IEEE Robotics & Automation Magazine - September 2011 - 92
automaton corresponding to /; this object jointly encodes
the system dynamics and hard specifications. It is defined
as a tuple (SA , SA0 , dA , xA , FA ), where SA is the set of
states, SA0 the set of initial states, dA : SA 7! 2SA the transition function, xA : SA 3 SA 7! Rþ a positive-valued weight
function inherited from TP , and FA the set of accepting
states. We denote by d(pi , pj ) the shortest distance (based
on the weight of the transitions) from state pi 2 SA to
pj 2 SA . The shortest distance between two states can be
efficiently computed by a shortest path algorithm such as
Dijkstra's algorithm [15].
The cost assignment (Note that the costs associated
with each state are computed offline and then saved in a
look-up table to be used later in conjunction with the realtime controller.) for all states pi 2 SA is defined as
&
J(pi ) ¼
minpj 2FA? d(pi , pj ),
0,
if pi 62 FAÃ
,
if pi 2 FAÃ
(4)
where FA? is the largest self-reachable subset of FA . This
cost encodes the minimum distance from states in A to the
set FA? . We use this cost to keep our controller returning
to these states since acceptance of a run through A requires
that the run intersect this set infinitely often.
We introduce rewards (positive real numbers) to the
states and transitions indicated by S 1 and S 2 , respectively.
When the collections S 1
* and S2 are not empty,
the states and transitions
A hard specification
satisfying these sets have
the potential to produce
incorporates physical
rewards. The rewards
are generated in a timeconstraints and aesthetic
varying fashion, and the
appearance and disaprequirements that the
pearance of rewards and
their values are ranrobotic system must satisfy. domized. Upon visiting
* a state or taking a transition, the corresponding
reward is collected and subsequently consumed. By maximizing the collected rewards, we encourage the system to
visit desired states and take preferred transitions more frequently. We make the natural assumption that, at each
time instant, the system can foresee only a few steps ahead
(analogous to a dancer who is performing a free, improvised solo and planning just a few steps ahead at a time).
Thus, the run will contain infinite repetitions of finite runs
that maximize the total reward collected over the finite
runs with path length not more than a planning horizon
length L.
The controller is a combination of a precontroller and
a postcontroller. The precontroller drives the system
from any state p 62 FA? to a state p0 , where J(p0 ) < J(p),
while maximizing the rewards collected locally. The
repeated executions of the precontroller will drive the
92
*
IEEE ROBOTICS & AUTOMATION MAGAZINE
*
SEPTEMBER 2011
system from a state p 62 FA? to a state in FA? in a finite
number of transitions. The postcontroller drives the system from a state in FA? to a state outside FA? , while also
maximizing the local rewards. The controller switches
between the precontroller and postcontrollers (depending
on the current state) infinitely many times so that the
accepting states of A is visited infinitely many times, and
thus the produced infinite run is accepted by A. Since A
accepts all and only the runs of TP whose words satisfy /,
the run produced by the proposed controller can be used
to obtain a run on TP that favors the desired states and
transitions and generates a word satisfying / (i.e., a
desired movement phrase).
Generating Ballet Phrases
The hard specifications, which are applied to both case
studies and ensure that the basic style objectives are met,
are given according to a collection of physically and aesthetically driven rules (which are presented here with
minor adjustments from those in [12]) converted to LTL
formulas as follows:
1) We first ensure that no physically infeasible joint poses
are entered by the system. Such poses include those that
i) imply impossible jumping positions, i.e., state (q6 , q7 ),
ii) are impossible to hold because they are neither jumping poses nor feasible standing poses, i.e., both legs
attempting to touch the ankle of the other as in state
(q4 , q4 ), and iii) the geometry of the biped's hips will not
easily allow and thus form awkward poses that are ugly in
the eyes of the traditional ballet choreography, i.e., state
(q1 , q3 ). Thus, sets of poses are disallowed:
i) Always avoid both legs off ground except poses in
which two legs are in the same position, and poses
satisfying p7 ^ p10 , p5 ^ p8 , or p4 ^ p5 .
G :(Roffground ^ Loffground ^ (:Spose)^
:(p7 ^ p10 ) ^ :(p5 ^ p8 )) ^ :(p4 ^ p5 )):
ii) Always avoid both legs in poses p4 , p6 , and p10 .
G :(Spose ^ (p4 _ p6 _ p10 )):
iii)Always avoid both legs in poses satisfying p1 ^ p2 ,
p1 ^ p3 , or p2 ^ p3 .
G :((p1 ^ p2 ) _ (p1 ^ p3 ) _ (p2 ^ p3 )):
2) Next, we prevent the system from switching to a pose
which, given a current pose, would cause a biped to fall
over. We restrict transitions from states that have only
one leg supporting the body; these include i) jumping
from a leg in releve-this would require just the toe
joint of one leg to lift the entire body weight into the air,
i.e., (q3 , q8 ) to (q8 , q5 ), ii) extending the supporting leg
when it is in plie, i.e., (q1 , q5 ) to (q8 , q5 ), and iii) extending the supporting leg when it is on flat, i.e., (q2 , q5 ) to
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