A plan that orients the part up to symmetry is an example of a guaranteed plan. Although we have shown here that a guaranteed plan always exists, the guaranteed plan may be overly conservative if a shorter plan exists that works most of the time. With a model of probability and cost, we can define stochastic plans that optimize expected performance [6].
For example, consider the two-step plan shown in
Figure 4.
Now consider a one-step plan that consists of only the first step. The
one-step plan will align the major axis with the gripper unless the part's
major axis is initially almost orthogonal to the jaws. If we had a
probabilistic model of the part's initial orientation, then we could
compute the probability that the major axis will be aligned with the
gripper after only one step. If this probability were, say, , then we
may be willing to accept the one-step plan rather than the more
conservative two-step plan.
How can we compare a one-step plan that succeeds with probability and
a two-step plan that succeeds with probability
? We expect that, on
average, we will have to execute the one-step plan
times
until the part is oriented. On the other hand, we only have to execute the
two-step plan once to orient the part. If every step in the plan requires
one time unit then the expected time for the one-step plan is
units and the expected time for the two-step plan is
time units.
Under these conditions we might prefer the one-step plan.
Goldberg [6] shows how the planning algorithm can
be modified to find stochastically optimal plans in time
.