wrench contact for robots

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Proceedings

o f the 1003 IEEE

loternational Cooferenee on Robot ics &Automat ion

Taipei, Taiwan.

September

14-19,

1003

FSW Feasible Solution of Wrench) for Multi-legged Robots

SAIDA

Takao,

Yasuyoshi YOKOKOHJI,

Tsuneo

YOSHIKAWA

Department of Mechanical Engineering, Graduate School of Engineering,

Kyoto University, Kyoro, 606-8501JAPAN.

st [email protected], yoko koji ,yoshi} @m [email protected]

Abstract

In

this paper; w e focuse

on

a problem how to confirm a

feasible condition of applied farce to a multi-legged robot

on rough terrain. The problem often appe als as a subject

o

walking stabiliry criterion fo r a legged robot. ZMP (Zero

Moment P oint) is one of the well-known criteria.

It

shows

the condition as footprints o f a legged robot,

but

it cannot

be defined on the rough terrain . Therefore, we suggest a

new criterion FSW (Feasible Solution o Wrench), which

give s the feasible condition even on the mus h terrain from

rhe viewpoint o “wrench” a special representation of

forc e screw. And we present two short examples of FSWfo r

a biped robot, how ro analyse the validity of ZMP on stairs

and how to design a forc e trajectory on rough terrain.

1. Introduction

Multi-legged robots are expected to find ways into var-

ious works and to collaborate with humans in the future.

But, as yet

so

far, it is a difficult problem of them to ac-

complish their stabile mobilities in

any

environment. ZMP

(Zero Moment Point) [I] is one of famous stability crite-

ria for a walking robot [2-5]. Its physical fea ture is known

as “ COP (center of pressure)” between the ground and the

feet of the robot [6]. But it cannot he defined when the

robot moves on plural contact planes, for example, going

up stairs or opening

a

door.

To

avoid the disadvantage of

COP

(ZMP) in such situation, Kogami [7] suggested the

enhanced ZMP constraint and Sugihara [SI proposed the

Z M P on the virtual horizontal plane. But their validities or

physical meanings have not been confirmed.

On

the other hand, to analyse singularities of paral-

lel

robot manipulators or to measure qualities of multi-

fingered robot hands, another force criterion wrench

is

often adopted [9-131. Its definition is not depending

on

geometric features.

From the veiwpoint of wrench, we propose

a

new cri-

terion

FSW (Feasible Solation of Wrench )

to confirm feasi-

ble conditions of applied forces to multi-legged robots even

on rough terrain as well as on a single plane. This paper

is organized as follows. Sec.2 introduces some important

features

of

ZM P and wrench, and then, S ec.3 defines FSW.

Sec.4 describes som e characteristics of FSW. Finally, Sec.5

shows two examples of FSW.

fE

T r a n s f o r m a t i o n

-

2. ZMP and Wrench

In

this section, we introduce two important indices, ZMP

and wrench.

To

discuss following derivations simply, let

p ,

f,

and denote position, force, momen t and direction

cosine

R”’),

espectively.

2.1. ZMP Zero Moment Point)

ZM P (Zero Moment Point) [I ] is a special point where

the applied mo ment is parallel to the normal of the contact

plane (Fig.1). Now, suppose that force f E and moment

n E

are applied at point

p , on

the contact plane. From this

configuration, ZMP

pmp

is specified as

where f ZMP and

n m p

are the force and the moment ob-

served

at

the ZMP. ez is the unit normal of the contact

plane. f, E E’) and e E are the magnitude and the di-

rect ion cos ine of f E .

In

addition, the symbol “ x ” and the

upper-right subscript

“ T

mean the cross product and the

transpose of vector, respectively.

ZMP is equivalent to “COP (center of pressure)” on the

contact plane [6]. It is only available within the minim um

convex hull enclosing the contact region between the end-

effector (foot) and the external object (ground). But its def-

inition has following problems:

0-7803-7736-2/03/ 17.0002003 IEEE

381

5

mailto:[email protected]






1 It cannot be clearly defined on rough terrain because

ez

cannot be identified in such condition.

2) It is not explicitly sensitive to any shearing force on

the contact plane because the dominator of Eq n.(l) is

insensitive against the shearing force.

2.2. Wmnch

“Wrench” is a special representation of force screw

[9,

101.

Its force and mom ent are parallel to each other (Fig.1).

Its line of action is referred to as “wrench axis”, p,. From

f E , nE

and pE , the axis

pw

is specified as follows:

where

f

and

n w

are the force and the mom ent observed

on the wrenc h axis p,. s an arbitrary scalar parame ter.

“Pitch”, pw, is another fundamental component of the

wrench. It gives the ratio between f nd

~ L W .

Substituting Eqn.(7) into

5) .

we get

nw

=

P w f w . (8)

Furthermore, wrench has following characteristics:

1) It can be defined even on plural contact planes.

2) The nearest point of its axis from the observation

point can evaluate the effect of applied shearing

forces

on the

contact plane.

3 Its axis passes through

COP ZMP)

if and only if

e,

is

parallel to

ez

or pw is zero.

4)

Its moment has minimum Euclidean norm in the

equivalen t representatio ns of the forc e screw.

3. FSW(Feasib1e Solution of Wrench)

In this

section, we suggest a new criterion

FSW(Feasib1e

Solution o Wrench).

FSW is derived from axis and pitch

of wrench. It gives an insight for a multi-legged robot

to

keep its dynamic equilibrium condition on rough terrain.

3.1. Definitions

of FSW

Suppose that

f

E ,

n E

are the resultant force and mo-

ment at the observation point

p,,

respectively (Fig.2). To

discuss simply, let

p,

locate at the origin of a coordinate

system. Then, f

E

and n E are specified as

M u l t i - l e g g e d

R o b o t

C o m p O n e n t

A x i s N ’ ’

Fig. 2.

point.

and

wrench.

Component

forces,

resultant force

and

mornen1

at observation

where f and n E % enote the applied force and moment

to the multi-legged robot at the 2-th application point p E t.

Substituting Eqn.(9),(10) into 4)- 6), we get the

wrench for the resultant force screw.

In Eqn.(4),

let

pFsw

be pWls=o. The point pFsw an be regarded as the inter-

cept where a plane cuts out the wrench axis p,. The plane

has

e B

as its normal, and it passes through pE .

So

that, we

can give two definitions about wrench.

Def I-Wrench Plane. Wrench Intercept;

“Wrench

Plane” is the plane which contains an observation point

of

resultant

force

and

moment.

Its normal

equals

to

the

wrench axis of the force screw. “Wrench Intercept” is the

intercept where the plane cuts through the axis.

Substituting Eqn.(9) and (10) into pFS w,we get

eExpE ) + ’ O

(11)

P n-F S W f

Pt.FSW

+ P m ~ F S W P O

where p,.Fsw,p,.Fsw,p~.Fsw are the first, second and third

element in the right side of pFsw, respectively. In the sam e

way,

let

~ F S W

be the pitch at

pFsw.

Substituting Eqn.(9)

and (10) into (7).

mSw

s specifided as

Where e E is also the direction cosine of the resultant force

f E

in Eqn.(9). In addition, we d efine the bias filter matrix

3816



E E= [ 3x3

~ e g ]

R3 x 3 .t gives

f E s y i = E E f E i (13)

PE,,<

=EEPE~

T

T

T

f E i i = e E f Eir

(14)E Z i

= e E p E < ,

nE z i = e E n E i , n E D y i= E E n E i (15)

where

IsX3

s the RsY3dentity matrix.

Then, we define FSW, the intercept

p,,,

its compo-

nen ts p..Fsw,p,~Fsw,p,Fsw, and the pitch PFSW as follows.

Def:

2-FSW

“FSW (Feasible Solution of Wrench)” is

the set of the feasible reaction wrenches which

can

act as

the external forces on a multi-legged robot. The wrench is

specified from the component forces and mom ents acting

between external objects and the end-effectors of the robot.

It is not restricted from force-transmission mechanisms be-

tween the objects and the robot, for instance, point contact,

flat contact, plural surface contact or mag netic field action.

But of course, it must keep constraints between them, for

example, torque limit or frictional condition. And FSW is

composed of following “I-FSW and “p-FSW.

Def: 3-i-FSW “i-FSW (intercept FSW)” is the set of

feasible wrench intercepts. Its element is denoted as pFsw.

Def:4-n-FSW; t-FSW;

m-FSW

“n-FSW (normal force

FSW)”, “t-FSW (tangential force FSW)” and “m-FSW

(moment FSW)” are components of i-FSW. They denote

the effects of normal component forces, tangential compo-

nent forces and component moments on the wrench plane.

Their elements are denoted as p,~, , , ,

p,.,,,, p,

The

sum of them represents p,,,.

Def:

5-p-FSW:

“p-FSW (pitch FSW)” is the set of fea-

sible wrench pitches. Its element is denoted as

p ~ s w .

4. Characteristics

of

FSW

4.1. COP ZMP) Domain Compatiblility

In Eqn.(l I , p,~,, in n-FSW is a kind

of

COP ZMP).

Its equation is similar to the problem to solve the center of

gravity in distributed mass systems.

If there is no fEii

f E Z j

< O V i , j ) , p,.Fsw s the COP

on the wrench plane. And the point is in the minimum

convex hull which encloses the footprints projected onto

the wrench plane (Fig.3). But, however, if there exists

f~~~E l j ( i j ) , p,.,, is an external dividing point.

Its domain is not closed. This case usually takes place in

the grasping operation of multi-fingered robots.

Unless the resultant force f E is cance led, n-FSW exists.

4.2. Effect

of

Shearing Forces and Steps

p,.,, in t-FSW indicates the effect that shearing forces

act at uneven heights on the feet. This kind of domain does

not appear in the set of

COP (ZMP).

s ts special condi-

tion,

if

there exists

p~~

such that

p~~ =

p ~ ~ ~ V i ) ,r.FSw

R e s u l t a n t

Fig.

3.

n-FSW of

a

multi-leggedrobot is

similar

to the

domain

of ZMP

i f th ere i s n ofEI i lEI ,

< O ( ’ d i , j ) inE qn. ( l l ) .

becomes

0

Substitute Eqn.(9),(13)into(lI), we get

E E f E e E

e o .

(16)

We define Eqn.(lb) is “zero t- FS W . It also represents a

special condition between the resultant force and the com-

ponent forces. This problem will be mentioned

in

Sec.S.2.

4.3. Normal Vector of Each Contact Plane

Now,

we assume that each contact surface is flat between

the external object and the end-effector of the multi-legged

robot. And we also assume that each application point p,,

is

COP (ZMP)

n the surface. Let the normal vector of the

each contact surface be eE, .Then, the applied component

moment n E z can be simplified as

PE=

f E

~ -__

n E l = n E l e E 2 . (17)

Eqn.(l7) and

( I

1) simplify

p,.,,

as

This shows the relationship between the resultant force and

the topographic feature of the external object. And

p,,,,

can be independent of

p,.,,

an d P,.~,,.

4.4.

Observation Invariance

FSW is invariant for the ObSeNatiOn point

po

because

the wrench the element of FSW s invariant for the

point. It is shortly proved as follows. In Eq n.(l ) and (2),

the dominator marks its maximum when the normal

ez

is

parallel to the resultant force

f

E . In Eqn.(4) and (51, the

wrench always keeps such condition because the normal of

the wrench plane is defined parallel to the force.

So

hat, the

domain of FS W keeps its shape even if

po

was changed.

381

7



To

comp ute the two application points, i-FSW

pFsw

nd

virutal ZMP

pNp,

et

f E z v = f EzVl = - fEzyZ.

Then,

substituting

f E r y

into Eq n.( ll) denotes

pFsw s

2

P F S W = ( z % P E $ y i ) + P O

+ PE12 - P E z I ) -

f E z y )

.

(19)

To compute the virtual Z M P

[7,8]

(let it he

ljzMp).

eglect

the height of the application points on the target virtual

plane. In this application, it means to neglect

p ~ ~ i

rom

Eqn.(l). Then,

ljZMps

specified

as

f E

a n d

T a r g e t

L' t e .1 Z M P

2

l j Z M F = ( 5 2 P E z y i )

+Po (20)

where we assumed that the direction of resultant force e E ,

the nonnal of stair eEl and the no rmal of the target virtual

plane

dz

lie in the sa me direction.

In Eqn.(lY) and (20), there

is

a deviation between

pFsw

and

ljzMPn

the target plane. But it is inconsistent on the

force screw s at the points. Th is problem is specified as

Fig. 4. Going

up

stairs with

a

couple

of

two reciprocal shearing forces.

4.5.

Stabi l i ty and Projected Footprints

Although

pFsw

s out

of

the footprints projected onto the

wrench plane, it does not mean any locomotion instabil-

fw = f z ~ p

=

f ~ r n W =

e L M P (21)

ity due to the remainder of t-FSW and m-FSW. But, how-

ever, even if

pFSw

s within the i-FSW, the legged robots

might be falling down when they break the restriction of

the p-FSW. The detail of this subject is our future work.

5.

Application

of

FSW

As

mentioned in Sec.2.1, ZMP cannot be defined on

rough terrain. There fore, instead of ZMP, we will apply

FSW and show two examples for a biped -goin g up stairs

and walking on rough terrain.

5.1.

Going

up/down stairs

To

go up stairs with

a

ZM P criterion, biped robots need

some techniques. On e of the convenient techniqu es is

to

make a virtual Z M P on a virtual plane

[7,8].

But the

ZMP cannot estimate any effect of shearing force, and it

i s

not invariant for the plane.

In this section, we show the difference betw een FSW an d

virtual ZMP o n stairs in terms of tumbling en or m oment.

In Fig.4, suppose. following ite ms

on

each foot:

1)

Each foot h as flat contact on each stair.

2)

eEi i s

the normal vector of e ach stair.

3) Application point

pEi

s

COP

(ZMP)

and fixed.

4) Moment

n E i

equals to 0.

5 )

Shearing force

f E z y i

is fixed.

6)

Two shearings are canceled (fE zy l+ fE my 2

= O .

in other words, the following tumbling moment A n does

not appear in the performance of

pM P

A n

= (PFSW - B M P

x f~

= PE22 P E z l )

( f E r y

x eE) .

(22)

Note that even if the biped

is

under con trol with such virtual

ZM P criterion,

A n

must he com pensated only just by local

feedback control of each joint. But FSW provides

a

good

criterion even if the robot walks over the undulatio ns of the

stairs.

5.2.

I r regu la r Te r ra in

ZM P control scheme can effectively stabilize the loco-

motion of biped robots on single plane [ 2 4 1 but not on

rough terrain. Mainly, this problem lies on the design

of

force trajectory with the robot.

So

that, in this section, we

show an application of the force trajectory desig n

as

an in-

verse problem of

pFsw

n

f

and

nE.

Now, we assume follow ings to solve the inverse problem

on the rough terrain by

FSW:

1 Each support foot lies flat on the rough terrain.

2) Application point pEi s COP

(ZMP)

and fixed.

3)

egi is

the normal vector of each contact plane and it

4) Coulomb friction model is adopted between the feet

is outward from the rough terrain.

and the terrain. And the friction model is static one.

And the schematic view of this subject is shown in Fig S.

3818



M o v i n g D i r e c t i o n

--a

Fig.

5. Schematic view

of

walking

robot

on ough rerain

with

FSW.

Let i EPX e ( f E , / f E ) . Then,

i

specifies e E .

p,Fswand

pD.Fsw

n Eqn.( 11) as follows:

e E

f

E / f E = f E z / f E =

4i

(23)

i

P,.FsW = -PC E+~ (24)

i

PwFSW = C p E i z y i e Z i . 25)

With respect to the biped robot, the inverse problem of

i

Eqn.(24) with (23) for 4i can be solved

as

X e E pc Fsw

(26)

while X is an arbitrary scalar parameter. Th e parameter can

be specified from another parameter

6:

P E z 2

PE21 E1 2

[ 4] Al eE l (PE z1 P Ea 2) -PtGF.SW

(27)

=

where { represents the virtual height of application point

from the wrench plane. Therefore,

X

shows the dividing

ratio between the tw o application points

p , ,

and

p E l .

=

p ~ ~ 2

it means p , ~ , , , =

0

as

mentioned in Sec.4.2), +< an be solved in Eqn.(26) and

X

still conserves the meaning of the ratio. And also, the next

equation (2 8) is always satisfied.

Substituting Eqn.(26) into (25),pmrSws simplified as

Note that even if

Pn-FSW = P E z y l + 1 p E r y 2 . (28)

After all, Eqn.(28) means that, whatever

p,.,,,

is, only

X

explicitly decides pn~Fsw.

c o m p o n e n t

o r c e

E <

4 fE

c o n t a c t

n e

Fig. 6 . Parameters of

frictional

cones on he conta~l lane.

As mentioned at the top of this section, we assume

Coulomb friction model

on

each foot. The m odel restricts

the applied forces on the feet. This condition is shown in

Fig.6. Co nsider static translational friction at p E i , he com-

ponent force

f

E i

can be specified as

-

f E t

=

f E @ i = f E i ( e E i + P a i e E z y i ) (29)

-

where e E Z y i s the unit orthogonal vector to e E i . f E i is the

pushicg intensity.

is the translational friction coefficient.

And f E i should be J E ~ 0, ai should be 0 5 cy, 5 1.

Substitute Eqn.(26) into (29), we get the frictional con-

dition

on

the each foot

as

follows:

30)

<

=

{ A i e E

PE^^ P, . FS W}

P

e

{ X i e E + APE,^

P,FSW}

where

APE.^ = P E Z I Ezz) , APE,Z

( P E z 2 - P E a l ) .

Now, we assum e another item to solve simply the inverse

X I = A , Xz = (1 A .

problem in this application.

It simplifies &, E i and

ai

inEqn.(26), (9) and 30)

as:

p,.,, = 0 (zero t-FSW).

f i

= X i e E

(31)

si

X i f E

h f E eE

(32)

where 8, is the angle of friction. It interprets the limit of a.

into

0 cos 8i 5

cos

(tan-' P ) .

2 in Eqn.(29), it requires that

X i

must be

0 5

i

5

1 n

Eqn.(32). So

that, { should be

To keep

m p PE l i

5

5 m y E i i

(34)

This constraint and Eqn.(28) set

p,.,,

within the min-

imum convex hull of the footprints of the biped robot.

Moreover, Eqn.(32) requires that the resultant force

f

E

should be parallel with the each component force f

Ei

act-

ing on the foot of the

robot (Fig.7).

381

9



Fig. 7.

applied forces are parallel tu each other.

Relationship between force distribution and parameter

A. The

Next, consider static rotational friction at

p, ; .

The com-

ponent m oment

nEi

cting on the each foot can be speci-

fied as

n E i

=

D i v f E i e E i

(35)

where, U is the rotational friction coefficie nt.

&

should he

-1

5

fl i

5

1.

And remember that we assumed the point

p Ei

as the

CoP(ZMP)

on the each foot.

Eqn.(32) gives

n E i

as

(36)

n E i

=

X i fl i f E ( v e E i e E , e E )

where

j E i

=

X i e e E

is easily specified fro m Eqn.(29)

and (32). Immediately, as to the resultant mom ent of the

wrench,

nw

in Eqn.(S), we get

= C X i P i f E v e E ( e g i e E ) ' )

. (37)

i

In the sam e way, in

Qn.

11) and (IZ), p,,,, and p ~ s wre

specified as follows:

i

Fig.8 shows the schematic view of p, and

n w .

Th e resu lts specifyp,,,,

f w, w.

t means that we can

treat a lot of cases on trajectory designs for biped robots on

rough terrain. And for example, to employ the results, let

X be

a

function of time and then solve a locom otion of the

robot on

X t).

Its scheme is sim ilar to the for ce distribution

planning in 2D locom otion proposed by Pratt

[SI.

Whereas,

a simulation or an experiment of this application will be our

future work.

Fig. 8. Relationship hetween moment distribution and parameter

X

6. Conclusion

In this paper, we have suggested a new criterion FSW

(Feasible Solut ion of Wrench)

for multi-legged robots. The

criterion shows the feasible condition of forces applied to

the robot even on rough terrain, from the viewpoint of

wrench. FSW is composed of some domains which have

different physical meanings on the wrench plane: n-FSW

(pressure), t-FSW (shearing and step), m-FSW (m oment)

and p-FSW (force-moment ratio).

We have shown the usefulness of FSW by explaining two

typical applic ations for a biped robot on rough terrain

go-

ing up stairs and walking on the terrain. O ur future research

will be focused on how to gene rate and stabilize the motion

of multi-legged robots with FSW over rough terrain.

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Auto matio n. pp.52E-531 (199 5).

(2001).

3820

wrench contact for robots

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