A Qualitative Representation of Trajectory Pairs

There has been much research in temporal and spatial reasoning, both quantitative and qualitative, the latter being of particular interest from a cognitive viewpoint [2]. Attempts to combine both spatial and temporal relationships include [1,7,9]. A database approach to specifying spatial relationships that hold between moving objects during a particular interval is [10]. An approach that combines topological relationships between regions in 2D space with temporal relationships between convex intervals is [1]. However, we believe the question remains of how to describe motion adequately within a qualitative calculus. Motion can be divided into change of location, i.e. translation, and change in orientation, i.e. rotation. We focus on the former aspect here. A thorough investigation into mereotopological spatio-temporal continuous change has been conducted in [6], though there has been little work on describing relative motion of disconnected objects. However, it is clear that mobile disconnected objects (animals, vehicles,...) are prevalent in many domains and it would be highly desirable to be able to describe their motion in a qualitative manner. One move in this direction is the extension of qualitative physics to handle relative positions of objects in 2D [11], but this relies on projecting positions to x and y axes and does not provide a calculus with a set of jointly exhaustive and pairwise disjoint (JEPD) relationships. A simple calculus for describing traffic events is [4]. The work presented in this paper can be viewed as a continuation of these strands of previous research; i.e. it is an exploration of trajectories of moving (point like) objects. If objects do not change their form during the movement and we focus on the representation of spatially disjoint objects, then we can take an arbitrary point (e.g. the centroid) as the spatial location of an object. Therefore in this paper objects are represented simply as points. Moving objects can be partitioned in those having a free trajectory and those with a constrained trajectory [8]. A free trajectory means that there are no significant restrictions on the movement of a point in an nD space, such as an airplane traveling through the sky. A constrained trajectory means that the movement of an object in space is strongly restricted. A 1D representation can provide a useful abstraction for many free trajectory applications; e.g. even though a prey and a predator move in nD, the vital question is whether or not the predator catches the prey (represented by their Eucledian distance apart). Positional information is determined by the orientation and the distance relation [3]. Based on this and the notion of mode space (in which a space is subdivided in homogenous clusters) [6], the movement or transition between two objects at an instant can be qualitatively represented using three functions: i. movement of the 1st object wrt the 2nd object’s position ii. movement of the 2nd object wrt the 1st object’s position iii. relative speed of the 1st object wrt the 2nd object Since we are interested in a qualitative calculus, we can represent the values of each of these functions by “+”, “0” or “−” (cf [13]).