trol. Inverse optimal control is also an
active research area in machine learning. In the context of reinforcement
16, 20 inverse reinforcement
learning constitutes another resolution
paradigm based on Markov decision
processes with spectacular results on
challenging problems such as helicopter control.
7 The method corpus comes
from stochastic control (see Kober et
20 and references therein.)
Computation: A Practical
or Theoretical Problem?
In computer animation optimization-based motion generation is experienced
as giving excellent results in terms of
realism in mimicking nature. For instance, it is possible to use numerical
optimization to simulate very realistic
walking, stepping, or running motions
for human-like artifacts. These complex
body structures include up to 12 body
segments and 25 degrees of freedom.
At first glance, the approach a priori
applies to humanoid robotics. Figure
4 (top) provides an example of the way
HRP2 steps over a very large obstacle.
However, robotics imposes physi-
cal constraints absent from the virtual
worlds and requiring computation per-
formance. Biped walking is a typical
example where the technological limi-
tation implies a search of alternative
formulations. The bottleneck is the ca-
pacity of the control algorithm to meet
the real-time constraints.
In the current model-based simulation experiments the time of computation is evaluated in minutes. Minute is
not a time scale compatible with real
time. For instance, computation time
upper-bounds of a few milliseconds
are required to ensure the stability of
a standing humanoid robot. So taking advantage of general optimization
techniques for the real-time control
necessary requires building simplified
models or to develop dedicated methods. The issue constitutes an active
line of research combining robot control and numerical optimization.
An example is given by the research
on walking motion generation for hu-
manoid robots. The most popular walk-
ing pattern generator is based on a sim-
plified model of the anthropomorphic
body: the linearized inverted pendulum
model. It was introduced in Kajita et al.
and developed for the HRP2 humanoid
robot. The method is based on two ma-
jor assumptions: ( 1) the first one sim-
plifies the control model by imposing a
constant altitude of the center of mass,
( 2) the second one assumes the knowl-
edge of the footprints. Assumption ( 1)
has the advantage to transform the origi-
nal nonlinear problem into a linear one.
The corresponding model is low dimen-
sioned and it is possible to address ( 1)
via an optimization formulation.
this formulation, assumption ( 2) is no
longer required. The method then gives
rise to an on-line walking motion genera-
tor with automatic footstep placement.
This is made possible by a linear model-
predictive control whose associated qua-
dratic program allows much faster con-
trol loops than the original ones in Kajita
17 Indeed, running the full quadratic
program takes less than 1ms with state-
of-the-art solvers. More than that, in this
specific context, it is possible to devise
an optimized algorithm that reduces by
100 the computation time of a solution.
An example of this approach is given
in Figure 4 (bottom) that makes the real
HRP2 step over an obstacle. The approach based on model reduction enables the robot to be controlled in real
time. However, the reduced model does
not make a complete use of the robot
dynamics. The generated movement
is less optimal than when optimizing
the robot whole-body trajectory. Consequently, it is not possible to reach the
same performances (in this case, the
same obstacle height): the whole-body
optimization enables the robot to reach
Figure 4. Two stepping movements obtained with (top) a whole-body trajectory optimization36 (courtesy from K. Mombaur) and (bottom)
a linearized-inverted-pendulum based walking pattern generator17 (courtesy from O. Stasse.
39). The whole-body optimization enables
the robot to reach higher performances but the numerical resolution is yet too slow to obtain an effective controller.