One obvious dsadvantage of sticking termination is that it is not complete. For example, a planner employing sticking termination exclusively will not be able to find strategies that require stopping in mid-air" even when such strategies would be feasible given the position sensing accuracy of the robot. Sticking termination requires all strategies to "run aground", that 'is, to be in contact (and in fact, .sticking) at termination time.
With more general position/force/time termination criteria, the requirement that motions must terminate in contact 'is relaxed. However, a forward-chaining planner (such as LIMITED) is still left with the problem of deciding where a motion should terminate in a multi-step strategy. That is, the decision problem involves existential quantification not only over the commanded directions, but also over all subsets of the forward projection corresponding to possible push-forwards. Put simply, a forward-chaining planner must not only guess the direction to command a motion, but must also guess where it terminates before chaining ahead to the next motion. While the space of commanded motions may be realistically quantized and searched, the space of push-forwards may not be searched in this manner.
While LIMITED is a forward-chaining planner, the problem of existential quan-tification over the push-forward is finessed by restricting LIMITED to a few very simple termination conditions (there are only three; see sec. 62), one of which is sticking. Given these termination types, it 'is possible to generate the corresponding
a priori push-forwards, and test them to see whether they yield an EDR strategy.
For example, the push-forward for contact termination 'is simply the obstacle edges in the forward projection. The push-forward for sticking termination 'is the a priori push-forward based on sticking, which was dscussed above.
More generally, it may be possible to define a parameterized family of termina-tion predicates, each with an associated a priori push-forward. Each push-forward could then be tested for distinguishability. For example, consider the class of ter-mination conditions
I "Terminate after t seconds." I t >
An associated family of push-forwards ight be the time-'indexed forward projec-tions
Fo(Rt) I t > 0.
However, the existential quantification over the push-forward in the decision problem for EDR planning is, in fact, an artifact of forward-chaining. We can see this by comparing and contrasting backchaining vs. forward-chaining in preim-age planners for guaranteed strategies. In a backward-chaining planner, this extra computation is eliminated. The difference is as follows. Consider how a guaranteed-strategy preimage planner would construct a motion guaranteed-strategy 91, - , On to achieve a goal G. 01 'is the first motion in the plan, On is the last. Consider the difference in how a forward-chaining planner and a backchaining planner would compute steps Oi and
Oi+,.-0 A forward-chaining planner must calculate where. motion Oi will terminate, since this termination region is the start region for the next motion, Oi+,.
Since this calculation involves some choice I it amounts to a formulation of the decision problem with existential quantification over the push-forward of motion Oi. In a back-chai'ng planner, where the motion must terminate has already been computed: it is the next preimage with respect to Oi+,, namely
p8i+1 p8i+2 ... P8n(G)) .))
-Thus we have seen why a back-chaining planner can (in pnciple) be complete for guaranteed strategies, while a forward-chaining planner cannot, unless it guesses push-forwards.
This suggests the following approach to EDR planning:
0 Use a back-chaining planner to find a guaranteed strategy for part of the start region. Then extend it to an EDR strategy using forward-chaining verification.
This appears to be a reasonable heuristic approach. However, for EDR plan-ning, it 'is still merely a halfway measure. While 'it removes from the EDR planner's responsibility the decision of where to terminate a motion within a subgoal, the
problem remains of deciding where wthin the EDR region H a motion should ter-minate. This is one of the key theoretical questions in EDR- 'it 'is addressed at some length later. The computational solution seems to involve quantifying over push-forwards even when a combination backward- and forward-chaining planner
is envisioned. LIMITED uses only forward-chaining for this reason. However, the combination back- and forward-chaining approach deserves more exploration. In
particular, the backchaining first stage could be used to suggest and guide the search for good candidate EDR strategies. Randy Brost has reported' a backchain-ing plannbackchain-ing algorithm which can generate multi-step plans in which each motion is a one-step EDR strategy.
7 [Personal Communication]. See also Brost's forthcoming Ph.D. thesis.