Z Combinatorial Filters: Sensor Beams, Obstacles, and Possible Paths

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1 Z Combinatorial Filters: Sensor Beams, Obstales, and Possible Paths BENJAMIN TOVAR, Northwestern University FRED COHEN, University of Rohester LEONARDO BOBADILLA, University of Illinois JUSTIN CZARNOWSKI, University of Illinois STEVEN M. LAVALLE, University of Illinois A problem in whih a moving body (robot, human, animal, vehile, and so on) travels among obstales and binary detetion beams that onnet between obstales or barriers is introdued. Eah beam an be viewed as a virtual sensor that may have many possible alternative implementations. The task is to determine the possible body paths based only on sensor observations that eah simply report that a beam rossing ourred. This is a basi filtering problem enountered in many settings, under a variety of sensing modalities. Appliations inlude robot exeution monitoring, sensor-based forensis, assisted living, seurity, and environmental monitoring. Filtering methods are presented that reonstrut the set of possible paths at three levels of resolution: 1) the possible sequenes of regions (bounded by beams and obstales) visited, 2) equivalene lasses of homotopi paths, and 3) the possible numbers of times the path winds around obstales. The simplest ase is addressed first, in whih all beams are disjoint (their detetion regions do not overlap), distinguishable (it is always ertain whih beam was rossed), and direted (the diretion of rossing, forward or bakward, is known for eah sensor observation). More omplex ases are then onsidered, allowing for any amount of beams overlapping, indistinguishability, and lak of diretional information. The method was implemented in simulation. An inexpensive, low-energy, easily deployable arhiteture was also reated, whih implements the beam model and validates the methods of the paper with experiments. Categories and Subjet Desriptors: I.2.9 [Robotis]: Sensors General Terms: Algorithms, Design, Experimentation, Theory Additional Key Words and Phrases: Filtering. Sensor fusion. Foundations of sensor networks. Robotis. Inferene. Topology. ACM Referene Format: Tovar, B., Cohen, F., Bobadilla, L., Czarnowski, J., LaValle, S. M., XXXX. Combinatorial Filters: Sensor Beams, Obstales, and Possible Paths. ACM Trans. Sens. Net. X, Y, Artile Z (Marh 1885), 32 pages. DOI = / INTRODUCTION Imagine installing a bunh of heap, infrared eye beams throughout a ompliated warehouse, offie, or shopping enter; see Figure 1. Just like the safety beam on a This work was supported in part by NSF grants (IIS Robotis) and (Cyberphysial Systems), DARPA SToMP grant HR , and MURI/ONR grant N B. Tovar is with the Mehanial Engineering Department, Northwestern University. Evanston, IL USA. b-tovar@northwestern.edu. F. Cohen is with the Department of Mathematis, University of Rohester. Rohester, NY USA. fohen@rohester.edu. L. Bobadilla, J. Czarnowski, and S. M. LaValle are with the Department of Computer Siene, University of Illinois. Urbana, IL USA. {bobadil1,jzarno2,lavalle}@uiu.edu. Permission to make digital or hard opies of part or all of this work for personal or lassroom use is granted without fee provided that opies are not made or distributed for profit or ommerial advantage and that opies show this notie on the first page or initial sreen of a display along with the full itation. Copyrights for omponents of this work owned by others than ACM must be honored. Abstrating with redit is permitted. To opy otherwise, to republish, to post on servers, to redistribute to lists, or to use any omponent of this work in other works requires prior speifi permission and/or a fee. Permissions may be requested from Publiations Dept., ACM, In., 2 Penn Plaza, Suite 701, New York, NY USA, fax +1 (212) , or permissions@am.org ACM /1885/03-ARTZ $10.00 DOI / ACM Transations on Sensor Networks, Vol. X, No. Y, Artile Z, Publiation date: Marh 1885.

2 a f b e d Fig. 1. What an be determined about the path using only the word babdeeefe, whih indiates the sequene of sensor beams rossed? motorized garage door, a single bit of information is provided: Is the beam urrently obstruted? Now suppose that there are one or more moving bodies, whih ould be people, robots, animals, and so on. If the beams are distinguishable and we know the order in whih beams were rossed, what an we infer about the paths taken by the moving bodies? This may be onsidered as a filtering problem, but with minimal, ombinatorial information, in ontrast to popular Kalman filters and partile filters. We introdue the notion of ombinatorial filters, whih are minimalist analogs to ommon Bayesian filters. They instead redue omplexity by employing ombinatorial reasoning, whih lies at the heart of omputational geometry and some parts of omputational topology. The idea is to reason in an exat but disrete way about ontinuous spaes by identifying or representing ritial piees of information. This paper proposes the study of a family of inferene problems that arise from moving bodies rossing sensor beams among obstales. It turns out that the subjet is muh more general than the partiular senario just desribed. In addition to binary detetion beams or regions plaed around an environment, the mathematial model arises in other ontexts. For example, if a robot arries a amera and ertain image features ritially hange, then the event may be equivalent to rossing a virtual beam in the environment (see Setion 3). Our questions are inspired by many problems that soiety urrently faes. There is widespread interest in developing assisted living systems that use sensors to monitor the movements of people in their homes or hospitals. How muh an be aomplished with simple detetion beams, whih are affordable, robust, and respet privay? Alternatively, imagine the field of sensor-based forensis, in whih polie investigators or lawyers would like to orroborate or refute a testimony about how people moved at a rime sene (for related work, see [Yu and LaValle 2010] and referenes therein). A simple verifiation test based on the sequene of beam rossings might establish whether someone is lying. Other problems inlude traking wildlife movement for on- 2

3 Fig. 2. This paper onsiders sensors that are learly too weak to fully reonstrut the path taken by the body, but are nevertheless able to produe useful information regarding the path. Rather than approah the most ommon problem of omplete reonstrution (top of the diagram), we present filtering methods that determine information about the path at three levels of ambiguity. The result of eah level in the diagram ould be derived from the information diretly above it; however, the omplete path is never given. Only the sequene of rossed sensor beams is given. servation purposes, landmark-based navigation with outdoor vehiles, sensor-assisted safe hild are, and seurity. Suppose there there is one moving body and we reeive information that a partiular sequene of sensor beams was rossed. We present a family of methods for reonstruting information about the path taken by the body. See Figure 2, whih relates various forms of information that ould be obtained about the path, ordered by the amount of ambiguity. Considering the example in Figure 1, note that the beams and obstales partition the spae into regions in whih the body an move without being deteted. Setion 4 develops filtering methods that reonstrut the possible sequenes of regions traversed by the body. This is the tightest possible representation of the set of possible body paths that explain the observed data. In Setion 5, we then introdue the homotopy equivalene relation on the set of paths. In that ase, we reonstrut possible paths up to the resolution of homotopy lasses. This is a more oarse, and possibly more ompat, summary of the possible paths than the sequenes of possible regions. In Setion 6, an even more oarse haraterization of possible paths is made, in terms of winding numbers: How many times does the path effetively wrap around eah obstale? Our approah is inspired by works in mathematis, algorithms, robotis, and sensor networks. From mathematis, our reonstrutions based on homotopy and winding numbers inspired by topology, and are motivated by homotopy and homology, respetively. Furthermore, they are losely related to word problems in ombinatorial group theory, in whih it must be determined whether two words or group presentations are equivalent. In the general group-theoreti setting, suh questions go bak to 1910 with Dehn s fundamental problems [Dehn 1987]; deidability and omplexity results are reviewed in [Epstein et al. 1992]. Regarding algorithms, some of the most losely related works are algorithms that deide whether two paths in a puntured plane are homotopi [Cabello et al. 2002; 3

4 Efrat et al. 2006]. These algorithms are based on extending vertial lines from eah of the puntures (whih are interior obstales in our paper). The vertial lines serve two purposes: First, any given path is represented by the sequene of vertial rays that it intersets. Seond, they onnet the different fibers of a overing spae of the puntured plane. In this ontext, two paths are homotopi if and only if they have the same endpoints when they are lifted to the universal overing spae. Our work draws inspiration from this; however, we start with sensor words and must first onvert them into path desriptions. This represents an inverse problem that is onstrained by the geometry and topology of the sensors and obstales. In the ontext of robotis and sensor networks, we draw inspiration primarily from works that fous on minimalism and also on ombinatorial reasoning, whih is prevalent in omputational geometry. This has led, for example, to the notion of a relational sensor [Guibas 2002]. Similarly, works suh as [Kim et al. 2005; Shrivastava et al. 2006b; Singh et al. 2007] propose binary proximity sensors to detet and ount targets. The binary proximity sensors an be onsidered as overlapping beams that go off when a body is in range. Target traking is often aomplished with a partile filter, in whih eah partile is a andidate trajetory of a target. In robotis, the use of partile filters has been very suessful in solving tasks suh as simultaneous loalization and mapping (SLAM) [Castellanos et al. 1999; Choset and Nagatani 2001; Dissanayake et al. 2001; Montemerlo et al. 2002; Parr and Eliazar 2003; Thrun et al. 1998]. Traditionally, the fous of SLAM approahes is the prodution of an environment representation based on metri information. From a sensing perspetive, algorithms suh as the ones used in SLAM, are onerned with the problem of sensor fusion, in whih several sensors are added to inrease the auray of a solution. In ontrast, others have studied the minimal sensing requirements to solve a partiular task[erdmann and Mason 1988; Goldberg 1993; O Kane and LaValle 2007; Tovar et al. 2007b]. This typially involves a haraterization and simplifiation of the information spae assoiated with the task [LaValle 2006;?], whih onsiders the whole histories of ommands given to the atuators and sensing observations. From an information spae perspetive, in [Yu and LaValle 2008] the loation of moving bodies is inferred from ombinatorial hanges in sensing observations. Suh ombinatorial hanges may orrespond to visual events [Durand 1999], whih an be abstrated in our work as sensor beams. An example of this is presented in Setion 3, in whih the ombinatorial hanges orrespond to the rossing of landmarks within the field of view [Tovar et al. 2007a]. The areful onsideration of visual events is the basis for solutions to problems suh as loalization [Dudek et al. 1998; Guibas et al. 1995] and visibility-based pursuit-evasion [Gerkey et al. 2004; Guibas et al. 1999; Kameda et al. 2006; Lee et al. 1999; Suzuki and Yamashita 1992]. This paper is an expanded and updated form of [Tovar et al. 2009]. 2. PROBLEM FORMULATION LetW R 2 bethelosureofasimplyonneted(ontratible)openset.aommonase is W = R 2. Let O be a set of n pairwise-disjoint obstales, whih are eah the losure of asimplyonnetedopenset.letx bethefreespae,whihistheopensubsetofw that has every o O removed. Let B be a set of m beams, eah of whih is an open linear subset of X. It is possible to generalize B to allow nonlinear beams without affeting the results in this paper; however, this will be avoided in the presentation to improve larity. The model is defined to allow the ases of both W bounded or unbounded. If W is bounded,theneverybeamisalinesegmentwithbothendpointsontheboundaryofx. Figure3(a)showsanexampleinwhihW isboundedandtherearefourobstales.note that beams may onnet an obstale boundary to itself, another obstale s boundary, or 4

5 a r r 3 1 f b e d r 2 (a) r 1 a f r 7 r 8 b r 6 e e r 2 a r 5 d b r 3 r 4 (b) Fig. 3. (a)asimpleexample,whihinludes6beamsandthreeregions,r 1,r 2,andr 3,inwhihthebodyan move without being deteted. (b) Beams may be diretional, may interset, and may be indistinguishable. The positive beam diretion is indiated by an arrow plaed along the beam. There are 8 regions. 5

6 the boundary of W; also, a beam may onnet the boundary of W to itself. If W is unbounded, then some beams may be infinite rays that emanate from the boundary of an obstale, and others may even be infinite lines that are ontained in the interior of W. It is possible to generalize B to allow nonlinear beams without affeting the results in this paper; however, this will be avoided in the presentation to improve larity. Regions. The olletion of obstales and beams indues a deomposition of the free spae X into onneted ells. If the beams in B are pairwise disjoint, then eah B B is a 1-ell and the 2-ells are maximal regions bounded by 1-ells and portions of the boundary of X. If beams interset, then the 1-ells are maximal segments between any beam intersetion points or boundary elements of X; the 2-ells follow aordingly. Every 2-ell will be alled a region. The regions are shown for the examples in Figure 3. It is assumed that beams are arranged in general position so that if a pair of beams intersetions, then the intersetion must our at one point. Body path. Suppose that a body moves along a state trajetory or path, x : [0,1] X, in whih [0,1] is imagined as a time interval, but time saling is unimportant to our questions. (Alternatively, [0,t f ] ould be allowed for any t f > 0.) Sensormodel. Ifthebodyrossesabeam,whatexatlyisobserved?Assumethatthe set of possible x is restrited so that: 1) every beam rossing is transverse (the body annot touh a beam without rossing it and the body annot move along a beam) and 2) the body never rosses an intersetion point between two or more beams (if any suh intersetions exist). Let L be a finite set of labels. Suppose eah beam is assigned a unique label by some bijetion α : B L. The sensor model depited in Figure 1 an be obtained by a sensor mapping h : X Y, in whih Y = L {#} is the observation set. If x(t) B for some B B, then h( x(t)) = α(b); otherwise, h( x(t)) = #, whih is a speial symbol to denote no beam. This is referred to as the undireted beam model beause it indiates that the beam was rossed, but we do not know the diretion. To obtain a direted beam model, let D = { 1,1} be a set of diretions, in whih 1 is alled the positive and 1 is alled the negative diretion, respetively. In this ase, the observation set is Y = (L D) {#}, and the sensor mapping yields the orientation of eah beam rossing. Note that in addition to x(t), the domain of the sensor mapping h must inlude open subsets of X so that the side of the beam that the body was shortly before time t an be measured. So far, the beams have been fully distinguishable beause α is a bijetion. It is possible to make L < m (the number of beams) and obtain some indistinguishable beams, in whih ase α : B L is not bijetive. It might seem odd that beams annot be distinguished, but this an our frequently in pratie; see Setion 3. If a olletion B of beams is disjoint, distinguishable, and direted, the ase will be referred to as DDD beams, whih is the most ideal situation. We onsider the most general ases, however, in whih these onditions are not met. Figure 3(b) shows one suh example. Sensor words. What observations are aumulated after x is exeuted? Sine there is no preise timing information in our problems, all# observations an be safely ignored, resulting in a sequene ỹ, alled the sensor word, of the remaining observations (ỹ is a kind of observation history [LaValle 2006]). For the example in Figure 1, L = {a,b,,d,e,f} and the sensor word is babdeeefe. For a direted beam with label λ, the symbol λ is used to denote traversal in the positive diretion, and λ is used to denote traversal in the negative diretion. A sensor word for direted beams may then appear as aba bb ab, for example. 6

7 a b (a) (b) Fig. 4. (a) The left-to-right traversal of the two undireted, nearby, parallel beams yields the sensor word ab. (b) The beams effetively simulate a direted beam, in whih is observed when the original word is ab and when the original word is ba. The triangle along the beam indiates the positive diretion. Note that if two parallel, undireted beams are plaed losely together(see Figure 4), then they an simulate one direted beam. This requires an assumption that the pair must always be rossed transversely, rather than rossing one beam and returning. Inferene and filtering. Let Ỹ be the set of all possible sensor words and let X be the set of all possible paths. Let φ : X Ỹ denote the mapping that produes the sensor word ỹ = φ( x). Suppose that ỹ has been obtained with no additional information. What an be inferred about x? Let φ 1 (ỹ) denote the preimage of ỹ: φ 1 (ỹ) = { x X ỹ = φ( x)}. (1) The main task in this paper is to haraterize the preimage(or olletion of plausible paths) from a given sensor word. Setion 4 will provide a simple and effiient algorithm that provides an exat haraterization of φ 1 (ỹ) in terms of sequenes of regions that may have been traversed. Furthermore, the method works inrementally as a ombinatorial filter by updating its information state [Kuhn 1953; von Neumann and Morgenstern 1944] effiiently as eah new observation ours. This proess is analogous to the popular Kalman filter, whih omputes the next mean and ovariane based on the new sensor observation and the previous mean and ovariane [Kalman 1960; Kumar and Varaiya 1986]. The Kalman filter falls under the general family of Bayesian filters, to whih partile filtering tehniques are usually applied [Thrun et al. 2005]. For a sensor word ỹ k of length k, let κ(ỹ k ) denote an information state, whih ould, for example, be the set of possible urrent regions. A ombinatorial filter effiiently omputes κ(ỹ k+1 ) using only κ(ỹ k ) and y k+1, in whih y k+1 is the last (most reent) letter in ỹ k+1. This implies that ỹ k does not need to be stored in memory; only κ(ỹ k ) is needed. Simple filters of this form will be presented in Setion 4. In addition to simple region-based filters, we provide methods that diretly transform the sensor word into topologial path desriptions. Setion 5 onverts the sensor word into one or more elements of the fundamental group. Eah element orresponds to a lass of homotopially equivalent paths that are possible given the sensor word ỹ. This provides a more oarse haraterization of the preimage φ 1 (ỹ) than the methods of Setion 4. The information is useful in many ontexts, inluding searh-based planning for navigation [Bhattaharya et al. 2011]. Setion 6 produes information about 7

8 (a) (b) () (d) Fig. 5. Some examples simple systems to implement sensor beams:(a) Indutive oils are plaed in roads to detet ar rossings. (b) Passive infrared sensors detet movement within a speifi zone. () Infrared garage beams are plaed under garage doors for safety, but an be used in many other settings. (d) Pressure mats or ables an be plaed along or into the ground. φ 1 (ỹ) that is even more oarse. Given ỹ, a method is presented that alulates the winding number with respet to eah obstale. In other words, it determines the number of times, positive or negative, that the body path wrapped around eah obstale ounterlokwise. 3. MODEL MOTIVATION Before moving on to omputing desriptions of φ 1 (ỹ), this setion motivates the general formulation of Setion 2 to illustrate the wide range of settings to whih it applies. In many settings, sensors may be plaed in the environment to diretly obtain the beam detetion behavior. Some examples are shown in Figure 5. In Setion 7.1, we desribe our own beam detetion system, whih osts around $5 US per beam. Using simple sensors, virtually any beam model from Setion 2 an be implemented. It is possible, however, to obtain the sensor beam model in a more indiret way. For example, imagine that a robot moves in a large field, in whih several landmarks (e.g., radio towers) are visible using an omnidiretional amera. This an be modeled by W = R 2 and O as a set of point obstales. Suppose that the landmarks are fully distinguishable and some simple vision software indiates when a pair of landmarks are ontopofeahother intheimage.inotherwords,therobotandtwolandmarksare ollinear, with one of the two landmarks in the middle. The result is mathematially 8

9 (a) (b) Fig. 6. (a) Imagine how the landmarks (small diss) appear in an omnidiretional amera while traversing the shown trajetory. The arrows show eah pereived virtual beam. (b) Here the virtual beams are based on passing diretly north of a landmark. equivalent to plaing n(n 1) beams as shown in Figure 6(a), in whih rays extend outward along lines passing through eah pair of landmarks. Several interesting variations are possible based on preisely what is deteted in the image.iftheonlyinformationisthato i ando j rossedeahotherintheimage,thenall beams are undireted and the two beams assoiated with o i and o j are indistinguishable. If we know whether o i passes in front of or behind o j, then the two beams beome distinguishable from eah other. If we know whether o i passes to the left or right of o j in the image, then the beams even beome direted. Senarios suh as these motivated the onsideration of beam labels and indistinguishability in Setion 2. For another example, onsider hanging the landmark sensing so that instead of deteting pairwise landmark rossings, the robot simply knows when some landmark is diretly south. This ould be ahieved by using a ompass to align the vehile and noting when a landmark rosses a fixed spot on the image plane or windshield. Figure 6(b) shows virtual beams that are obtained in this way. Direted and undireted beam models are possible, based on whether the sensor indiates the left-right diretion that the landmark moves as it rosses the fixed spot. An important property of this model is that the beams do not interset (assuming the points are not ollinear). Setion 5 utilizes this property to reonstrut the path up to homotopy equivalene. 4. REGION FILTERS In this setion, we present a simple method to keep trak of the possible regions in whih the body might be after obtaining the sensor word. The possible sequenes of regions traversed an also be omputed. Setion 4.1 overs the ase of a single body, and Setion 4.2 provides some extensions to multiple bodies One Body SupposethatnobstalesO andmbeamsb aregivenalongwithlandα(beamlabels). Furthermore, assume that W and O are represented in a way that enables exat, effiient omputations. For example, it is suffiient to assume all sets are polygonal. The beams may interset, may or may not be direted, and some beams may be indistinguishable. Using standard representations of subdivisions (suh as the half-edge data struture) and planar deomposition algorithms, the regions and their onnetivity in- 9

10 (a) (b) Fig. 7. (a) Two interseting beams: a is direted and b is undireted; (b) the orresponding multigraph G; multiple edges are ompressed into a single edge with a list of the labels that ause the transition. Fig. 8. The multigraph G orresponding to Figure 3(b). formation an be easily omputed (see [de Berg et al. 2000] for an overview of suh methods). Let R 0 denote the set of possible regions that initially ontain the body, before any sensor data is observed. Let R k denote the smallest set of possible regions that ontain the body after a sensor word ỹ k, of length k, has been obtained. The task is to design a ombinatorial region filter, whih is a funtion φ of the form R k+1 = φ(r k,y k+1 ) (2) that an be effiiently omputed. In the first appliation of φ, R 1 is omputed from R 0 and y 1. Eah subsequent R k is similarly omputed. Before implementing the filter φ, a speial graph is ontruted in a preproessing phase. Let G be a direted multigraph that possibly ontains self-loops. Eah vertex of G is a region, and a direted edge is made from region r 1 to region r 2 if the orresponding regions are adjaent (an interval along a beam lies on the boundary of r 1 and 10

11 r 2 ). The edge is labeled aording to the sensor observation that is reeived when the body rosses the shared beam from r 1 to r 2. A self-loop in G is made if it is possible to ross a beam and remain in the same region. Figure 7 provides a simple example. Figure 8 shows G for the example in Figure 3(b). One again, G an be omputed from well-known deomposition algorithms [de Berg et al. 2000]; even thousands of beams would present little omputational hallenge. The region filter (2) is implemented over G in a way similar to the simulated operation of a nondeterministi finite automaton. Let V denote the set of all verties of G, whih orresponds bijetively tothe setof allregions. Let m k : V {0,1} be afuntion that is omputed as eah stage k. Let m k (v) = 1 mean that the body might be in the region that orresponds to v, and let m k (v) = 0 mean that the body is ertainly not in that region. Initially, m 0 is omputed by assigning m 0 (v) = 1 to eah vertex that orresponds to a region in R 0. For all others, m 0 (v) = 0. For the inremental operation of the filter, assume that R k has been already alulated from ỹ k. This means that m k (v) = 1 for eah orresponding region in R k and no others. Suppose that y k+1 is observed, whih extends the sensor word by one observation.initially,assignm k+1 (v) = 0forallv V.Foreahvertexv V forwhihm k = 1, assign m k+1 (v) = 1 for every outgoing edge that has y k+1 as its label. After this is ompleted, the set of all regions for whih m k+1 (v) = 1 represents R k+1, whih is the desired result (2). Note that R k+1 may be larger than R k beause multiple outgoing edges may math y k+1 at eah vertex. Also, this approah works for the ase of partially distinguishable beams beause the math is based on the observed beam label y k+1, rather than the partiular beam. Now suppose that after omputing R k, we would like to know the possible sequenes ofregionstraversedbythebody.construta(k+1)-partitegraphh inwhihtheithset of verties in the partition is denoted by V i (there are no edges between verties in V i ). EahV i orrespondsbijetivelytotheregionsinr i.theedgesinh areformeddiretly from the omputations of the region filter. For every r R i+1, if for r R i, there is an edge in G from r to r and it is labeled with y i, then an edge from the orresponding vertex is made in H. This edge is formed from the vertex in V i orresponding to r to the vertex in V i+1 orresponding to r. One H has been omputed, a ompat enoding of the set of all possible region sequenes given ỹ k is obtained: It is the set of all paths in H from any vertex in V 0 to any vertex in V k. Note that from some v V 0 a path might not even exist to any v V k beause it was learned from later observations that the body must not have initially been in the region orresponding to v. In other words, observations gained at later stages an refine our knowledge about what might have ourred several stages earlier. Note that the sequenes of possible regions provides a tight haraterization of the set of possible paths given ỹ k : It is the set of all paths that traverse any one of the omputed possible region sequenes. Any paths that traverse a region sequene not listed as a possible sequene must not be inluded beause it would have yielded a different sensor word. TheregionfilterrunsintimeO( V + E )foreahupdatefromk tok+1,inwhih V and E are the numbers of verties and edges in G, respetively. The bipartite graph H is extended with no additional overhead. Note that the number of omputations grows with the number of verties for whih m k (v) = 1, whih reflets the amount of unertainty about the urrent region. In some ases the number of marked verties annot inrease, as in the ase of DDD beams. To illustrate the region filter, we present simulations for the onrete senario of beams oming from rossings of landmarks. We omputed two ases. In the first one, all beams are distinguishable and direted (see Figure 9). In the seond, the only information available at a rossing is the orresponding pair of landmarks. Therefore, 11

12 the beams are not direted, and eah beam label an be reported by exatly two beams (see Figure 10). In these examples, a beam is identified with the pair of landmarks that produe it Multiple Bodies The formulation in Setion 2 an be naturally extended by allowing more than one body to move in X. In this ase, suppose that the sensor beams annot distinguish between bodies. They simply indiate the beam label whenever rossed. Furthermore, assume that bodies never ross beams simultaneously. The task is to reonstrut as muh information as possible about what path they might have taken. Figure 11(a) shows a simple example of this, in whih there is one obstale, two bodies, and three undireted beams. This yields a set of three regions: R = {r 1,r 2,r 3 }. Using(r,r )todenotetheregionthatontainsthefirstandseondbodies,respetively, there are nine ombinations of region assignments: (r 1,r 1 ), (r 1,r 2 ), (r 1,r 3 ), (r 2,r 1 ), (r 2,r 2 ), (r 2,r 3 ), (r 3,r 1 ), (r 3,r 2 ), and (r 3,r 3 ). Consider I = pow(r R), whih is the set of all subsets of possible region assignments. A region filter an be made over this set in the form ι k+1 = φ(ι k,y k+1 ), (3) in whih ι k represents the set of all possible region assignments after ỹ k has been observed. Initially, some ι 0 I is given. After eah observation is reeived, ι k+1 I is omputed from I k and y k+1. The method of Setion 4.1 an be easily extended to ompute (3). Let G 2 be the multigraph formed by taking the Cartesian produt G G in the sense that the verties orrespond to all ordered pairs of regions. Eah edge in G 2 is formed if a transition from one ordered pair to another is possible after a single observation y k+1. One G 2 is formed, the method alulating m k from Setion 4 an be easily extended. If there are n bodies in the environment, then the method an be extended by forming an n-fold Cartesian produt of R to obtain I and a n-fold produt of G to obtain G n. Although oneptually simple, the number of region assignments grows exponentially in the number of bodies, and the power set needed to obtain I is exponentially larger. Therefore, an important diretion of future researh on region filters is to identify ways to redue omplexity from the task desription. For example, onsider the following question for the example in Figure 11(a): Are the two bodies together in a region, or are they separated by a beam? Consider designing the simplest region filter that orretly answers this question. Figure 11(b) shows a surprisingly simple ombinatorial filter that answers the question for any sensor word and uses only four information states. The T information state means they are together in some room. Eah D x information state means they are in different rooms, with beam x separating them. The set of all information states is I = {T,D a,d b,d }, and a filter of the form (3) in niely obtained. With only two bits of memory, arbitrarily long sensor words an be digested to produe the answer to the question, in onstant time during eah iteration. It is required, however, to know the initial information state in I. Under what other onditions an suh dramati redutions be made? The example in Figure 11(b) an be extended to more regions and bodies by asking the question of whethereahregionontainsanoddorevennumberofbodies.inthisase,afilteran be made that reords only one bit per region (the parity). It is hallenging, however, to find more useful settings in whih simple region filters exist for multiple bodies. One important point to note about the problem in Figure 11 is that it did not distinguish between the bodies. If there are n bodies and the task does not require distinguishing between then, the number of region assignments is redued from m n to 12

13 r 6 r 6 r 7 r 1 r 2 r 7 r 1 r 2 11 d r 3 r 10r a b r 4 11 d r 3 r 10r a b r 4 r 9 r 8 r 5 r 9 r 8 r 5 R 0 = {r 1,r 2,...,r 10,r 11 } R 1 = {r 3,r 9 }, rossing d r 6 r 6 r 7 r 1 r 2 r 7 r 1 r 2 11 d r 3 r 10r a b r 4 11 d r 3 r 10r a b r 4 r 9 r 8 r 5 r 9 r 8 r 5 R 2 = {r 8 }, rossing ad R 3 = {r 3 }, rossing a r 6 r 6 r 7 r 1 r 2 r 7 r 1 r 2 11 d r 3 r 10r a b r 4 11 d r 3 r 10r a b r 4 r 9 r 8 r 5 r 9 r 8 r 5 R 4 = {r 1 }, rossing db R 5 = {r 2 } rossing da Fig. 9. Region filter simulation. Crossings of landmarks for direted and distinguishable beams. 13

14 r 6 r 6 r 7 r 1 r 2 r 7 r 1 r 2 11 d r 3 r 10r a b r 4 11 d r 3 r 10r a b r 4 r 9 r 8 r 5 r 9 r 8 r 5 R 0 = {r 1,r 2,...,r 10,r 11 } R 1 = {r 2,r 3,r 9,r 10,r 11 }, rossing d r 6 r 6 r 7 r 1 r 2 r 7 r 1 r 2 11 d r 3 r 10r a b r 4 11 d r 3 r 10r a b r 4 r 9 r 8 r 5 r 9 r 8 r 5 R 2 = {r 1,r 8 }, rossing ad R 3 = {r 3,r 6 }, rossing a r 6 r 6 r 7 r 1 r 2 r 7 r 1 r 2 11 d r 3 r 10r a b r 4 11 d r 3 r 10r a b r 4 r 9 r 8 r 5 r 9 r 8 r 5 R 4 = {r 1,r 5,r 7 }, rossing bd R 5 = {r 2 }, rossing ad Fig. 10. Region filter simulation. Crossings of landmarks for undireted beams and partially distinguishable beams. Eah beam label an be reported by exatly two beams. 14

15 D a a b b a D T a b D b (a) (b) Fig. 11. (a) A three-region problem with two bodies; (b) a tiny ombinatorial filter that determines whether the bodies are together in a room. a b a b a b Fig. 12. Keeping trak of only the number of bodies per region, rather than the partiular region assignments, dramatially redues the omplexity. In this simple example, there are only two bodies and three regions. Rather than obtain 2 3 = 9 possible region assignments, only ( 3 2 ) = 6 are needed if we only are about the number per region. ), in whih there are m regions. Figure 12 shows a simple example; only the number of bodies per region matters. It is the lassial balls and urns problem from ombinatoris. Only the number of bodies per region needs to be maintained in the filter. ( m+n 1 n 5. RECONSTRUCTION UP TO HOMOTOPY In this setion, the task is to use the sensor word ỹ to reonstrut a desription of possible paths x X up an equivalene lass of homotopi paths. Assume that all body paths start and stop at some fixed basepoint x 0 X whih lies in the interior of some region; this assumption is lifted at the end of the setion. Two paths, x and x are 15

16 alled homotopi if there exists a ontinuous funtion h : [0,1] [0,1] X for whih the following four onditions are met: (1) (Start with first path) h(s,0) = x(s) for all s [0,1]. (2) (End with seond path) h(s,1) = x (s) for all s [0,1]. (3) (Hold starting point fixed) h(0,t) = h(0,0) for all t [0,1]. (4) (Hold ending point fixed) h(1,t) = h(1,0) for all t [0,1]. The parameter t an be interpreted as a knob that is turned to gradually deform the path from x into x without jumping over obstales. This definition indues an equivalene relation on the set X of all paths. Note that paths that follow different sequenes of regions ould possibly be equivalent homotopially. It is natural to ask: Under what onditions an be it guaranteed that if two paths traverse the same region sequene, then they are homotopi? We would like to argue that a desription of the path up to homotopy equivalene is oarser than a desription up to the region sequene. This will be answered later in the setion. Desribing the equivalenes lasses of paths up to homotopy equivalene is part of basi algebrai topology. An algebrai group an be formed by defining the following binary operation on X. Let x 1 and x 2 be two loop paths with the same basepoint x 0. Their produt x = x 1 x 2 is defined as x(t) = { x1 (2t) if t [0,1/2) x 2 (2t 1) if t [1/2,1]. (4) This results in a ontinuous loop path beause x 1 terminates at x 0, and x 2 begins at x 0. In a sense, the two paths are onatenated end-to-end. The operation forms a group on the spae of all paths. Using the homotopy equivalene relation, we do not want to distinguish between paths that equivalent. Therefore, a quotient group is formed by starting with the operation and applying to the homotopy equivalent lasses. The result is alled the fundamental group, denoted by π 1 (X), on the spae of all equivalene lasses of paths [Hoking and Young 1988]. The struture of this group reveals muh about the topologial struture of the spae X. In our ase, the topology of X is somewhat simple in that it depends only on the number n of holes, whih from Setion 2 is the number of obstales in O. For this ase, the fundamental group π 1 (X) is isomorphi to F n, in whih F n is alled the free group on n letters. It is alled free beause it an be presented using generators and no relations (other than the ones appearing in the group axioms). The group F n an be desribed as follows. Start with an alphabet Σ of n letters. For every letter σ Σ, make an inverse letter denoted by σ 1. This simply doubles the size of the alphabet to make 2n symbols. The elements of F n are simply the set of all finite-length words (or strings) that an be formed from the 2n symbols. For example, suppose O onsists of three obstales. The fundamental group is F 3, the free group on 3 letters. Let the alphabet be Σ = {σ 1,σ 2,σ 3 }. The group F 3 onsists of all words that an be formed from σ 1, σ 2, σ 3, σ1 1, σ 1 2, and σ 1 3. An example is σ 3 σ2 1 σ 1σ 1 σ 2 F n. This is not the omplete story, however, beause the group axioms ontain relations foridentitiesandinverses,eveninafreegroup.letεdenotetheidentityelementinf n. It must be true for any element σ Σ that εσ = σε = σ and also that σσ 1 = σ 1 σ = ε. This indues an equivalene relation on the set of all words formed from 2n symbols. For example, σ 3 σ2 1 σ 2σ 1 σ 2 = σ 3 σ 1 σ 2. An arbitrary word an often be simplified into an equivalent, shorter word by apply these relations. The shortest word in the equivalene lass is alled a redued word. 16

17 Fig. 13. There are three obstales and the fundamental group is F 3. Three loop paths are hosen as representatives that orrespond to σ 1, σ 2, and σ 3 in Σ. In this way, eah word σi k niely represents a lass of paths that wraps k times ounterlokwise around the ith obstale. A familiar problem in group theory is determining whether two group presentations are equivalent, in other words, their orresponding groups are isomorphi. Reall from linear algebra that there are many ways to define and transform bases for a vetor spae. A similar but more ompliated situation exists for F n. If we say that F n is the fundamental group of X, what do the symbols σ Σ atually represent? Eah σ should orrespond to an equivalene lass of homotopi paths. Within an equivalene lass, there exists the ommon issue of piking a representative path for the whole lass. However, the situation is more ompliated than this beause the lass that σ represents is somewhat arbitrary. Figure 13 shows a simple way to represent the symbols for F 3. This way is ommon for explaining the fundamental group in textbooks. All elements of F 3 an be onstruted using and it seems the topi is finished. However, sine we are reonstruting paths from sensor data, we may be fored to work with other representative elements of F n and transform them bak into easy-to-interpret representations. Even without the ompliation of paths, F n itself an be represented in many ways. Suppose it is presented using the alphabet Σ = {σ 1,...,σ n }. Consider a funtion g : Σ F n, whih replaes every σ Σ with a word in F n. In some ases, g yields an automorphism (isomorphism to itself) of F n. The group Aut(F n ) of all automorphisms of F n is remarkably ompliated. One way to desribe them is to perform Nielsen transformations on the alphabet and use them to generate Aut(F n ) [Magnus et al. 1976]. Our goal in the remainder of this setion is to arefully sidestep ompliated issues regarding automorphisms of F n while nevertheless reonstruting simple representations of equivalene lasses of paths Perfet Beams We now handle the ase that is oneptually as simple as Figure 13. More ompliated ases are built upon it. Let a beam be alled outer if it is either an infinite ray (possible onlyifx isunbounded)oritisafinitesegmentthatonnetsanobstaletotheboundary of W. For a set of n obstales, let a perfet olletion of beams mean that there are exatly m = n DDD beams (reall that this means disjoint, distinguishable, and direted beams), with exatly one outer beam attahed to eah obstale. For onveniene, further assume that all beams in a perfet olletion are oriented so that a ounterlokwise traversal orresponds to the positive diretion, as shown in Figure 14. If they are not, then we an easily transform our solution to reverse the orresponding diretions. Suppose that a perfet olletion of beams is given with label set L. Let f : L Σ denote any bijetion, in whih Σ denotes an alphabet as used in the definition of F n. We obtain the following proposition. 17

18 Fig. 14. A perfet olletion of beams and a loop path. b b 2 b b b σ σ 2 σ σ σ x 0 Fig. 15. With this olletion of obstales and beams, the sensor word ỹ diretly transforms into an element of the fundamental group F n by renaming symbols. Proposition 5.1 For any sensor word ỹ for a perfet olletion of beams, if the transformation f is applied to every element, the resulting transformed word is the orresponding element of F n, under the basis of one ounterlokwise loop per obstale (as depited in Figure 15). Proof: While preserving homotopy equivalenes, the obstales and basepoint an be moved into a anonial form as shown in Figure 15. This orresponds to hoosing a partiular basis of F n in whih eah generator σ i is exatly a ounterlokwise loop around one obstale. It does not matter what loop in partiular is hosen, provided that exatly one obstale is enirled for eah generator. Eah b i orresponds to a beam and letter in L. A word w F n is formed as follows. Let y k denote the kth observation in ỹ. Thekthelementofw isdefinedasf(y k ).Eahbeamrossingthenorrespondsdiretly to a generator of F n under the hosen basis. This onverts every ỹ into an element of F n that represents the loop path that was traversed using basepoint x 0. 18

19 l b e a m d n r g o i k f p q e b a d g i k f s h j h j (a) (b) Fig. 16. (a) A suffiient olletion of DDD beams; (b) a minimally suffiient olletion of DDD beams forms trees that eah ontain one outer beam. This example is obtained by removing beams from the the figure on the left. Reall that arbitrary words in F n an be simplified to redued words by applying the group axioms. Note that this simplifiation an even be performed diretly on the sensor word, before the transformation into F n, without hanging the result. For example, b 1 b 2 b 2b 3 an be simplified to b 1 b 3 beause σ 2 σ2 1 = ε would ause a anellation anyway after the transformation f is applied Suffiient DDD Beams What if the olletion of beams is not perfet? For some arrangements of obstales, it might not even be possible to design a perfet olletion (unless beams are allowed to be nonlinear). Suppose that a olletion B of m n DDD beams is given. It is alled suffiient if all of the resulting regions are simply onneted; see Figure 16(a). Note that any suffiient olletion must ontain at least one outer beam. Also, any perfet olletion is also suffiient. Reall the previous question regarding whether two paths must be homotopi if the region sequene they traverse is the same. We obtain the following proposition: Proposition 5.2 If the olletion of beams is suffiient and two paths x and x traverse the same sequene of regions, then they must be homotopi. Proof: For a suffiient olletion, they must be homotopi beause if they are not, then it implies that there exists an obstale o O that bloks the deformation of one path into another. This would be possible only if o lies in the interior of a region; however, o must border two or more regions beause in a suffiient olletion, a beam is attahed to o. Therefore, the homotopy map would ause a hange in the region sequene, whih violates that assumption that it is the same for both paths. The impliation of Proposition 5.2 is that path reonstrution up to homotopy is more oarse than reonstrution up to region sequenes. The region filters provide the tightest possible desription of the path, but homotopy equivalene allows some paths that traverse different region sequenes to be delared as being the same. Suppose that a sensor word ỹ is obtained for a suffiient olletion B of beams. The first step in desribing the path as an element of F n is to disregard redundant beams. 19

20 Fig. 17. Forming a basis using a suffiient tree of beams. To ahieve this, let B B be a minimal subset of B that is still suffiient. Suh a olletion is alled minimally suffiient and an be omputed using linear-time spanning tree algorithms suh as depth-first or breadth-first searh. An example is shown in Figure 16(b). Reall the bijetive transformation f : L Σ. The funtion an be applied to obtain elements of F n as was done for Proposition 5.1; however, a different representation is obtained in omparison to the nie one in Figure 13. Proposition 5.3 For a minimally suffiient olletion B of DDD beams, the sensor word ỹ maps diretly to the orresponding element of F n by simply applying f to eah letter. Proof: A basis for the fundamental group is defined as follows. For eah tree of beams in B, a olletion of loop paths an be formed as shown in Figure 17. Eah loop must ross transversely the interior of exatly one beam and enlose a unique nonempty set of obstales. Suh loops always exist and an be onstruted indutively by first enlosing the leaves of the tree and then progressing through parents until a loop is obtained that traverses the outer beam. Sine there is only one region, it is possible to indutively onstrut suh a olletion of loops for every tree of beams in B. The total olletion of loops forms a basis of F n, whih an be related to the basis in Proposition 5.1 via Tietze or Nielsen transformations [Magnus et al. 1976]. The mapping f from ỹ to f(ỹ) F n is one again obtained by mapping eah letter in ỹ to its orresponding unique loop that traverses the beam. Using Proposition 5.3, a simple algorithm is obtained. Suppose that any suffiient olletion B of DDD beams is given and a sensor word ỹ is obtained. A spanning tree B B of beams is omputed, whih is minimally suffiient. Let L L denote the orresponding set of beam labels. To ompute the element of F n, the first step is to delete from ỹ any letters in L \ L. This yields a redued word ỹ for whih eah letter an be mapped diretly to a loop using the proof of Proposition 5.3 to obtain a representation of the orresponding path in F n. One again, redutions based on the identity and inverses in F n an be performed before or after the mapping is applied. 20

21 5.3. The General Case We finally return to most general olletion of beams, whih inludes examples suh as Figure 3(b). Consider a olletion B of beams in whih some may interset, some may be undireted, and some may even be indistinguishable. The olletion is nevertheless assumed to be suffiient, whih means that all of the orresponding regions are simply onneted. Rather than worry about making a minimal subset of B, the method for the general ase works by inventing a olletion of fititious beams that happens to be minimally suffiient. Sine the ambiguity may be high enough to yield a set of possible paths, the region filter from Setion 4 is used. There are two phases to the omputation. In the first phase, a set of imaginary DDD beams is onstruted from the original olletion of obstales and beams. In the seond phase, the sensor word ỹ is proessed and a representation of possible path lasses is given in terms of imaginary beams. For the first phase, suppose W, O, B, L, and α (beam labels) are given in a way that one again failitates exat omputation (as Setion 4.1). The algorithm proeeds as follows: (1) Compute the arrangement of regions and multigraph G from Setion 4. (2) For eah vertex in G hoose a sample point in its orresponding region. (3) For eah direted edge e in G, ompute a pieewise-linear sample path that: i) starts at the sample point of the soure vertex of e, ii) ends at the sample point of the destination vertex of e, and iii) rosses the beam assoiated with e in a manner onsistent with its label. The sample path must be hosen so that it does not interset additional beams. The sample paths an be omputed using standard motion planning tehniques, suh as trapezoidal deomposition or triangulations (see [LaValle 2006]). The result is an embedding of G into the free spae X, as depited in Figure 18(a). (4) Construt any minimally suffiient olletion B I of imaginary DDD beams. The hoie does not depend at all on previous steps. A onvenient hoie is to make all imaginary beams vertial, one from eah obstale. See Figure 18(b). (5) For all omputed sample paths from Step 3, ompute their intersetions with the imaginary beams of Step 4 and reord the order in whih they our. Now suppose that a sensor word ỹ is given. The following steps onstrut the possible paths up to homotopy equivalene: (1) Apply the region filter of Setion 4 is used to determine the set of possible region sequenes. (2) Eah region sequene orresponds to a yli walk through G. Using the embedding of G into W from the first phase of omputation (reall Figure 18(b)), a loop path x is obtained by onatenating the orresponding sample points and sample paths in G. (3) Using the onstrution in the proof of Proposition 5.3, ỹ is mapped diretly to an element of F n. (4) TheobtainedwordinF n issimplifiedtoobtainthereduedword.asbefore,simplifiations an be applied to ỹ in advane or to its image in F n and the same result is obtained. (5) In the final step, dupliate redued words are removed from the olletion obtained for eah sequene produed by the region filter. Asusual,redutionsanbeappliedtoỹ oritsimageinf n.oneelementsoff n are omputed and redued for eah possible region sequene, dupliates are removed to obtain the omplete set of possible homotopially distint paths based on the sensor word ỹ. 21

22 r 1 a f r 7 r 8 r 2 b r 5 d r 6 b e e a r 4 r 3 (a) (b) Fig. 18. (a) This shows the embedding of the multigraph G into the free spae X for the example of Figure 3(b). (b) A minimally suffiient olletion of DDD beams forms trees that eah ontain one outer beam. This example is obtained by removing beams from the figure on the left. 22

23 Note that B I essentially gives the user the freedom to define whatever basis of F n is desired to express the result. By extending the method to nonlinear beams, the solution an even be expressed in terms of perfet beams, instead of the unusual loop paths produed by the method of suffiient DDD beams. Proposition 5.4 The given algorithm reonstruts from ỹ the omplete set, up to homotopy equivalene, of paths that ould have possibly produed ỹ in terms of a ommon basis for F n. Proof: The sensor word ỹ is given. By applying the region filter from Setion 4, all possible sequenes of regions traversed by the path are obtained. Construting the walk through G and onverting it into a sample path provides one representative from the orret homotopy lass. This is true beause the sample path was onstruted by traversing the same region sequene and all paths that traverse the same region sequene are homotopi, as established by Proposition 5.2. It is now simple a matter of piking a basis for F n to desribe the homotopy lass. The olletion of imaginary beams is a suffiient DDD olletion. Proposition 5.3 therefore implies that the orresponding element of F n is obtained by the diret mapping f. This establishes that the omputed representation is indeed the orret homotopy lass that orresponds to the region sequene. Sine the lass is orretly haraterized for every possible region sequene, the omplete set of paths is haraterized up to homotopy equivalene, as required. Furthermore, all lasses are desribed with respet to the same basis of F n beause the imaginary beams are fixed in advane and all homotopy lasses are expressed in terms loops that enompass them. We now remove the assumption that only loop paths are exeuted using a basepoint x 0. If all paths start at some x 0 and terminate at some x 1, then a fixed path segment that onnets x 1 bak to x 0 an be hosen. This path may interset some beams, whih is false information; however, the possible body paths are nevertheless haraterized orretly up to homotopy equivalene. If instead we allow either of the path endpoints to vary, then all paths beome trivial (homotopi to a onstant path) by ontinuously deforming eah path to be a onstant funtion into one basepoint. One possibility is to assume that there are several possible fixed points based on the starting and final regions produed in eah sequene from the region filter. In this way, possible paths an at least be ompared if their starting and terminating regions math. 6. PATH WINDING NUMBERS The setion provides a reonstrution of the path at a level that is more oarse than homotopi equivalene. Suppose that the body travels along a loop path. There is an integer winding number v i Z for eah o i O, in whih m i is defined as the number of times the path wraps ounterlokwise around o i after deleting all other obstales and pulling the path tight around o i using homotopy. If there are n obstales, then a vetor of n winding numbers is obtained. Two paths are alled homologous if and only if their vetors of winding numbers are idential. This notion of equivalene is ruial to algebrai topology, and in partiular homology theory [Hather 2002], in whih a homology group is omputed. The first homology group H 1 (X) an be onsidered as the abelianized version of the fundamental group π 1 (X). For our problem, H 1 (X) is obtained by applying the equivalene relation σ 1 σ 2 = σ 2 σ 1 over all words in F n, for all σ 1,σ 2 Σ Perfet Beams In the ase of perfet beams, the winding numbers are obtained by diretly abelianizing the sensor word ỹ. Using Proposition 5.1, the sensor word maps diretly to a word 23

24 F n by applying f to every letter. For example, onsider the sensor word: After applying f to eah symbol, the word ỹ = b 1 b 2 b 1b 2 b 2 b 1 b 2b 2b 1 b 2b 2b 2. (5) w = σ 1 σ 2 σ1 1 σ 2σ 2 σ 1 σ2 1 σ 1 2 σ 1σ2 1 σ 1 2 σ 1 2 F n (6) is obtained. By applying the ommutativity relation, we simply sort the symbols in the word, perform anellations, and ount the resulting number of eah. For the example, the result is w =σ 1 σ 2 σ1 1 σ 2σ 2 σ 1 σ2 1 σ 1 2 σ 1σ2 1 σ 1 2 σ 1 2 =σ 1 σ 1 σ 1 σ1 1 σ 2σ 2 σ2 1 σ 1 2 σ 1 2 σ 1 2 σ 1 2 (7) =σ1σ The winding numbers are simply the exponents; for (7) we obtain v = (2, 3). The appliationoff wasunneessaryinthisasetoobtaintheresult.theoperationsanbe performed diretly on ỹ due to the simpliity of f. Note that the winding numbers an be omputed in time O( ỹ ) without atually sorting by simply maintaining n ounters, one for eah letter in λ L. San aross ỹ and inrement or derement eah ounter, basedonwhetherλorλ,isenountered,respetively.notethatthismakesaonstanttime ombinatorial filter of the form v k+1 = φ(v k,y k+1 ), by omputing the winding numbersv k+1 atstepk+1fromthewindingnumbervetorv k andthelastobservation y k+1, whih is the last letter of ỹ k Suffiient DDD Beams One a minimally suffiient olletion is determined (reall Figure 16), the winding numbers an be alulated in the same way as in the perfet beams ase. However, the result needs to be transformed to obtain the orret result beause the path may wind around multiple obstales simultaneously. Reall the basis from Figure 17 and suppose that for a path, the sensor word is ỹ = aabbd bdbd ab b a b aaab b b b d. (8) Applying the simple method from the perfet beams ase suggests winding numbers (5, 3,4,1). For example, there are 5 more as than a s in ỹ. This means that the path wraps 5 times around o 3, but it also wraps 5 times around o 1, o 2, and o 4. Likewise, it wraps 3 times around o 1 and o 2. A ounter is made for eah obstale and eah omputed exponent raises or lowers some ounters. After being performed for eah omponent, the orret result is obtained. For the sensor word in (8), the orreted winding numbers are (3,2,5,9). Note that if the positive diretion of a beam is in the lokwise diretion, then the omputed winding number needs to be multiplied by The General Case Now suppose that a suffiient olletion of general beams has been given, whih is the model used in Setion 5.3. A straightforward approah is to first run the algorithm of Setion 5.3. After the suffiient olletion of imaginary beams has been plaed and the elements of F n have been omputed, eah an be abelianized to obtain winding numbers using the method just desribed. This yields a set of vetors of winding numbers beause there might not be enough sensor data to infer the exat numbers. In some ases, this approah omputes more information than is needed to obtain the winding numbers. For example, suppose that a suffiient olletion of beams is given that is not neessarily disjoint, but all beams are direted and distinguishable. Sine the winding number essentially ignores all other beams, an approah an be developed 24

25 Fig. 19. A simple ommutator example that yields sensor word aba b, group element σ 1 σ 2 σ 1 1 σ 1 2 F 2, and winding numbers v = (0,0), but orresponds to path that is not homotopi to a onstant path. by piking a minimally suffiient olletion of beams that is not neessarily disjoint. The beam intersetions do not interfere with the alulation of winding numbers. For a given sensor word, any letters that do not appear in the minimally suffiient olletion an simply be deleted. The method then proeeds as in the ase of DDD beams Higher Order Winding Numbers There is atually a way to provide path desriptions that are more oarse than homotopy equivalene but are finer than homology equivalene. The winding numbers give a measure of how many times a body irles around a given obstale, but ignores how the body weaves in between obstales. They are insensitive to paths suh as a ommutator around obstales, as shown in Figure 19. There are higher order winding numbers whih keep trak of how a body does in fat interweave through different obstales. They ount the umulative number of times that a ommutator was ahieved. It is also possible, however, to ount ommutators that are build from other ommutators. The notion of higher order arises from the levels of nesting of ommutators. These strutures an be aptured by a Lie algebra in whih the Lie braket is the group ommutator: [σ 1,σ 2 ] = σ 1 σ 2 σ1 1 σ 1 2. These higher order winding numbers in the ase above arise in two lassial ways refleting the interplay between geometry and a free group. These are the Lie algebras whih arise from either(i) the desending entral series of a free group, or(ii) prinipal ongruene subgroups of level p r in the group of SL(2,Z). These Lie algebras provide measures of omplexity in addition to higher order winding numbers and it remains an open problem to develop omputation methods that haraterize them for a given sensor word. It remains an open problem to effiiently ompute higher order winding numbers for problems presented in this paper. 7. EXPERIMENTAL IMPLEMENTATION In this setion, we present an inexpensive hardware arhiteture that implements some simple, reliable, low-ost beams, both direted and undireted. We onduted several experiments that involved reonstruting the path of moving bodies in a laboratory setting by applying the region filters from Setion 4. Sine the method was validated for these filters, they would also learly work for the methods of Setions 5 and 6 beause they are derived from the same sensor observations. Many more details regarding our hardware hoies, their osts, and our experiments appear in [Czarnowski 2011]. 25

26 (a) Fig. 20. (a) An emitter-detetor pair with an body rossing the beam; (b) a simple Analog to Digital Conversion board using the ubiquitous LM339 omparator. (b) 7.1. Hardware Arhiteture We implemented the beam sensor using optial emitter-detetors pairs. One pair was used to make an undireted beam. To make a direted beam, two pairs were plaed lose together using the idea in Figure 4, and some simple logi iruitry was used to determine the rossing diretion. We hose presentation laser pointers due to their low ost (about $3 US eah) and beause they an be aimed easily. The pointers were modified to use external battery paks using three AA batteries. Inexpensive and easily obtainable photodiodes (about $2 US eah) were plaed on the opposite side to detet body rossings. A hange in voltage aross the photodiode is observed when a body rosses the beam, thereby bloking the laser light from reahing the photodetetor. This hange in voltage is deteted by a basi ADC iruit using the LM339 omparator (about $0.20 US eah). The threshold voltage for the beams an be set using a potentiometer. For the purposes of this experiment, simple iruit boards were fabriated to aommodate the ADC iruit (Figure 20 (b) ). Eah board an handle four inputs from the outputs of the photodetetors. The outputs of the ADC board are onneted to the digital I/O pins of a Complex Programmable Logi Devie (CPLD). The CPLD is a programmable logi devie that takes are of debouning and denoising the inputs. It then outputs an ASCII harater orresponding to the beam label over a serial port. The CPLD was designed using the Verilog hardware desription language. There are several reasons why we hose a CPLD for use in our system: (1) Prie: The least expensive Altera MAX IIZ CPLD osts less than $7 US. (2) Energy Consumption: The Altera MAX IIZ CPLD an run on as little as 25µA. (3) Reonfigurability: The iruit implemented on a CPLD an easily be reonfigured in-iruit by hanging the Verilog ode and reprogramming the devie with a PC. Additionally, the CPLD design software reports the hardware resoures (inluding logi elements) that are used for a given design. Thus, it is possible to quikly estimate the hardware ost of a mass-produed (ASIC-based) produt. For example, the CPLD usage for a six-beam implementation uses 78 logi elements and 21 total pins. Thus, this design easily fits in the sub-$7 devie mentioned above, whih has 240 logi elements and 54 IO pins. Detailed analysis of the resoure usage suggests that a designinorporatingatleast20beamsanfitineventhissmallcpld.ifalargersystem is desired, the Verilog design an be seamlessly migrated to a larger CPLD. 26

27 Fig. 21. A physial implementation of the six-beam example from Figure 1. Theost ofasix-beam deployment isunder $30 US.One properly setup, thebeams do not miss any body rossings. Furthermore, our system has low energy onsumption. When using three AA alkaline batteries, the urrent draw was measured to be 21.7 ma. Using a standard AA alkaline with 2700 mah apaity, a beam an be powered for over 120 hours. The CPLD board an be powered with a simple rehargeable battery pak or through an USB port. Using Altera s energy onsumption estimator, the Altera MAX IIZ would use ma when using a six-beam implementation. Thus, the CPLD ould theoretially run for around 37,000 hours using only a set of three alkaline batteries. Wireless ommuniation an be easily added to our arhiteture if needed. We have experimented with 2.4GHz XBee modules that implement the protool. For under $25 US, these devies allow very reliable and simple wireless ommuniation. Eah of this Zig-bee modules is apable of handling up to six beams diretly using the on-board ADC iruitry. To the best of our knowledge, our hardware implementation ompares favorably in terms of ost with previous implementations for traking using binary sensors [Kim et al. 2005; Shrivastava et al. 2006a]. First of all, we used heaper sensors than Passive Infrared Sensors (PIR) or aousti sensors. Furthermore, instead of using a full sensor mote that may exessive for simple omputations, we determined the preise omputational requirements for solving our task and implemented them in hardware. This leads to an overall derease in prie and energy onsumption Reonstruting the Motion of a Single Body We will illustrate how the arhiteture works with a simple example. As illustrated in Figure 21, we deployed six sensor beams in a 1.67m by 1.3m environment. The outer boundary of the free spae is formed from inder bloks and the obstales made of briks. This physial setup, shown in Figure 21, was designed to math the example in Figure 1. After reating the environment, we let an unpreditable rolling ball, alled a Weaselball, to wander for several minutes as the moving body. (See [Bobadilla et al. 2011a] for more information on Weaselball motions.). We used an overhead amera and algorithm implementations in OpenCV to extrat the ground truth path of the body (Figure 22(a)). For the sake of larity, we show only the first several seonds of the ground truth traking. In Figure 22 (b), we show the reonstrution of the path that 27

28 (a) a f b e d (b) Fig. 22. A physial implementation of the example in Figure 1: (a) The ground truth path (ausing ỹ = bbee) of the body; (b) the reonstrution of the path over about 15 seonds of motion. A sample path is shown based on the reonstruted region sequene. was obtained from the omputed region sequene. Note that the atual experiment ran for threee minutes, with 62 deteted beam rossings and suessful reonstrution of the path up to the sequene of regions traversed by the body. In this experiment and others, the beam did not fail to detet any of the body rossings. This was observed in dozens of experiments that were performed, with hundreds of beam rossings. The only type of mistake made was that the system reported a rossing when the body entered the beam but did not ross through it. This violates the assumption in Setion 2 that the body must ross transversely. If desired, this 28

29 Fig. 23. A physial implementation of the environment and filter desribed in Figure 11. The beam loations are shown with green tape. Wires onneting to the photodiodes are visible. problem an be alleviated by the use of two adjaent beams to ensure the omplete rossing of the body before reporting an observation The Two-Body, Three-Region Filter We implemented the two body problem illustrated in Figure 11(a), and the orresponding experimental version appears in Figure 23. A group of small briks serves as the barrier (the enter of the ring), whereas larger paving briks form the outer boundary of the free spae. The ring is then partitioned into three setions by our laser-detetor pairs. Reall the simple filter shown in Figure 11(b), whih keeps trak of whether the two bodies are in separate regions. We implemented the simple automaton on the Altera MAX II CPLD. The CPLD resoure usage for the three room design is as follows: 52 logi elements, 18 total pins. Thus, the design is easily implemented on the inexpensive MAXIIZCPLDmentionedabove.Thetwomovingballsintheenvironmentandthefilter suessfully kept trak of whether they were together for long periods of time. One again, the errors ourred only when a body entered a laser beam but did not omplete the rossing. The use of two emitter-detetor pairs mounted lose to eah other should alleviate this issue. Other multibody traking experiments appear in [Bobadilla et al. 2011b] and [Czarnowski 2011]. 8. CONCLUSIONS AND OPEN QUESTIONS In this paper we identified a basi inferene problem based on bodies moving among obstales and detetion beams. Reall from Setion 3 that the beams may diretly model physial sensors or they may arise virtually from a variety of other sensing models. Therefore, the region filter, homotopi reonstrution, and winding-number omputations provide basi information that arises in numerous settings suh as robotis, seurity, forensis, environmental monitoring, and assisted living. Reall that the region filter provides the tightest haraterization of the set of possible paths given the sensor word ỹ. Charaterization up to homotopy equivalene is more oarse than pro- 29