaloptimum. (In a sense, our result means that the evaluationlandscape of our problem is relatively smooth.) Nevertheless,our method is still advantageous because it indicates function-ality of each hinge which is useful to tune up the stiffnessdistribution. We believe this is very important in practicalapplications.

Some extensions of the method also can be considered. Forinstance, we optimize only a single SDC string for each of therolling and stopping motion, however it is better to prepareseveral strings and apply them depending on phases, such asstanding motion from standstill, acceleration or decelerationand steady movement, of a desired locomotion.

VI. CONCLUSION

In the present work we propose a new method of locomotioncontrol, named “Stiffness Distribution Control (SDC)”, torealize various locomotion of a closed link robot with me-chanical softness. SDC directly defines the reference stiffnesscoefficient at each hinge by SDC string which represents thedistribution of the stiffness. This simple method does not needcostly calculation at all and thus is suitable for controllinga closed link robot. We build BIYOn as the prototype ofclosed link robots, the variable stiffness hinge (VSH) ofwhich is newly designed. SDC is experimented on BIYOnand generate a rolling motion in practice. We also develop asystematic optimization method of stiffness distribution based

3148 steps, 15.74 sec

3156 steps, 15.78 sec

3160 steps, 15.80 sec

3164 steps, 15.82 sec

3152 steps, 15.76 sec

3168 steps, 15.84 sec

3172 steps, 15.86 sec

3176 steps, 15.88 sec

3180 steps, 15.90 sec

3184 steps, 15.92 sec

Fig. 16. Fast and smooth rolling motion obtained by optimization.

4000 steps, 10.00 sec

4048 steps, 10.12 sec

4096 steps, 10.24 sec

4144 steps, 10.36 sec

4192 steps, 10.48 sec

4240 steps, 10.60 sec

4288 steps, 10.72 sec

4336 steps, 10.84 sec

4384 steps, 10.96 sec

4432 steps, 11.08 sec

Fig. 17. Sudden stopping motion obtained by optimization. SDC string isA0A2A2D8 D8A0A0A5. By a large deformation, the robot stops suddenly.

on collective descent method and succeed in optimization toobtain a fast and smooth rolling motion and a sudden stoppingmotion.

SDC can apply only planar closed link robots now. Someextensions are necessary that we utilize this method in variouskinds of closed link robots. Nonetheless, the newly developedcontrol method will allow us to simplify a problem of con-trolling closed link robots significantly.

ACKNOWLEDGMENT

This research was partially supported by the Ministry ofEducation, Science, Sports and Culture, Grant-in-Aid for Ex-ploratory Research, 16650036, 2003.

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