IEEE Robotics & Automation Magazine - September 2011 - 94

Frequency of Motion Occurrence

10
(q1,q5) → (q1,q8)
(q2,q5) → (q2,q8)
(q3,q5) → (q3,q8)

9
8
7
6
5
4
3
2
1
0

0

1

2
3
Case Number

4

Figure 5. Case 0 shows the average of ten runs without any
soft specification, showing the tendency of the system without
any encouragement from soft specifications. The occurrences of
developpe in plie, on flat, and on releve are shown for the
remaining four cases in blue, green, and red, respectively. Each
sample path contains 52 steps.

rewards directly to our composed transition system. These
soft specifications encourage specific movements to occur
and are not phrased in LTL statements like the hard specifications-instead, we employ our receding-horizon controller to collect the rewards on transitions and thus produce a
desired output sequence. Namely, in each case we encourage
the transitions coupo , passo , and deveo when the standing
leg is in the desired pose. For Case 1, this corresponds to six
transitions: (q2 , q2 ) ! (q1 , q4 ), (q1 , q4 ) ! (q1 , q5 ), (q1 , q5 )
! (q1 , q8 ), and the three transitions that correspond to performing the movement on the other leg. The soft specification can be summarized as in (3).
Second, we present a case study where, by modifying
the hard specifications, we encourage full sequences of

Frequency of Motion Occurrence

5
4.5
4

(q7,q7) → (q5,q8)
(q7,q7) → (q8,q8)

3.5
3
2.5
2
1.5
1
0.5
0

0

1
Case Number

Figure 6. Occurrences of the transitions satisfying the soft
specification. Each output sequence contained 100 steps.
Without the specification, neither sequence showed up in the
 pas de
output sequence. With the specification, glissade plie
veloppe
 (blue) appeared four times and glissade plie

chat de
 (red) appeared five times.
grand jete

94

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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SEPTEMBER 2011

desired movements rather than single poses as in [12] or
transitions as presented above. Specifically, we add another
rule that translates to an LTL formula as given as follows
and encourage the pose ðq7 ; q7 Þ:
"Any time a glissade happens, follow it with plie grand
jete or plie pas de chat developpe."
G((p7 ^ Spose) ! X((p2 ^ Spose) ^ X((p1 ^ Spose)^
X((p2 ^ Spose) ^ X(((p2 ^ p4 ) ^ X((p4 ^ p5 )^
X(p5 ^ p8 ))) _ (p8 ^ Spose)))))).
The encouraged sequences of movement, glissade plie
grand jete, corresponds to (q7 , q7 )(q2 , q2 )(q1 , q1 )(q2 , q2 )
(q8 , q8 ) and the other, glissade plie pas de chat developpe,
corresponds to (q7 , q7 )(q2 , q2 )(q1 , q1 )(q2 , q2 )(q2 , q4 )(q4 , q5 )
(q5 , q8 )(q8 , q8 ). Note that the specification is symmetric;
hence, a sequence with the left and right leg roles reversed
is also encouraged.
With this added hard specification, we are able to ensure
that one of these movement sequences will occur every time
both legs are in pose 7. Encouraging this pose in the soft
specification will thus encourage the frequency of the desired
sequences. The results from this case study are presented in
Figure 6. The resulting sequences exhibit frequent use of this
series of jumps that may comprise the virtuousic displays that
characterize certain sections of evening length ballets.
Conclusions and Outlook
In this article, we have specified a grammar for leg positions in
ballet movements restricted to the coronal plane. Furthermore, our specification is guaranteed to follow certain principles, both physical and aesthetic, through the expressive
power of LTL. This primary contribution may lead to systems
(humanoid or otherwise) that behave in a way that is natural
for its given context. A deterministic program can limit the
flexibility of a system's ability to cope and adapt to its environment. When implemented on a robotic system, a behavior, as
outlined here, provides one or more states (perhaps a preferred state) that the system may enter given its current state.
A novel venue for movement and style analysis is the
second contribution of this model. The movement analysis
required to produce a system capable of automatically generating movement phrases in the style of classical ballet
results in a quantitative phrasing of the rules that govern
this somewhat curious example of human behavior. A
quantitative survey of specific dance styles (between genres, i.e., flamenco versus hip hop, and choreographers i.e.,
Mark Morris versus Twyla Tharp) would bolster and perhaps corroborate years of qualitative dance study that hold
that specific movement patterns evoke different aesthetic
and emotive effects in dance choreography.
From a dancer's perspective, the lack of tools for representing human movement induces quite an imbalance: a
musician works with notes arranged in octaves, chords,
and scales; an artist paints with a palette of colors known
to span the space of light that is perceivable to humans;
and writers knit nouns and verbs into clauses, sentences,
and paragraphs. Movements are certainly classified and
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