JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
1
The Static Center of Pressure Sensitivity:
a further Criterion to assess Contact Stability
and Balancing Controllers
Francesco Romano, Daniele Pucci, Silvio Traversaro,
Francesco Nori
Abstract—Legged locomotion has received increasing attention from
the robotics community. In this respect, contact stability plays a critical
role in ensuring that robots maintain balance, and it is a key element for
balancing and walking controllers. The Center of Pressure is a contact
stability criterion that defines a point that must be kept strictly inside
the support polygon in order to ensure postural stability. In this paper,
we introduce the concept of the sensitivity of the static center of pressure:
roughly speaking, the rate of change of the center of pressure with respect
to the system equilibrium configurations. This new concept can be used
as an additional criterion to assess the robustness of the contact stability.
We show how the sensitivity of the center of pressure can also be used as
a metric to assess balancing controllers by considering two state-of-the-
art control strategies. The analytical analysis is performed on a simplified
model, and validated during balancing tasks on the iCub humanoid robot.
Index Terms—Humanoid Robots, Legged Robots, Redundant Robots,
Posture Equilibrium Analysis.
I. INTRODUCTION
In the last decades, legged robots have become increasingly more
important. Climbing stairs, avoiding obstacles, and exploiting multi-
ple contacts are only a few advantages that humanoid robots have over
wheeled systems. These advantages have been the main motivations
behind the attention they have received from the robotics community.
Legged locomotion, on the other hand, arises interesting issues, which
must be solved before humanoids can be successfully deployed in
the human environment. The recent DARPA Robotics Challenge has
clearly highlighted these difficulties [1]: most of the failures during
trials were due to a non perfect balancing or walking. This paper
introduces a new metric, called sensitivity of the static center of
pressure, which can be further used to analyse contact stability.
Nowadays there exist two main methodologies to implement bal-
ancing and walking controllers on bipedal robots: the first class is
based robot simplified models [2], [3], [4] complemented with the
use of inverse kinematics to bridge the gap between the simplified
model and the real one [5], [6], [7]. The second class is based on the
full dynamic model of the robot [8], [9], [10], [11] and it has been
successfully exploited for the synthesis of instantaneous controllers.
In this regard, a special attention is paid to controlling the interaction
forces between the robot and the environment [8], [12].
Given the complex nature of humanoid robots, researchers have
progressively shifted from using classic nonlinear control techniques
to adopting optimisation-based solutions [13], [14], [15], [16]. These
optimisation-based control techniques allow one to specify, at high
level, the tasks to be accomplished by the robot: the underlining solver
is then in charge of finding a feasible solution to the optimisation
problem. One disadvantage of this process is that properties of
the solution can be more difficult to find. If some properties have
to be enforced, they can be specified by means of constraints or
optimisation criteria. But how this high-level criteria influence the
solution is not trivial and must be analysed on a case-by-case basis.
*This work was supported by the FP7 EU project CoDyCo (No. 600716
ICT 2011.2.1 Cognitive Systems and Robotics) and Koroibot (No. 611909
ICT-2013.2.1 Cognitive Systems and Robotics).
The authors are with the iCub Facility department, Istituto Italiano di
Tecnologia, Via Morego 30, Genoa, Italy. name.surname@iit.it
Manuscript submitted October 3, 2016.
In this paper, we analyse two state-of-the-art balancing controllers
commonly found in literature ([8], [16], [17], [18], [10] and [19],
[20]), and we analyse them by means of the center of pressure
sensitivity here introduced: we show how these criteria yield different
performance when the robot balances on its two feet.
Maintaining the support foot (or feet) stable is of paramount impor-
tance for balancing and walking. As a result, various postural stability
criteria analysing the stance foot dynamics have been proposed in the
robotics and human locomotion literature. These criteria focus on the
force and torques exchanged at the ground-foot interface.
The Ground projection of the Center of Mass (GCoM) can be
applied if the robot dynamics is at equilibrium: it states that to assess
the postural stability of the biped, the projection on the ground of
the CoM must lie inside the convex hull of the support feet.
The Center of Pressure (CoP), also known as Zero Moment Point
(ZMP) [21], [22], can be applied as soon as the static assumption is
relaxed. The center of pressure can be viewed as the point on the
ground surface where the resultant of the normal contact force acts.
Because of the unilaterality of the normal component of the contact
forces, the center of pressure must lie inside the support polygon.
It is worth noting that the CoP coincides with the GCoM when the
stationarity hypothesis is satisfied [22].
Finally, the Foot Rotation Indicator (FRI) point has been recently
introduced [22]: when the foot does not rotate, the FRI point coincides
with the CoP, and lies inside the support polygon. Differently than
the CoP, the FRI can exit the support polygon, which occurs when
the foot starts rotating.
All the previously described postural stability criteria are not
directly linked to a specific robot dynamical model or balancing
controller implementation, thus being useful for a multitude of control
solutions. On the other hand, one of these criteria (e.g. CoP, FRI, etc)
must be taken into account during the synthesis of postural controllers
to ensure stable foot planting and postural stability of the whole robot.
In this work, we introduce the concept of the sensitivity of the
static center of pressure: roughly speaking the rate of change of the
center of pressure versus the system equilibrium configuration. We
show how this new concept can further characterise the stability of
a contact, thus complementing the notion of center of pressure.
We also show that the sensitivity of the center of pressure can be
used to evaluate the performances of different balancing controllers.
In this regard we analyse two state-of-the-art controllers: they both
stabilise the robot linear and angular momentum as control objective,
assuming the contact forces and torques as virtual control inputs.
Because the system is overdetermined in presence of more than one
contact, the resulting force redundancy can be exploited in different
ways. In particular, the two state-of-the-art controllers differ in the
criterion used to resolve the redundancy: the minimisation of the
contact wrenches norm [8], [16], [17], [18], [10], [23], [24] in one
case and the minimisation of the internal joint torques norm [19],
[20] in the other: we show that the two criteria exhibit different static
center of pressure sensitivities. We can thus exploit the introduced
metric to discriminate between two, apparently similar, controllers.
To simplify the analytical analysis, we model the lower body
of a humanoid robot while balancing as a planar four-bar linkage
mechanism and we further extend and apply the analysis on the iCub
humanoid robot so as to assess the validity of the simplified model.
The paper is structured as follows. Section II introduces the
dynamical model of a free floating mechanical system and the
notation used throughout the paper and Section III describes the
concept of sensitivity of static center of pressure. The sensitivity of
the static center of pressure is applied first to a four-bar linkage model
in Section IV and then to the iCub humanoid robot in Section V.
Finally Section VI draws the conclusions.
arXiv:1610.01495v2  [cs.RO]  29 May 2017
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
2
II. BACKGROUND
A. Notation
Throughout the paper we will use the following definitions.
• I denotes an inertial frame, with its z axis pointing against the
gravity. We denote with g the gravitational constant.
• ei ∈Rm is the canonical vector, consisting of all zeros but the
i-th component, which is one.
• Given two orientation frames A and B, and vectors of coordi-
nates expressed in these frames, i.e. Ap and Bp, respectively,
the rotation matrix ARB is such that Ap = ARB
Bp.
• 1n ∈Rn×n is the identity matrix of size n; 0m×n ∈Rm×n is
the zero matrix of size m × n and 0n = 0n×1.
• We denote with S(x) ∈R3×3 the skew-symmetric matrix such
that S(x)y = x×y, where × denotes the cross product operator
in R3. If x = (˜x⊤, 0)⊤and y = (˜y⊤, 0)⊤, with ˜x, ˜y ∈R2 one
has S(˜x)˜y = −˜x⊤S˜ye3, with S =
"
0
−1
1
0
#
.
B. System modelling
We assume that the robot is composed of n+1 rigid bodies – called
links – connected by n joints with one degree of freedom each. In
addition, we also assume that the multi-body system is free floating,
i.e. none of the links has an a priori constant pose with respect to
the inertial frame. This implies that the multi-body system possesses
n + 6 degrees of freedom. The configuration space of the multi-body
system can then be characterised by the position and the orientation
of a frame attached to a robot’s link – called base frame B – and the
joint configurations. More precisely, the robot configuration space is
defined by
Q = R3 × SO(3) × Rn.
An element of the set Q is then a triplet q = (IpB, IRB, qj), where
(IpB, IRB) denotes the origin and orientation of the base frame
expressed in the inertial frame, and qj denotes the joint angles. The
velocity of the multi-body system can then be characterised by the
algebra V of Q defined by:
V = R3 × R3 × Rn.
An element of V is then a triplet ν = (I ˙pB,I ωB, ˙qj), where IωB
is the angular velocity of the base frame expressed w.r.t. the inertial
frame, i.e. I ˙RB = S(IωB)IRB.
We also assume that the robot is interacting with the environment
through nc distinct contacts. Applying the Euler-Poincar´e formalism
to the multi-body system yields the following equations of motion
[25, Ch. 13.5]1:
M(q) ˙ν + C(q, ν)ν + G(q) = Bτ +
nc
X
k=1
J⊤
Ckfk
(1)
where M ∈Rn+6×n+6 is the mass matrix, C ∈Rn+6×n+6 is the
Coriolis matrix, G ∈Rn+6 is the gravity term, B = (0n×6, 1n)⊤is
a selector matrix, τ are the internal actuation torques, and fk denotes
an external wrench applied by the environment to the link of the k-
th contact. We assume that the external wrench is associated with
a frame Ck, which is attached to the robot’s link where the wrench
acts on. Then, the external wrench fk is expressed in a frame whose
orientation coincides with that of the inertial frame I, but whose
origin is the origin of Ck.
1The Euler-Lagrange’s formulation can be applied only to mechanical
systems evolving in vector spaces. The Euler-Poincar´e equations, instead, are
valid for mechanical systems evolving in arbitrary Lie groups.
CoM
Joint
x
y
λ
ρ
fR
fL
z
I
ξ
Fig. 1: Planar four-bar linkage exploiting two rigid contact to stand
The Jacobian Jk = Jk(q) is the map between the robot’s velocity
ν and the linear and angular velocity IvCk := (I ˙pCk,I ωCk) of the
frame Ck, i.e.
IvCk = JCk(q)ν.
(2)
The Jacobian has the following structure.
JCk(q) =
h
Jb
Ck(q)
Jj
Ck(q)
i
∈R6×n+6,
(2a)
Jb
Ck(q) =
"
13
−S(IpCk −IpB)
03×3
13
#
∈R6×6.
(2b)
Lastly, we assume that holonomic constraints may be enforced
on the robot due to rigid contacts with the environment. These
constraints are modelled as a kinematic constraints that forbid any
motion of the frame Ck, i.e. JCk(q)ν = 0.
III. SENSITIVITY OF THE STATIC CENTER-OF-PRESSURE
This section discusses a property of any center of pressure associ-
ated with rigid, flat contacts and quasi-static robot’s configurations.
We shall see in Section IV that this property – called static center
of pressure sensitivity – can be used as an element of evaluation of
contact stability and balancing controllers for humanoid robots.
A. The sensitivity of the static center of pressure in a case study: the
four-bar linkage case
To provide the reader with an introduction to the sensitivity of the
static center of pressure, focus on Figure 1. This picture depicts a
four-bar linkage standing on two flat, rigid contacts. In this configura-
tion, the mechanical system possesses one degree of freedom, i.e. the
variable ξ, which represents the minimal coordinate of the constrained
mechanical system. At the equilibrium configuration, the system is
hyperstatic because the equilibrium conditions of the Newton-Euler
equations are not sufficient to determine all the (internal) joint torques
τ. Since the joint torques are assumed to be a control input, however,
one can assume that their value is given, with the only requirement
that the system remains at the equilibrium configuration.
Assume that the joint torques depend only on the minimal coordi-
nate ξ. Then, the center of pressures at each static robot configuration,
i.e. ˙ξ = 0, depend only on the minimal coordinate ξ. This poses the
following interesting question.
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
3
What is the rate-of-change of the center of pressures over the
quasi-static constrained robot’s motion?
Different control laws for balancing the four-bar linkage may
be associated with different rate-of-changes of the static center of
pressure over the constrained robot’s motion. Clearly, the higher this
rate-of-change, the higher the likelihood to hit the boundaries of the
support polygon of the foot, which causes breaking the associated
contact and, eventually, a fall of the robot. The above rate-of-change
of the center of pressure is what we call static center of pressure
sensitivity, and the following generalises this concept to any multi-
body system constrained by some rigid, flat contacts.
B. The formal definition
Assume that the robot makes nc rigid contacts with the environ-
ment, and that no external wrench acts on the robot other than gravity
and contact wrenches. Let J denote the Jacobian obtained by stacking
all contact Jacobians JCk, i.e.
J =


JC1
...
JCnc

∈Rk×n+6,
(3)
with k = 6nc. Consequently, all rigid contacts imply the following
constraint on the floating base system (1): Jν = 0, the time derivative
of which is given by
˙Jν + J ˙ν = 0.
(4)
By combining the above constraint with Eq. (1) and by defining
f := (f ⊤
1 , . . . , f ⊤
nc)⊤∈Rk,
one can find the explicit expression of all contact forces, i.e.
f = (JM −1J⊤)−1 h
JM −1C(q, ν)ν+G(q)−Bτ


−˙Jν
i
.
(5)
The hidden assumption for the above equality to hold is that J is
full row rank. Once the obtained expression of f is substituted into
Eq. (1), one obtains the following equations of motion:
M(q) ˙ν + J⊤(JM −1J⊤)−1 ˙Jν = N

Bτ −C(q, ν)ν −G(q)

(6)
with N given by
N = 1n+6 −J⊤(JM −1J⊤)−1JM −1.
Evaluating Eq. (6) at the equilibrium condition (ν, ˙ν) = (0, 0) yields
N

Bτ −G(q)

= 0.
(7)
Eq. (7) defines the values that the joint torques must take to satisfy the
equilibrium condition (ν, ˙ν) = (0, 0). Then, we make the following
assumption.
Assumption 1. The holonomic constraints due to contacts restrain
the configuration space Q into the set Rn+6−k, i.e. there exists a
differentiable mapping χ : Rn+6−k →Q such that
∀q ∈Q
∃ξ ∈Rn+6−k
:
q = χ(ξ).
(8)
Furthermore, the joint torques τ at the equilibrium configuration
(ν, ˙ν) = (0, 0) depend only on the robot constrained configuration
space, i.e. τ = τ(ξ).
In view of the above assumption, the contact wrenches at the
equilibrium configuration, i.e.
f e =


f e
1
...
f e
nc

= (JM −1J⊤)−1JM −1 (G−Bτ) ,
(9)
depend only on the robot constrained motion, i.e. f e = f e(ξ). In
light of the above, we can now define the static center of pressure
sensitivity as follows.
Definition 1. Let SCoPj ∈R2 denote the center of pressure of the
j-th rigid contact at the equilibrium (ν, ˙ν) = (0, 0), i.e.
SCoPj =
1
e⊤
3 f e
j
"
−e⊤
5 f e
j
e⊤
4 f e
j
#
.
(10)
The static center of pressure sensitivity ηj is defined as the rate-of-
change of (10) over the constrained robot motion, i.e.
ηj := ∂SCoPj
∂ξ
.
(11)
In general, the static center of pressure sensitivity is a matrix
belonging to R2×n+6−k, and characterises how fast the center of
pressure associated with a rigid, planar contact moves depending on
the constrained robot’s motion. This sensitivity may then characterise
“the stability” of a contact associated with a robot’s configuration.
Of particular importance are the cases when the sensitivity ηj tends
to infinity. Clearly, these configurations must be avoided by any
controller associated with the robot, since they may cause abrupt
system’s responses and, eventually, a robot fall.
Remark 1. The internal torques at the equilibrium configuration
must satisfy Eq. (7) that induces, in general, multiple solutions
when solved for τ. The proposed sensitivity may be then used to
compare different balancing controllers, since lower values of ηj are
associated with smaller variations of the center of pressures, which
may increase the likelihood of not breaking the contact.
Remark 2. Note that, by its definition in Eq. (11), the actual value
of the sensitivity depends on the choice of the parametrisation ξ.
Different choices of parametrisation may lead to different values of
the sensitivity.
IV. THE SENSITIVITY FOR ASSESSING BALANCING
CONTROLLERS PERFORMANCE:
THE FOUR-BAR LINKAGE CASE STUDY
This section studies how the sensitivity of the static center of
pressure may vary depending on different balancing controllers
synthesised for the same system. What follows presents the analytical
analysis on a four-bar linkage when standing on two, flat, rigid
contacts. For balancing purposes of the four-bar linkage, we choose
two state-of-the-art momentum-based controllers that differ in the
criterion adopted for the contact forces redundancy resolution. More
precisely, we show next that minimising the joint torques when the
Newton-Euler equations are at the equilibrium decreases the static
center of pressure sensitivity, thus improving the stability of the
contact. As the experimental results in Section V show, the four-bar
linkage case study is representative of a humanoid robot standing on
two feet.
A. Modelling
Consider the four-bar linkage shown in Figure 2, where l and d
denote the lengths of the leg and of the upper rod, respectively. Each
leg possesses a mass of ml while the upper rod has mass mb. The
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
4
x
y
λ
ρ
B
q1
q2
q3
q4
z
ϑ
I
CoM
Joint
d
l
Fig. 2: Free floating planar four-bar linkage
inertial frame I is chosen so as the y axis opposes gravity, the z
axis exits from the plane, and the x axis completes the right-hand
base. Being a planar model, only translations in the x −y plane and
rotations about the z axis are allowed. The vectors
λ =
h
λ1
λ2
i⊤
∈R2,
ρ =
h
ρ1
ρ2
i⊤
∈R2
connect the center of mass to the left and right foot frames respec-
tively. Let IpB ∈R2 denote the position of the origin of the frame B
– which is located at the center of the upper link – expressed in the
inertial frame. The configuration space is then parametrised by the
following coordinates
q = (IpB, θ, q1, q2, q3, q4) ∈R7.
Being a vector space, we use the Lagrange formalism to derive the
equations of motion [26, Ch. 4], i.e.
d
dt
∂L
∂˙q + ∂L
∂q =
"
03
τ
#
,
(12)
with L := T −V the Lagrangian, and T and V respectively the
kinetic and potential energy of the multi-body system.
Hypothesis 1. For the four-bar linkage model, it holds that:
i) Each link is approximated as a point mass at its center of mass.
ii) The mass of the left leg equals the mass of the right leg.
iii) The upper body mass, mb, is twice the leg mass, i.e. mb = 2ml.
It is worth noting that hypothesis ii) and iii) are in general satisfied
in humanoid robots, e.g. in the iCub robot used for the experiments.
For the sake of simplicity, the complete dynamic model obtained
by applying Eq. (12) is not presented here. We present only the
terms of the dynamic and kinematic model that are necessary for the
comprehension of the remainder of the section. In particular, partition
the mass matrix M as follows
M =
"
Mb
Mbj
M ⊤
bj
Mj
#
.
Let sin(qi) = si and cos(qi) = ci. One can verify that
Mbj = ml
2


ls1
0
ls3
0
−lc1
0
−lc3
0
l2s2
1−lc1(d−lc1)
2
0
l2s2
3+lc3(d+lc3)
2
0


(13)
The feet Jacobians are given by the following expression
JCL =


0
0
0
12
R(θ)
"
ls1
d
2 −lc1
#
R(θ)
"
ls1
−lc1
#
0
0
0
0
0
1
1
1
0
0


JCR =


0
0
0
12
R(θ)
"
ls3
−d
2 + lc3
#
0
0
R(θ)
"
ls3
−lc3
#
0
0
0
1
0
0
1
1

.
To represent a classical balancing context of humanoid robots, we
assume that the mechanism makes contact with the environment at
the two extremities. Also, we assume that the distance between the
contacts equals d, i.e. the length of the upper rod – see Figure 1. Then,
the rigid constraint acting on the system takes the form (4), with J =
h
J⊤
CL
J⊤
CR
i⊤
∈R6×7, and ν = ˙q. The associated contact wrenches
are f = (fL, fR) ∈R6. As a consequence of these constraints, the
mechanism possesses one degree of freedom when standing on the
extremities.
With the aim of evaluating the sensitivity of the static center of
pressure we define the minimal coordinate ξ as the angle between
the upper link and one of the legs (see Figure 1). Observe that the
mapping q = χ(ξ) in (8) is given by:


Ipb
θ
q1
q2
q3
q4


=


1
2de1 + l sin(ξ)e2 + l cos(ξ)e1
0
ξ
π −ξ
ξ
π −ξ


.
(14)
B. Two balancing controllers inducing different sensitivities
Remark that in order to evaluate the sensitivity of the static center
of pressure, we need an expression for the contact forces at the
equilibrium configuration – see Definition 1. What follows proposes
an analysis of the sensitivity of the static center of pressures evaluated
with contact wrenches obtained from two state-of-the-art criteria.
1) The contact wrenches ensure the equilibrium of the centroidal
dynamics written in the plane. The three dimensional redun-
dancy of the contact wrenches minimises the norm of the contact
wrenches2 |f| [8], [16], [17], [18], [10].
2) The contact wrenches ensure the equilibrium of the centroidal
dynamics. The three dimensional redundancy of the contact
wrenches minimises the norm of the joint torques |τ| [19], [20].
The following lemma presents the results when the first criterion
is applied.
Lemma 1. Assume that Hypothesis 1 holds. If the contact wrenches
redundancy is chosen to minimise the norm of the contact wrenches,
i.e. criterion 1), then the induced contact wrenches are given by the
2It is worth noting that the norm of a wrench is not well defined and highly
dependent on the chosen metric. Nevertheless, the minimisation of the contact
wrench norm is something extensively adopted in literature and it has been
used in the manuscript only for comparison purposes.
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
5
-15
-10
-5
0
5
10
15
-60
-40
-20
0
20
40
60
(a) Minimum wrenches norm solution
-15
-10
-5
0
5
10
15
-60
-40
-20
0
20
40
60
(b) Minimum joint torque norm solution
Fig. 3: Left and right foot CoPs together with the CoM for different configuration of ξ. In (b) the asymptotes are in correspondence of the
vertical component of the forces going to zero
-15
-10
-5
0
5
10
15
0
0.2
0.4
0.6
0.8
1
Fig. 4: Comparison of the vertical force component of the right foot
for the two contact forces choice. The vertical force is normalised
w.r.t. the total mass of the mechanism, i.e. divided by 1/mg
following expression:
Lf L = mg


0
1
2

1+ d2
4
 + d 5d−6l cos(ξ)
10(d2+4)
6l cos(ξ)
5(d2+4)


,
Rf R = mg


0
1
2

1+ d2
4
 + d 5d+6l cos(ξ)
10(d2+4)
6l cos(ξ)
5(d2+4)


.
(15)
Furthermore, the static center of pressure sensitivity is given by:
ηL = −60l sin(ξ)
d2 + 4
(5d2 + 6ld cos(ξ) + 20)2 ,
ηR = −60l sin(ξ)
d2 + 4
(5d2 −6ld cos(ξ) + 20)2 .
(16)
The proof is in the Appendix.
It is worth noting that if d is small enough, then Eq. (15) points
out that one has an almost constant vertical force independently of
ξ. In this case, the vertical force is approximately equal to mg/2,
independently of the center-of-mass position.
Figure 3a shows the static CoP, referred to as SCoP, of the left
and right foot in correspondence of the CoM motion. We can notice
how SCoPL (the SCoP of the left foot) and SCoPR (the SCoP of the
right foot) moves together with the CoM.
Let us now present analogous results of Lemma 1 when applying
the criterion 2), i.e. the choice of the force redundancy minimises the
joint torques.
Lemma 2. Assume that Hypothesis 1 holds. If the contact wrenches
redundancy is chosen to minimise the norm of the internal joints
torques, i.e. criterion 2), the induced wrenches are given by the
following expressions.
LfL=mg







0
1
2
−λ1
2

+


3l cos(ξ)2
8d sin(ξ)
3l cos(ξ)
8d
3l cos(ξ)
8d

λ2
cos(ξ)
sin(ξ) −λ1

+ d
4






, (17a)
RfR=mg







0
1
2
−ρ1
2

−


3l cos(ξ)2
8d sin(ξ)
3l cos(ξ)
8d
3l cos(ξ)
8d

ρ2
cos(ξ)
sin(ξ) −ρ1

+ d
4






. (17b)
Furthermore, the static center of pressure sensitivity is given by:
ηL = −
45d2l sin(ξ)
(10d + 3l cos(ξ))2 ,
ηR = −
45d2l sin(ξ)
(10d −3l cos(ξ))2 .
(18)
The general expression of the contact wrenches (17) when Hy-
pothesis 1-iii) does not hold is given in the Appendix.
Focus on the contact wrenches in (17). Differently from Eq. (15),
the vertical components of the contact forces
LfL and
LfR in
equations (17a) and (17b) do vary with the variable ξ, see Figure
4. In particular, when the center of mass of the system is on top of
one of the feet, the corresponding vertical force is equal to mg, i.e.
the foot supports the whole weight. As a consequence, the force on
the other foot is null.
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
6
-15
-10
-5
0
5
10
15
0
0.5
1
1.5
2
Fig. 5: Comparison of the norm of the sensitivity of the static center
of pressure of the right foot for the two contact forces choice
The behaviour of the static centers of pressure can be observed in
Figure 3b. Interestingly, while the minimum torques norm solution
exhibits more “flat” CoPs around the centered position and on the
support foot, we can notice a divergent behaviour around ∆ξ =
±14[deg]. This is due to the behaviour of the vertical components of
the force previously described, i.e. to the fact that with this solution,
the force on one foot tends to zero as the weight is moved towards
the other foot. It is worth noting that the center of pressure showing
the divergent behaviour is the one of the non-supporting foot.
We can compare the sensitivity of the static centers of pressure
induced by criteria 1) and 2) when the CoM moves from left to
right. Figure 5 compares the norm of the right foot static center of
pressure sensitivities obtained by applying the two criteria. Notice
that the second criterion, i.e. the minimum joint torques norm
solution, provides less sensitive solutions at the initial, symmetrical
configuration, i.e. ∆ξ = 0. Furthermore, the sensitivity decreases
when the foot is increasingly supporting all the weight, i.e. toward
∆ξ = 14[deg] in the considered case.
Remark 3. A criterion similar to the minimisation of the norm
of the contact wrenches that is sometimes found in literature is
the minimisation of the tangential components of the forces [11].
The following lemma states the equivalence of the two solutions
when planar contacts are considered. The proof can be found in
the Appendix.
Lemma 3. The solution exploiting the force redundancy to minimise
the tangential component of the contact wrenches is equivalent to the
minimum contact wrench norm solution (Lemma 1).
Remark 4. The proposed analysis focuses on the sensitivity of the
center of pressure, that is, on its rate of change with respect to a
parametrisation of the minimal coordinates of the constrained system.
The use of inequalities on the contact wrenches, usually enforced by
the use of QP solvers, limits the actual value of the center of pressure.
The two concepts are thus complementary.
V. EXPERIMENTS
In this section, we validate the analysis performed on the four-bar
linkage model on the full iCub humanoid robot. We apply the same
principles stated in criterion 1) and 2) in Section IV-B to achieve
balancing.
Our test platform is the iCub robot, a state-of-the-art 53 degrees
of freedom humanoid robot [27]. For the purpose of the present
experiment, we consider only the principal 23 degrees of freedom,
located in the legs, torso and upper arms. These degrees of freedom
are supplied of a low level torque control that runs at 1kHz and
is responsible of stabilising desired joint torques. As a comparison
with the simplified model, the hips are about 50cm high and the feet
are kept, during the experiments, at about 14cm of distance between
each other. Contact forces with the ground are estimated by proper
estimation algorithm that exploits six six-axes force torque sensors
installed within iCub [10, Sec. 3.2].
The controller is a dynamic controller composed of a strict hier-
archy of two control objectives. The first, and with highest priority,
is responsible of tracking a desired robot momentum through the
use of the contact wrenches. Note that the desired robot momentum
contains feedback terms on the center of mass position and velocity.
The second one, instead, is responsible of stabilising the resulting
zero dynamics. Discussion about the stability of the zero dynamics is
out of the scope of the present paper and we refer the interested reader
to [19] for further details on the zero dynamics stability problem.
Assuming only two contacts located at the feet, the dynamics of
the robot momentum is given by the following equation:
˙H = XLfL + XRfR + m¯g
(19)
where H ∈R6 is the robot linear and angular momentum,
XL =
"
13
03×3
S(xL −xCoM)
13
#
, XR =
"
13
03×3
S(xR −xCoM)
13
#
are transformation matrices, xCoM is the position of the center of mass
and xL, xR are the positions of the left and right foot respectively.
Finally m¯g is the wrench due to gravity.
Solutions to Eq. (19) yield the desired fL and fR to be applied
by the robot, which are in turn generated by the actuation torques so
as to satisfy the first control objective, i.e. tracking a desired robot
momentum. To obtain the torques to be applied, we can invert Eq.
(5). If we denote with f ∗= [f ∗⊤
L , f ∗⊤
R ]⊤a possible solution to Eq.
(19), we obtain the following expression for the control variable τ:
τ = (JM −1B)†[JM −1(Cν + G −J⊤f ∗) −˙Jν],
(20)
where (JM −1B)† denotes the Moore-Penrose pseudoinverse of the
matrix JM −1B. The choice of f ∗, and as a consequence the torques
applied by the robot, depends on the criterion chosen to obtain a
solution to Eq. (19). Similarly as we did for the four-bar linkage
system we apply criterion 1) and 2) (see Sec. IV-B) on the real robot
and we compare the obtained results.
In the experiments, we move the robot along the frontal plane3 with
very low velocity so as to simulate the quasi-static scenario presented
in the analysis in Section IV. We begin from the initial, symmetrical
configuration, moving the robot towards the configuration where the
CoM lies on the left or right foot frame origin. In particular, we
impose a sinusoidal reference for the center of mass y-component,
being the y-axis the axis along the robot frontal plane, and we fix the
reference amplitude and frequency to be equal for the control laws
associated with both criteria. When we apply the criterion 2), we
manage to track a full period of the reference sine, for the criterion
1) we have to perform two different experiments, i.e. one moving the
CoM towards the left foot and the other towards the right one.
We first tested the minimum wrenches norm solution, i.e. criterion
1). As we showed in the analysis in the previous section we expect
that the vertical components of the forces do not change so much
from the half weight of the robot. This is confirmed also on the real
robot. As we can see in Figure 6a the forces start symmetrical. When
3We define with frontal plane any vertical plane that divides the robot into
anterior and posterior portions, similarly as in humans anatomy.
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
7
0
5
10
15
0
50
100
150
200
250
300
(a) Minimum wrenches norm solution
0
10
20
30
40
50
0
50
100
150
200
250
300
(b) Minimum joint torque norm solution
Fig. 6: Vertical components of the contact forces for left and right foot during the robot quasi-static motion. The experiment in (a) depicts
the forces when the CoM moves towards the left foot. In (b) the CoM tracks a full sinusoidal reference period instead.
0
5
10
15
-5
0
5
Left
Right
L. Foot Limit
R. Foot Limit
(a) CoM towards left foot
0
5
10
15
-5
0
5
Left
Right
L. Foot Limit
R. Foot Limit
(b) CoM towards right foot
Fig. 7: Center of pressure for the left and right foot when the minimisation of the contact wrenches criterion is applied. In (a) we show the
experiment with the CoM moving towards the left foot, while in (b) the CoM moves towards the right foot instead.
the CoM starts moving towards the left foot we notice that both the
forces change of only a limited quantity. The same behaviour was
verified when we moved the CoM towards the right foot. We then
tested criterion 2), i.e. we exploited the redundancy of the forces
to optimise for the joint torques. Figure 6b shows the forces during
the experiment for a full period of the CoM sinusoidal reference,
highlighting the dependence of the forces upon the center of mass
position.
We then analysed the center of pressure behaviour during the
experiments. Figure 7 shows the left and right foot center of pressure
when we applied the first criterion. As mentioned previously we had
to perform two different experiments, one where the CoM moves
towards the left foot, in Fig. 7a, and one where the CoM moves
towards the right foot, in Fig. 7b. As the analysis on the four-bar
linkage showed the two centers of pressure move almost in parallel
regardless of which foot supports most of the weight.
When we apply the control law obtain by satisfying criterion 2),
i.e. minimisation of the internal torques norm, the left and right
foot centers of pressure behave differently, with the CoP of the foot
supporting most of the weight moving less that the other one, as
Figure 8 clearly highlights.
Remark 5. As last step in our experimental validation, we compare
how the two different criteria behave with respect to the minimal
coordinate variables. Given the kind of movements performed by
the robot, we assume as minimum coordinate variable ξ the left hip
roll joint, which corresponds to the joint q1 in the four-bar linkage
model shown in Figure 2. Having performed this choice, we can
now properly compare the CoPs obtained with the two force criteria.
Fig. 9 shows the aforementioned comparison considering only the
left foot, where Fig 9a shows the analytical results on the four-bar
linkage model and Fig. 9b shows the results collected on the iCub
humanoid robot. It is worth noting the similarity between the two
results, thus proving the validity of the reduced model we adopted
for the theoretical analysis.
VI. CONCLUSIONS
In this paper we introduced the concept of static center of pressure
sensitivity as an additional tool to analyse and evaluate the stability of
a contact and to benchmark the performance of balancing controllers.
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
8
0
10
20
30
40
50
-5
0
5
Left
Right
L. Foot Limit
R. Foot Limit
CoM towards left
| CoM towards right
Fig. 8: Center of pressure for the left and right foot when the
minimisation of the joint torques norm criterion is applied.
We first analysed the newly introduced metric on a planar four-bar
linkage, which approximates the lower body of a biped robot when
balancing. Subsequently, we verified the analysis on the full iCub
humanoid robot, obtaining similar results, thus proving the power of
the approximate model. The effect on the stability of the contacts of
two different choices of contact forces are then shown and compared.
The analysis we performed, both analytically and numerically,
shows how different criteria imply very different properties on the
contact wrenches, and in turn, on the resulting center of pressures.
In particular the wrenches obtained by minimising the internal joint
torques produces a lower sensitivity w.r.t. the minimum wrench norm
solution.
We want to stress the fact that we do not advocate for the use of the
proposed static center of pressure sensitivity as a substitute to other
criteria such as the center of pressure or the foot rotation indicator. On
the contrary, the newly introduced metric can be used as an additional
tool to discriminate between apparently similar balancing controllers.
During the analysis we noticed an additional, interesting property
of the solution which minimises the joint torques. Differently from
the other solution, the vertical components of the contact force on
one foot tends to zero when the center of mass of the robot moves
towards the opposite foot. While a proper theoretical analysis should
be put into effort, this solution can help minimise discontinuities due
to a change of the contact set.
APPENDIX
A. Proof of Lemma 1
Consider the equilibrium condition of the centroidal dynamics [28]
of the mechanism, i.e. the balance of the external wrenches acting
on the mechanism when written w.r.t. a frame centered in the center
of mass and with the orientation of the inertial frame.
0 = XL
Lf L + XR
Rf R −mge2 = Xf −mge2
(21)
where mge2 is the wrench due to gravity and X :=
h
XL
XR
i
,
XL =
"
12
02×1
(Sλ)⊤
1
#
,
XR =
"
12
02×1
(Sρ)⊤
1
#
,
λ =
h
λ1
λ2
i⊤
∈R2 and ρ =
h
ρ1
ρ2
i⊤
∈R2 the vectors
connecting the center of mass to the left and right foot frames
respectively, see Figure 1.
Since X ∈R3×6 by construction is always full row rank, (21)
admits infinite solutions, i.e.
f = X†mge2 + Nff0
(22)
where X† ∈R6×3 is the Moore-Penrose pseudoinverse of X, Nf ∈
R6×6 is the projector onto the nullspace of X and f0 ∈R6 is a
free variable. The unique minimum norm solution corresponds to the
choice f0 = 0. The actual expression of the contact wrenches are
thus the ones given in Eq. (15).
The center of pressure in the planar case possesses only one
coordinate, which is given by the following expressions
LSCoPL = e⊤
3 f
e⊤
2 f ,
RSCoPR = e⊤
6 f
e⊤
5 f
which evaluated with the expression of the wrenches in Eq. (15) yield
LSCoPL =
12l cos(ξ)
5d2 + 6ld cos(ξ) + 20,
RSCoPR =
12l cos(ξ)
5d2 −6ld cos(ξ) + 20.
To evaluate the static center of pressure sensitivity, we can differ-
entiate the above expression w.r.t ξ, that is
ηL = ∂SCoPR
∂ξ
= −60l sin(ξ)
d2 + 4
(5d2 + 6ld cos(ξ) + 20)2 ,
ηR = ∂SCoPR
∂ξ
= −60l sin(ξ)
d2 + 4
(5d2 −6ld cos(ξ) + 20)2 .
B. Proof of Lemma 2
Consider the balance of the external wrenches acting on an
articulated rigid body in the general 3D case:
XL
Lf L + XR
Rf R −mge3 = 0
(23)
where XL, XR ∈R6×6 are used to express the wrenches respectively
in L and R at the frame located at center of mass with the orientation
of the inertial frame. Impose the following expression for the contact
wrenches:
Lf L = X−1
L
mg
2 e3 + X−1
L ∆
Rf R = X−1
R
mg
2 e3 −X−1
R ∆
(24)
where ∆∈R6 is a free variable. We can notice that with the
above choice, (23) is always satisfied independently of the choice
of ∆. In what follows, the redundancy ∆is used to minimize
the joint torques. To simplify the calculations we perform a state
transformation as described in [29]. In particular, the dynamics in
the new state variables is decoupled in the acceleration, i.e. the mass
matrix is block diagonal, with the two blocks being the terms relative
to the base and the joints. We refer the reader to [29] for additional
details, and we report here only the relevant transformations. To avoid
confusion, all the dynamic quantities in the new state variables are
denoted with the overline.
The new base frame is chosen to correspond to a frame with the
same orientation of the inertial frame I and the origin instantaneously
located at the center of mass. We denote with cXB the corresponding
6D transformation. Of particular interest is the form of the Jacobian
of the left (right) foot frame L (R) in the new state variables:
JL =
h
JL,b
JL,j
i
=
h
X⊤
L
JL,j −1
mX⊤
L ΛMbj
i
,
JR =
h
JR,b
JR,j
i
=
h
X⊤
R
JR,j −1
mX⊤
R ΛMbj
i
,
Λ := 16 −P(16 + cX−1
B P)−1cX−1
B .
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
9
-10
-5
0
5
10
-4
-2
0
2
4
6
(a) Four-bar linkage model
-10
-5
0
5
10
-4
-2
0
2
4
6
Min. Wrenches
Min. Joint Torque
(b) Robot
Fig. 9: Comparison of the center of pressure of the left foot for the two criteria on the simplified model (a) and on the real robot (b). The
dotted blue lines in Figure (b) denote the convex hull of the left foot.
P has the following expression:
P :=
"
03×3
03×3
S(pCoM −pB)
I
m −13
#
where I is the 3D inertia of the robot expressed w.r.t. the orientation
of I and the center of mass as origin and pCoM −pB is the distance
between the center of mass and the base frame before the state
transformation.
The relation between the joint torques and the contact wrenches
is given by Eq. (9). So, by inverting this relation, we evaluate the
effects of the contact wrenches redundancy ∆on the joint torques at
the equilibrium configuration, i.e.
τ = (JjM −1
j )†JM −1(mge3 −J⊤f).
(25)
We can now substitute the definition of f as in (24) into the obtained
expression of τ = τ(f) in (25), i.e.
τ = (JjM −1
j )†JM −1[(2 −J⊤
LX−1
L
−J⊤
RX−1
R )mg
2 e3
−(J⊤
LX−1
L
−J⊤
RX−1
R )∆].
(26)
Now, it is easy to verify that
J⊤
LX−1
L
+ J⊤
RX−1
R
=
"
2 16
J⊤
L,jX−1
L
+ J⊤
R,jX−1
R −2
M⊤
bj
m Λ⊤
#
J⊤
LX−1
L
−J⊤
RX−1
R
=
"
06×6
J⊤
L,jX−1
L
−J⊤
R,jX−1
R
#
.
Observe also that
P ⊤e3 =
"
03×3
−S(r)
03×3
I
m −13
#
e3 = 06 ⇒Λ⊤e3 = 16.
Then (26) becomes
τ = −

JjM −1
j
†
JjM −1
j

J⊤
L,jX−1
L
+ J⊤
R,jX−1
R
−2M ⊤
bj
m
!
mg
2 e3 +

J⊤
L,jX−1
L
−J⊤
R,jX−1
R

∆

.
(27)
If we consider the four-bar linkage planar model, we can notice
that the number of rows of Jj are greater than the DoFs of the
system and thus the product of the first two terms simplifies into the
identity matrix. As a consequence, the vector ∆that minimizes the
joint torques is given by:
∆= −mg
2

J⊤
L,jX−1
L
−J⊤
R,jX−1
R
†
 
J⊤
L,jX−1
L
+ J⊤
R,jX−1
R −2M ⊤
bj
m
!
e2
(28)
which, by using the actual expression for the Jacobians, boils down
to
∆= mg
2


ml+mb
m
l cos2(ξ)
d sin(ξ)
ml+mb
m
l cos(ξ)
d
d
2

.
(29)
The complete expression of the contact wrenches can be obtained
by substituting the expression of ∆in Eq. (29) into the following
equation
LfL = X−1
L
mg
2 e2 + X−1
L ∆,
RfR = X−1
R
mg
2 e2 −X−1
R ∆,
yielding the following final expression for the contact wrenches:
LfL = mg






0
1
2
−λ1
2


+


mb+ml
m
l cos(ξ)2
2d sin(ξ)
mb+ml
m
l cos(ξ)
2d
mb+ml
m
l cos(ξ)
2d

λ2
cos(ξ)
sin(ξ) −λ1

+ d
4






,
(31a)
RfR = mg






0
1
2
−ρ1
2


−


mb+ml
m
l cos(ξ)2
2d sin(ξ)
mb+ml
m
l cos(ξ)
2d
mb+ml
m
l cos(ξ)
2d

ρ2
cos(ξ)
sin(ξ) −ρ1

+ d
4






.
(31b)
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
10
Note that by assuming also Hypothesis 1-iii) one obtains the expres-
sions in Eq. (17).
We can now compute the center of pressures SCoPL and SCoPR.
Taking the forces expression in (24), the static CoPs are thus:
SCoPL = −1
2ρ1 + ρ2∆1 −ρ1∆2 + ∆3
1
2 + ∆2
,
SCoPR = −1
2ρ1 −ρ2∆1 + ρ1∆2 −∆3
1
2 −∆2
.
(32)
By plugging the expression of ∆in Eq. (29) we obtain the
expression of the static center of pressure as a function of the free
variable ξ, which is
SCoPL =
9ld cos(ξ)
20d + 6l cos(ξ),
SCoPR =
9ld cos(ξ)
20d −6l cos(ξ).
(33)
To obtain the expression of the sensitivity of the static center of
pressure we can differentiate Eq. (33) w.r.t ξ, finally obtaining
ηL = ∂SCoPL
∂ξ
= −
45d2l sin(ξ)
(10d + 3l cos(ξ))2
ηR = ∂SCoPR
∂ξ
= −
45d2l sin(ξ)
(10d −3l cos(ξ))2 .
C. Proof of Lemma 3
Consider Eq. (22). We want to use f0 to minimize the tangential
component of the contact wrenches, i.e. we want find the solution to
"
e⊤
1
e⊤
4
#
f = 0 ⇐⇒
"
e⊤
1
e⊤
4
#
Nff0 +
"
e⊤
1
e⊤
4
#
X†mge2 = 0.
Because of the expression of the nullspace projector and of the
pseudoinverse, the above equation reduces to
1
2
"
1
0
0
−1
0
0
−1
0
0
1
0
0
#
f0 = 0
which has the minimum norm solution in f0 = 0.
REFERENCES
[1] DRC-Teams, “What happened at the DARPA Robotics Challenge?”
www.cs.cmu.edu/∼cga/drc/events, 2015.
[2] S. Kajita, F. Kanehiro, K. Kaneko, K. Yokoi, and H. Hirukawa, “The
3D Linear Inverted Pendulum Mode: A simple modeling for a biped
walking pattern generation,” in Intelligent Robots and Systems, 2001.
Proceedings. 2001 IEEE/RSJ International Conference on, vol. 1, 2001,
pp. 239–246 vol.1.
[3] S. H. Lee and A. Goswami, “Reaction Mass Pendulum (RMP): An
explicit model for centroidal angular momentum of humanoid robots,”
in Proceedings 2007 IEEE International Conference on Robotics and
Automation, April 2007, pp. 4667–4672.
[4] J. Pratt, J. Carff, S. Drakunov, and A. Goswami, “Capture point: A step
toward humanoid push recovery,” in 2006 6th IEEE-RAS International
Conference on Humanoid Robots, Dec 2006, pp. 200–207.
[5] S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Harada, K. Yokoi,
and H. Hirukawa, “Biped walking pattern generation by using preview
control of zero-moment point,” in Robotics and Automation, 2003.
Proceedings. ICRA ’03. IEEE International Conference on, vol. 2, Sept
2003, pp. 1620–1626 vol.2.
[6] B. Stephens, “Humanoid push recovery,” in 2007 7th IEEE-RAS Inter-
national Conference on Humanoid Robots, Nov 2007, pp. 589–595.
[7] J. Pratt, T. Koolen, T. de Boer, J. Rebula, S. Cotton, J. Carff,
M.
Johnson,
and
P.
Neuhaus,
“Capturability-based
analysis
and
control
of
legged
locomotion,
Part
2:
Application
to
M2V2,
a lower-body humanoid,” The International Journal of Robotics
Research, vol. 31, no. 10, pp. 1117–1133, 2012. [Online]. Available:
http://ijr.sagepub.com/content/31/10/1117.abstract
[8] A. Herzog, L. Righetti, F. Grimminger, P. Pastor, and S. Schaal, “Bal-
ancing experiments on a torque-controlled humanoid with hierarchical
inverse dynamics,” in Intelligent Robots and Systems (IROS 2014), 2014
IEEE/RSJ International Conference on, Sept 2014, pp. 981–988.
[9] L. Sentis, J. Park, and O. Khatib, “Compliant control of multicontact
and center-of-mass behaviors in humanoid robots,” IEEE Transactions
on Robotics, vol. 26, no. 3, pp. 483–501, June 2010.
[10] F. Nori, S. Traversaro, J. Eljaik, F. Romano, A. Del Prete, and
D. Pucci, “iCub Whole-Body Control through Force Regulation on
Rigid Non-Coplanar Contacts,” Frontiers in Robotics and AI, vol. 2,
p. 6, 2015. [Online]. Available: http://journal.frontiersin.org/article/10.
3389/frobt.2015.00006
[11] L. Righetti, J. Buchli, M. Mistry, M. Kalakrishnan, and S. Schaal,
“Optimal distribution of contact forces with inverse-dynamics control,”
The International Journal of Robotics Research, vol. 32, no. 3, pp.
280–298, 2013. [Online]. Available: http://ijr.sagepub.com/content/32/3/
280.abstract
[12] C. Ott, M. A. Roa, and G. Hirzinger, “Posture and balance control for
biped robots based on contact force optimization,” in 2011 11th IEEE-
RAS International Conference on Humanoid Robots, Oct 2011, pp. 26–
33.
[13] ——, “Posture and balance control for biped robots based on contact
force optimization,” in Humanoid Robots (Humanoids), 2011 11th IEEE-
RAS International Conference on, Oct 2011, pp. 26–33.
[14] T. Koolen, S. Bertrand, G. Thomas, T. de Boer, T. Wu, J. Smith,
J.
Englsberger,
and
J.
Pratt,
“Design
of
a
Momentum-Based
Control Framework and Application to the Humanoid Robot Atlas,”
International Journal of Humanoid Robotics, vol. 13, no. 01, p.
1650007, 2016. [Online]. Available: http://www.worldscientific.com/
doi/abs/10.1142/S0219843616500079
[15] P. M. Wensing and D. E. Orin, “Generation of dynamic humanoid be-
haviors through task-space control with conic optimization,” in Robotics
and Automation (ICRA), 2013 IEEE International Conference on, May
2013, pp. 3103–3109.
[16] M. A. Hopkins, D. W. Hong, and A. Leonessa, “Compliant locomotion
using whole-body control and divergent component of motion tracking,”
in 2015 IEEE International Conference on Robotics and Automation
(ICRA), May 2015, pp. 5726–5733.
[17] S.-H. Lee and A. Goswami, “Ground reaction force control at each
foot: A momentum-based humanoid balance controller for non-level and
non-stationary ground,” in Intelligent Robots and Systems (IROS), 2010
IEEE/RSJ International Conference on, Oct 2010, pp. 3157–3162.
[18] M. Liu and V. Padois, “Reactive whole-body control for humanoid
balancing on non-rigid unilateral contacts,” in Intelligent Robots and
Systems (IROS), 2015 IEEE/RSJ International Conference on, Sept 2015,
pp. 3981–3987.
[19] G. Nava, F. Romano, F. Nori, and D. Pucci, “Stability analysis and
design of momentum-based controllers for humanoid robots,” in 2016
IEEE/RSJ International Conference on Intelligent Robots and Systems
(IROS), Oct 2016, pp. 680–687.
[20] D. Pucci, F. Romano, S. Traversaro, and F. Nori, “Highly dynamic
balancing via force control,” in 2016 IEEE-RAS 16th International
Conference on Humanoid Robots (Humanoids), Nov 2016, pp. 141–141.
[21] M. VUKOBRATOVI and B. BOROVAC, “Zero-moment point
thirty
five years of its life,” International Journal of Humanoid Robotics,
vol.
01,
no.
01,
pp.
157–173,
2004.
[Online].
Available:
http:
//www.worldscientific.com/doi/abs/10.1142/S0219843604000083
[22] A. Goswami, “Postural Stability of Biped Robots and the Foot-
Rotation Indicator (FRI) Point,” The International Journal of Robotics
Research, vol. 18, no. 6, pp. 523–533, 1999. [Online]. Available:
http://ijr.sagepub.com/content/18/6/523.abstract
[23] S. Hyon, J. Hale, and G. Cheng, “Full-body compliant human-humanoid
interaction: Balancing in the presence of unknown external forces,”
Robotics, IEEE Transactions on, vol. 23, no. 5, pp. 884–898, Oct 2007.
[24] P. M. Wensing, G. Bin Hammam, B. Dariush, and D. E. Orin,
“Optimizing foot centers of pressure through force distribution in
a humanoid robot,” International Journal of Humanoid Robotics,
vol.
10,
no.
03,
p.
1350027,
2013.
[Online].
Available:
http:
//www.worldscientific.com/doi/abs/10.1142/S0219843613500278
[25] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symme-
try: A Basic Exposition of Classical Mechanical Systems.
Springer
Publishing Company, Incorporated, 2010.
[26] L. Sciavicco, B. Siciliano, and B. Sciavicco, Modelling and Control of
Robot Manipulators, 2nd ed. Secaucus, NJ, USA: Springer-Verlag New
York, Inc., 2000.
[27] G. Metta, L. Natale, F. Nori, G. Sandini, D. Vernon, L. Fadiga, C. von
Hofsten, K. Rosander, M. Lopes, J. Santos-Victor, A. Bernardino, and
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
11
L. Montesano, “The iCub humanoid robot: An open-systems platform
for research in cognitive development,” Neural Networks, vol. 23, no.
89, pp. 1125 – 1134, 2010, social Cognition: From Babies to Robots.
[28] D. Orin, A. Goswami, and S.-H. Lee, “Centroidal dynamics of a
humanoid robot,” Autonomous Robots, vol. 35, no. 2-3, pp. 161–176,
2013. [Online]. Available: http://dx.doi.org/10.1007/s10514-013-9341-4
[29] S. Traversaro, D. Pucci, and F. Nori, “A Unified View of the
Equations of Motion used for Control Design of Humanoid Robots,”
Multibody System Dynamics, 2017, Submitted. [Online]. Available:
https://traversaro.github.io/preprints/changebase.pdf
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
1
The Static Center of Pressure Sensitivity:
a further Criterion to assess Contact Stability
and Balancing Controllers
Francesco Romano, Daniele Pucci, Silvio Traversaro,
Francesco Nori
Abstract—Legged locomotion has received increasing attention from
the robotics community. In this respect, contact stability plays a critical
role in ensuring that robots maintain balance, and it is a key element for
balancing and walking controllers. The Center of Pressure is a contact
stability criterion that defines a point that must be kept strictly inside
the support polygon in order to ensure postural stability. In this paper,
we introduce the concept of the sensitivity of the static center of pressure:
roughly speaking, the rate of change of the center of pressure with respect
to the system equilibrium configurations. This new concept can be used
as an additional criterion to assess the robustness of the contact stability.
We show how the sensitivity of the center of pressure can also be used as
a metric to assess balancing controllers by considering two state-of-the-
art control strategies. The analytical analysis is performed on a simplified
model, and validated during balancing tasks on the iCub humanoid robot.
Index Terms—Humanoid Robots, Legged Robots, Redundant Robots,
Posture Equilibrium Analysis.
I. INTRODUCTION
In the last decades, legged robots have become increasingly more
important. Climbing stairs, avoiding obstacles, and exploiting multi-
ple contacts are only a few advantages that humanoid robots have over
wheeled systems. These advantages have been the main motivations
behind the attention they have received from the robotics community.
Legged locomotion, on the other hand, arises interesting issues, which
must be solved before humanoids can be successfully deployed in
the human environment. The recent DARPA Robotics Challenge has
clearly highlighted these difficulties [1]: most of the failures during
trials were due to a non perfect balancing or walking. This paper
introduces a new metric, called sensitivity of the static center of
pressure, which can be further used to analyse contact stability.
Nowadays there exist two main methodologies to implement bal-
ancing and walking controllers on bipedal robots: the first class is
based robot simplified models [2], [3], [4] complemented with the
use of inverse kinematics to bridge the gap between the simplified
model and the real one [5], [6], [7]. The second class is based on the
full dynamic model of the robot [8], [9], [10], [11] and it has been
successfully exploited for the synthesis of instantaneous controllers.
In this regard, a special attention is paid to controlling the interaction
forces between the robot and the environment [8], [12].
Given the complex nature of humanoid robots, researchers have
progressively shifted from using classic nonlinear control techniques
to adopting optimisation-based solutions [13], [14], [15], [16]. These
optimisation-based control techniques allow one to specify, at high
level, the tasks to be accomplished by the robot: the underlining solver
is then in charge of finding a feasible solution to the optimisation
problem. One disadvantage of this process is that properties of
the solution can be more difficult to find. If some properties have
to be enforced, they can be specified by means of constraints or
optimisation criteria. But how this high-level criteria influence the
solution is not trivial and must be analysed on a case-by-case basis.
*This work was supported by the FP7 EU project CoDyCo (No. 600716
ICT 2011.2.1 Cognitive Systems and Robotics) and Koroibot (No. 611909
ICT-2013.2.1 Cognitive Systems and Robotics).
The authors are with the iCub Facility department, Istituto Italiano di
Tecnologia, Via Morego 30, Genoa, Italy. name.surname@iit.it
Manuscript submitted October 3, 2016.
In this paper, we analyse two state-of-the-art balancing controllers
commonly found in literature ([8], [16], [17], [18], [10] and [19],
[20]), and we analyse them by means of the center of pressure
sensitivity here introduced: we show how these criteria yield different
performance when the robot balances on its two feet.
Maintaining the support foot (or feet) stable is of paramount impor-
tance for balancing and walking. As a result, various postural stability
criteria analysing the stance foot dynamics have been proposed in the
robotics and human locomotion literature. These criteria focus on the
force and torques exchanged at the ground-foot interface.
The Ground projection of the Center of Mass (GCoM) can be
applied if the robot dynamics is at equilibrium: it states that to assess
the postural stability of the biped, the projection on the ground of
the CoM must lie inside the convex hull of the support feet.
The Center of Pressure (CoP), also known as Zero Moment Point
(ZMP) [21], [22], can be applied as soon as the static assumption is
relaxed. The center of pressure can be viewed as the point on the
ground surface where the resultant of the normal contact force acts.
Because of the unilaterality of the normal component of the contact
forces, the center of pressure must lie inside the support polygon.
It is worth noting that the CoP coincides with the GCoM when the
stationarity hypothesis is satisfied [22].
Finally, the Foot Rotation Indicator (FRI) point has been recently
introduced [22]: when the foot does not rotate, the FRI point coincides
with the CoP, and lies inside the support polygon. Differently than
the CoP, the FRI can exit the support polygon, which occurs when
the foot starts rotating.
All the previously described postural stability criteria are not
directly linked to a specific robot dynamical model or balancing
controller implementation, thus being useful for a multitude of control
solutions. On the other hand, one of these criteria (e.g. CoP, FRI, etc)
must be taken into account during the synthesis of postural controllers
to ensure stable foot planting and postural stability of the whole robot.
In this work, we introduce the concept of the sensitivity of the
static center of pressure: roughly speaking the rate of change of the
center of pressure versus the system equilibrium configuration. We
show how this new concept can further characterise the stability of
a contact, thus complementing the notion of center of pressure.
We also show that the sensitivity of the center of pressure can be
used to evaluate the performances of different balancing controllers.
In this regard we analyse two state-of-the-art controllers: they both
stabilise the robot linear and angular momentum as control objective,
assuming the contact forces and torques as virtual control inputs.
Because the system is overdetermined in presence of more than one
contact, the resulting force redundancy can be exploited in different
ways. In particular, the two state-of-the-art controllers differ in the
criterion used to resolve the redundancy: the minimisation of the
contact wrenches norm [8], [16], [17], [18], [10], [23], [24] in one
case and the minimisation of the internal joint torques norm [19],
[20] in the other: we show that the two criteria exhibit different static
center of pressure sensitivities. We can thus exploit the introduced
metric to discriminate between two, apparently similar, controllers.
To simplify the analytical analysis, we model the lower body
of a humanoid robot while balancing as a planar four-bar linkage
mechanism and we further extend and apply the analysis on the iCub
humanoid robot so as to assess the validity of the simplified model.
The paper is structured as follows. Section II introduces the
dynamical model of a free floating mechanical system and the
notation used throughout the paper and Section III describes the
concept of sensitivity of static center of pressure. The sensitivity of
the static center of pressure is applied first to a four-bar linkage model
in Section IV and then to the iCub humanoid robot in Section V.
Finally Section VI draws the conclusions.
arXiv:1610.01495v2  [cs.RO]  29 May 2017
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
2
II. BACKGROUND
A. Notation
Throughout the paper we will use the following definitions.
• I denotes an inertial frame, with its z axis pointing against the
gravity. We denote with g the gravitational constant.
• ei ∈Rm is the canonical vector, consisting of all zeros but the
i-th component, which is one.
• Given two orientation frames A and B, and vectors of coordi-
nates expressed in these frames, i.e. Ap and Bp, respectively,
the rotation matrix ARB is such that Ap = ARB
Bp.
• 1n ∈Rn×n is the identity matrix of size n; 0m×n ∈Rm×n is
the zero matrix of size m × n and 0n = 0n×1.
• We denote with S(x) ∈R3×3 the skew-symmetric matrix such
that S(x)y = x×y, where × denotes the cross product operator
in R3. If x = (˜x⊤, 0)⊤and y = (˜y⊤, 0)⊤, with ˜x, ˜y ∈R2 one
has S(˜x)˜y = −˜x⊤S˜ye3, with S =
"
0
−1
1
0
#
.
B. System modelling
We assume that the robot is composed of n+1 rigid bodies – called
links – connected by n joints with one degree of freedom each. In
addition, we also assume that the multi-body system is free floating,
i.e. none of the links has an a priori constant pose with respect to
the inertial frame. This implies that the multi-body system possesses
n + 6 degrees of freedom. The configuration space of the multi-body
system can then be characterised by the position and the orientation
of a frame attached to a robot’s link – called base frame B – and the
joint configurations. More precisely, the robot configuration space is
defined by
Q = R3 × SO(3) × Rn.
An element of the set Q is then a triplet q = (IpB, IRB, qj), where
(IpB, IRB) denotes the origin and orientation of the base frame
expressed in the inertial frame, and qj denotes the joint angles. The
velocity of the multi-body system can then be characterised by the
algebra V of Q defined by:
V = R3 × R3 × Rn.
An element of V is then a triplet ν = (I ˙pB,I ωB, ˙qj), where IωB
is the angular velocity of the base frame expressed w.r.t. the inertial
frame, i.e. I ˙RB = S(IωB)IRB.
We also assume that the robot is interacting with the environment
through nc distinct contacts. Applying the Euler-Poincar´e formalism
to the multi-body system yields the following equations of motion
[25, Ch. 13.5]1:
M(q) ˙ν + C(q, ν)ν + G(q) = Bτ +
nc
X
k=1
J⊤
Ckfk
(1)
where M ∈Rn+6×n+6 is the mass matrix, C ∈Rn+6×n+6 is the
Coriolis matrix, G ∈Rn+6 is the gravity term, B = (0n×6, 1n)⊤is
a selector matrix, τ are the internal actuation torques, and fk denotes
an external wrench applied by the environment to the link of the k-
th contact. We assume that the external wrench is associated with
a frame Ck, which is attached to the robot’s link where the wrench
acts on. Then, the external wrench fk is expressed in a frame whose
orientation coincides with that of the inertial frame I, but whose
origin is the origin of Ck.
1The Euler-Lagrange’s formulation can be applied only to mechanical
systems evolving in vector spaces. The Euler-Poincar´e equations, instead, are
valid for mechanical systems evolving in arbitrary Lie groups.
CoM
Joint
x
y
λ
ρ
fR
fL
z
I
ξ
Fig. 1: Planar four-bar linkage exploiting two rigid contact to stand
The Jacobian Jk = Jk(q) is the map between the robot’s velocity
ν and the linear and angular velocity IvCk := (I ˙pCk,I ωCk) of the
frame Ck, i.e.
IvCk = JCk(q)ν.
(2)
The Jacobian has the following structure.
JCk(q) =
h
Jb
Ck(q)
Jj
Ck(q)
i
∈R6×n+6,
(2a)
Jb
Ck(q) =
"
13
−S(IpCk −IpB)
03×3
13
#
∈R6×6.
(2b)
Lastly, we assume that holonomic constraints may be enforced
on the robot due to rigid contacts with the environment. These
constraints are modelled as a kinematic constraints that forbid any
motion of the frame Ck, i.e. JCk(q)ν = 0.
III. SENSITIVITY OF THE STATIC CENTER-OF-PRESSURE
This section discusses a property of any center of pressure associ-
ated with rigid, flat contacts and quasi-static robot’s configurations.
We shall see in Section IV that this property – called static center
of pressure sensitivity – can be used as an element of evaluation of
contact stability and balancing controllers for humanoid robots.
A. The sensitivity of the static center of pressure in a case study: the
four-bar linkage case
To provide the reader with an introduction to the sensitivity of the
static center of pressure, focus on Figure 1. This picture depicts a
four-bar linkage standing on two flat, rigid contacts. In this configura-
tion, the mechanical system possesses one degree of freedom, i.e. the
variable ξ, which represents the minimal coordinate of the constrained
mechanical system. At the equilibrium configuration, the system is
hyperstatic because the equilibrium conditions of the Newton-Euler
equations are not sufficient to determine all the (internal) joint torques
τ. Since the joint torques are assumed to be a control input, however,
one can assume that their value is given, with the only requirement
that the system remains at the equilibrium configuration.
Assume that the joint torques depend only on the minimal coordi-
nate ξ. Then, the center of pressures at each static robot configuration,
i.e. ˙ξ = 0, depend only on the minimal coordinate ξ. This poses the
following interesting question.
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
3
What is the rate-of-change of the center of pressures over the
quasi-static constrained robot’s motion?
Different control laws for balancing the four-bar linkage may
be associated with different rate-of-changes of the static center of
pressure over the constrained robot’s motion. Clearly, the higher this
rate-of-change, the higher the likelihood to hit the boundaries of the
support polygon of the foot, which causes breaking the associated
contact and, eventually, a fall of the robot. The above rate-of-change
of the center of pressure is what we call static center of pressure
sensitivity, and the following generalises this concept to any multi-
body system constrained by some rigid, flat contacts.
B. The formal definition
Assume that the robot makes nc rigid contacts with the environ-
ment, and that no external wrench acts on the robot other than gravity
and contact wrenches. Let J denote the Jacobian obtained by stacking
all contact Jacobians JCk, i.e.
J =


JC1
...
JCnc

∈Rk×n+6,
(3)
with k = 6nc. Consequently, all rigid contacts imply the following
constraint on the floating base system (1): Jν = 0, the time derivative
of which is given by
˙Jν + J ˙ν = 0.
(4)
By combining the above constraint with Eq. (1) and by defining
f := (f ⊤
1 , . . . , f ⊤
nc)⊤∈Rk,
one can find the explicit expression of all contact forces, i.e.
f = (JM −1J⊤)−1 h
JM −1C(q, ν)ν+G(q)−Bτ


−˙Jν
i
.
(5)
The hidden assumption for the above equality to hold is that J is
full row rank. Once the obtained expression of f is substituted into
Eq. (1), one obtains the following equations of motion:
M(q) ˙ν + J⊤(JM −1J⊤)−1 ˙Jν = N

Bτ −C(q, ν)ν −G(q)

(6)
with N given by
N = 1n+6 −J⊤(JM −1J⊤)−1JM −1.
Evaluating Eq. (6) at the equilibrium condition (ν, ˙ν) = (0, 0) yields
N

Bτ −G(q)

= 0.
(7)
Eq. (7) defines the values that the joint torques must take to satisfy the
equilibrium condition (ν, ˙ν) = (0, 0). Then, we make the following
assumption.
Assumption 1. The holonomic constraints due to contacts restrain
the configuration space Q into the set Rn+6−k, i.e. there exists a
differentiable mapping χ : Rn+6−k →Q such that
∀q ∈Q
∃ξ ∈Rn+6−k
:
q = χ(ξ).
(8)
Furthermore, the joint torques τ at the equilibrium configuration
(ν, ˙ν) = (0, 0) depend only on the robot constrained configuration
space, i.e. τ = τ(ξ).
In view of the above assumption, the contact wrenches at the
equilibrium configuration, i.e.
f e =


f e
1
...
f e
nc

= (JM −1J⊤)−1JM −1 (G−Bτ) ,
(9)
depend only on the robot constrained motion, i.e. f e = f e(ξ). In
light of the above, we can now define the static center of pressure
sensitivity as follows.
Definition 1. Let SCoPj ∈R2 denote the center of pressure of the
j-th rigid contact at the equilibrium (ν, ˙ν) = (0, 0), i.e.
SCoPj =
1
e⊤
3 f e
j
"
−e⊤
5 f e
j
e⊤
4 f e
j
#
.
(10)
The static center of pressure sensitivity ηj is defined as the rate-of-
change of (10) over the constrained robot motion, i.e.
ηj := ∂SCoPj
∂ξ
.
(11)
In general, the static center of pressure sensitivity is a matrix
belonging to R2×n+6−k, and characterises how fast the center of
pressure associated with a rigid, planar contact moves depending on
the constrained robot’s motion. This sensitivity may then characterise
“the stability” of a contact associated with a robot’s configuration.
Of particular importance are the cases when the sensitivity ηj tends
to infinity. Clearly, these configurations must be avoided by any
controller associated with the robot, since they may cause abrupt
system’s responses and, eventually, a robot fall.
Remark 1. The internal torques at the equilibrium configuration
must satisfy Eq. (7) that induces, in general, multiple solutions
when solved for τ. The proposed sensitivity may be then used to
compare different balancing controllers, since lower values of ηj are
associated with smaller variations of the center of pressures, which
may increase the likelihood of not breaking the contact.
Remark 2. Note that, by its definition in Eq. (11), the actual value
of the sensitivity depends on the choice of the parametrisation ξ.
Different choices of parametrisation may lead to different values of
the sensitivity.
IV. THE SENSITIVITY FOR ASSESSING BALANCING
CONTROLLERS PERFORMANCE:
THE FOUR-BAR LINKAGE CASE STUDY
This section studies how the sensitivity of the static center of
pressure may vary depending on different balancing controllers
synthesised for the same system. What follows presents the analytical
analysis on a four-bar linkage when standing on two, flat, rigid
contacts. For balancing purposes of the four-bar linkage, we choose
two state-of-the-art momentum-based controllers that differ in the
criterion adopted for the contact forces redundancy resolution. More
precisely, we show next that minimising the joint torques when the
Newton-Euler equations are at the equilibrium decreases the static
center of pressure sensitivity, thus improving the stability of the
contact. As the experimental results in Section V show, the four-bar
linkage case study is representative of a humanoid robot standing on
two feet.
A. Modelling
Consider the four-bar linkage shown in Figure 2, where l and d
denote the lengths of the leg and of the upper rod, respectively. Each
leg possesses a mass of ml while the upper rod has mass mb. The
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
4
x
y
λ
ρ
B
q1
q2
q3
q4
z
ϑ
I
CoM
Joint
d
l
Fig. 2: Free floating planar four-bar linkage
inertial frame I is chosen so as the y axis opposes gravity, the z
axis exits from the plane, and the x axis completes the right-hand
base. Being a planar model, only translations in the x −y plane and
rotations about the z axis are allowed. The vectors
λ =
h
λ1
λ2
i⊤
∈R2,
ρ =
h
ρ1
ρ2
i⊤
∈R2
connect the center of mass to the left and right foot frames respec-
tively. Let IpB ∈R2 denote the position of the origin of the frame B
– which is located at the center of the upper link – expressed in the
inertial frame. The configuration space is then parametrised by the
following coordinates
q = (IpB, θ, q1, q2, q3, q4) ∈R7.
Being a vector space, we use the Lagrange formalism to derive the
equations of motion [26, Ch. 4], i.e.
d
dt
∂L
∂˙q + ∂L
∂q =
"
03
τ
#
,
(12)
with L := T −V the Lagrangian, and T and V respectively the
kinetic and potential energy of the multi-body system.
Hypothesis 1. For the four-bar linkage model, it holds that:
i) Each link is approximated as a point mass at its center of mass.
ii) The mass of the left leg equals the mass of the right leg.
iii) The upper body mass, mb, is twice the leg mass, i.e. mb = 2ml.
It is worth noting that hypothesis ii) and iii) are in general satisfied
in humanoid robots, e.g. in the iCub robot used for the experiments.
For the sake of simplicity, the complete dynamic model obtained
by applying Eq. (12) is not presented here. We present only the
terms of the dynamic and kinematic model that are necessary for the
comprehension of the remainder of the section. In particular, partition
the mass matrix M as follows
M =
"
Mb
Mbj
M ⊤
bj
Mj
#
.
Let sin(qi) = si and cos(qi) = ci. One can verify that
Mbj = ml
2


ls1
0
ls3
0
−lc1
0
−lc3
0
l2s2
1−lc1(d−lc1)
2
0
l2s2
3+lc3(d+lc3)
2
0


(13)
The feet Jacobians are given by the following expression
JCL =


0
0
0
12
R(θ)
"
ls1
d
2 −lc1
#
R(θ)
"
ls1
−lc1
#
0
0
0
0
0
1
1
1
0
0


JCR =


0
0
0
12
R(θ)
"
ls3
−d
2 + lc3
#
0
0
R(θ)
"
ls3
−lc3
#
0
0
0
1
0
0
1
1

.
To represent a classical balancing context of humanoid robots, we
assume that the mechanism makes contact with the environment at
the two extremities. Also, we assume that the distance between the
contacts equals d, i.e. the length of the upper rod – see Figure 1. Then,
the rigid constraint acting on the system takes the form (4), with J =
h
J⊤
CL
J⊤
CR
i⊤
∈R6×7, and ν = ˙q. The associated contact wrenches
are f = (fL, fR) ∈R6. As a consequence of these constraints, the
mechanism possesses one degree of freedom when standing on the
extremities.
With the aim of evaluating the sensitivity of the static center of
pressure we define the minimal coordinate ξ as the angle between
the upper link and one of the legs (see Figure 1). Observe that the
mapping q = χ(ξ) in (8) is given by:


Ipb
θ
q1
q2
q3
q4


=


1
2de1 + l sin(ξ)e2 + l cos(ξ)e1
0
ξ
π −ξ
ξ
π −ξ


.
(14)
B. Two balancing controllers inducing different sensitivities
Remark that in order to evaluate the sensitivity of the static center
of pressure, we need an expression for the contact forces at the
equilibrium configuration – see Definition 1. What follows proposes
an analysis of the sensitivity of the static center of pressures evaluated
with contact wrenches obtained from two state-of-the-art criteria.
1) The contact wrenches ensure the equilibrium of the centroidal
dynamics written in the plane. The three dimensional redun-
dancy of the contact wrenches minimises the norm of the contact
wrenches2 |f| [8], [16], [17], [18], [10].
2) The contact wrenches ensure the equilibrium of the centroidal
dynamics. The three dimensional redundancy of the contact
wrenches minimises the norm of the joint torques |τ| [19], [20].
The following lemma presents the results when the first criterion
is applied.
Lemma 1. Assume that Hypothesis 1 holds. If the contact wrenches
redundancy is chosen to minimise the norm of the contact wrenches,
i.e. criterion 1), then the induced contact wrenches are given by the
2It is worth noting that the norm of a wrench is not well defined and highly
dependent on the chosen metric. Nevertheless, the minimisation of the contact
wrench norm is something extensively adopted in literature and it has been
used in the manuscript only for comparison purposes.
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
5
-15
-10
-5
0
5
10
15
-60
-40
-20
0
20
40
60
(a) Minimum wrenches norm solution
-15
-10
-5
0
5
10
15
-60
-40
-20
0
20
40
60
(b) Minimum joint torque norm solution
Fig. 3: Left and right foot CoPs together with the CoM for different configuration of ξ. In (b) the asymptotes are in correspondence of the
vertical component of the forces going to zero
-15
-10
-5
0
5
10
15
0
0.2
0.4
0.6
0.8
1
Fig. 4: Comparison of the vertical force component of the right foot
for the two contact forces choice. The vertical force is normalised
w.r.t. the total mass of the mechanism, i.e. divided by 1/mg
following expression:
Lf L = mg


0
1
2

1+ d2
4
 + d 5d−6l cos(ξ)
10(d2+4)
6l cos(ξ)
5(d2+4)


,
Rf R = mg


0
1
2

1+ d2
4
 + d 5d+6l cos(ξ)
10(d2+4)
6l cos(ξ)
5(d2+4)


.
(15)
Furthermore, the static center of pressure sensitivity is given by:
ηL = −60l sin(ξ)
d2 + 4
(5d2 + 6ld cos(ξ) + 20)2 ,
ηR = −60l sin(ξ)
d2 + 4
(5d2 −6ld cos(ξ) + 20)2 .
(16)
The proof is in the Appendix.
It is worth noting that if d is small enough, then Eq. (15) points
out that one has an almost constant vertical force independently of
ξ. In this case, the vertical force is approximately equal to mg/2,
independently of the center-of-mass position.
Figure 3a shows the static CoP, referred to as SCoP, of the left
and right foot in correspondence of the CoM motion. We can notice
how SCoPL (the SCoP of the left foot) and SCoPR (the SCoP of the
right foot) moves together with the CoM.
Let us now present analogous results of Lemma 1 when applying
the criterion 2), i.e. the choice of the force redundancy minimises the
joint torques.
Lemma 2. Assume that Hypothesis 1 holds. If the contact wrenches
redundancy is chosen to minimise the norm of the internal joints
torques, i.e. criterion 2), the induced wrenches are given by the
following expressions.
LfL=mg







0
1
2
−λ1
2

+


3l cos(ξ)2
8d sin(ξ)
3l cos(ξ)
8d
3l cos(ξ)
8d

λ2
cos(ξ)
sin(ξ) −λ1

+ d
4






, (17a)
RfR=mg







0
1
2
−ρ1
2

−


3l cos(ξ)2
8d sin(ξ)
3l cos(ξ)
8d
3l cos(ξ)
8d

ρ2
cos(ξ)
sin(ξ) −ρ1

+ d
4






. (17b)
Furthermore, the static center of pressure sensitivity is given by:
ηL = −
45d2l sin(ξ)
(10d + 3l cos(ξ))2 ,
ηR = −
45d2l sin(ξ)
(10d −3l cos(ξ))2 .
(18)
The general expression of the contact wrenches (17) when Hy-
pothesis 1-iii) does not hold is given in the Appendix.
Focus on the contact wrenches in (17). Differently from Eq. (15),
the vertical components of the contact forces
LfL and
LfR in
equations (17a) and (17b) do vary with the variable ξ, see Figure
4. In particular, when the center of mass of the system is on top of
one of the feet, the corresponding vertical force is equal to mg, i.e.
the foot supports the whole weight. As a consequence, the force on
the other foot is null.
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
6
-15
-10
-5
0
5
10
15
0
0.5
1
1.5
2
Fig. 5: Comparison of the norm of the sensitivity of the static center
of pressure of the right foot for the two contact forces choice
The behaviour of the static centers of pressure can be observed in
Figure 3b. Interestingly, while the minimum torques norm solution
exhibits more “flat” CoPs around the centered position and on the
support foot, we can notice a divergent behaviour around ∆ξ =
±14[deg]. This is due to the behaviour of the vertical components of
the force previously described, i.e. to the fact that with this solution,
the force on one foot tends to zero as the weight is moved towards
the other foot. It is worth noting that the center of pressure showing
the divergent behaviour is the one of the non-supporting foot.
We can compare the sensitivity of the static centers of pressure
induced by criteria 1) and 2) when the CoM moves from left to
right. Figure 5 compares the norm of the right foot static center of
pressure sensitivities obtained by applying the two criteria. Notice
that the second criterion, i.e. the minimum joint torques norm
solution, provides less sensitive solutions at the initial, symmetrical
configuration, i.e. ∆ξ = 0. Furthermore, the sensitivity decreases
when the foot is increasingly supporting all the weight, i.e. toward
∆ξ = 14[deg] in the considered case.
Remark 3. A criterion similar to the minimisation of the norm
of the contact wrenches that is sometimes found in literature is
the minimisation of the tangential components of the forces [11].
The following lemma states the equivalence of the two solutions
when planar contacts are considered. The proof can be found in
the Appendix.
Lemma 3. The solution exploiting the force redundancy to minimise
the tangential component of the contact wrenches is equivalent to the
minimum contact wrench norm solution (Lemma 1).
Remark 4. The proposed analysis focuses on the sensitivity of the
center of pressure, that is, on its rate of change with respect to a
parametrisation of the minimal coordinates of the constrained system.
The use of inequalities on the contact wrenches, usually enforced by
the use of QP solvers, limits the actual value of the center of pressure.
The two concepts are thus complementary.
V. EXPERIMENTS
In this section, we validate the analysis performed on the four-bar
linkage model on the full iCub humanoid robot. We apply the same
principles stated in criterion 1) and 2) in Section IV-B to achieve
balancing.
Our test platform is the iCub robot, a state-of-the-art 53 degrees
of freedom humanoid robot [27]. For the purpose of the present
experiment, we consider only the principal 23 degrees of freedom,
located in the legs, torso and upper arms. These degrees of freedom
are supplied of a low level torque control that runs at 1kHz and
is responsible of stabilising desired joint torques. As a comparison
with the simplified model, the hips are about 50cm high and the feet
are kept, during the experiments, at about 14cm of distance between
each other. Contact forces with the ground are estimated by proper
estimation algorithm that exploits six six-axes force torque sensors
installed within iCub [10, Sec. 3.2].
The controller is a dynamic controller composed of a strict hier-
archy of two control objectives. The first, and with highest priority,
is responsible of tracking a desired robot momentum through the
use of the contact wrenches. Note that the desired robot momentum
contains feedback terms on the center of mass position and velocity.
The second one, instead, is responsible of stabilising the resulting
zero dynamics. Discussion about the stability of the zero dynamics is
out of the scope of the present paper and we refer the interested reader
to [19] for further details on the zero dynamics stability problem.
Assuming only two contacts located at the feet, the dynamics of
the robot momentum is given by the following equation:
˙H = XLfL + XRfR + m¯g
(19)
where H ∈R6 is the robot linear and angular momentum,
XL =
"
13
03×3
S(xL −xCoM)
13
#
, XR =
"
13
03×3
S(xR −xCoM)
13
#
are transformation matrices, xCoM is the position of the center of mass
and xL, xR are the positions of the left and right foot respectively.
Finally m¯g is the wrench due to gravity.
Solutions to Eq. (19) yield the desired fL and fR to be applied
by the robot, which are in turn generated by the actuation torques so
as to satisfy the first control objective, i.e. tracking a desired robot
momentum. To obtain the torques to be applied, we can invert Eq.
(5). If we denote with f ∗= [f ∗⊤
L , f ∗⊤
R ]⊤a possible solution to Eq.
(19), we obtain the following expression for the control variable τ:
τ = (JM −1B)†[JM −1(Cν + G −J⊤f ∗) −˙Jν],
(20)
where (JM −1B)† denotes the Moore-Penrose pseudoinverse of the
matrix JM −1B. The choice of f ∗, and as a consequence the torques
applied by the robot, depends on the criterion chosen to obtain a
solution to Eq. (19). Similarly as we did for the four-bar linkage
system we apply criterion 1) and 2) (see Sec. IV-B) on the real robot
and we compare the obtained results.
In the experiments, we move the robot along the frontal plane3 with
very low velocity so as to simulate the quasi-static scenario presented
in the analysis in Section IV. We begin from the initial, symmetrical
configuration, moving the robot towards the configuration where the
CoM lies on the left or right foot frame origin. In particular, we
impose a sinusoidal reference for the center of mass y-component,
being the y-axis the axis along the robot frontal plane, and we fix the
reference amplitude and frequency to be equal for the control laws
associated with both criteria. When we apply the criterion 2), we
manage to track a full period of the reference sine, for the criterion
1) we have to perform two different experiments, i.e. one moving the
CoM towards the left foot and the other towards the right one.
We first tested the minimum wrenches norm solution, i.e. criterion
1). As we showed in the analysis in the previous section we expect
that the vertical components of the forces do not change so much
from the half weight of the robot. This is confirmed also on the real
robot. As we can see in Figure 6a the forces start symmetrical. When
3We define with frontal plane any vertical plane that divides the robot into
anterior and posterior portions, similarly as in humans anatomy.
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
7
0
5
10
15
0
50
100
150
200
250
300
(a) Minimum wrenches norm solution
0
10
20
30
40
50
0
50
100
150
200
250
300
(b) Minimum joint torque norm solution
Fig. 6: Vertical components of the contact forces for left and right foot during the robot quasi-static motion. The experiment in (a) depicts
the forces when the CoM moves towards the left foot. In (b) the CoM tracks a full sinusoidal reference period instead.
0
5
10
15
-5
0
5
Left
Right
L. Foot Limit
R. Foot Limit
(a) CoM towards left foot
0
5
10
15
-5
0
5
Left
Right
L. Foot Limit
R. Foot Limit
(b) CoM towards right foot
Fig. 7: Center of pressure for the left and right foot when the minimisation of the contact wrenches criterion is applied. In (a) we show the
experiment with the CoM moving towards the left foot, while in (b) the CoM moves towards the right foot instead.
the CoM starts moving towards the left foot we notice that both the
forces change of only a limited quantity. The same behaviour was
verified when we moved the CoM towards the right foot. We then
tested criterion 2), i.e. we exploited the redundancy of the forces
to optimise for the joint torques. Figure 6b shows the forces during
the experiment for a full period of the CoM sinusoidal reference,
highlighting the dependence of the forces upon the center of mass
position.
We then analysed the center of pressure behaviour during the
experiments. Figure 7 shows the left and right foot center of pressure
when we applied the first criterion. As mentioned previously we had
to perform two different experiments, one where the CoM moves
towards the left foot, in Fig. 7a, and one where the CoM moves
towards the right foot, in Fig. 7b. As the analysis on the four-bar
linkage showed the two centers of pressure move almost in parallel
regardless of which foot supports most of the weight.
When we apply the control law obtain by satisfying criterion 2),
i.e. minimisation of the internal torques norm, the left and right
foot centers of pressure behave differently, with the CoP of the foot
supporting most of the weight moving less that the other one, as
Figure 8 clearly highlights.
Remark 5. As last step in our experimental validation, we compare
how the two different criteria behave with respect to the minimal
coordinate variables. Given the kind of movements performed by
the robot, we assume as minimum coordinate variable ξ the left hip
roll joint, which corresponds to the joint q1 in the four-bar linkage
model shown in Figure 2. Having performed this choice, we can
now properly compare the CoPs obtained with the two force criteria.
Fig. 9 shows the aforementioned comparison considering only the
left foot, where Fig 9a shows the analytical results on the four-bar
linkage model and Fig. 9b shows the results collected on the iCub
humanoid robot. It is worth noting the similarity between the two
results, thus proving the validity of the reduced model we adopted
for the theoretical analysis.
VI. CONCLUSIONS
In this paper we introduced the concept of static center of pressure
sensitivity as an additional tool to analyse and evaluate the stability of
a contact and to benchmark the performance of balancing controllers.
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
8
0
10
20
30
40
50
-5
0
5
Left
Right
L. Foot Limit
R. Foot Limit
CoM towards left
| CoM towards right
Fig. 8: Center of pressure for the left and right foot when the
minimisation of the joint torques norm criterion is applied.
We first analysed the newly introduced metric on a planar four-bar
linkage, which approximates the lower body of a biped robot when
balancing. Subsequently, we verified the analysis on the full iCub
humanoid robot, obtaining similar results, thus proving the power of
the approximate model. The effect on the stability of the contacts of
two different choices of contact forces are then shown and compared.
The analysis we performed, both analytically and numerically,
shows how different criteria imply very different properties on the
contact wrenches, and in turn, on the resulting center of pressures.
In particular the wrenches obtained by minimising the internal joint
torques produces a lower sensitivity w.r.t. the minimum wrench norm
solution.
We want to stress the fact that we do not advocate for the use of the
proposed static center of pressure sensitivity as a substitute to other
criteria such as the center of pressure or the foot rotation indicator. On
the contrary, the newly introduced metric can be used as an additional
tool to discriminate between apparently similar balancing controllers.
During the analysis we noticed an additional, interesting property
of the solution which minimises the joint torques. Differently from
the other solution, the vertical components of the contact force on
one foot tends to zero when the center of mass of the robot moves
towards the opposite foot. While a proper theoretical analysis should
be put into effort, this solution can help minimise discontinuities due
to a change of the contact set.
APPENDIX
A. Proof of Lemma 1
Consider the equilibrium condition of the centroidal dynamics [28]
of the mechanism, i.e. the balance of the external wrenches acting
on the mechanism when written w.r.t. a frame centered in the center
of mass and with the orientation of the inertial frame.
0 = XL
Lf L + XR
Rf R −mge2 = Xf −mge2
(21)
where mge2 is the wrench due to gravity and X :=
h
XL
XR
i
,
XL =
"
12
02×1
(Sλ)⊤
1
#
,
XR =
"
12
02×1
(Sρ)⊤
1
#
,
λ =
h
λ1
λ2
i⊤
∈R2 and ρ =
h
ρ1
ρ2
i⊤
∈R2 the vectors
connecting the center of mass to the left and right foot frames
respectively, see Figure 1.
Since X ∈R3×6 by construction is always full row rank, (21)
admits infinite solutions, i.e.
f = X†mge2 + Nff0
(22)
where X† ∈R6×3 is the Moore-Penrose pseudoinverse of X, Nf ∈
R6×6 is the projector onto the nullspace of X and f0 ∈R6 is a
free variable. The unique minimum norm solution corresponds to the
choice f0 = 0. The actual expression of the contact wrenches are
thus the ones given in Eq. (15).
The center of pressure in the planar case possesses only one
coordinate, which is given by the following expressions
LSCoPL = e⊤
3 f
e⊤
2 f ,
RSCoPR = e⊤
6 f
e⊤
5 f
which evaluated with the expression of the wrenches in Eq. (15) yield
LSCoPL =
12l cos(ξ)
5d2 + 6ld cos(ξ) + 20,
RSCoPR =
12l cos(ξ)
5d2 −6ld cos(ξ) + 20.
To evaluate the static center of pressure sensitivity, we can differ-
entiate the above expression w.r.t ξ, that is
ηL = ∂SCoPR
∂ξ
= −60l sin(ξ)
d2 + 4
(5d2 + 6ld cos(ξ) + 20)2 ,
ηR = ∂SCoPR
∂ξ
= −60l sin(ξ)
d2 + 4
(5d2 −6ld cos(ξ) + 20)2 .
B. Proof of Lemma 2
Consider the balance of the external wrenches acting on an
articulated rigid body in the general 3D case:
XL
Lf L + XR
Rf R −mge3 = 0
(23)
where XL, XR ∈R6×6 are used to express the wrenches respectively
in L and R at the frame located at center of mass with the orientation
of the inertial frame. Impose the following expression for the contact
wrenches:
Lf L = X−1
L
mg
2 e3 + X−1
L ∆
Rf R = X−1
R
mg
2 e3 −X−1
R ∆
(24)
where ∆∈R6 is a free variable. We can notice that with the
above choice, (23) is always satisfied independently of the choice
of ∆. In what follows, the redundancy ∆is used to minimize
the joint torques. To simplify the calculations we perform a state
transformation as described in [29]. In particular, the dynamics in
the new state variables is decoupled in the acceleration, i.e. the mass
matrix is block diagonal, with the two blocks being the terms relative
to the base and the joints. We refer the reader to [29] for additional
details, and we report here only the relevant transformations. To avoid
confusion, all the dynamic quantities in the new state variables are
denoted with the overline.
The new base frame is chosen to correspond to a frame with the
same orientation of the inertial frame I and the origin instantaneously
located at the center of mass. We denote with cXB the corresponding
6D transformation. Of particular interest is the form of the Jacobian
of the left (right) foot frame L (R) in the new state variables:
JL =
h
JL,b
JL,j
i
=
h
X⊤
L
JL,j −1
mX⊤
L ΛMbj
i
,
JR =
h
JR,b
JR,j
i
=
h
X⊤
R
JR,j −1
mX⊤
R ΛMbj
i
,
Λ := 16 −P(16 + cX−1
B P)−1cX−1
B .
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
9
-10
-5
0
5
10
-4
-2
0
2
4
6
(a) Four-bar linkage model
-10
-5
0
5
10
-4
-2
0
2
4
6
Min. Wrenches
Min. Joint Torque
(b) Robot
Fig. 9: Comparison of the center of pressure of the left foot for the two criteria on the simplified model (a) and on the real robot (b). The
dotted blue lines in Figure (b) denote the convex hull of the left foot.
P has the following expression:
P :=
"
03×3
03×3
S(pCoM −pB)
I
m −13
#
where I is the 3D inertia of the robot expressed w.r.t. the orientation
of I and the center of mass as origin and pCoM −pB is the distance
between the center of mass and the base frame before the state
transformation.
The relation between the joint torques and the contact wrenches
is given by Eq. (9). So, by inverting this relation, we evaluate the
effects of the contact wrenches redundancy ∆on the joint torques at
the equilibrium configuration, i.e.
τ = (JjM −1
j )†JM −1(mge3 −J⊤f).
(25)
We can now substitute the definition of f as in (24) into the obtained
expression of τ = τ(f) in (25), i.e.
τ = (JjM −1
j )†JM −1[(2 −J⊤
LX−1
L
−J⊤
RX−1
R )mg
2 e3
−(J⊤
LX−1
L
−J⊤
RX−1
R )∆].
(26)
Now, it is easy to verify that
J⊤
LX−1
L
+ J⊤
RX−1
R
=
"
2 16
J⊤
L,jX−1
L
+ J⊤
R,jX−1
R −2
M⊤
bj
m Λ⊤
#
J⊤
LX−1
L
−J⊤
RX−1
R
=
"
06×6
J⊤
L,jX−1
L
−J⊤
R,jX−1
R
#
.
Observe also that
P ⊤e3 =
"
03×3
−S(r)
03×3
I
m −13
#
e3 = 06 ⇒Λ⊤e3 = 16.
Then (26) becomes
τ = −

JjM −1
j
†
JjM −1
j

J⊤
L,jX−1
L
+ J⊤
R,jX−1
R
−2M ⊤
bj
m
!
mg
2 e3 +

J⊤
L,jX−1
L
−J⊤
R,jX−1
R

∆

.
(27)
If we consider the four-bar linkage planar model, we can notice
that the number of rows of Jj are greater than the DoFs of the
system and thus the product of the first two terms simplifies into the
identity matrix. As a consequence, the vector ∆that minimizes the
joint torques is given by:
∆= −mg
2

J⊤
L,jX−1
L
−J⊤
R,jX−1
R
†
 
J⊤
L,jX−1
L
+ J⊤
R,jX−1
R −2M ⊤
bj
m
!
e2
(28)
which, by using the actual expression for the Jacobians, boils down
to
∆= mg
2


ml+mb
m
l cos2(ξ)
d sin(ξ)
ml+mb
m
l cos(ξ)
d
d
2

.
(29)
The complete expression of the contact wrenches can be obtained
by substituting the expression of ∆in Eq. (29) into the following
equation
LfL = X−1
L
mg
2 e2 + X−1
L ∆,
RfR = X−1
R
mg
2 e2 −X−1
R ∆,
yielding the following final expression for the contact wrenches:
LfL = mg






0
1
2
−λ1
2


+


mb+ml
m
l cos(ξ)2
2d sin(ξ)
mb+ml
m
l cos(ξ)
2d
mb+ml
m
l cos(ξ)
2d

λ2
cos(ξ)
sin(ξ) −λ1

+ d
4






,
(31a)
RfR = mg






0
1
2
−ρ1
2


−


mb+ml
m
l cos(ξ)2
2d sin(ξ)
mb+ml
m
l cos(ξ)
2d
mb+ml
m
l cos(ξ)
2d

ρ2
cos(ξ)
sin(ξ) −ρ1

+ d
4






.
(31b)
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
10
Note that by assuming also Hypothesis 1-iii) one obtains the expres-
sions in Eq. (17).
We can now compute the center of pressures SCoPL and SCoPR.
Taking the forces expression in (24), the static CoPs are thus:
SCoPL = −1
2ρ1 + ρ2∆1 −ρ1∆2 + ∆3
1
2 + ∆2
,
SCoPR = −1
2ρ1 −ρ2∆1 + ρ1∆2 −∆3
1
2 −∆2
.
(32)
By plugging the expression of ∆in Eq. (29) we obtain the
expression of the static center of pressure as a function of the free
variable ξ, which is
SCoPL =
9ld cos(ξ)
20d + 6l cos(ξ),
SCoPR =
9ld cos(ξ)
20d −6l cos(ξ).
(33)
To obtain the expression of the sensitivity of the static center of
pressure we can differentiate Eq. (33) w.r.t ξ, finally obtaining
ηL = ∂SCoPL
∂ξ
= −
45d2l sin(ξ)
(10d + 3l cos(ξ))2
ηR = ∂SCoPR
∂ξ
= −
45d2l sin(ξ)
(10d −3l cos(ξ))2 .
C. Proof of Lemma 3
Consider Eq. (22). We want to use f0 to minimize the tangential
component of the contact wrenches, i.e. we want find the solution to
"
e⊤
1
e⊤
4
#
f = 0 ⇐⇒
"
e⊤
1
e⊤
4
#
Nff0 +
"
e⊤
1
e⊤
4
#
X†mge2 = 0.
Because of the expression of the nullspace projector and of the
pseudoinverse, the above equation reduces to
1
2
"
1
0
0
−1
0
0
−1
0
0
1
0
0
#
f0 = 0
which has the minimum norm solution in f0 = 0.
REFERENCES
[1] DRC-Teams, “What happened at the DARPA Robotics Challenge?”
www.cs.cmu.edu/∼cga/drc/events, 2015.
[2] S. Kajita, F. Kanehiro, K. Kaneko, K. Yokoi, and H. Hirukawa, “The
3D Linear Inverted Pendulum Mode: A simple modeling for a biped
walking pattern generation,” in Intelligent Robots and Systems, 2001.
Proceedings. 2001 IEEE/RSJ International Conference on, vol. 1, 2001,
pp. 239–246 vol.1.
[3] S. H. Lee and A. Goswami, “Reaction Mass Pendulum (RMP): An
explicit model for centroidal angular momentum of humanoid robots,”
in Proceedings 2007 IEEE International Conference on Robotics and
Automation, April 2007, pp. 4667–4672.
[4] J. Pratt, J. Carff, S. Drakunov, and A. Goswami, “Capture point: A step
toward humanoid push recovery,” in 2006 6th IEEE-RAS International
Conference on Humanoid Robots, Dec 2006, pp. 200–207.
[5] S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Harada, K. Yokoi,
and H. Hirukawa, “Biped walking pattern generation by using preview
control of zero-moment point,” in Robotics and Automation, 2003.
Proceedings. ICRA ’03. IEEE International Conference on, vol. 2, Sept
2003, pp. 1620–1626 vol.2.
[6] B. Stephens, “Humanoid push recovery,” in 2007 7th IEEE-RAS Inter-
national Conference on Humanoid Robots, Nov 2007, pp. 589–595.
[7] J. Pratt, T. Koolen, T. de Boer, J. Rebula, S. Cotton, J. Carff,
M.
Johnson,
and
P.
Neuhaus,
“Capturability-based
analysis
and
control
of
legged
locomotion,
Part
2:
Application
to
M2V2,
a lower-body humanoid,” The International Journal of Robotics
Research, vol. 31, no. 10, pp. 1117–1133, 2012. [Online]. Available:
http://ijr.sagepub.com/content/31/10/1117.abstract
[8] A. Herzog, L. Righetti, F. Grimminger, P. Pastor, and S. Schaal, “Bal-
ancing experiments on a torque-controlled humanoid with hierarchical
inverse dynamics,” in Intelligent Robots and Systems (IROS 2014), 2014
IEEE/RSJ International Conference on, Sept 2014, pp. 981–988.
[9] L. Sentis, J. Park, and O. Khatib, “Compliant control of multicontact
and center-of-mass behaviors in humanoid robots,” IEEE Transactions
on Robotics, vol. 26, no. 3, pp. 483–501, June 2010.
[10] F. Nori, S. Traversaro, J. Eljaik, F. Romano, A. Del Prete, and
D. Pucci, “iCub Whole-Body Control through Force Regulation on
Rigid Non-Coplanar Contacts,” Frontiers in Robotics and AI, vol. 2,
p. 6, 2015. [Online]. Available: http://journal.frontiersin.org/article/10.
3389/frobt.2015.00006
[11] L. Righetti, J. Buchli, M. Mistry, M. Kalakrishnan, and S. Schaal,
“Optimal distribution of contact forces with inverse-dynamics control,”
The International Journal of Robotics Research, vol. 32, no. 3, pp.
280–298, 2013. [Online]. Available: http://ijr.sagepub.com/content/32/3/
280.abstract
[12] C. Ott, M. A. Roa, and G. Hirzinger, “Posture and balance control for
biped robots based on contact force optimization,” in 2011 11th IEEE-
RAS International Conference on Humanoid Robots, Oct 2011, pp. 26–
33.
[13] ——, “Posture and balance control for biped robots based on contact
force optimization,” in Humanoid Robots (Humanoids), 2011 11th IEEE-
RAS International Conference on, Oct 2011, pp. 26–33.
[14] T. Koolen, S. Bertrand, G. Thomas, T. de Boer, T. Wu, J. Smith,
J.
Englsberger,
and
J.
Pratt,
“Design
of
a
Momentum-Based
Control Framework and Application to the Humanoid Robot Atlas,”
International Journal of Humanoid Robotics, vol. 13, no. 01, p.
1650007, 2016. [Online]. Available: http://www.worldscientific.com/
doi/abs/10.1142/S0219843616500079
[15] P. M. Wensing and D. E. Orin, “Generation of dynamic humanoid be-
haviors through task-space control with conic optimization,” in Robotics
and Automation (ICRA), 2013 IEEE International Conference on, May
2013, pp. 3103–3109.
[16] M. A. Hopkins, D. W. Hong, and A. Leonessa, “Compliant locomotion
using whole-body control and divergent component of motion tracking,”
in 2015 IEEE International Conference on Robotics and Automation
(ICRA), May 2015, pp. 5726–5733.
[17] S.-H. Lee and A. Goswami, “Ground reaction force control at each
foot: A momentum-based humanoid balance controller for non-level and
non-stationary ground,” in Intelligent Robots and Systems (IROS), 2010
IEEE/RSJ International Conference on, Oct 2010, pp. 3157–3162.
[18] M. Liu and V. Padois, “Reactive whole-body control for humanoid
balancing on non-rigid unilateral contacts,” in Intelligent Robots and
Systems (IROS), 2015 IEEE/RSJ International Conference on, Sept 2015,
pp. 3981–3987.
[19] G. Nava, F. Romano, F. Nori, and D. Pucci, “Stability analysis and
design of momentum-based controllers for humanoid robots,” in 2016
IEEE/RSJ International Conference on Intelligent Robots and Systems
(IROS), Oct 2016, pp. 680–687.
[20] D. Pucci, F. Romano, S. Traversaro, and F. Nori, “Highly dynamic
balancing via force control,” in 2016 IEEE-RAS 16th International
Conference on Humanoid Robots (Humanoids), Nov 2016, pp. 141–141.
[21] M. VUKOBRATOVI and B. BOROVAC, “Zero-moment point
thirty
five years of its life,” International Journal of Humanoid Robotics,
vol.
01,
no.
01,
pp.
157–173,
2004.
[Online].
Available:
http:
//www.worldscientific.com/doi/abs/10.1142/S0219843604000083
[22] A. Goswami, “Postural Stability of Biped Robots and the Foot-
Rotation Indicator (FRI) Point,” The International Journal of Robotics
Research, vol. 18, no. 6, pp. 523–533, 1999. [Online]. Available:
http://ijr.sagepub.com/content/18/6/523.abstract
[23] S. Hyon, J. Hale, and G. Cheng, “Full-body compliant human-humanoid
interaction: Balancing in the presence of unknown external forces,”
Robotics, IEEE Transactions on, vol. 23, no. 5, pp. 884–898, Oct 2007.
[24] P. M. Wensing, G. Bin Hammam, B. Dariush, and D. E. Orin,
“Optimizing foot centers of pressure through force distribution in
a humanoid robot,” International Journal of Humanoid Robotics,
vol.
10,
no.
03,
p.
1350027,
2013.
[Online].
Available:
http:
//www.worldscientific.com/doi/abs/10.1142/S0219843613500278
[25] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symme-
try: A Basic Exposition of Classical Mechanical Systems.
Springer
Publishing Company, Incorporated, 2010.
[26] L. Sciavicco, B. Siciliano, and B. Sciavicco, Modelling and Control of
Robot Manipulators, 2nd ed. Secaucus, NJ, USA: Springer-Verlag New
York, Inc., 2000.
[27] G. Metta, L. Natale, F. Nori, G. Sandini, D. Vernon, L. Fadiga, C. von
Hofsten, K. Rosander, M. Lopes, J. Santos-Victor, A. Bernardino, and
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
11
L. Montesano, “The iCub humanoid robot: An open-systems platform
for research in cognitive development,” Neural Networks, vol. 23, no.
89, pp. 1125 – 1134, 2010, social Cognition: From Babies to Robots.
[28] D. Orin, A. Goswami, and S.-H. Lee, “Centroidal dynamics of a
humanoid robot,” Autonomous Robots, vol. 35, no. 2-3, pp. 161–176,
2013. [Online]. Available: http://dx.doi.org/10.1007/s10514-013-9341-4
[29] S. Traversaro, D. Pucci, and F. Nori, “A Unified View of the
Equations of Motion used for Control Design of Humanoid Robots,”
Multibody System Dynamics, 2017, Submitted. [Online]. Available:
https://traversaro.github.io/preprints/changebase.pdf