Qualitative Spatial Reasoning for Topological Map Learning

SPATIAL COGNITION AND COMPUTATION, vol(issue), 1–end c YYYY, Lawrence Erlbaum Associates, Inc. Copyright

Qualitative Spatial Reasoning for Topological Map Learning Jan Oliver Wallgrün University of Bremen In this article we investigate the application of qualitative spatial reasoning methods for learning the topological map of an unknown environment. We develop a topological mapping framework that achieves robustness against ambiguity in the available information by tracking all possible graph hypotheses simultaneously. We then exploit spatial reasoning to reduce the space of possible hypotheses. The considered constraints are qualitative direction information and the assumption that the map is planar. We investigate the effects of absolute and relative direction information using two different spatial calculi and combine the approach with a real mapping system based on Voronoi graphs. Keywords: qualitative spatial reasoning, robot navigation, topological maps, spatial representation, constraint satisfaction, planarity, direction relations, multi-hypothesis tracking, Voronoi graphs

1

Introduction

The problem of learning and maintaining a spatial model of an initially unknown environment has attracted a lot of attention from several disciplines. For instance, psychologists and geographers have studied general properties of human spatial knowledge as well as the process of acquiring this knowledge (Lynch, 1960; Downs & Stea, 1973; Siegel & White, 1975; Hirtle & Jonides, 1985; McNamara, 1986; Montello, 1992; Tversky, 1992, 1993; Golledge, 1999). Similarly, from investigations on the navigation behavior of animals we know what kind of spatial cues are used by different species including those with very limited brain capac-

Correspondence concerning this article should be addressed to Jan Oliver Wallgrün, Department for Mathematics and Informatics, University of Bremen, P.O. Box 330 440, 28334 Bremen, Germany; email [email protected].

¨ 2 JAN OLIVER WALLGRUN ities like bees (F. C. Dyer, 1996) or ants (Wehner, 1999). Based on these cues different navigation strategies can be realized ranging from align-and-approach over guidance to place-recognition-triggered-response and to topological navigation (Trullier, Wiener, Berthoz, & Meyer, 1997). In robotics and AI, the problem of learning an environmental model is studied under terms like the simultaneous mapping and localization (SLAM) problem (Leonard & Durrant-Whyte, 1991), or simply the robot mapping or map learning problem. While a lot of recent research on the robot mapping problem is driven by the goal of finding general technical solutions to the overall task, several researchers have explicitly endeavored to develop map learning systems for mobile robots that directly imitate or model biological systems (Kortenkamp, Weymouth, Chown, & Kaplan, 1992; Kuipers, 1983, 2000; Franz, Schölkopf, Mallot, & Bülthoff, 1998; Yeap & Jefferies, 1999). From an abstract perspective, the general task in map learning is to integrate local spatial information gathered over time into a coherent overall model which correctly reflects the spatial properties of a section of the external world. Hence, map learning is also related to other fields dealing with spatial data fusion like 3D reconstruction (C. R. Dyer, 2001) and remote sensing (Pohl & Genderen, 1998). Typically, integration during the map learning process is supposed to proceed incrementally, meaning that new information is incorporated into the agent’s internal model and then discarded, in contrast to going through the input data multiple times. The problem is challenging because the available information (for instance stemming from sensor measurements) is typically erroneous, noisy, imprecise, and ambiguous. A large part of the problem is to adequately deal with perceptual aliasing (the fact that different places in the environment may look the same) and to establish the correct correspondences between the currently observed entities which make up the robot’s current local observation and the memorized entities in the robot’s spatial model (for simplicity also just called the robot’s map herein after). In this article, we look at the map learning problem from the perspective of spatial reasoning. In particular, we are interested in investigating how much established formal spatial reasoning methods can contribute to resolving the ambiguities existing in the input information. Obviously, when rich and precise sensor information is available, perceptual aliasing is reduced, spatial reasoning becomes more effective, and many potential hypotheses about the layout of the environment can be ruled out, as long as the information can be seen as reliable. However, in practice there is a trade-off between the level of precision and the reliability of sensor information. Precise metric measurements are typically afflicted with a significant level of uncertainty. To investigate agents with more human-like map learning abilities, our approach in this paper is to look at the other extreme and to make rather weak assumptions about the general sensor abilities of the involved spatial agent. We focus on coarse spatial relations for which we can assume that they can be perceived reliably. The spatial reasoning approaches we employ stem

SPATIAL REASONING FOR TOPOLOGICAL MAP LEARNING 3 from the area of qualitative spatial reasoning (QSR) (Cohn & Hazarika, 2001; Cohn & Renz, 2007). More specifically we use so-called qualitative constraint calculi (Renz & Nebel, 2007) which define operations for reasoning with sets of spatial relations. These calculi have been developed to deal with different aspects of space like topology (Egenhofer, 1989; Randell, Cui, & Cohn, 1992), orientation and direction (Freksa, 1992; Frank, 1991; Ligozat, 1993, 1998; Moratz, Renz, & Wolter, 2000; Moratz, 2006; Renz & Mitra, 2004) or direction and distance (Hernandez, 1994; Moratz, Nebel, & Freksa, 2003). Even when focusing on coarse but reliably perceivable categories for spatial information, the properties and performance of map learning still depend on which kind of spatial information is available to the agent. In particular, map learning becomes significantly easier with access to sensors or tools that provide absolute spatial information which in contrast to egocentric relative information reduces the effects of error accumulation. We will investigate this aspect by directly comparing absolute and relative forms of spatial information for one exemplary spatial aspect which is especially well-suited for our purpose, namely direction information. Absolute direction information coming in the form of cardinal direction relations (for instance provided by a compass) will be modeled using the qualitative cardinal direction calculus (Ligozat, 1998). Relative information will be modeled using a second qualitative calculus, the OPRA2 calculus (Moratz, 2006). Naturally, the map learning processes and the underlying spatial representation approach used to formulate the spatial model are intricately linked. Hence, the framework we will describe in the following is similarly tailored to a particular way of conceptualizing and representing an environment (although our approach can certainly be adopted for other ways to represent space). During the last decades, work on mobile robots has been concentrated on coordinate-based spatial representations, i.e., representations resembling floor plans. Examples are occupancy grids as illustrated in Fig. 1(a) and feature-based representations illustrated in Fig. 1(b) (cf. (Thrun, 2002; Thrun, Burgard, & Fox, 2005) for an overview). An alternative to these representation approaches—the one that we will be concerned with in this work—are graph-based representations often referred to as topological maps (Kuipers, 2000; Remolina & Kuipers, 2004). Inspired by results on environmental representations in humans (Siegel & White, 1975), these approaches conceptualize the environment as a route graph (Werner, Krieg-Brückner, & Herrmann, 2000) consisting of nodes that stand for distinctive places or navigational decision points and edges that stand for the distinctive paths connecting neighbored places. An example of such a topological map or route graph representation is shown in Fig. 1(c), overlayed on a floor plan of the actual environment. Route graph representations are particular suited to represent indoor environments or path networks. Solving the map learning problem for such a topological map representation implies that we have to develop techniques to determine the correct graph struc-

¨ 4 JAN OLIVER WALLGRUN

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Figure 1.: Different representations of space: (a) occupancy grid, (b) feature map, (c) topological map

ture from a sequence of local observations collected while moving through the environment. Although it is known that humans employ a diverse set of spatial features during navigation and learning, especially landmark information (Denis, 1997; Sorrows & Hirtle, 1999; Lovelace, Hegarty, & Montello, 1999), we have chosen to study the topological map learning problem in its purest form: The environment itself is seen as a graph embedded into the plane and the exploration experience of the agent consists of two kinds of “observations” providing information about the spatial layout of the nodes and edges, (1) the observation of a node with leaving edges and (2) the traversal of an edge. The same approach of studying topological mapping as graph exploration has been taken by several authors (Dudek, Jenkin, Milios, & Wilkes, 1991; Bender, Fernández, Ron, Sahai, & Vadhan, 1998; Rekleitis, Dujmovic, & Dudek, 1999). One interesting theoretical result given in (Dudek et al., 1991) is that successful map learning cannot be guaranteed without the help of at least one movable marker. However, in this work it was assumed that no additional spatial information about the nodes and edges is available that could be exploited to infer the layout of the environment. When employing the topological mapping paradigm in practice based on real sensor data, the predominantly chosen approach has been to maintain a single map hypothesis during the exploration (Kortenkamp et al., 1992; Franz et al., 1998; Yeap & Jefferies, 1999; Choset, Walker, Eiamsa-Ard, & Burdick, 2000). However, as a consequence of this commitment to a single hypothesis the incremental map construction process lacks robustness; it tends to fail as soon as a wrong decision is made in the face of ambiguity. A more promising approach, for instance suggested in (Dudek, Freedman, & Hadjres, 1996) and (Kuipers, Modayil, Beeson, MacMahon, & Savelli, 2004), is to deal with ambiguity by keeping track of all possible graph hypotheses simultaneously. However, this approach can increase the computational costs dramatically as the number of possible topological map hypotheses can grow exponentially with the number of observations. Hence, it becomes crucial to exploit available spatial information as effectively as possible in order to eliminate as many hypotheses as possible and make the

SPATIAL REASONING FOR TOPOLOGICAL MAP LEARNING 5 multi-hypothesis tracking approach feasible, which is exactly the goal of the work described here. In the following, we extend multiple-hypothesis topological mapping and earlier work on qualitative spatial reasoning in topological maps (Moratz et al., 2003; Moratz & Wallgrün, 2003) in which reasoning was employed to identify individual spatial features (landmarks or junctions) but not to reason about the complete layout of the environment. We develop a general multi-hypothesis tracking framework for topological mapping which exploits spatial constraints and spatial reasoning to prune the space of possible map hypotheses. This overall framework can be adapted for different sorts of spatial constraints which can be of two general kinds: (1) constraints stemming from the individual observations and (2) constraints stemming from background assumptions. For the first kind of constraints we will use the relations of the previously mentioned two qualitative direction calculi to describe the directions of leaving edges (or hallways) perceived at a node (or junction). To exploit the direction information we make use of spatial reasoning techniques for consistency checking (Mackworth, 1977; Montanari, 1974). As a background constraint we make the assumption that the environment is planar, a constraint that has already been investigated in (Savelli & Kuipers, 2004). We also discuss and investigate two alternative representation variants for the graph hypotheses in our mapping approach. Besides developing the theoretical topological mapping framework (Sects. 2 and 3) which also turns out to be a valuable test-bed to evaluate and compare spatial reasoning approaches, we demonstrate that the approach can be applied successfully in practice by combining it with a topological mapping approach for real robots operating in indoor environments (Sect. 4). The approach uses generalized Voronoi graphs as a means to derive the topological map (Wallgrün, 2005). We report on several experiments performed to evaluate the theoretical framework, to compare different settings with regard to the available spatial information and the employed representation variant, and to test the Voronoi-based approach using simulated environments as well as real exploration data from a mobile robot (Sect. 5). The experiments yield several interesting results and demonstrate the benefits of employing spatial reasoning while also revealing shortcomings of existing spatial calculi and reasoning methods.

2

Multi-Hypothesis Topological Mapping

Let us consider the following scenario: A robot is roaming through a graph-like environment like the one shown in Fig. 2. The environment consists of straight hallways connected by junctions. For every passed junction, the robot stores a junction observation Ji consisting of a cyclically ordered set of leaving hallways [J ] [J ] [J ] hl1 i , l2 i , ..., ln i i and a spatial description consisting of spatial relations over the set of observed leaving hallways. For instance, for junction observation J1


l [J1 1] B

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Figure 2.: (a) Walk of a robot through a graph-like environment and (b) the first four junction observations J1 to J4 for this walk [J ]

[J ]

in Fig. 2 the spatial description could be {southwest(l1 1 ), south(l2 1 )} using [J ] [J ] cardinal directions or, alternatively, it could be {obtuse(l1 1 , l2 1 )} when using some qualitative categories for the angles formed by pairs of hallways. Junction observations are connected by hallway traversal actions consisting of leaving the current junction via one of the observed leaving hallways and arriving at the next junction via one of the leaving hallways belonging to the next [J ] [J ] junction observation, e.g., l2 1 → l1 2 for traversing the hallway connecting A [J2 ] and C where l1 is the observed leaving hallway leading north in J2 . Hence, [J ] [J ] l2 1 and l1 2 correspond to different endings of the same physical hallway. A list hJ1 , T1 , J2 , T2 , ..., Tn−1 , Jn i of alternating junction observations Ji and hallway traversals Tj forms the history of one particular exploration run through the graph environment. The goal of a topological mapping algorithm now is to incrementally process the history of observations and actions and for each step determine one or all route graph hypotheses describing different topological structures of the environment that can be considered valid explanations of the information processed so far. A route graph hypothesis H consists of the following: • an undirected graph GH = (VH , EH ) in which the nodes in VH represent the junctions of the environment and the edges EH represent the hallways • a combinatorial embedding of GH into the plane (e.g., represented by specifying the cyclic order of incident edges for each node in VH ) • the starting position SH of the robot at the beginning of the exploration run (e.g., specified by the node corresponding to J1 and the edge corresponding [J ] to l1 1 ) A route graph hypothesis makes an assumption about which junction observations correspond to the same physical junction and which observed leaving hall-

SPATIAL REASONING FOR TOPOLOGICAL MAP LEARNING 7 ways correspond to the same physical hallway. Junction observations assigned to the same node in the hypothesis need to be compatible, meaning that the perceived spatial relations over the leaving hallways match (this aspect will be further discussed in Sect. 3.2). Given the combinatorially embedded graph GH and the starting position SH of a route graph hypothesis H, a history unambiguously induces a corresponding walk through GH which directly yields the nodes and edges associated with each junction observation and hallway traversal. In practice, it is convenient to also store the current position CH of the robot by recording the node corresponding to the last processed junction observation and the edge corresponding to the last hallway traversal. Whenever we depict a route graph hypothesis, we have to choose one of the infinitely many possible ways to draw the graph into the plane and if possible we do it in a way that preserves the spatial relations contained in the junction observations. However, we have to keep in mind that a route graph hypothesis does not specify a geometric embedding into the plane but rather restricts possible geometric embeddings via its combinatorial embedding and the associated spatial relations. An important design decision is whether we want the route graph hypotheses to model only those junctions that have been visited and perceived or also make predictions about how perceived but not traversed leaving hallways are connected (resulting in additional junctions). For instance, given junction observations J1 and J2 in the example from Fig. 2, one approach would be to construct hypotheses that predict whether the hallway leaving southwest in J1 and the one leaving west from J2 meet at the same junction or not, or we could construct the hypotheses in a way that leaves this aspect completely unspecified. The first representation variant has the advantage that it makes predictions about unseen parts which are known to exist, but doing so also leads to an increased complexity. In this work, we examine the feasibility of both variants and we will further discuss them in Sect. 2.3. For the examples in the rest of this section, we assume that we are employing the first variant and explicitly model unperceived junctions. During exploration, a hypothesis that has been plausible so far may turn out to be invalid when the next junction observation is processed. Hence, instead of committing to a single hypothesis, our approach is to track all valid hypotheses simultaneously. That means the hypotheses we consider during the mapping process form a search tree like the one sketched in Fig. 3. The currently considered hypotheses form the leafs of the search tree. When a new hallway traversal action Ti together with a new junction observation Ji+1 are processed, successor hypotheses are generated for every hypotheses Hj corresponding to a leaf in the search tree by performing the following two steps: 1. The current position CHj of the robot within the graph GHj is updated in accordance with Ti . 2. Ji+1 is matched with the node in EHj that now corresponds to the updated


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Figure 3.: Part of the search space of valid route graph hypotheses for the example from Fig. 2(a)

position. Information about new hallways in Ji+1 is used to update GHj by adding new edges. The fact that there may exist multiple ways of how the new edges can be connected to existing nodes in GHj (or to a completely new node), means that there can be multiple successor hypotheses to Hj in the search tree. In addition, if no way exists to match Ji+1 with the current position of the robot, there will be no successor hypotheses and Hj becomes a dead branch in the search tree. In Fig. 3 the top row shows three possible hypotheses for the situation where the robot has just arrived at junction G in the example from Fig. 2 and has observed J1 to J3 . We here assume that the junction observations are given in terms of qualitative cardinal direction relations from Ligozat’s cardinal direction calculus (Ligozat, 1998), north, northwest, west, etc. (see Fig. 7(a) and Sect. 3.2 for the precise meaning of these relations). Black nodes in the figure stand for junctions that have been observed, while white nodes are introduced for the end points of hallways that have not been traversed so far and which therefore may be complemented with additional edges later. When moving on to F and processing the new observation J4 , the first hypothesis H1 can be complemented in two different ways leading to two successors H4 and H5 in the search tree. Similarly, the third hypothesis H3 has five successors. For the second hypothesis H2 , however, the new observation leads to a contradiction: no hallway leading northeast is observed and, hence, this hypothesis can be discarded completely based on the direction information. Let us briefly consider some of the things that would happen when the robot moves on to junction B and observes J5 and then returns to C observing J6 .

SPATIAL REASONING FOR TOPOLOGICAL MAP LEARNING 9 For H4 , J5 is compatible with node a but a new edge needs to be added leading southwest. This edge could lead to node c or to a new node. Hence, we would get two successor hypotheses. Pretty much the same would happen to H6 and H7 but all other hypotheses will have successor hypotheses as well because they also do not contradict observation J5 . Moving back to C and observing J6 will not cause any contradictions for all successors of H4 , H6 , and H7 and J2 and J6 will be correctly mapped to the same node b. In all other current hypotheses for which J6 does not cause any contradictions, J2 and J6 will be incorrectly mapped to different nodes. Nevertheless, they are still valid explanations of the robot’s observation history. Obviously, the previous example exploits the assumption that all hallways are straight. As a matter of fact, the complete theoretical framework developed in this section and Sect. 3 is based on this assumption which allows us to draw conclusions about the locations of junctions from the perceived directions of the leaving hallways. To employ the theoretical framework in practice where the environment itself will not be as graph-like as the environment used in the example, the framework will have to be combined with a concrete topological representation approach which realizes the abstraction from the real environment to the discrete graph structure of the corresponding topological map so that information about perceived nodes and edges can be fed into our mapping framework. In this case, the straight hallway (or edge) assumption will often not be satisfied. In Sect. 4, we therefore discuss how this assumption can be replaced with weaker assumptions which still allow us to constrain the locations of junctions and how our basic framework needs to be extended for this. An experimental evaluation of how the resulting modifications affect the map learning can be found in Sect. 5.1.5. For now we continue by discussing the details of the theoretical mapping framework.

2.1

Minimal Route Graph Model Finding

The approach sketched above performs an exhaustive search through the tree of possible hypotheses. A modification of this approach proposed by Kuipers (Kuipers et al., 2004) is to prefer among all valid hypotheses the one that offers the simplest explanation. In this article, we interpret simplest as meaning a hypothesis that contains a minimal number of nodes. Hence, using |VH | to refer to the number of nodes in a route graph hypothesis H and HE for the set of all plausible hypotheses for a given exploration history E, we can formally define the ∗ subset HE of hypotheses we are interested in as: ∗ HE = {H ∈ HE | ∀H 0 ∈ HE : |VH 0 | ≥ |VH |} ∗ We will call the elements of HE minimal route graph models. While the choice of looking at the number of nodes seems natural, other criteria for minimality are ∗ conceivable as well and could easily be incorporated into the definition of HE .

¨ 10 JAN OLIVER WALLGRUN In the following, we assume that we are interested in finding one minimal route ∗ graph model and do not require the complete set HE . The number of nodes in the graph hypotheses grows monotonically with increasing depth in the search tree because new nodes and edges will be added but never removed when new observations are incorporated and successor hypotheses are formed. As a result, we can search for a minimal route graph model in a best-first manner, keeping all current hypotheses in a queue sorted by their node numbers and always expanding the currently minimal hypothesis. This means that in the example from Fig. 3 the successors of the hypothesis H3 would not be generated as our current minimal hypothesis H4 would only contain 7 nodes while H3 itself already contains 7 nodes and, hence, cannot lead to better solution. We will refer to the resulting mapping algorithm as the minimal (route graph) model finding algorithm.

2.2

Valid Route Graph Models

The search tree depicted in Fig. 3 only contains valid route graph hypotheses in the sense that we can assign coordinates to the junction nodes so that the available spatial information is reproduced correctly. In addition, we might demand that additional constraints stemming from background knowledge need to be satisfied. Exploiting these kind of constraints is crucial for the minimal model finding algorithm to counteract the exponential growth of the search tree with the length of the exploration history and the otherwise high degree of ambiguity. As discussed in the introduction, our goal is to investigate map learning based on coarse but reliably observable spatial information and we assume that the available information consists of qualitative direction information about the leaving hallways. This information can come in either absolute or relative form and we want to study the effects this has on the map learning performance. As background knowledge we make the assumption that the environment is planar. Other kind of spatial or general constraints could be incorporated into the overall framework. In our case, a hypothesis then has to satisfy three conditions to be considered valid: 1. Repeating the sequence of actions specified in the history within the hypothetical route graph yields a sequence of node degrees identical to the original sequence of leaving hallway numbers (structural constraint). 2. There must exist a way to draw the hypothetical route graph into the plane without crossing edges that is in accordance with the specified combinatorial embedding (planarity constraint). 3. There needs to be a drawing satisfying condition 2 that at the same time also reproduces the direction relations provided by the original junction observations when repeating the actions in the given exploration history (direction constraints).

SPATIAL REASONING FOR TOPOLOGICAL MAP LEARNING 11

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Figure 4.: Two invalid hypotheses for the history from Fig. 2(a) As a first step of generating the successor hypotheses of a hypothesis in the search tree, we construct all hypotheses which satisfy the structural constraint. In addition, we take into account that two junction observations can only correspond to the same node in a hypothesis if the perceived directions match. As a result, we can simply store the direction information as constraints to the edges in the graphs. The second step then is to check if these hypotheses also satisfy the planarity constraint and the direction constraints. In Fig. 4 we show two examples of invalid hypotheses, for the walk depicted in Fig. 2(a) up to J5 . Both hypotheses are depicted by one particular drawing of the route graph into the plane and both satisfy the structural constraint. The drawing of the first hypothesis would also reproduce the observed direction relations. However, it has crossing edges and, more importantly, no drawing without crossing edges exists that is in accordance with the underlying combinatorial embedding. This is a direct consequence of the fact that the combinatorial embedding itself is not planar which is easy to verify (see Sect. 3.1). The drawing of the second hypothesis is planar but the positions assigned to the nodes do not reproduce the direction information correctly as the hallway that directly connects the junctions labeled J2 and J4 is supposed to lead east from J2 and arrive at J4 from the west. Hence, J4 would have to be to the east of J2 . However, from the knowledge that the hallway connecting J2 with J3 leads south and the hallway connecting J3 with J4 leads to the west it can be concluded that J4 has to be somewhere to the west of J2 . As a result of this reasoning, we know that no drawing satisfying the direction constraints for this hypothesis can exist at all because the contained direction information is inconsistent. The overall minimal route graph model finding problem we have described here is a combinatorial optimization problem. For deciding whether a given hypothesis is valid or not, we need to determine whether a drawing exists that satisfies the planarity constraint as well as the direction constraints. Such a drawing exists, if we can assign coordinates to the junctions such that all constraints are satisfied. Hence, we have to solve constraint satisfaction problems (CSPs) over infinite domains (points in the plane) which is a computationally challenging task. As there are infinitely many possible ways to assign coordinates to the junctions,

¨ 12 JAN OLIVER WALLGRUN the standard methods for deciding consistency of discrete CSPs based on domain reduction and search cannot be applied. On the positive side, efficient methods for checking the planarity of an combinatorial embedding and for checking the consistency of qualitative spatial relations individually exist, which—as the two previous examples demonstrate—is already sufficient to rule out many structurally valid but overall invalid hypotheses generated during the search process. For qualitative spatial relations, the respective techniques have been developed in the area of qualitative spatial reasoning (Cohn & Hazarika, 2001; Cohn & Renz, 2007; Renz & Nebel, 2007). They are based on algebraic operations defined on the spatial relations and are able to decide consistency of a spatial CSP despite the fact that the domains are infinite. Our approach is to employ the individual consistency checking methods to discard as many invalid hypotheses as soon as possible. As a result, testing a single hypothesis is comparatively fast. However, the downside of this procedure is that the individual hypothesis check is incomplete in the sense that it may not recognize all invalid map hypotheses: There may exist a drawing for a given map hypothesis that is planar and one that is compliant with the direction constraints but none that is both. Such an invalid hypothesis will not be recognized by our approach but may still lead to contradictions later on. Results on how well this approach works in practice will be given in Sect. 5 on experimental evaluation. The details of incorporating the constraints into the minimal model finding algorithm will be explained in Sect. 3 on rejection based on spatial constraints.

2.3

Two Representation Variants: CompEnv and VisOnly

As mentioned at the beginning of Sect. 2, the decision whether junctions forming the end points of perceived but never traversed hallways should be modeled in the route graph hypotheses or not leads to two variants of our mapping approach. The first variant is clearly more complex than the second but the second variant can similarly benefit from reasoning and consistency checking in the same way as discussed in the previous examples. We will investigate and compare both representation variants and refer to them as CompEnv (for complete environment because it models all junctions and hallways that are known to exist) and VisOnly (for visited only because it only models junctions which have been perceived), respectively. In the CompEnv variant, each edge in a route graph hypothesis connects two nodes. In the VisOnly variant, we allow edges for which the end node is not specified which is depicted by an open line ending. Fig. 5(b) illustrates the differences between the two variants for the exploration run shown in Fig. 5(a) in which the robot traverses a small loop and arrives back at starting junction A. After perceiving J1 to J4 , there are five different hypotheses explaining the observation history in the case of CompEnv. For VisOnly, there


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Figure 5.: Comparison of the CompEnv and VisOnly variants for the exploration run shown in (a): While there exist five hypotheses using CompEnv, there are only two when using VisOnly

exist only two hypotheses: H1 of VisOnly states that observations J1 and J4 correspond to the same physical junction (hence, it corresponds to H1 of CompEnv), while H2 states that J1 and J4 correspond to different junctions (hence, it subsumes H2 –H5 of CompEnv).

3

Rejection Based on Spatial Constraints

In the following, we describe how checking of planarity and, in particular, of consistency of the direction constraints using the absolute cardinal direction calculus (Ligozat, 1998) and the relative OPRA2 calculus (Moratz, 2006) are realized in our mapping approach.

3.1

Planarity Constraint

Each graph hypothesis for which the cyclic order information derived from the cyclic orders of perceived hallways does not describe a planar embedding can be immediately discarded. Checking whether a combinatorial embedding is planar takes O(n) time (Hopcroft & Tarjan, 1974; Lempel, Even, & Cederbaum, 1967). The criterion for deciding whether a general graph with a combinatorial embedding describes a planar embedded graph is that its genus is 0. The genus of an undirected graph G = (V, E) is given by Euler’s formula: genus(G) = (|E| + 2c − |V | − i − f )/2 where c is the number of connected components in the graph, i is the number of nodes of degree 0 (isolated nodes), and f is the number of faces formed by traversing edges in accordance with the cyclic ordering information. We only consider


Figure 6.: Two combinatorially embedded graphs: the embedding of the left graph is planar, while that of the right one is not connected graphs without isolated nodes here and thus the formula becomes: genus(G) = (|E| − |V | − f )/2 + 1 Fig. 6 shows two combinatorial embedded graphs in which each undirected edge is depicted by two directed ones. The faces of the graphs are depicted by the dashed arrows. The left graph has three faces, while the right one with an additional (undirected) edge has only two. As a consequence, according to the formula above the first graph has genus of 0 and the second graph has a genus of 1. Hence, only the combinatorial embedding of the first graph is planar. We employ an incremental approach of planarity checking which is similar to the one described in (Savelli & Kuipers, 2004). Planarity checking is integrated into our search algorithm by representing the route graph hypotheses as bidirected graphs (Mehlhorn et al., 1999) as illustrated in Fig. 6 and by updating the information about faces of the embedding whenever we modify the graph structure. When the genus becomes non-zero, the hypothesis at hand can be discarded as the planarity constraint is violated.

3.2

Qualitative Direction Information

To incorporate direction constraints, we formulate observed directions by using the relations from a qualitative constraint calculus. Fig. 7(a) illustrates the base relations of the absolute cardinal direction calculus which relates two point objects. The calculus distinguishes nine base relations, the eight direction relations n, nw, w, etc. and the eq relation for the case where both points are equal. The relative OPRA2 calculus, which describes the relative orientation of two oriented point objects, is illustrated in Fig. 7(b). For each of the two objects the plane is divided into eight sectors (four linear ones and four planar ones as in the case of the cardinal direction calculus) and each relation states which sector of A contains B and which sector of B contains A. For instance, the relation A ∠17 B means that B lies in sector 1 of A and A lies in sector 7 of B. In addition, there are eight

SPATIAL REASONING FOR TOPOLOGICAL MAP LEARNING 15 n

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(b) OPRA2 relation A ∠17 B stating that B lies in sector 7 of A while A lies in sector 1 of B and OPRA2 relation A ∠7 B stating that A and B coincide and the orientation arrow of B lies in sector 7 of A

Figure 7.: The employed spatial calculi: (a) the cardinal direction calculus and (b) the OPRA2 calculus relations for cases in which A and B coincide (cf. (Moratz, 2006) for a detailed description of OPRA2 ). In our approach absolute direction information is exploited to enforce three conditions: 1. Valid direction orderings: When adding a new edge to a node A, it can only be inserted into the cyclic edge order of A in a way that the edges remain ordered in accordance to the inherent cyclic order of cardinal directions. 2. Valid junction matchings: When associating a new junction observation J with a particular node A, correspondences between the leavings hallways of J and edges of A need to be established in a way that the direction relations of corresponding leaving hallways of J and edges of A are compatible (which in general means they have a non-empty intersection) while at the same time preserving the cyclic edge orderings. 3. Global consistency: There needs to be a way of assigning coordinates to the nodes such that all direction constraints are satisfied (see Sect. 2.2). An example of the first condition is shown in Fig. 8(a): The edge with direction northwest of B can only be inserted into the cyclic edge order of A between the edges with directions south and east. If A would have an additional edge leading southeast, there would be two possibilities, either directly before or after that edge. The second condition is illustrated in Fig. 8(b): A here is a node which has no junction observation associated with it yet. The existing edges of A and their directions are merely the result of previous assumptions made in this particular hypothesis. Hence, while every edge of A needs to have a corresponding leaving hallway in J, not every perceived leaving hallway needs to have a corresponding edge. In addition, based on the last hallway traversal action it is already known


n w

nw

A

e nw s

(a)

B

(b)

nw nw w sw sw

A J

e e

Figure 8.: Exploiting direction information: (a) enforcing valid direction orderings when inserting a new edge at A and (b) enforcing valid junction matchings when associating a new junction observation J with node A

that the edge leading east corresponds to the leaving hallway leading east. Under these conditions, there exist two possibilities of mapping the remaining edges of A to the remaining leaving hallways of J in a way that the direction relations are compatible while preserving the cyclic orderings: The edge leaving southwest has to correspond to the leaving hallway leading southwest, the leaving hallway leading west does not correspond to any edge of A (and hence will result in a new edge being added to A), and the edge leading northwest can either belong to the first or the second leaving hallway leading northwest (the other one will result in another new edge being added to A). For nodes which already have a junction observations associated with, the mapping between the edges and leaving hallways needs to be bijective because no new edges can be added. Hence, there either exists one or no valid junction matching in this case. For a relative calculus like OPRA2 , valid direction matching cannot be employed because there is no inherent cyclic order between the base relations. However, the valid junction matching condition and global consistency condition are still applicable. Enforcing the first two conditions when constructing new hypotheses is straightforward. For the global consistency check, we first extract a so-called constraint network from the given route graph hypothesis which explicitly represents the directional constraints. We then need to determine whether this constraint network is consistent meaning that it is possible to assign objects from the domain of the calculus (e.g., points in the plane) to the variables which for instance stand for the junctions in the hypothesis. Unfortunately, the domain in this kind of constraint satisfaction problem is infinite. However, research on qualitative spatial reasoning has produced special techniques to deal with this kind of problem. One of them is the algebraic closure (or path-consistency) algorithm (Mackworth, 1977; Montanari, 1974). The algebraic closure algorithm is a standard technique for deciding satisfiability of a constraint network based on operations defined on the relations

SPATIAL REASONING FOR TOPOLOGICAL MAP LEARNING 17 A

A {s}

{ sw } B

C

B

{w}

C

Figure 9.: Cardinal direction constraint network for a route graph hypothesis

of the calculus at hand. The most important operation is the composition operation which yields the relation holding between objects A and C when the relations holding between A and B and between B and C are given. The algebraic closure algorithm runs in O(n3 ) time where n is the number of variables in the constraint network. We employ our qualitative spatial reasoning toolbox SparQ (Wallgrün, Frommberger, Wolter, Dylla, & Freksa, 2007) to perform the consistency check using the algebraic closure algorithm. SparQ provides an efficient implementation of this algorithm and supports a wide range of different spatial calculi. In Fig. 9 we see an exemplary constraint network for the cardinal direction calculus. The extracted constraint network contains one variable for each node in the route graph hypothesis and the constraints holding between them are directly derived from the direction relations in the junction obervations associated with each node. For all pairs of variables not connected by an arrow, the constraint here is not simply the universal relation U but actually U without the equal relation eq because different junctions are not allowed to coincide. For OPRA2 , which describes the relative orientation of two oriented points, the constraint network is constructed as follows (see Fig. 10): One oriented point variable is introduced for each pair of node and incident edge (meaning one for every leaving hallway). Hence, we end up with 2 × n variables where n is the number of edges in the hypothesis. The observed relative directions of leaving hallways directly yield the constraints holding between the variables belonging to the same junction. In addition, we need to state that the oriented points corresponding to the same hallway (e.g., AB and BA) face each other (relation ∠00 ). All other constraints (again omitted in the figure) are set to U minus those relations in which the respective oriented point objects would coincide. Employing both mentioned calculi allows us to compare the effects of absolute and relative direction information. However, both calculi have their individual shortcomings. The cardinal direction calculus, on the one hand, does not allow for expressing the cyclic order information about the leaving edges in the route graph. As a result, it can happen that a constraint network deemed consistent by the consistency check only has solutions for which the cyclic order information is not preserved. In addition, cardinal direction information is less accessible

¨ 18 JAN OLIVER WALLGRUN A AB

{ 1} AB

AC

{ BA B

CA

BA BC

CB

C

AC {

0} 0

{ 7} BC

{ 2} {

0} 0

0 0}

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Figure 10.: OPRA2 constraint network for a route graph hypothesis because special equipment such as a compass is needed. On the positive side, checking consistency can be done efficiently because a large tractable subset exists for which the algebraic closure algorithm decides consistency (Ligozat, 1998). In particular, this subset contains all relation that can occur in our application. OPRA2 , on the other hand, can express the cyclic order information. However, for OPRA2 algebraic closure does not decide consistency even for atomic constraint networks (Frommberger, Lee, Wallgrün, & Dylla, 2007). Hence, some inconsistent hypotheses may not be discovered by the global consistency check. On the positive side, the relative OPRA2 calculus is applicable for agents with rather limited sensor abilities because only rough angles estimates between simultaneously perceived hallways are required. In Table 1 we summarize the most important properties of both employed calculi. Overall, we have chosen these two calculi because to our knowledge no direction or orientation calculus currently exists which at the same time has good computational properties, is expressive enough so that it will rule out many hypotheses, can express the cyclic edge ordering, and defines relations which are easily and reliably identifiable by an autonomous agent. In addition, the calculi offer a similar level of granularity which is advantageous for comparing the two.

reference system a-closure decides consistency can expressiv cyclic order accessibility expressivity network size

cardinal directions

OPRA2

absolute yes1 no needs compass good2 |nodes|

relative no yes no tools required good2 2 × |edges|

Table 1: Comparison of the cardinal direction calculus and the OPRA2 calculus 1 The 2 see

maximal tractable subset contains all relations that can occur in the context of this work. experimental results in Sect. 5


4

Application to Voronoi Graph Representations

The previous two sections described our basic minimal model finding topological mapping framework. We will report on several experiments performed to evaluate this theoretical framework in Sects. 5.1.1–5.1.4. To apply and evaluate this framework in practice, it needs to be combined with a concrete topological representation approach which achieves the actual abstraction from the real environment to the topological network. What we need is a procedure that yields information about the nodes and edges of the topological map based on local information about the robot’s immediate surroundings extracted from sensor information. Depending on the chosen approach, this information could simply be a single node (corresponding to a junction observation in our framework) or a single edge (corresponding to our hallways), or it could be a small local subgraph of the complete topological map comprising multiple nodes and edges. In both cases, the local information collected over time needs to be put together to derive the global map and for that we want to use our minimal model finding approach. For a successful combination, it might become necessary to extend or adapt the basic mapping approach to accommodate the specific properties of the particular abstraction approach. In this section, we describe the combination of our framework with a concrete topological representation approach developed for indoor environments perceived via range sensors described in (Wallgrün, 2005), together with the extensions and modifications necessary to facilitate this combination. The representation approach is based on the idea of employing the generalized Voronoi diagram (GVD) to derive a route graph representation from sensor data (see for instance (Choset et al., 2000)). As shown in Fig. 11(a), the GVD is a retraction of free space to a network of one-dimensional curve segments (the Voronoi curves) which meet at so-called meet points. The GVD can be abstracted into an undirected graph called the generalized Voronoi graph (GVG) as depicted in Fig. 11(b). To increase its suitability as a spatial representation, the GVG is annotated with additional information, e.g. a combinatorial embedding into the plane, local node descriptions, and relative geometric information. The resulting combined mapping system is sketched in Fig. 12. Using the Voronoi-based approach from (Wallgrün, 2005), the local topological information is provided in the form of local GVGs extracted from small metric grid maps of the robot’s immediate surroundings. The local GVGs have to be put together correctly. We use a technique also described in (Wallgrün, 2005) to determine those nodes that are most relevant for navigation and, hence, correspond best to real junctions. This in principle yields a simplified version of the GVG. Information about observed nodes and traversed edges in this simplified GVG forms the history information that is passed on to our multi-hypothesis mapping module which updates the search tree and computes a new current hypothesis.


(a)

(b)

Figure 11.: (a) The generalized Voronoi diagram (fine lines) derived from a polygonal 2D environment, (b) the corresponding generalized Voronoi graph Hypotheses tree Metric mapper

Matching & history extraction

GVG extraction

local grid map

local GVG

Multi-hypothesis topological mapper

node observations + edge traversals

currently best hypothesis

Figure 12.: The employed topological mapping system based on local Voronoi graphs used for the experiment described in Sect. 5.2

The information provided to our minimal model finding algorithm is this combined mapping system deviates in several aspects from the theoretical setting studied in Sects. 2 and 3. To accommodate these changes we made the following adaptations: 1. Multiple connections between two nodes are allowed. 2. Observed local GVGs can contain multiple relevant nodes and edges which are translated into history information without actually traversing the edges. 3. In practice, it may not be possible to reliably determine the exact direction relations. Therefore, we utilize disjunctions of base relations when the perceived direction is a linear relation or lies close to the boundary of a relation sector (e.g., {ne, n, nw} for observed relation {n}). 4. Voronoi curves are typically not straight line connections. Hence, we only employ direction constraints in the global consistency check if the direction can be assessed reliably which is the case when both connected Voronoi


(a)

(b)

(c)

Figure 13.: Random graph environments used for the experiments: (a) irregular graphs, (b) grids, (c) 3-regular pseudo-GVGs nodes are perceived simultaneously. Otherwise, we only use information about the direction in which an edge leaves a node for matching junction observations. The step from straight hallways in our theoretical framework to Voronoi edges which can be curved obviously reduces the effectivity of the global spatial consistency checking. However, this is partially compensated by the fact that we will often perceive neighbored relevant nodes which allows us to still determine the direction between them. The exact effects of the modifications to the basic theoretical mapping framework described here will be experimentally evaluated in Sect. 5.1.5. In addition, the results of applying the overall Voronoi-based mapping system on the data set from a real world exploration run can be found in Sect. 5.2.

5

Experimental Evaluation

To evaluate our theoretical minimal model finding framework developed in Sect. 2 and Sect. 3, the application of qualitative spatial reasoning, and the effects of the two different representation variants (CompEnv and VisOnly) and the two different forms of direction information (absolute and relative), we performed several simulated exploration experiments in randomly generated graph environments of varying size using random walks through the graphs (see Sects. 5.1.1–5.1.4). The graph environments were of the three types shown in Fig. 13: irregular graphs, grids, and 3-regular graphs similar to GVGs. Using these different kinds allowed us to base the evaluation on a good mixture of environments with different properties, especially regarding the degree of perceptual aliasing in the environment. The difficulty of a particular problem instance depends on several factors, in particular the size of that part of the graph environment which is covered by the random walk of the simulated agent and the length of the walk which determines how often each edge is traversed on average. For the diagrams depicting the results

¨ 22 JAN OLIVER WALLGRUN of the experiments in the following, we have chosen to group trials based on the number of nodes in the correct hypothesis (referred to as the size s of the problem instance) whereas the length of the walk was systematically varied between 0.5×s and 2 × s. In addition to the random graph experiments, we evaluated the adaptations discussed in Sect. 4 (see Sect. 5.1.5), in particular the deviation from the straight hallway assumption, and the combination of our approach with the mapping system based on Voronoi graphs using a data set from a real-world exploration run (see Sect. 5.2).

5.1

Simulation Experiments

In the simulation experiments, we investigated several aspects of our approach: • the effects on the solution quality • the effects on the size of the search space • the differences between relative and absolute direction information • the overall computational costs • the effects of the modifications required for Voronoi graphs (see Sect. 4) The main results of the individual experiments are summarized below. 5.1.1

Solution Quality

We first investigated how much the planarity constraint and qualitative direction information help in order to improve the solution quality by ruling out incorrect hypotheses and, as a result, increase the frequency in which the correct solution is found by the minimal model approach. To measure the quality of a solution, we use a simple error measure: We count how often either two junction observations that correspond to different junctions have been mapped to the same node or two observations that correspond to the same junction have not been unified. To ensure that even searching without pruning is possible in reasonable time, we used rather small problem instances varying the size between 4 and 16 nodes. Table 2 shows the overall results of 15600 trials performed for each representation variant of the minimal model algorithm, CompEnv and VisOnly, and for each of the following settings: (1) only structural constraint, (2) structural constraint and planarity constraint, (3) structural constraint and cardinal direction constraints, (4) structural constraint, planarity constraint, and cardinal directions. In addition, Fig. 14 illustrates how the average error distances grow with increasing size of the correct model throughout the experiment.


25

CompEnv, structural CompEnv, structural+planarity CompEnv, structural+cardinal directions CompEnv, structural+planarity+cardinal directions

20

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error distance

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Correct model found

Average error distance

CompEnv

structural only structural, planarity structural, card. dir. structural, planarity, card. dir.

4.77% 5.97% 50.62% 50.92%

9.86 7.27 1.41 1.18

VisOnly


59.00% 64.77% 97.92% 98.15%

5.63 4.07 0.20 0.17

Table 2: Experimental results regarding the solution quality

As the average error distances show, the planarity constraint and in particular the direction constraints significantly improve the solution quality. For the CompEnv variant, the planarity constraint achieves a 26.27% reduction of error distance, while direction information decreases the error distance by 85.70%. Combining both planarity and direction constraints only gives slightly better results than without applying the planarity constraint. The application of the constraints is highly beneficial but in most cases is not sufficient to resolve all ambiguities. For the VisOnly case in which unvisited junctions are not included in the model, the improvements are even more drastic. When using the cardinal direction information, the correct model is found in 98.15% of all trials and the average error distance becomes extremely low with only 0.20, or even only 0.17 when combined with the planarity constraint.

¨ 24 JAN OLIVER WALLGRUN The experiment shows that the planarity constraint and in particular the cardinal direction constraints are able to resolve most of the model ambiguities remaining on the structural level leading to a largely increased solution quality. 5.1.2

Pruning Efficiency

To investigate the effects of the individual settings on the size of the hypothesis space that has to be searched, we performed another set of random graph experiments. To examine this aspect individually, we always performed the search until all hypotheses up to the same size of the correct model have been considered, instead of stopping at the first and hence smallest consistent hypothesis. Otherwise, a setting with bad solution quality in general would look better than it actually is because it only searches to a lesser depth in the search tree. We recorded (1) the number of expansion steps in which successors of a hypothesis are generated, (2) the average branching factor in the search tree, and (3) the maximal queue size occurring during the search. The number of expansion steps and the average branching factor give a good indication of the computational costs involved, while the maximal queue size tells us how many hypotheses were tracked simultaneously during the search and, hence, reflects the space-consumption. The result of this experiment are summarized in Table 3. Fig. 15 shows how the number of expansions grows with increasing size of the environment (logarithmic scale is used for the y-axes). Setting

Expansions Branch. factor

Max. queue size

CompEnv


2407.09 284.97 39.84 21.85

4.49 2.38 2.48 1.64

833.96 86.17 13.58 6.10

VisOnly


790.61 254.25 20.72 20.11

3.19 2.00 1.18 1.15

160.88 47.87 2.95 2.76

Table 3: Results regarding the pruning efficiency We clearly see that the CompEnv variant of the minimal model finding problem is much more complex than the VisOnly variant. The planarity constraint leads to an 88.16% decrease in expansion steps for CompEnv and 67.84% for VisOnly. The average branching factor has been decreased by 46.99% to 2.38 (CompEnv) and by 37.30% to 2.00 (VisOnly). For the cardinal direction constraints, we see a very high reduction of expansion steps of 98.35% for CompEnv and 97.38% for

SPATIAL REASONING FOR TOPOLOGICAL MAP LEARNING 25 Expanded nodes over size for CompEnv

10000

CompEnv, structural CompEnv, structural+planarity CompEnv, structural+cardinal directions CompEnv, structural+planarity+cardinal directions

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Figure 15.: Comparison of expansion steps depending on the size of the correct model for CompEnv and VisOnly (different scales are used for the x-axes and y-axes are logarithmically scaled)

VisOnly. By combining both, an extreme reduction in expansion steps of 99.99% was achieved for CompEnv which corresponds to an average branching factor of 1.64. For VisOnly the cardinal direction constraints yields a 99.97% reduction and an average branching factor of 1.18. We conclude that the planarity assumption and the coarse direction information given by the qualitative cardinal relations lead to a much increased efficiency of the minimal model finding approach which would otherwise only be feasible for very small problem instances. 5.1.3

Absolute vs. Relative Direction Information

One of the goals of our analysis was to compare the effects of employing absolute direction information (e.g., relations from the cardinal direction calculus) and relative direction information (e.g., OPRA2 relations). Therefore, we repeated the experiments for determining solution quality and pruning efficiency for both calculi using all constraints. Besides the previously considered parameters, we distinguished the exact reasons for rejecting a hypothesis. The reasons are (1) direction ordering violation, (2) junction matching violation, and (3) global consistency violation as distinguished in Section 3.2. As also discussed there, direction ordering only plays a role for absolute direction information. In addition, it can only occur at unvisited junctions and, hence, when using the CompEnv variant. Fig. 16 shows the diagrams for error distance and expansions steps for both CompEnv and VisOnly. With regard to solution quality, the change from absolute to relative direction information increased the average error distance from 1.64 to 1.82 for CompEnv and from 0.44 to 0.68 for VisOnly. The average number of expansion steps increased from 49.12 to 82.60 (branching factor from 1.33 to

20

¨ 26 JAN OLIVER WALLGRUN Error distance over size for CompEnv

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1.42) for CompEnv and from 12.79 to 13.93 (branching factor from 1.21 to 1.24) for VisOnly. The observed decrease in performance is not surprising as relative direction information in general allows for more perceptual aliasing, while absolute direction counteracts the accumulation of uncertainty. Taking this into account, the decrease in performance seems to be rather modest, especially for the VisOnly variant, and still much lower than for the less constrained settings investigated in the previous experiments. This means that our minimal model finding framework facilitates successful map learning even for an unaided agent without tools that provide absolute spatial information. When we look at the reasons for rejection that are summarized in Table 4 and Table 5, we see that direction ordering violation does not play an important role at all. Junction matching violations and global consistency violations show about the same rejection ratios for absolute and relative direction information in the case of

SPATIAL REASONING FOR TOPOLOGICAL MAP LEARNING 27 Setting

Violation

% of rejected hypotheses

cardinal direction calculus

direction ordering junction matching global consistency

0.59% 23.84% 75.57%

OPRA2 relations


— 23.09% 76.91%

Table 4: Reasons for rejection involving the direction information (CompEnv)

Setting

Violation

% of rejected hypotheses

cardinal direction calculus


— 78.73% 21.27%

OPRA2 relations


— 55.59% 44.41%

Table 5: Reasons for rejection involving the direction information (VisOnly)

the CompEnv variant. Global consistency violation occurs much more often than rejection caused by junction matching violations. For VisOnly, the picture changes significantly. While for relative direction information global consistency violations still are the reason for about 44.41% of all rejections, only 21.27% are caused by global inconsistencies in the case of absolute direction information. The general increase of junction matching violations clearly results from the fact that for VisOnly complete information about all junctions is available. This allows rejection of many hypotheses early before global consistency is even tested. The difference between absolute and relative direction information in the case of VisOnly shows that absolute direction information reduces perceptual aliasing to a much higher degree and, hence, increases the predominance of rejections based on junction matching violations. Overall, as expected relative direction information is inferior to absolute direction information in terms of solution quality and pruning efficiency. The main advantage of relative information is that it often can be obtained more easily and more reliably.


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5.1.4

Overall Computational Costs

When investigating the pruning efficiency, we restricted ourselves to small problem instances which allowed to apply the model finding approach even without planarity and direction constraints. In addition, we focused on the effects of the different settings on the search space. One result was that the CompEnv variant is significantly more complex than the VisOnly one. When performing the experiment for larger problem instances using the complete minimal model finding approach with full planarity and directional constraints, we made two observations: First, as shown by the diagrams for the node expansions in Fig. 17, it was confirmed that for the CompEnv variant the approach does not scale very well to larger environments whereas for VisOnly the situation looks much better. The CompEnv variant seems currently limited to scenarios with a rather small number of junctions in which the ability to predict the structure of unvisited parts is worth the increased computational costs (for instance for a hallway system in a building). Second, we noticed that for the relative OPRA2 calculus the computational costs quickly become excessive, making this approach infeasible for large environments for both variants. By comparing the overall computation times with the times spent on the consistency check in SparQ in Fig. 18, we see that the sharp rise can largely be attributed to the consistency checking. In some cases it makes up 90% of the overall computation time. There seem to be two issues involved that contribute to this explosion in computational costs: The first one is the large number of base relations in the OPRA2 calculus which makes it impossible to store the complete composition table of general relations and has the potential of slowing down the algebraic closure algorithm significantly. The second one is the fact that, as mentioned, the number of variables in the constraint networks


7000

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grows faster for relative direction information (2× the number of edges) which then enters cubicly into the time complexity of the algebraic closure algorithm. Overall, it seems that when employing this particular set of spatial constraints, at present the VisOnly variant in combination with absolute direction information is the only one that scales sufficiently well to large environments. For relative direction information, we currently lack adequate reasoning formalisms that are at the same time expressive and efficient enough to deal with this kind of task. Developing these kind of formalisms poses an important challenge for future research. 5.1.5

Adaptions for Voronoi Graphs

To evaluate the effects of the modifications discussed in Sect. 4 (deviation from straight hallway assumption, disjunctions of base relations, etc.) for applying the theoretical topological mapping framework to Voronoi graph representations based on real sensor data, we employed complete GVGs produced from real environmental data. We then generated random paths through the GVGs and simulated the sensor limitations by assuming that a neighbor of the current Voronoi node is simultaneously perceived if both are less than 2.5m apart. If this is not the case, no direction relation is derived for this connection. Disjunctions of relations were used when the perceived direction was closer than 5◦ to the sector boundary between two relations. In the experiment, we restricted ourselves to the VisOnly variant and compared the adapted version with the original version without adaptations for both, cardinal direction constraints and OPRA2 constraints. The results of the experiment are summarized in Fig. 19. It shows that the required adaptations only lead to

¨ 30 JAN OLIVER WALLGRUN Error distance over size for card. directions

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a minor degradation of performance with regard to solution quality as given by the average error distance and with regard to pruning efficiency given by the average number of node expansions. Overall, the average error distance increased from 0.79 to 1.48 for the cardinal direction calculus and from 2.11 to 3.21 for OPRA2 . The average number of node expansions increased from 54.87 to 59.25 and from 110.74 to 142.29, respectively. These results allow the conclusion that the effects of perceiving entire connections sufficiently compensate for the overall weaker direction information. Another factor certainly is that the VisOnly variant relies to a lesser degree on rejections caused by global consistency violations as discussed previously.

5.2

Real-World Experiment

In the final experiment, we tested the combination of the minimal model finding approach with the Voronoi graph representation described in Sect. 4 using the


inner connection

outer loop

(a)

(b)

Figure 20.: (a) Environment of the real-world experiment, (b) computed route graph

complete mapping system shown in Fig. 12. We used an existing data set freely available on the web3 . The environment and the trajectory of the robot during the experiment are shown in Fig. 20(a). In this data set, the robot first drove three times around the main outer loop of the environment and then another time with also entering the individual rooms. Finally, it traversed the inner connection. Due to the results from the previous experiments we only used the VisOnly variant. When employing the cardinal direction information, the local maps were manually aligned as the data set does not contain compass readings. Fig. 20(b) shows the minimal model computed using the cardinal direction calculus which is indeed the correct graph model for this exploration run. For OPRA2 , the resulting model was correct as well except for two wrongly merged nodes in a room that was entered via two different doors. However, while the computation took 16 seconds using cardinal directions, it took over 10 hours for OPRA2 because of the issues already described in Sect. 5.1.4. Fig. 21 shows how the error distance of the currently best solution and the number of expansion steps develop over the 150 exploration steps for both spatial calculi. The diagram for the error distance shows that the variant using cardinal directions immediately settles for the correct hypothesis after the robot has traversed the main loop of the environment for the first time in step 23, while this takes almost the second traversal of the main loop for OPRA2 . Another increase in error distance occurs later when the robots starts to enter individual rooms. We also see that the number of expansion steps required and the number of tracked alternative hypotheses is significantly higher for OPRA2 . However, the main 3 The data set has been recorded at the Intel Research Lab, Seattle, and is available at (http:// radish.sourceforge.net/), courtesy of D. Hähnel.

¨ 32 JAN OLIVER WALLGRUN Error distance over step number

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reason for the hugely increased computation time again was the time spent on the global consistency checking.

6

Conclusions

We considered the task of learning a topological map of an known environment as the problem of finding a minimal route graph model that explains a sequence of observations and actions. Our solution is a multi-hypothesis tracking approach which can be seen as a search through a tree of possible graph hypotheses. We compared two representation variants, one which focuses on modeling the junctions that have been visited, while the other also models how hallways that have been perceived but never traversed are connected. In both cases, the approach strongly depends on the exploitation of spatial constraints derived from qualitative direction information and an underlying planarity assumption. We considered two forms of direction information, absolute direction given in the form of relations from the cardinal directions calculus and requiring special tools like a compass,

SPATIAL REASONING FOR TOPOLOGICAL MAP LEARNING 33 and relative direction information available even to an unaided agent and given in terms of relations from the OPRA2 calculus. Checking the consistency of the qualitative direction information is a crucial building block of our approach. The experimental evaluation using random graph environments showed that our approach leads to a significantly reduced search space and improved solution quality. However, it also turned out that for the CompEnv representation variant this is still not sufficient to make the approach scale well to larger environments. Comparing absolute and relative information showed that in general relative information is only slightly inferior but has the advantage of being more accessible. We also showed how the theoretic framework can be adapted for concrete topological mapping approaches based on real sensor data by combining it with a mapping system based on Voronoi graphs. In contrast to typical topological mapping approaches only tracking a single hypothesis, our multi-hypothesis approach achieves a much higher degree of robustness as demonstrated by the successful mapping experiment for a large and complex indoor environment. From the perspective of exploiting spatial information and qualitative spatial reasoning, topological map learning has turned out to be an interesting problem domain, well-suited to serve as benchmark problem to evaluate different calculi, different reasoning techniques, and also different inference engines—tasks which are becoming more and more important (Nebel & Wölfl, 2009). The results with regard to spatial consistency checking based on qualitative direction information can be seen as a challenge for future research as none of the currently existing directional calculi ideally fits the demands arising from the problem. The cardinal direction calculus, for instance, while having good computational properties cannot express cyclic ordering information and requires some kind of compass. For relative directional calculi like OPRA2 , the standard constraint reasoning techniques based on algebraic closure are not sufficient to decide consistency. In addition, even the application of the standard algebraic closure algorithm quickly became infeasible for OPRA2 which demonstrates the need for improved reasoning techniques and spatial calculi. Other promising directions for future research ensuing from this work are the investigation of other kinds of spatial constraints and the development of new search approaches which for instance limit the number of simultaneously tracked hypotheses simulating human-like capacity limitations. Moreover, we want to make the step from a passive map learning approach to active decision making by considering wayfinding decision problems arising within our multi-hypothesis approach. For instance, strategies for navigating under uncertainty and for actively resolving map ambiguities (Chown, 1999; Duckham, Kulik, & Worboys, 2003) lend themselves to be adapted and studied within our framework and be used to improve its applicability and performance.


Acknowledgements The author would like to thank Christian Freksa, Benjamin Kuipers, Diedrich Wolter, Frank Dylla, and Lutz Frommberger for valuable feedback and fruitful discussions regarding the work presented here. Several anonymous reviewers and participants of the AAAI Spring Symposium 2009 and the COSIT 2009 conference provided helpful feedback on earlier versions of this material (Wallgrün, 2009b, 2009a). Funding by the Deutsche Forschungsgemeinschaft (DFG) under grants SFB/TR 8 Spatial Cognition and GRK 1498 Semantic Integration of Geospatial Information is gratefully acknowledged.

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