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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
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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
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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.
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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.
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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).
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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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