An Analysis of Time-Dependent Planning

This paper presents a framework for exploring issues in time-dependent planning: planning in which the time available to respond to predicted events varies, and the decision making required to formulate effective responses is complex. Our analysis of time-dependent planning suggests an approach based on a class of algorithms that we call anytime algorithms. Anytime algorithms can be interrupted at any point during computation to return a result whose utility is a function of computation time. We explore methods for solving time-dependent planning problems based on the properties of anytime algorithms. Time-dependent planning is concerned with determining how best to respond to predicted events when the time available to make such determinations varies from situation to situation. In order to program a robot to react appropriately over a range of situations, we have to understand how to design effective algorithms for time-dependent planning. In this paper, we will be concerned primarily with understanding the properties of such algorithms, and providing a precise characterization of time-dependent planning. The issues we are concerned with arise either because the number of events that the robot has to contend with varies, and, hence, the time allotted to deliberating about any one event varies, or the observations that allow us to predict events precede the events they herald by varying amounts of time. The range of planning problems in which such complications occur is quite broad. Almost any situation that involves tracking objects of differing velocities will involve time-dependent planning (e.g., vehicle monitoring [Lesser and Corkill, 1983; Durfee, 19871, signal processing [Chung et al., 19871, and juggling [Donner and Jameson, 19861). S t t i ua ions where a system has to dynamically reevaluate its options [Fox and Smith, 1985; Dean, 19871 or delay committing to specific options until critical information arrives [Fox and Kempf, 19851 generally can be cast as time-dependent planning problems. To take a specific example, consider the problem faced by a stationary robot assigned the task of recognizing and intercepting or rerouting objects on a moving conveyor *This work was supported in part by the National Science Foundation under grant IRI-8612644 and by an IBM faculty development award. belt. Suppose that the robot’s view of the conveyor is obscured at some point by a partition, and that someone on the other side of this partition places objects on the conveyor at irregular intervals. The robot’s task requires that, between the time each object clears the partition and the time it reaches the end of the conveyor, it must classify the object and react appropriately. We assume that classification is computationally intensive, and that the longer the robot spends in analyzing an image, the more likely it is to make a correct classification. One can imagine a variety of reactions. The robot might simply have to push a button to direct each object into a bin intended for objects of a specific class; the time required for this sort of reaction is negligible. Alternatively, the robot might have to reach out and grasp certain objects and assemble them; the time required to react in this case will depend upon many factors. One can also imagine variations that exacerbate the timedependent aspects of the problem. For instance, it might take more time to classify certain objects, the number of objects placed on the conveyor might vary throughout the day, or the conveyor might speed up or slow down according to production demands. The important thing to note is, if the robot is to make optimal use of its time, it should be prepared to make decisions in situations where there is very little time to decide as well as to take advantage of situations where there is more than average time to decide. This places certain constraints on the design of the algorithms for performing classification, determining assembly sequences, and handling other inferential tasks. Traditional computer science concerns itself primarily with the complexity and correctness of algorithms. In most planning situations, however, there is no one correct answer, and having the right answer too late is tantamount to not having it at all. In dealing with potentially intractable problems, computer scientists are sometimes content with less than guaranteed solutions (e.g., answers that are likely correct and guaranteed computed in polynomial time (Monte Carlo algorithms), answers that are guaranteed correct and likely computed in polynomial time (Las Vegas algorithms), answers that are optimal within some factor and computed in polynomial time (approximation algorithms). While we regard this small concession to reality as encouraging, it doesn’t begin to address the problems in time-dependent planning. For many planning tasks, polynomial performance is not sufficient; we need algorithms that compute the best answers they can in the time they have available. Planning is concerned with reasoning about whether to act and how. Scheduling is concerned with reasoning about Dean and Boddy 49 From: AAAI-88