Multiple Balance Strategies
From One Optimization Criterion
Christopher G. Atkeson and Benjamin Stephens
Robotics Institute, Carnegie Mellon University
www.cs.cmu.edu/
˜
cga, cga@cmu.edu, bstephens@cmu.edu
Abstract—Multiple strategies for standing balance have been
observed in humans, including using the ankles to apply torque
to the ground, using the hips and/or arms to generate horizontal
ground forces, and using the knees and hips to squat. This
paper shows that multiple strategies can arise from the same
optimization criterion. It is likely that humanoid robots will
exhibit the same balance strategies as humans.
I. INTRODUCTION
This paper addresses two questions: 1) Will humanoid
robots show the same multiple strategies for standing balance
as seen in humans? and 2) If so, do these multiple strategies
arise from independent design and control processes, or a
single design and control process? This paper demonstrates
that it is likely that humanoid robots with backdrivable joints
will exhibit the same behavioral strategies seen in humans.
A design process using a single optimization criterion results
in a controller using multiple strategies. A small perturbation
is handled using one strategy, while a large perturbation is
handled using another strategy.
Studies of human standing balance have revealed several
strategies to compensate for perturbations: the ankle strategy,
in which torque about the ankle joints is used to balance
and the rest of the body is held in a fixed posture, the hip
strategy, in which torque about the hip joints is used to
generate horizontal ground forces moving the center of mass,
the squat strategy, in which the knees and hips are flexed
to lower the center of mass [1], and the step strategy, in
which a step is taken [2], [3]. These multiple strategies reflect
the mechanical constraints faced by humans and humanoid
robots. One question this paper addresses is whether each
strategy is controlled by a separate controller, as in human eye
movement control, or whether a single controller and design
process can be used to generate all strategies. In studies of
humans, the ankle strategy seems to be used for small and
slow perturbations on flat rigid support surfaces, while the hip
strategy seems to be used for large or fast perturbations and
on narrow or compliant support surfaces [4].
This paper focuses on two balance strategies that do not
involve stepping: the ankle and the hip strategies. A future
paper will attempt to include stepping. From a humanoid
robotics point of view, the ankle strategy turns the body into an
inverted pendulum, balanced upright using ankle torque. Hip
torque is applied only to keep the hip joint in a fixed position.
The hip strategy is that of a two link acrobot [5], where only
hip torque is applied and the ankle is unactuated. The acrobot
uses hip torque to generate horizontal ground forces, which
keeps the center of mass over the foot on average.
Hemami and colleagues analyze the ankle strategy [1]. Sev-
eral researchers provide examples of humanoid robot standing
balance implemented using two hand designed or optimized
controllers, one for the ankle strategy and one for the hip
strategy, including [6], [7]. Guibard and Gorce present a
classifier to select between ankle and hip strategies. Each
strategy is separately optimized using different criteria and/or
constraints [8]. Kudoh and colleagues choose between an
optimized strategy and a predesigned feedback control strategy
based on the current state. The optimization finds the best
acceleration on the current step (a local or greedy optimiza-
tion) rather than creating the best response over time [9].
Kuo designs Linear Quadratic Regulators for each perturbation
to generate controllers [10]. Different response strategies are
generated by changing the optimization criterion based on
the size of the perturbation. In Kuo’s work the system is
linearized. In our work we use a single optimization criterion
for all perturbation sizes. Thus, we do not need to “recognize”
a perturbation in order to select the appropriate response.
We also find that a nonlinear controller outperforms linear
controllers and linear controllers with constrained outputs.
Abdallah and Goswami explore a balance approach in which
two strategies are used in temporal sequence [11].
II. THE ONE LINK MODEL
Figure 1 shows a one link inverted pendulum responding to
a perturbation in the sagittal plane (fore/aft motion only). In
this simple model all joints except the ankle are held in a fixed
position. This model of standing balance has only one strategy
available to it, applying torque at the ankles. The amount of
ankle torque is limited by the size of the feet. We will use this
example to describe our optimization approach.
The model is facing to the right. The one link model is 2
meters high and has a total body weight of 70kg. The ankle
angle was bounded by −0.4 < θa < 0.8 radians. θa = 0 is
upright. We assume that in standing the center of pressure is
at the center of the foot. We therefore use a symmetric foot 0.2
meters long in our model. This results in a maximum ankle
torque of approximately ±70 Newton-meters.
In this case we model perturbations as impulses applied to
the middle of the torso (1.5 meters above the ankle). In this
example we present only perturbations that push the model
forwards, to simplify our figures. For this model perturbations