body with sufficient density. Another simplification we make is to
penalize any relative velocity at contacting end effectors (see (2)),
which results in trajectories that do not have any noticeable slip-
ping. Instead, in a low friction environment character moves overly
conservatively, making sure contact forces do not travel outside the
friction cone and is unable to exploit possible slipping when plan-
ning motions.
The style of the motions was not the focus of this work, and could
use a lot of improvement, particularly for low-energy motions such
as walking where humans use every bit of physiology (which we
do not model) to their advantage. Since our method performs long-
horizon trajectory optimization, we should be able to incorporate
biomechanically-inspired cost terms from [Wang et al. 2009] to
shape the stylistic aspects of the motion.
In all our examples, standard methods for local gradient-based opti-
mization were able to find good solutions efficiently. This is one of
the key advantages of our framework. Still, the use of global opti-
mization could provide more robust exploration of the space of mo-
tions, especially for tasks such as getting up that have a wide range
of possible and equally good solutions. In such cases we would
prefer to produce multiple solutions, and select the ideal one.
In the examples presented here the number and duration of phases
was fixed. Generally, we have found no problems in overestimat-
ing the number of movement phases required to complete an ac-
tion. The character typically uses up extraneous movement phases
by keeping still or sitting down before or after completing the task.
Underestimating the number of phases is more problematic and can
result in very energy inefficient or completely unphysical leaps in
the motion. However, a fixed number of phases would not be as
much of an issue if the task costs were reformulated as running
costs and the system was used in model-predictive, or online replan-
ning setting. In that case the number of phases would correspond
to a future planning horizon, and not dictate the total duration of
the motion. The number and duration of phases could also be opti-
mized, although we have not tested this.
Perhaps the most exciting direction for future work is applying CIO
to a full physics model that takes into account the limb inertias
and non-linear interaction forces. Indeed we formulated the CIO
method so that it is directly applicable to such a model. We expect
this to significantly enhance the realism/style of the resulting move-
ments, particularly in behaviors where saving energy is important.
While we do not anticipate any significant obstacles, how efficient
the method will be in the context of these more challenging opti-
mization problems remains to be seen. It may turn out that a hybrid
approach is preferable, where we first use the present simplified
model to obtain a solution that already looks quite good, and then
optimize with respect to a full physics model to refine the solution.
Acknowledgments
We thank Tom Erez and Yuval Tassa for inspiring technical dis-
cussions. This research was supported in part by NSF, NIH, and
NSERC.
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A
Simplified Character Model
Our simple model specifies character’s state q(s) at a particular
time through a small number of features, rather than a full set of
joint angles.
x =
p
c
r
c
p
1...N
r
1...N
T
(12)
Where p
c
and r
c
are torso position and orientation, respectively,
and p
i
, r
i
are end effector positions and orientations for each limb
i (see figure 2). Rotations are represented with exponential map
[Grassia 1998] because of its suitability in trajectory optimization.
From the above features, we can reconstruct the actual character’s
pose, including limb base locations b
i
, which can be derived from
local location points on the torso. We assume character’s limbs
have two links, which allows us to analytically solve for middle
joint location m
i
and orientations of the two links. For limbs that
have more than two links, it would be necessary to use an iterative
inverse kinematics method to derive the individual joint locations.
We define the motion with positions and velocities of our features at
the boundaries between phases. Cubic splines with knots at phase
boundaries are used to define a continuous feature trajectory from
which positions, velocities, accelerations at any point in the trajec-
tory can be computed. Combining contact variables for the phase
into a vector c, the solution vector s ∈ R
(12(N +1)+N )K
that we
optimize is
s =
x
1...K
˙
x
1...K
c
1...K
T
(13)
The dynamics of our simple model correspond to those of a single
rigid body with multiple forces acting on it from rectangular contact
surfaces. In this setting, contact forces can efficiently be solved
using either the approach of [Stephens 2011] or [Lee and Goswami
2010].
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