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Ch19-I044963.fm Page 91 Tuesday, August 1, 2006 2:54 PM
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The number of crossings in the initial state is five, and that in the objective state is zero. We can reduce
the number of crossings from five to zero by applying two operations UOu and one operation UOi or
UOiv at least. Their possible sequences are described as follows:
OQi : UOu -> UO U -> UOi/UO IV, (1)
OQ 2 : UOu -> UOj/UOiv -> UOn, (2)
OQ 3 : UO,/UO| V -> UOu -> UOu. (3)
Then, we check whether the required process can be realized by applying these three uncrossing
operations to the object in the above orders or not. In Fig.4, the following sequences of state transitions
SQi, SQ2, and SQ3 correspond to operation sequences OQi, OQ2, and OQ3, respectively.
SQ, :S, -> S 2 -> S 5 (4)
SQ2 • Si ~~^ S2 ~~* S6 (5)
SQ 3 :S, -> S 3 -> S 6 Sn. (6)

Tf the required process can not be realized with two operations UOn and one operation UOi or UOiv,
we check that with one operation UOn and three operations UOi and/or UOiv. In general, we check
repeatedly whether a knot with n crossings can be unknotted by applying , Ti vC v combination of x
operations UOu and y operations UO, and/or UO IV with decreasing x and increasing y so that they
satisfy 2x+y=n until a sequence of operations to unknot it is found. Thus, we can efficiently derive
manipulation processes including fewer state transitions, that is, processes with lower JV, without
generating the whole graph including all possible processes.

(c) State S
(a) State S 1 (b) State S (c) State S 5 5
(b) State S 2 2

(d) Action 1 (e) Action 2 (f) Action 3
(e) Action 2
(f) Action 3
1
(d) Action
Figure 5: Consideration of changing times of grasping points
Next, we select adequate actions so that a manipulation process has fewer changing times N c of
grasping points. Let us consider sequence SQi. For the first transition from state Si to state S 2, assume
M
that segments 2L 5 and 4V3 are grasped from the front side as shown in Fig. 5 (a) and moved to perform
operation UOn. Then, grasped segments become equivalent to segment V^'s in state S2 as shown in
M
Fig.5(b). State S2 can be changed into state S5 by moving segment 2-L"j. After that, it is found that
segment 'iL in state S 5 is grasped from the front side by two manipulators as shown in Fig.5(c). There
are three actions to change the state into Sri- Action 1 is to regrasp segment '|L from the back side for