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-
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.
We thank Tom Erez and Yuval Tassa for inspiring technical dis-
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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
are torso position and orientation, respectively,
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
, which can be derived from
have two links, which allows us to analytically solve for middle
joint location m
and orientations of the two links. For limbs that
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
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