Quasi-Monotonic Segmentation of State Variable Behavior for Reactive Control

Real-world agents must react to changing conditions as they execute planned tasks. Conditions are typically monitored through time series representing state variables. While some predicates on these times series only consider one measure at a time, other predicates, sometimes called episodic predicates, consider sets of measures. We consider a special class of episodic predicates based on segmentation of the the measures into quasi-monotonic intervals where each interval is either quasi-increasing, quasi-decreasing, or quasi-flat. While being scale-based, this approach is also computational efficient and results can be computed exactly without need for approximation algorithms. Our approach is compared to linear spline and regression analysis. Reactive control and monitoring It has been recognized for some time that real-world agents have to monitor conditions as they execute planned tasks. It is rarely the case that the only changes that can occur in the world result from actions taken by the agent. Actions taken by other agents, uncontrollable events occurring in the physical world, and the presence of noise and uncertainty are the norm. Given this, an agent must be able to react to changes in the world. Moreover, the agent must be able to notice the changes in the first place. A typical description of this is a three-tiered architecture (Bonasso et al. 1997), with sensors and effectors at the “bottom” tier, and a high-level planner at the “top” tier. In between, the task execution system monitors for changing conditions and executes actions as appropriate based on the goals given it by the planner. Reactive task execution systems include EXEC (Pell et al. 1997), the RAP System (Firby 1989), PRS (Myers 1996), and Apex (Freed et al. 2003). Of course, the conditions to which an agent might imaginably react can be arbitrarily complex, and, for an agent acting in the real world, a “paralysis of analysis” prevents effective action. Part of an effective architecture for agent design is describing the kinds of useful conditions which are likely to be relevant to the agent making use of the kinds of sensors typically available to the agent. Furthermore, it must be feasible–that is, computational efficient–to recognize the Copyright c © 2005, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. In AAAI05, Pittsburg, USA, July 2005. relevant conditions, especially in concert with a variety of simultaneously executing tasks and monitors. Sensors nominally create time-stamped data, with readings sometimes done at constant intervals, sometimes not. Each reading provides a time-stamped measurement of a state variable, that is, an attribute of some object (often the agent, or a component of the agent), which changes over time. For example, an autonomous aircraft will typically have sensors for its own altitude, longitude and latitude, among other measurements. Tasks such as “raise-gear” might have a precondition that the measured altitude be ≥ a certain amount. Given the time series nature of such sensor data, it is natural to consider monitors being “live” for some interval of time. Within such an interval I , we have an ordered set of measurements for a given sensor, {xt}, ordered by time t. If P is a boolean predicate on x, and n is the cardinality of {xt}, then testing for a value in I upon which P is true is clearly at least O(n). In a typical case, P is of constant complexity (such as the example of ≥ given above), and so, testing for value upon which P is true is O(n). Predicates on individual measurement values are not the only possible boolean predicates of interest. For example, an agent might want to act when the average value of a measurement within an interval reaches a certain point. The presence of a condition based on multiple measurements (of the same or different state variable) is sometimes called an episode (Mannila, Toivonen, & Verkamo 1997). Episodic predicates can, of course, be arbitrarily complex, and thus be difficult or impossible for an agent to consider. In this paper, we discuss one particular class of episodic predicates, namely those based on segmenting an ordered set of measurements according to changes in monotonicity. For example, consider Fig. 1, which shows a flight path of an airplane (simulated) as it takes off, circles an airport, and then crashes. A reactive controller may monitor for nominal conditions, such as (in the present case) making the first turn. A controller may also monitor for off-nominal conditions, such as failing to make a turn, or (in the present case) “in danger of crashing.” In the nominal case, “making a turn” can be sensed by the appropriate increase in longitude and/or latitude. In the off-nominal example, “in danger of crashing” can be sensed by the an unexpected decrease in altitude. If sensor data were typically comprised of long monotone 0 200 400 600 80