Bottom-up Derivation of the Qualitatively Diierent Behaviors of a Car across Varying Spatio-temporal Scales: a Study in Abstraction of Goal-directed Motion 1 Levels of Motion Description

Driving a car involves considering it at diierent spatio-temporal scales, and somehow leads to behavior such as the parallel parking maneuver, the three-point turn, free Euclidean driving in a desert, following a road, and translationally passing other vehicles at high speed. In the study of autonomous systems, it is desirable to nd a representation in which such diierent behaviors of a single system can be related to each other, and to nd precisely how and under what conditions a change of representation and corresponding choice of motions occurs. In this paper, we formulate an abstraction mechanism based on approximations of ows of commutators of vector elds. We apply it to the goal-directed motion of a car and show how the environmental constraints induce, through this abstraction mechanism, a recognizable hierarchy of descriptions of the car's motion. Imagine that you are planning to visit some friends that live in B. You decide to go from A (your town) to B by car. At you kitchen table, you decide which route to take, using maps of rather coarse scale, choosing the quickest or most scenic route. You then go to the garage, maneuver the car onto the driveway and turn onto the road. You take your favorite route through town to the beginning of the freeway or scenic route, and drive along, passing other cars. When you arrive at B, some planning through that town is required to nd your friends' house. You pass it, make a three point turn, and parallel-park in front of their house. You straighten out the wheels, thereby ending a typical car trip. Now consider at what level of detail you think of your car's motions at each of these stretches. On the map, your car is a `directed point', and as you mentally move it over the map it behaves like a particle pushed through a tube (the road on the map), taking turns as the tube leads it. When you are actually on the freeway, passing cars, you consider your car as a translational device: the steering moves you sideways relative to the car in front of you, the accelerator moves you forward. Your heading is still determined by the road. During the local trips through the towns A and B, the turning of your car is essential, as it is for