Modeling and Control of Periodic Humanoid
Balance using the Linear Biped Model
Benjamin Stephens
Carnegie Mellon University
Pittsburgh, PA 15213, USA
Email: bstephens@cmu.edu
Christopher Atkeson
Carnegie Mellon University
Pittsburgh, PA 15213, USA
Email: cga@cmu.edu
Abstract—We present work on compliant control of dynamic
humanoid balance and walking. We use the Linear Biped Model
(LiBM) to model the dynamics of balance on two feet. To
achieve periodic motion, as in walking, we derive an orbital
energy controller for this model. We also present our methods
for applying this control to a torque-controlled humanoid robot,
which include estimating the center of mass state and generating
feed-forward torque commands.
I. INTRODUCTION
Humanoid robots are complex systems that are often studied
using simple models. One such model is the Linear Inverted
Pendulum Model (LIPM) [1]. Often, this model is used in
combination with desired foot trajectories and a trajectory for
the center of mass is created so that the center of pressure
is always within the base of support. Rather than following
pre-determined trajectories, we would like to utilize reactive
controllers that stabilize the system from large and unknown
perturbations.
We develop a similar model to the LIPM, which we call
the “Linear Biped Model” (LiBM) that models the forces on
the center of mass for a biped system, shown in Figure 1.
During single support, the dynamics are equivalent to a LIPM.
During double support, however, the dynamics are described
by two superimposed LIPMs. We present a analytic feedback
controller designed to regulate the periodic motion of the
LiBM which is inspired by the concept of orbital energy [2].
A lookahead footstep planner is also described that allows the
system to recover from larger perturbations.
The LiBM is used for controlling a torque-controlled hu-
manoid robot. The model is used to approximate the dynamics
and predict ground reaction forces. The torques and forces
predicted by the simplified model are used as feed-forward
controls. The linear model is also used for improved motion
estimation by combining various sensor measurements in a
Kalman Filter.
This paper is outlined as follows. First, in Section II, we
describe the Linear Biped Model and derive the linear dy-
namics. In Section III, we derive the orbital energy controller
for coronal balance. We describe how a stepping controller can
be written for the LiBM in IV. Then we focus on applying
this model to control of our humanoid robot in V.
Fig. 1. The Linear Biped Model consists of two superimposed Linear Inverted
Pendulum Models. It can be used for modeling, estimation and control of
biped systems, such as a humanoid robot.
A. Related Work
Standing balance of humanoids has been studied using many
techniques. Some researchers focus on simple models that
correspond to a lumped mass model centered at the center of
mass [1] [3] [4]. This model is sometimes modified to include
a rotational inertia term that models angular momentum, which
can add significant stability to the system [5] [6] [7] [8]. Such
simple systems can be used by high-level planners to decide
strategies, for example by considering the viability kernel [9].
Control of full body humanoid balance has been considered
more recently. Various techniques have been proposed for
solving the complex floating body dynamics and control prob-
lem, including inverse kinematics [10], inverse dynamics [11],
quadratic programming [12], approximate policy transfer from
simpler models [13], prioritized control [14], and passivity-
based gravity compensation [15]. However, these methods
generally rely on accurate dynamic models. In this paper we
employ a full-body force-control technique similar to virtual
actuator control [16], which was previously proposed for
control of biped robots [17].
Step recovery has also been studied using simple models
that predict the best footstep location [6] [18]. These models