IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 34, NO. 5, SEPTEMBER 2004 630
Forces Acting on a Biped Robot. Center of
Pressure—Zero Moment Point
Philippe Sardain and Guy Bessonnet
Abstract—In the area of biped robot research, much progress
has been made in the past few years. However, some difficulties re-
main to be dealt with, particularly about the implementation of fast
and dynamic walking gaits, in other words anthropomorphic gaits,
especially on uneven terrain. In this perspective, both concepts of
center of pressure (CoP) and zero moment point (ZMP) are obvi-
ously useful. In this paper, the two concepts are strictly defined, the
CoP with respect to ground-feet contact forces, the ZMP with re-
spect to gravity plus inertia forces. Then, the coincidence of CoP
and ZMP is proven, and related control aspects are examined. Fi-
nally, a virtual CoP-ZMP is defined, allowing us to extend the con-
cept when walking on uneven terrain. This paper is a theoretical
study. Experimental results are presented in a companion paper,
analyzing the evolution of the ground contact forces obtained from
a human walker wearing robot feet as shoes.
Index Terms—Biped robot, center of pressure (CoP), mechanical
feet, uneven terrain, walking, zero moment point (ZMP).
I. INTRODUCTION
T
WO FRENCH laboratories, LMS and INRIA Rhône-
Alpes, have designed and constructed an anthropomor-
phic biped robot, Bip. The goals and initial results of the project
are reported in [1] and [2] and the implementation of the pos-
tural motions and static walks achieved until now are described
in [3]. The current research is directed toward the generation
of anthropomorphic trajectories, and toward efficient ways for
the biped robot to control them. The robot is fitted with feet
equipped with sensors measuring the ground-foot forces, in
order to exploit the concepts of center of pressure (CoP) and
zero moment point (ZMP). The notion of ZMP has been known
about for more than thirty years, but is in no way old-fashioned,
since as long as gravity forces govern walking gaits, the ZMP
will be a significant dynamic equilibrium criterion. In addition,
this quite useful concept has not been completely explored, and
unfortunately some misinterpretations are sometimes encoun-
tered in the literature.
In this paper, we refer the point associated with contact forces
as CoP, while ZMP is considered to be related to gravity plus
inertia forces. Strict definitions of CoP and ZMP are specified
in Section II, and concise algebraic relationships for the com-
putation of both points are formulated. In Section III, before
Manuscript received December 11, 2002; revised February 25, 2004. This
work was supported in part by the Council of the Poitou-Charentes Region and
the Council of the Vienne Department, France, under Grant 95/RPC-R-94. This
paper was recommended by Associate Editor C. E. Smith.
The authors are with the LMS, Laboratoire de Mécanique des Solides, CNRS-
Université de Poitiers, BP 30179, 86962 Futuroscope Chasseneuil Cedex,
France (e-mail: Philippe.Sardain@lms.univ-poitiers.fr; Guy.Bessonnet@lms.
univ-poitiers.fr).
Digital Object Identifier 10.1109/TSMCA.2004.832811
discussing the relevance of the CoP-ZMP as a means of con-
trolling the dynamic equilibrium of bipeds, the coincidence of
the two points is proven. Finally, Section IV suggests an ex-
tension of the CoP-ZMP concept when the feet of the biped do
not lie in the same plane. Indeed, the concept holds only with
respect to a single plane, and cannot be directly applied when
the biped walks across an uneven terrain. The understanding of
the demonstrations on the one hand, and the physical interpre-
tation of the phenomena on the other hand are facilitated by the
notion of the central axis of moment field, which is recalled in
Appendix I.
This theoretical study is aimed at clarifying and extending the
CoP-ZMP concepts. As a perspective, to go further toward the
knowledge of walking gaits with mechanical feet, specific data
are required. The tests carried out to validate the instrumenta-
tion of the Bip feet have been performed by a human walker
wearing the Bip rigid soles as shoes [4]. This experimental ap-
proach has proved to be a relevant way of studying the influence
of “mechanical” feet on the gait of the subject. The results are
presented and analyzed in the companion paper [5].
II. CoP V
ERSUS ZMP
The concepts of CoP and ZMP are quite useful for the control
of the dynamic equilibrium of bipeds, but first the exact meaning
we attach to these notions has to be defined.
A. Definitions
The forces acting on a walker can be separated in two cat-
egories: 1) forces exerted by contact and 2) forces transmitted
without contact (gravity and, by extension, inertia forces). The
CoP is linked to the former, and the ZMP to the latter.
1) CoP Definition: Let us consider a biped when one single
foot is in contact with the ground. The field of pressure forces
(normal to the sole) is equivalent to a single resultant force, ex-
erted at the point where the resultant moment is zero. This point
is termed CoP.
At any point
of the sole [see Fig. 1(a)], the elementary con-
tact force
can be split up into two components, a normal
component
, which is the local pressure force, and a tan-
gential component
which is the local frictional force
(1)
with
being the unit vector normal to the sole , oriented
toward the foot, outwards from the support surface.
1083-4427/04$20.00 © 2004 IEEE
631 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 34, NO. 5, SEPTEMBER 2004
Fig. 1. Contact forces and moments acting on the sole.
Consequently, the resultant of these elementary contact forces
appears as the sum, on the one hand, of the local pressure forces
(2)
whose moment about any point
is
(3)
and, on the other hand, of the local friction forces
(4)
whose moment about
is
(5)
with the superscripts
and denoting pressure and friction
components, respectively.
Relationship (3) shows that the moment of the pressure forces
is always perpendicular to the normal vector
, whatever the
point
. Consequently, as on the other hand, the resultant of
the pressure forces is directed along
, then one axis ex-
ists, where the moment
vanishes at every point of this axis.
One can notice that
is the central axis of the pressure force
wrench, as defined in Appendix I-A. The CoP
which is de-
fined as the point of the sole where
is the intersection
between the axis
and the plane of the sole [see Fig. 1(b)].
Relationship (5) shows that the moment of the friction forces
is parallel to the normal vector
if the point belongs to the
plane of the sole. This is true at the level of
, because the CoP
is a point on the sole. Therefore, the CoP
can also be defined
as the point on the sole where the moment of the contact forces
is perpendicular to the sole
(6)
(7)
where the superscript
denotes contact force and moment.
Considering the contact forces, i.e., pressure plus friction
forces, one axis
exists where the moment at every
point
of this axis is parallel to (and ). The CoP
can finally be defined as the intersection between the axis
and the plane of the sole [see Fig. 1(b)]. One can remark
that
is a noncentral axis of the contact force wrench, as
defined in Appendix I-A.
It should be noted that if the two feet are in contact with a
plane ground, then all previous assumptions hold. The area
to be considered in order to define the CoP is the total area where
contact forces appear. The CoP is a point of the supporting plane
of the two feet, it falls within the convex hull of the two contact
areas.
2) ZMP Definition: The ZMP is the point on the ground
where the tipping moment acting on the biped, due to gravity
and inertia forces, equals zero, the tipping moment being de-
fined as the component of the moment that is tangential to the
supporting surface.
The ZMP concept was introduced and developed in [6]
and [7]. It has been exhaustively reviewed in [8]. It is a key
point in the control of the Honda biped robots [9]. The Honda
Co. has patented many detailed implementation developments
[10]–[12].
It should be noted that the term ZMP is not a perfectly exact
expression because the normal component of the moment gen-
erated by the inertia forces acting on the biped is not necessarily
zero. If we bear in mind, however, that ZMP abridges the exact
expression “zero tipping moment point,” then the term becomes
perfectly acceptable.
The resultant of the gravity plus inertia forces (superscript
)
may be expressed as
(8)
and the moment about any point
as
(9)
where
is the total mass, is the acceleration of the gravity,
is the center of mass (CoM) of the biped, is the accel-
eration of
, and is the rate of angular momentum at .
More developed expressions of (8) and (9) are proposed in Ap-
pendix II-A.
One axis
exists, where the moment is parallel to the
normal vector
, about every point of the axis. The ZMP,
“zero tipping moment point,” whose moment is by definition
directed along the normal vector
, necessarily belongs to this
axis. So,
, the ZMP, can be defined as the intersection between
the axis
and the ground surface (see Fig. 2), such that
(10)
It should be noted that this particular axis does not pass
through the global CoM
, although some authors draw as
such in their diagrams (as in [8] and [9] ), and although some
stability criteria are based on the position of the projection of
the CoM along
(see [13] for a bibliography). The position
of the axis
—and by extension —is clearly estab-
lished in Appendix I-B.
SARDAIN AND BESSONNET: FORCES ACTING ON A BIPED ROBOT. CENTER OF PRESSURE—ZERO MOMENT POINT 632
Fig. 2. Gravity-inertia forces-moments at the CoM
G
, and the same
transported at the ZMP
D
.
B. Expressions of the CoP and ZMP
The CoP and the ZMP, as defined above, can be computed as
follows.
1) CoP Computation: Let us consider
, a point of the sole
(generally the normal projection of the ankle), and
, the unit
normal vector directed outwards from the support surface.
Knowing the expression of the pressure forces about the point
, in other words knowing , the problem consists in
determining the position
of the CoP.
By definition,
is the point where the moment of the pressure
forces vanishes. Therefore,
(11)
and consequently the vector
can be expressed as
(12)
It is straightforward to establish that
can be formulated
as the function
(13)
of the total contact forces
about the point .
2) ZMP Computation: The Newton–Euler equations of the
global motion of the biped can be written as
(14)
(15)
and considered under the form
(16)
(17)
to put together the gravity plus inertia forces so that (14) and
(15) may be rewritten as
(18)
(19)
These equations express the fact that the biped is dynamically
balanced if the contact forces and the gravity-inertia forces are
strictly opposite.
Because of the opposition between the gravity-inertia forces
and the contact forces, the ZMP
is defined by an expression
similar to (13)
(20)
The widely favored formulations often met in literature,
which one can sum up as
(21)
are only true if the ground is horizontal, i.e., if
with
. Else, if the contact surface is inclined, then (20) has
to be used. Developments are given in Appendix II-B.
III. C
OP
AND ZMP C
OINCIDENCE
The fact that the two points coincide has some importance: on
the one hand for the understanding of the dynamic equilibrium
of walking gaits, on the other hand for the control of walking
robots.
A. Demonstration of the Coincidence
Because of (18) and (19), it is obvious that the axes
and coincide (indeed, they are noncentral axes defined
from two opposite wrenches). We have shown that the CoP and
the ZMP are the intersections of these axes with the supporting
plane ground surface. Therefore, the CoP and the ZMP are the
same point, called
, and obviously, as for any point of the
supporting plane ground surface, one gets
(22)
This fact does not admit any discussion, it is true whatever
the stability of the balance is, and in particular it is true when
the walker is falling down, as long as a contact exists with the
ground. Nevertheless, this evidence is sometimes discussed in
the case of the fall [14], and that provokes some misinterpreta-
tions. When the CoP reaches the edge of the support polygon,
the system as a whole rotates about the foot edge. In this case,
the system becomes under-actuated. Yet, (18) and (19) represent
the (trivial and obvious) proof that CoP and ZMP coincide.
A difficulty appears when the two feet are lying on two distinct
surfaces. Indeed, the concept of CoP and that of ZMP use in their
fundamental definitions the notion of vector normal to the con-
tact surface and the notion of intersection between an axis and
this surface. So the concepts are intrinsically related to walking
on a single plane surface (whatever the form of the contacts, flat
or no, for instance the rear foot can be heel-off and the front foot
heel-strike). Consequently, the definitions are unfounded, for the
double support phase, if the biped feet are contacting two nonco-
incident surfaces of an uneven terrain. Section IV proposes in this
case the notions of virtual equivalent surface and pseudo-ZMP.
Then, a pseudo-CoP exists, and the fundamental principles of
dynamics, expressed by (18) and (19), lead again to the fact that
pseudo-CoP and pseudo-ZMP coincide.
633 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 34, NO. 5, SEPTEMBER 2004
B. Interpretation of the Coincidence
With regard of our definitions, the coincidence of CoP and
ZMP is not surprising since they are two interpretations of
acting force-moment between the ground and the first link of
a kinematic chain. One interpretation (ZMP) is related to the
accumulated inertia and gravity force-moment of the chain.
The other (CoP) is linked to the ground reaction force-moment,
which equilibrates with the accumulated one. The same under-
standing is repeatedly used in dynamics computation such as
the Newton–Euler recursive inverse dynamics (in the case of
opened kinematic chains).
So, to conclude this section, in order to delete the misinter-
pretations appearing in literature and subsisting in some minds,
we can say that, as long as all the ground-sole contacts appear
in a single plane surface, then the CoP and the ZMP are ab-
solutely and definitely the same point, that consequently we
call CoP-ZMP. In case of two noncoincident contact surfaces, if
one defines a virtual equivalent surface, then the corresponding
pseudo-CoP and pseudo-ZMP are still the same point.
C. Control Aspects
The unilaterality of the foot-ground contact is a major con-
straint of legged locomotion. The fact that the pressure force is
oriented toward the foot, outwards from the support surface, im-
plies that the CoP lies within the support polygon (convex hull
of the contact points or surfaces). Because of the CoP and ZMP
coincidence, the latter, assumed to be linked to the gravity-in-
ertia forces, is submitted to the same constraint.
Of course, when the biped is flying, the support polygon dis-
appears, and consequently the CoP-ZMP is not defined. A con-
trario, the CoP-ZMP can be used as a control criterion when a
support area exists. This is precisely the definition of walking (a
support area always exists), and thus the CoP-ZMP criterion is
perfectly relevant for characterizing the tipping equilibrium of
walking bipeds.
The major advantage of the CoP-ZMP concept is that this
point can be measured: measuring the contact pressure force-
moment allows the CoP to be reconstructed, and the ZMP by
coincidence, and therefore the corresponding part of the gravity-
inertia forces. The CoP-ZMP concept has been used explicitly
by several authors, and implicitly by many others, those who
directly monitor the contact forces [15].
For a complete bibliography, see [8] and [13]. Of the earlier
works related to control carried out with the ZMP concept, only
one is referred to here: in the Honda biped robots, an applica-
tion of the CoP-ZMP control has been implemented, showing
that the CoP notion is related to contact forces, and that of the
ZMP to gravity plus inertia forces [9]. Indeed, the authors use
two types of control, the first being assumed acting on the CoP
(denoted as C-ATGR in their paper) by “lowering” the heel or
the toe of the foot, the second being assumed acting on the ZMP
by increasing the magnitude of the inertia forces (accelerating
the trunk position). The distinction goes in a direction analog
to our definitions of CoP and ZMP: the CoP (C-ATGR) is in-
voked when the actuation directly affects the foot (i.e., the con-
tact forces), the ZMP when the trunk acceleration is modified
(consequently the gravity plus inertia forces). What the authors
Fig. 3. The matter of two noncoincident contact surfaces, according to Honda
US Patent no. 5 357 433.
do not say is that a modification of the CoP causes instanta-
neously a ZMP modification, and vice versa, because the two
points coincide. An action about the ankle modifies the contact
forces, certainly, but also the biped configuration. An accelera-
tion of the trunk increases the inertia forces, certainly, but mod-
ifies the contact forces too. In other words, the modifications
happen simultaneously.
IV. A
DAPTATION OF THE
COP-ZMP CONCEPT
IN THE
CASE OF UNEVEN
TERRAIN
The CoP and ZMP concepts use in their fundamental defi-
nitions the vector normal to the ground surface (for the defini-
tion of axes
and ), and the ground plane itself (which
intersects the axes). So, the concepts are intrinsically related to
walking on one single plane surface (associated to one single
normal vector). The definitions are unsuitable, for the double
support phase when the biped feet are contacting two noncoin-
cident surfaces.
A. Virtual Surface and Virtual CoP-ZMP
The Honda US Patent [10] tackles the matter of irregular ter-
rain by defining a virtual ZMP and a virtual surface varying con-
tinuously from the first to the second surface during the weight
transfer from one foot to the other (see Fig. 3).
According to the authors, the virtual ZMP
is a weighting
function of the local ZMP’s
and , such that
(23)
where
is a function varying continuously from 0 to 1. The
authors have chosen a function proportional to the duration of
the double support phase,
.
The virtual surface is defined by the ZMP
and by the
normal vector
, resulting of an identical weighting function as
that of
, such that
(24)
The idea is very good, because it provides the continuity of
the contact forces (and consequently of the gravity-inertia ones)
from the single support to the double one. However, a detailed
study proves that these weighting functions are not in adequacy
with the ZMP concept, requiring that the moment of the contact
forces is perpendicular to the surface. Indeed, one can show that
if the point
is defined by (23), then the contact moment
is not directed along defined by (24).
SARDAIN AND BESSONNET: FORCES ACTING ON A BIPED ROBOT. CENTER OF PRESSURE—ZERO MOMENT POINT 634
Fig. 4. Pseudo-CoP-ZMP
C
and virtual surface
5
(proposition of this paper).
B. Proposition Respecting the CoP-ZMP Concept
We suggest to define a virtual surface
equivalent to the two
real surfaces
and , then to chose the pseudo-CoP-ZMP
lying in this surface such that the moment of the contact forces
is perpendicular to
.
When the surface
is defined (and consequently its normal
vector
), then an axis exists such that the contact moment
is parallel with , whatever the points of are (see
demonstration in Appendix I). The pseudo-CoP-ZMP
is the
intersection point between
and , as Fig. 4 shows.
We suggest that the surface
passes through the intersection
line of the two real surfaces
and , i.e., through line ,
where
(25)
and that its normal vector is defined as a weighting function of
and , the weight factors being respectively proportional to
the local pressure contact forces
and , such that
(26)
The computation of the virtual CoP-ZMP
is then carried
out with
(27)
where
and are the resultant contact force and moment,
such that
(28)
(29)
where
and are the local contact force and moment at
foot no.
.
A general expression of
if proposed in Appendix III,
where the weight factors are such that
(30)
and where vector
is decomposed such that
(31)
One must notice that, whatever the weight factors
and
are, one neither gets
(32)
nor
(33)
The previous remark corroborates the assertion of Sec-
tion IV-A, claiming that the simultaneous use of (23) and (24)
is incompatible with the CoP-ZMP concept.
A contrario, the method we propose to define the virtual sur-
face and the pseudo-ZMP is based on the definition of the CoP-
ZMP. Moreover, the weight factors we suggest in (26) give good
results, in accord with (32) and (33), as it will be discussed in
Appendix III.
The generalization of the ZMP concept would be actu-
ally complete if we could defined what is the pseudosup-
port-polygon, a certain projection of the three-dimensional
(3-D) convex hull (built from the two real support areas) onto
the virtual surface
, inside which the pseudo-ZMP stays. The
ideas developed in [16] are very clever and seem to be a good
inspiration source for a research in this direction. Nevertheless,
we did not achieve an exact demonstration, considering the
unilaterality of contacts on the one hand, and the friction on the
other hand. This question is not crucial if the angle between
the two planes
and is not too large, as the experiments
presented in paper [5] will show. A contrario, if the two planes
are very angled, then it is in our mind necessary to monitor the
whole contact forces, as pointed out in Section III-C (see [15]).
C. Case of Noncoincident Parallel Surfaces (Stairs)
In stairs, the feet are supported by parallel surfaces that have
different elevations. During the double support phase, the virtual
surface is naturally parallel to the others. The pseudo-CoP-ZMP
is a weighting function of the local CoP-ZMP’s and ,
such that
where
(34)
In this case, whatever the weight factors are, the CoP-ZMP
concept (moment of the contact forces perpendicular to the sur-
face), is respected. From analogy with the previous case, a good
choice is to take the weight factors proportional to the local pres-
sure forces
and , such as
(35)
