Stepping Stabilization Using a Combination of
DCM Tracking and Step Adjustment
Majid Khadiv∗†§, Sebastien Kleff†, Alexander Herzog†,
S. Ali. A. Moosavian∗, Stefan Schaal†‡, and Ludovic Righetti†
∗Department of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran
†Autonomous Motion Department, Max-Planck Institute for Intelligent Systems, Germany
‡CLMC Lab, University of Southern California, Los Angeles, USA
§Corresponding author: mkhadiv@mail.kntu.ac.ir, majid.khadiv@tuebingen.mpg.de
Abstract—In this paper, a method for stabilizing biped robots
stepping by a combination of Divergent Component of Motion
(DCM) tracking and step adjustment is proposed. In this method,
the DCM trajectory is generated, consistent with the predeﬁned
footprints. Furthermore, a swing foot trajectory modiﬁcation
strategy is proposed to adapt the landing point, using DCM
measurement. In order to apply the generated trajectories to the
full robot, a Hierarchical Inverse Dynamics (HID) is employed.
The HID enables us to use different combinations of the DCM
tracking and step adjustment for stabilizing different biped
robots. Simulation experiments on two scenarios for two different
simulated robots, one with active ankles and the other with
passive ankles, are carried out. Simulation results demonstrate
the effectiveness of the proposed method for robots with both
active and passive ankles.
Keywords— Biped robots; Divergent Component of Motion;
Hierarchical inverse dynamics
I. INTRODUCTION
In order to be able to take part in our future daily life, hu-
manoid robots should be capable of safely performing various
tasks in highly dynamic environments. This capability requires
a real-time walking pattern generation unit to produce feasible
walking patterns in complicated environments. Furthermore,
the generated patterns should be robust against uncertainties
and disturbances.
Exploiting the whole dynamics of a biped robot for generat-
ing walking patterns demands high computation burden, while
convergence to the global minimum is not guaranteed [1], [2].
However, simple models may be employed to generate walking
patterns in real-time. The Linear Inverted Pendulum Model
(LIPM) [3], reduces the Center of Mass (CoM) dynamics of
a biped robot to a linear model by assuming a constant CoM
height and negligible angular momentum. This model has been
very successfully used for the design of walking controllers
for complex biped robots. Kajita et al. [4] proposed a preview
control of the Zero Moment Point (ZMP) to generate a CoM
trajectory based on a predeﬁned ZMP trajectory. Wieber [5]
improved the performance of this approach in the presence
of relatively severe pushes. He proposed to recompute the
trajectories in a Model Predictive Control (MPC) framework
without any predeﬁned ZMP trajectory, using feedback from
the current state of the robot. Furthermore, other formulations
of the walking pattern generation as an MPC problem have
been suggested, for example by considering step locations in
the optimization procedure [6].
Analytical methods have been presented which do not rely
on optimization algorithms [7], [8]. In these methods, position
and velocity of the CoM are considered as states of the
system. Then, the trajectory for the CoM is obtained, using
the analytical solution of the LIPM and continuity constraints
at the control points. Although considering the position and
velocity of the CoM as states of the system enables us
to generate feasible walking patterns, the problem is over-
constrained. The reason is that by constraining the position
and velocity, both divergent and convergent parts of the LIPM
dynamics are constrained. Hence, Takaneka et al. [9] divided
the LIPM dynamics into its divergent and convergent parts.
Then, they just constrained the divergent part to generate
the Divergent Component of Motion (DCM) trajectory based
on predeﬁned footprints (ZMP trajectory). Before [9], the
divergent part of the LIPM dynamics had been used by Hof et
al. [10] to explain human walking properties under the name
of extrapolated Center of Mass (XCoM). This concept was
also developed in Pratt et al. [11] under the name of Capture
Point (CP), the point on which the robot should step to come
to rest.
By employing the DCM, Englsberger et. al [12] proposed
a method to control the unstable part of the CoM dynamics
without affecting the stable part. Although this controller can
react to the disturbances very fast, perfect DCM tracking
needs unconstrained manipulation of the CoP. For robots
with ﬁnite size feet and actuated ankles, the CoP can be
directly manipulated inside the support polygon, using the
ankle actuation. However, since the size of the feet is limited,
the actual control authority of the ankle joint is also limited.
This problem is more severe, when the robot has point contact
feet or ﬁnite size feet with passive ankles. For a robot with
point contact feet, the CoP is always located at the point of
contact during stepping. As a result, manipulation of the CoP
is not possible. For robots with passive ankles, the CoP cannot
be manipulated directly by the ankle joint torques and has to be
manipulated by more proximal joints which renders its control
more difﬁcult.
arXiv:1609.09822v1  [cs.RO]  30 Sep 2016
In addition to CoP manipulation, step adjustments consti-
tute a signiﬁcant tool for stabilizing biped robots and these
adjustments can be realized in a slower time scale (e.g. every
step). A constraint of step adjustment is that the step location
is limited by the reachable area, especially in very constrained
environments.
In order to map control policies generated with the simpli-
ﬁed LIPM dynamics to the full robot, a whole body controller
is employed. For robots with torque-control capability, meth-
ods based on inverse dynamics can be used [13], [14]. In this
context, Herzog et al. [15] proposed a method for solving
the Hierarchical Inverse Dynamics (HID) and demonstrated
its performance through various balance experiments on a
torque controlled biped robot. The HID enables us to take
into account equality and inequality constraints, and also to
prioritize tasks with respect to each other. The latter property
is especially useful for our work to control biped robots with
different structures.
In this work, we propose an approach for stabilizing biped
robots stepping by a combination of DCM tracking and step
adjustment strategies (Fig. 1). We employ different combi-
nations of the DCM tracking and step adjustment in the
HID, to stabilize biped robots with active and passive ankles.
Furthermore, for step adjustment, a swing foot trajectory
modiﬁcation method is proposed. The rest of this paper is
organized as follows: in section II the desired trajectory for the
DCM is generated. Section III proposes a method for swing
foot trajectory modiﬁcation. Section IV introduces the HID
and the considered tasks. In section V the simulation results
are presented and discussed. Finally, section VI concludes the
ﬁndings.
II. TRAJECTORY GENERATION FOR THE DCM
The block diagram of the proposed walking controller is
shown in Fig. 1. As it may be observed, both desired feet
and DCM trajectories are generated. Based on feedback from
the current state of the robot, the foot trajectory is modiﬁed
to compensate for the DCM tracking error. The generated
trajectories, then, are fed into the HID to generate desired
actuating torques of the joints consistent with the speciﬁed
tasks and hierarchy.
The LIPM constrains motion of the CoM on a plane
(horizontal plane for walking on a ﬂat surface), by using a
telescopic massless link connecting the CoP to the CoM [3].
The dynamics of this system may be formulated as:
¨x = ω2
0(x −u)
(1)
in which x is a 2-D vector containing CoM horizontal com-
ponents (the vertical component has a ﬁxed value z0), and u
is the CoP vector (u = [CoPx, CoPy]T ). Furthermore, ω0 is
the natural frequency of the pendulum (ω0 =
p
g/z0, where g
is the gravity constant, and z0 is the CoM height).
 
Hierarchical Inverse Dynamics 
(Section IV) 
 
Robot 
DCM trajectory 
(Section II) 
Foot trajectory 
(Section III-A) 
 
Desired 
footprints 
Step adjustment 
(Equation 8) 
 
Current stance (𝑢𝑖) 
𝜉𝑚𝑒𝑎 
Next footprint (𝑢𝑖+1) 
Fig. 1: The proposed control framework
By considering CoM (x) and DCM (ξ = x + ˙x/ω0) as the
state variables, the LIPM dynamics in the state space form
may be speciﬁed as:
˙x = ω0(ξ −x)
˙ξ = ω0(ξ −u)
(2)
Equation (2) decomposes the LIPM dynamics into its stable
and unstable parts, where the CoM converges to the DCM and
the DCM is pushed away by the CoP. Hence, in order to have
a stable walking pattern, it is enough to constrain the DCM
motion during walking, without restricting the other state of
the system.
Constraining the DCM motion can be achieved in different
ways, i.e. applying the DCM boundary condition to have cyclic
motion [9], specifying the DCM at the end of a previewed
number of steps to be at the top of foot print [12], and etc. In
this paper, we use the method in [12], and set the DCM (its
projection on the ground) coincident with the footprint at the
end of a previewed number of steps.
By solving the second equation of (2), the DCM trajectory
based on the natural dynamics of the LIPM can be obtained:
ξ(t) = (ξ0 −u)eω0t + u
(3)
where ξ0 and u are the DCM and CoP at the beginning
of a step. In the last step in the previewed period, we force
the DCM to be coincident with the last footprint. Then, we
compute the required initial condition for the DCM:
ξ0,n−1 = (un −un−1)e−ω0T + un−1
(4)
where n is the number of previewed steps, and T is the
step period. Then, as it is illustrated in Fig. 2, we compute the
DCM boundary conditions in a recursive fashion:
ξ0,i−1 = (ξ0,i−ui−1)e−ω0T +ui−1
,
i = 1, ..., n−1. (5)
Using this method, DCM boundary conditions which con-
strain the DCM motion are computed. It should be mentioned
that the CoP is considered ﬁxed during each step. Furthermore,
since this recursive procedure is carried out at each step, the
next footprints can be changed during the previewed period.
In other word, the boundary conditions are updated at each
step, based on the updated footprints.
III. STEP ADJUSTMENT
The desired DCM trajectory should be tracked to realize
the generated walking pattern. In other word, deviations from
the desired DCM trajectory may cause instability. However,
robots with different structures have different authorities to
manipulate the CoP. For example, a robot with passive ankles
has to use joints above the knee in order to modulate the CoP,
whereas robots with actuated ankles can apply ground forces
directly at the foot. As a result, relying on the DCM tracking
is not enough for stepping stabilization of robots with different
structures in the presence of disturbances and uncertainties. In
this section, we propose a method for swing foot trajectory
modiﬁcation, by using the DCM feedback. The aim of this
real-time swing foot trajectory modiﬁcation is to realize the
adapted landing point at the end of the step.
A. Swing Foot Trajectory
The trajectory for the swing foot is generated to ensure that
the foot lands without impact or slip. For the horizontal com-
ponents of the swing foot, we consider ﬁfth order polynomial
to satisfy position, velocity and acceleration constraints at the
start and end of the swing phase. For the vertical direction,
beside these constraints, the maximum height of the swing
foot at the midpoint of the swing phase is speciﬁed. As a
result, a sixth order polynomial is employed to satisfy all
the constraints in the vertical direction. These trajectories are
computed at the start of each step for a previewed period.
B. Real-time Modiﬁcation
Figure 2 illustrates the adaptation procedure that we propose
for the swing foot trajectory. In this procedure, the goal is
to modify the swing foot landing in the current step (using
measured DCM) such that the new footprint brings the DCM
to the desired DCM boundary condition at the end of the
next step (Fig. 2). Using this method, if there are no more
disturbances, after correction the swing foot will land on the
predeﬁned footprint, in the next steps.
By rearranging (3), and setting the current DCM measure-
ment as the initial state, the DCM position at the end of the
current step can be obtained as follows:
ξ0,i+1,es = (ξmea −ui)eω0(t−T ) + ui
,
0 ≤t ≤T
(6)
in which ξmea is the measured DCM, and ξ0,i+1,es is the
estimated DCM at the end of the current step. It is important to
note that in the case of perfect DCM tracking, ξ0,i+1,es would
coincide with ξ0,i+1 computed from the recursive equation of
(5).
Now, we aim at ﬁnding the next footprint which brings
the DCM from ξ0,i+1,es to ξ0,i+2 , for the next step. Again,
rearranging (3) and solving it for the desired next footprint
yields:
ui+1,es = ξ0,i+2 −ξ0,i+1,es eω0T
1 −eω0T
,
0 ≤t ≤T
(7)
 
Previewed footprints (n) 
𝑢𝑖+1,𝑒𝑠 
𝑢𝑖+1 
𝜉𝑚𝑒𝑎 
𝑢𝑛 
𝜉0,𝑛−1 
Fig. 2: the DCM boundary condtition and landing point adjustment
where ui+1,es is the estimated next footprint for compensating
the tracking error of the DCM and bringing it to its predeﬁned
boundary condition in the next step.
Now, the swing foot modiﬁcation can be computed:
mod = ui+1,es −ui+1
(8)
This modiﬁcation is added to the swing foot trajectory in
the horizontal directions, at each control cycle. In our step
adjustment method, the trajectory of the swing foot in the
vertical direction is not modiﬁed. As a result, at the start of
each step, the trajectories for the feet are generated for the
previewed period. Then, the modiﬁcation from (8) is computed
in real-time based on DCM measurement, and added to the
horizontal components of the swing foot trajectory.
IV. WHOLE BODY CONTROL
In order to obtain feasible joint torque commands from the
generated trajectories in the last sections,we use a hierarchical
inverse dynamics controller. In HID framework, constraints
and tasks are written as afﬁne functions of joint and ﬂoating-
base accelerations, actuation torques, and interaction forces
and moments. In fact, these variables constitute the design
variables which are optimized through a series of Quadratic
Programs (QP), at each control cycle. The goal of the con-
troller is to ﬁnd the design variables that satisfy different
control objectives with different priorities, while the highest
priority in the hierarchy is set to ensure physical consistency.
In lower priorities, various tasks with different ranks are
speciﬁed, and tasks in the same priority can be weighted with
respect to each other [15].
For all the tasks, we put the equations of motion and the
limits for the actuation in the highest priority. The equations
of motion can be written as:
M(q)¨q + N(q, ˙q) = ST τ + JT
c λ
(9)
in which M is the inertia matrix, q is the vector of
generalized coordinates, and N groups together the Coriolis,
centrifugal and gravitational effects. S represents the joint
selection matrix, τ is the vector of actuation torques, Jc is
the contact jacobian, and λ is the vector of contact forces and
moments. In the following, we describe lower priority tasks
and constraints that are meant to track swing foot and DCM
proﬁles, using admissible contact forces.
A. Foot Trajectory Tracking
The modiﬁed trajectory for the swing foot in section III
should be tracked to realize the stepping task. This task can
be written as:
Jsw¨q + ˙Jsw ˙q = ¨X + Kd( ˙Xd −˙X) + Kp(Xd −X)
(10)
in which X and Xd are the actual and reference swing foot
posture vectors , and Jsw is the jacobian of the swing foot.
Furthermore, Kp and Kd are diagonal gain matrices. It should
be noted that the desired orientation components of the swing
foot is set to zero. For the stance foot, the task is:
Jst¨q + ˙Jst ˙q = 0
(11)
where Jst is the jacobian of the stance foot. This task keeps
the stance foot in a stationary contact with the ground surface.
B. DCM Tracking
For the DCM tracking task, we employ the following control
rule [12]:
˙ξ −˙ξd = −Kξ(ξ −ξd)
(12)
in which kξ is the DCM control gain, while ξ and ξd are
the measured and desired DCM, respectively. Substituting this
equation into the second equation of (2) yields:
udes = ξ + 1
ω0
(Kξ(ξ −ξd) −˙ξd)
(13)
where udes is the desired CoP location that ensures the
closed-loop behavior deﬁned in (12). It should be noted
that the desired CoP computed from (13) is not necessarily
feasible, and should be projected inside the support polygon
(CoP constraint, subsection C). This projection results in a
DCM tracking error that is compensated using step adjustment
(section III).
The obtained desired CoP from (13), can be related as a
CoM task using (1):
¨XCoM,ref =
ω2
0(x −udes)
0

= JCoM ¨q + ˙JCoM ˙q
(14)
where JCoM speciﬁes the CoM jacobian, and x is a 2-D vector
containing CoM horizontal components.
C. Contact Constraints
The CoP can be computed from the contact forces (λ). To
guarantee stationary contact, the CoP should lie inside the
support polygon, which introduces an inequality constraint.
Furthermore, the resultant interacting forces at the contact
points should stay inside the friction cones. The friction
cones are approximated by pyramids to yield linear inequality
constraints on the contact forces [15].
(a) 
(b) 
 
 
 
Fig. 3: The two Sarcos humanoid robots used in simulation (a)
Hermes, which has fully actuated ankles (Credit:Luke Fisher Pho-
tography) (b) Athena, which has passive ankles
D. Posture Control
Posture control is a task in the joint space. We use this task
to constrain the robot to be as much as possible in upright
posture during walking. This task can be written as:
¨q = Kp(qd −q) −Kd ˙q
(15)
In this equation, Kp and Kd are diagonal square gain
matrices, with the same dimension as the actuated joints.
E. Base Control
The base control task keeps the torso in an upright position.
It is deﬁned as::
Jbase¨q + ˙Jbase ˙q = Kp(Qd −Q) −Kd ˙Q
(16)
where Jbase is the rotational part of the ﬂoating base
jacobian. Furthermore, Q and Qd are the actual and desired
quaternion of the ﬂoating base.
V. RESULTS AND DISCUSSIONS
In this section, we present two scenarios in a simulation
environment to show the effectiveness of the proposed frame-
work. In these scenarios, we employ different hierarchies in
the HID consistent with the control authority in the ankles.The
ﬁrst scenario corresponds to implementing walking patterns
on a fully actuated biped robot (Fig. 3(a)). In this scenario,
we push the robot to show the robustness of the gaits in the
presence of disturbances. In the second scenario, we remove
the actuation of the ankles (Fig. 3 (b)), and obtain a stable
stepping in place. Then, we change the time of stepping and
demonstrate the effectiveness of step adjustment to stabilize
the robot motion.
A. Walking with Actuated Ankles
In the ﬁrst scenario, simulation of walking with the lower
part of Sarcos humanoid robot Hermes (Fig. 3(a)) is per-
formed. Since the robot is equipped with active ankle joints,
we put the DCM tracking task in the same rank as the foot
trajectory tracking task, in the ﬁrst rank in the hierarchy
(physical consistency constraints are at rank 0). Besides, in
 
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push
Fig. 4: First scenario: walking of Hermes with active ankles on a ﬂat surface (in simulation environment). Case (a): The sagittal and lateral
components of the DCM and feet without disturbance are shown. The desired DCM is tracked without need for step adjustment. Case (b):
the lateral components of the DCM and feet in the presence of a push are shown (the push force is 10 percent of the robot weight and its
duration is 0.3 second). The step adjustment could deal with the disturbance and stabilize the walking.
order to constrain the controller to exploit feasible forces for
the DCM tracking, we put the contact constraints (CoP and
friction constraints) in the ﬁrst rank, as well. The posture
control and base control tasks are put in the lower priority
to exploit the remaining degrees of freedom to keep the robot
in an upright posture as much as possible.
In Fig. 4 case (a), the desired and actual values of the DCM
and feet trajectories in both sagittal and lateral directions are
depicted. As it can be observed, in this case, since the DCM
trajectory is tracked, modiﬁcation of the swing foot trajectory
is not required. Hence, the desired trajectories of the feet
remain almost constant, during normal walking. In order to
show the robustness during walking, in the case (b), the robot
is pushed laterally (the push force is 10 percent of the robot
weight and its duration is 0.3 second). As it can be observed
in the left ﬁgure, the actual DCM diverges from the desired
trajectory. The reason is that in this case the forces required
for tracking the DCM are infeasible and projecting them into
feasible area causes tracking error. In order to compensate
for this tracking error, the swing foot trajectory is modiﬁed
(the right ﬁgure), using the algorithm described in section III.
The result of this modiﬁcation can be observed as the DCM
converges to its desired value in the next step.
B. Stepping in place with passive ankles
In the second scenario, we simulate stepping in place of the
Sarcos humanoid robot Athena (Fig. 5(b)), with passive ankles.
In this scenario, having the DCM tracking task in the highest
priority would be problematic since the controller would create
large arms and upper-body motions to change the CoP. It is
not a desired behavior. Furthermore, there is no constraint for
step adjustment, because the environment is not constrained.
As a result, we consider another hierarchy consistent with the
actuation in the ankles. We put the foot trajectory tracking and
the posture control in the highest priority. Because we put the
posture control in the highest priority, there is no more degrees
of freedom to control the DCM. As a result, in this case, we
rely just on step adjustment to stabilize the stepping.
The obtained results from this scenario are illustrated in
Fig. 5. We only present trajectories in the lateral direction
since the robot is stepping in place. For the case (a), the step
timing is selected such that following the natural dynamics of
the robot realizes the stepping. As it can be seen in the case
(a), the foot trajectory in the lateral direction is not modiﬁed.
However, the measured DCM does not diverge, and remains on
a limit cycle. Although the stepping in this case is stable, this
situation is fragile; because a slight change in the step timing
causes instability of the robot. To show this, we increase the
step timing by 50 percent in the second case. The obtained
results are shown in Fig. 5(b). As it can be observed, without
step adjustment , the DCM trajectory diverges. This divergence
causes instability of the robot.
In order to show the robustness of the stepping exploiting
step adjustment as a stabilizing tool, we repeat the case (b)
with step adjustment. As a result, the landing point of the
swing foot is modiﬁed to realize stable stepping. By comparing
the DCM trajectory in the cases (b) and (c), it is clear
that the DCM does not diverge in the case (c). In fact, by
modifying the swing foot trajectory and changing the landing
position consequently, the DCM trajectory remains on a limit
cycle. This scenario shows signiﬁcance of the proposed step
adjustment method for preserving stability of the robot.
VI. CONCLUSIONS
In this paper, a method for stabilizing biped robots stepping
based on a combination of DCM tracking and step adjustment
was proposed. This method is based on generating a DCM
 
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case (a): exact timing
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Fig. 5: Second scenario: Stepping in place of Athena with passive ankles (in simulation environment).In this scenario the lateral components
of the DCM and feet for stepping in place are shown. Case (a): the step timing is tuned to yield a stable stepping without controlling the
DCM. Case (b): the tuned step timing is disturbed by 50 percent, so the DCM trajectory diverges and the robot is unstable. Case (c): Step
adjustment is activated and stabilizes the robot for the disturbed case.
trajectory consistent with footprints, and adjusting step lo-
cations using DCM feedback. For step adjustment, a novel
strategy based on modifying the swing foot trajectory was
suggested. To use a combination of these two stabilizing
tools consistent with the robot control authority, a hierarchical
inverse dynamics controller was used. Two simulation scenar-
ios were performed to demonstrate the effectiveness of the
proposed method. In the ﬁrst scenario, a robot with active
ankles was able to stabilize walking using a combination of
DCM tracking and step adjustment. In the second scenario,
a robot with passive ankles stabilized stepping by using only
step adjustments without any explicit CoM or DCM tracking
task. The obtained results showed that the proposed method
can stabilize stepping of robots with passive and active ankles,
in the presence of disturbances.
ACKNOWLEDGMENT
This research was supported by the Max-Planck Society,
MPI-ETH center for learning systems and the European Re-
search Council under the European Union’s Horizon 2020 re-
search and innovation program (grant agreement No 637935).
REFERENCES
[1] A. Herzog, N. Rotella, S. Schaal, and L. Righetti, “Trajectory gener-
ation for multi-contact momentum control,” in Humanoid Robots (Hu-
manoids), 2015 IEEE-RAS 15th International Conference on, pp. 874–
880, IEEE, 2015.
[2] M. Khadiv, S. A. A. Moosavian, A. Youseﬁ-Koma, M. Sadedel, and
S. Mansouri, “Optimal gait planning for humanoids with 3d structure
walking on slippery surfaces,” Robotica, pp. 1–19, 2015.
[3] 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. Proceed-
ings. 2001 IEEE/RSJ International Conference on, vol. 1, pp. 239–246,
IEEE, 2001.
[4] 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. Pro-
ceedings. ICRA’03. IEEE International Conference on, vol. 2, pp. 1620–
1626, IEEE, 2003.
[5] P.-B. Wieber, “Trajectory free linear model predictive control for stable
walking in the presence of strong perturbations,” in 2006 6th IEEE-
RAS International Conference on Humanoid Robots, pp. 137–142, IEEE,
2006.
[6] A. Herdt, H. Diedam, P.-B. Wieber, D. Dimitrov, K. Mombaur, and
M. Diehl, “Online walking motion generation with automatic footstep
placement,” Advanced Robotics, vol. 24, no. 5-6, pp. 719–737, 2010.
[7] M. Morisawa, K. Harada, S. Kajita, K. Kaneko, F. Kanehiro, K. Fu-
jiwara, S. Nakaoka, and H. Hirukawa, “A biped pattern generation
allowing immediate modiﬁcation of foot placement in real-time,” in 2006
6th IEEE-RAS International Conference on Humanoid Robots, pp. 581–
586, IEEE, 2006.
[8] K. Harada, S. Kajita, K. Kaneko, and H. Hirukawa, “An analytical
method for real-time gait planning for humanoid robots,” International
Journal of Humanoid Robotics, vol. 3, no. 01, pp. 1–19, 2006.
[9] T. Takenaka, T. Matsumoto, and T. Yoshiike, “Real time motion gen-
eration and control for biped robot-1 st report: Walking gait pattern
generation,” in 2009 IEEE/RSJ International Conference on Intelligent
Robots and Systems, pp. 1084–1091, IEEE, 2009.
[10] A. L. Hof, “The extrapolated center of mass concept suggests a simple
control of balance in walking,” Human movement science, vol. 27, no. 1,
pp. 112–125, 2008.
[11] 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, pp. 200–207, IEEE, 2006.
[12] J. Englsberger, C. Ott, and A. Albu-Sch¨affer, “Three-dimensional bipedal
walking control based on divergent component of motion,” IEEE Trans-
actions on Robotics, vol. 31, no. 2, pp. 355–368, 2015.
[13] M. Mistry, J. Buchli, and S. Schaal, “Inverse dynamics control of ﬂoating
base systems using orthogonal decomposition.,” in ICRA, pp. 3406–
3412, Citeseer, 2010.
[14] 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.
[15] A. Herzog, N. Rotella, S. Mason, F. Grimminger, S. Schaal, and
L. Righetti, “Momentum control with hierarchical inverse dynamics
on a torque-controlled humanoid,” Autonomous Robots, vol. 40, no. 3,
pp. 473–491, 2016.