QRPC: A new qualitative model for representing motion patterns

Abstract

The Qualitative Rectilinear Projection Calculus (QRPC), a new representation model based on planar trajectories, is presented in this work for describing qualitatively motion patterns. Several models have been developed for this purpose and by comparing them with our proposal we show that the proposed model results in an intuitive approach for representing, in any context, the kinematical behavior of two objects in motion on the plane through the possible relationships among the rectilinear projection of their trajectories. The paper is centered on the formal definition of the set of geometric relations in terms of the front–back and left–right dichotomies, and how by the composition of these relations, it can be possible enumerate an exhaustive set of qualitative states and the possible transitions among them. The complete iconic representation of the set of qualitative states defines the Conceptual Neighborhood Graph of the QRPC model. In order to illustrate how this qualitative representation can be used to analyze and describe the relative motion of two objects, some examples extracted from the traffic engineering field have been studied.

Introduction

If anyone would have to explain how objects are moving around, sentences such as: “the object is currently to my left moving toward the right” or “at this instant of time it is crossing just in front of me, moving toward the right” may be used. All these kind of descriptions contain spatiotemporal information in terms of spatial and temporal statements and can be used for distinguishing more complex behaviors and making reasoning processes. For that reason, the definition of a qualitative representation becomes determinant if any of these kind of processes have to be tackled.

The qualitative formalism has been proved to be suitable since it can be used in any context, even when only partial or imprecise information or knowledge is available (Gottfried and Witte, 2007, Lücke et al., 2011). Thus, this modeling approach has been applied in Geographic Information Systems for representing spatiotemporal data (Van de Weghe, 2003) from a causal perspective (El-Geresy, Abdelmot, & Jones, 2002), in robotics for implementing the navigation process (Dylla et al., 2007, Frommberger, 2007, Frommberger, 2008, Frommberger et al., 2011, Musto et al., 1999, Müller et al., 2000, Peris-Broch et al., 2008, Wolter et al., 2007), in spatial reasoning for topological map learning (Wallgrün, 2010), in video databases for analyzing and describing motion of a multiple object system (Li, Özsu, & Szafron, 1997), in spatial scene modeling for ambient intelligence environments (Bhatt and Dylla, 2009, Dylla and Bhatt, 2008), in robocup league (Dylla et al., 2005) or in a traffic engineering context (Fernyhough et al., 1997, Fernyhough et al., 2000, Galata et al., 2002, Glez-Cabrera, 2009) and to describe a cognitive systems for guiding blind people (Álvarez-Bravo, Peris-Broch, Álvarez-Sánchez, & Escrig-Monferrer, 2006).

With regard to the qualitative formalism itself, several authors have worked previously on different models of qualitative representation. An exhaustive revision of these works is given in Table 1. For classifying purposes, it has been distinguished three main domains depending on the character of the qualitative facts represented by them: temporal (and non-spatial), spatial (and non-temporal) or spatiotemporal facts. The most of the spatial and spatiotemporal models are considered as oriented models since they are able to represent sentences about the position of oriented objects (objects which are assumed to be faced up to a direction in the plane). Table 1 includes other features of each model such as the type of objects handled by the formalism (time intervals, points, dipoles, oriented points, etc.), the number of relationships defined specifically by each model, the cardinality of these relationships, the type of the resulting regions associated to these relationships and, finally a short description of each model. For example, the SCC (Single Cross Calculus) (Freksa, 1992a) is an oriented model which considers the position of a point C with regard to the segment defined by other two points, A and B. The SCC model is able to distinguish the relative position among any three points in the plane, in terms of 11 possible relationships defined by the model (each one associated to a specific region/area on the plane).

However, in spite of all these works, there are few complete representation models for overcoming the difficulties of describing and reasoning properly about motion in any context and, furthermore, these ones do not consider some kinematical behaviors such as rotational motions. Therefore, from the models described in short in Table 1, we pay special attention to the models: QTC (Qualitative Trajectory Calculus) (Van de Weghe, Cohn, Maeyer, & Witlox, 2005a), Gottfried’s model (Gottfried, 2004) and OPRA (Oriented Point Relation Algebra) (Dylla & Wallgrün, 2007a) since some of the notions introduced by them have inspired or are suitable for the development of our current research.

The Qualitative Trajectory Calculus (QTC) is presented in Van de Weghe et al., 2005a, Van de Weghe et al., 2005b for representing and reasoning about motion. This representation model is based on the relationship between the trajectories of two moving point objects using for this purpose the double-cross reference system introduced by Freksa, 1992a, Freksa and Zimmermann, 1992, Zimmermann and Freksa, 1996. Under this approach, the qualitative relationships between trajectories are described in terms of two dichotomies defined with respect to the double-cross reference system: the front–back and left–right dichotomies.

Gottfried introduces a sentence-based model and a diagrammatic one for representing motion patterns (Gottfried, 2004). Under these two approaches, any motion is described through sixteen atomic patterns which represent how two objects in motion can be located with respect to each other during two instants of time. Gottfried shows how these two viewpoints are complementary and both ones are able to catch different aspects of the same kinematical system. Finally, in Gottfried and Witte (2007) a qualitative calculus for searching specific locomotion configuration when only imprecise data is available (for instance, a biological system consisting of cells) is presented. This qualitative abstraction is expressed in terms of twenty three relationships among oriented intervals in two dimensions (Gottfried, 2010).

A different approach is based on the notion of geometry-based orientation which considers directed points or dipoles (Moratz, 2006). Then the authors study the geometric alternatives derived from the relative position of two dipoles in the plane (Dylla and Moratz, 2004, Dylla and Moratz, 2005). As a result of this work, the authors propose a new tool (Dylla et al., 2006, Wallgrün et al., 2007) and a powerful algebra associated with it. The proposed algebra stands a generalization of the orientation information in terms of different levels of granularity (Moratz, Dylla, & Frommberger, 2005), and it also introduces the idea of “conceptual neighborhood” (Dylla and Wallgrün, 2007a, Dylla and Wallgrün, 2007b, Frommberger et al., 2007) for relations between objects.

In order to provide a new intuitive approach in this regard, a qualitative representation model based on the relationships between the trajectories of two objects in two dimensions is described in this paper. Unlike the above works, this novel model provides a richer description of motion characterizing the trajectories of the objects in terms of an oriented rectilinear projection. Thanks to this feature, the model is able to describe motions that have not been considered previously such as, for instance, the rotation of a object in motion with respect to itself. The Qualitative Rectilinear Projection Calculus (QRPC) is based on previous works (Glez-Cabrera, 2009, Álvarez-Bravo et al., 2006) which are also included in Table 1.

Along this work, it is assumed that the proposed qualitative model of representation would be embedded in the Detection and Prediction System which governs the autonomous behavior (motion) of a hypothetical intelligent agent (e.g. a robot) which is assumed to be the reference object. This system should pay special attention to the avoidance of obstacles which is also assumed as its main design requirement. Under these assumptions, the abstract model of representation developed here is aimed at predicting and detecting all possible geometric relationships between two point objects in motion, by taking into account the front–back and the left–right dichotomies which are associated with these objects. The resulting model is a set of 48 relationships or possible states which are described in terms of a collection of qualitative features about their spatial locations. These possible states have been found as meaningful for the motion prediction of both objects.

As it is sketched in Fig. 1, the Qualitative-based Detection and Prediction System comprises several functional modules whereas the proposed abstract model of representation is the formal basis of the embedded knowledge layer across the whole system. Briefly, the qualitative module consists of the constraint satisfaction algorithm and a composition table. The qualitative module receives as input a collection of measurements from different types of sensors of the intelligent agent to determine directly the QRPC states with regard to other objects which are present in his environment. After a pre-processing of the quantitative (and raw) measurements which is carried out by the hybrid module, only one of the possible states of the abstract model will be satisfy the constraints imposed by the physical measurements. Once the overall state of the agent with regard to the other agents is known, this information will be fed to the prediction and detection module to compute the qualitative output of the whole Detection and Prediction System. At this point we need to introduce the notion of inference. We assume that the hypothetical intelligent agent (where the Detection and Prediction System is deployed) lives in a distributed and cooperative environment. That is to say, it is possible that there are other similar agents in the environment (therefore all together conforms a multi-agent system where the processing is distributed and it is possible the communication among them in order to cooperate). In this context, an object in motion A can know (through its sensors) its relative position with respect to other object in motion B (the QRPC state of A with respect to B). Although object A cannot determine its state with respect to a third object C (since C is outside of the range of its sensors), object B can communicate to A, its relative position with respect to C (the QRPC state of B with respect to C). Then, assuming that the state of A with respect to B and the state of B with respect to C are known by object A, it can be possible to infer (by object A) the qualitative state of A with respect to C. This inference task will be the main functionality of the qualitative module and it will be supported by the composition table.

Once it is presented the related work and the framework of the current research, this work is focused on the formal definition of the qualitative relationships of the abstract model, and the rest of the paper is organized as follows: in Section 2 the qualitative representation model is described. In this regard, the complete table of qualitative relationships (primitives) describing the motion between two objects is presented. Additionally, the Conceptual Neighborhood Graph is also defined. In Section 3, in order to illustrate the potential of this qualitative representation model, some examples extracted from a traffic engineering context are analyzed. Finally in Section 4 conclusions about our major achievements and some prospects are presented.