{"title": "An Autonomous Robotic System for Mapping Abandoned Mines", "book": "Advances in Neural Information Processing Systems", "page_first": 587, "page_last": 594, "abstract": "", "full_text": "An Autonomous Robotic System\nFor Mapping Abandoned Mines\n\nD. Ferguson1, A. Morris1, D. H\u00a8ahnel2, C. Baker1, Z. Omohundro1, C. Reverte1\n\nS. Thayer1, C. Whittaker1, W. Whittaker1, W. Burgard2, S. Thrun3\n\n1The Robotics Institute\n\n2Computer Science Department\n\n3Computer Science Department\n\nCarnegie Mellon University\n\nPittsburgh, PA\n\nUniversity of Freiburg\n\nFreiburg, Germany\n\nStanford University\n\nStanford, CA\n\nAbstract\n\nWe present the software architecture of a robotic system for mapping\nabandoned mines. The software is capable of acquiring consistent 2D\nmaps of large mines with many cycles, represented as Markov random\n\u00a3elds. 3D C-space maps are acquired from local 3D range scans, which\nare used to identify navigable paths using A* search. Our system has\nbeen deployed in three abandoned mines, two of which inaccessible to\npeople, where it has acquired maps of unprecedented detail and accuracy.\n\nIntroduction\n\n1\nThis paper describes the navigation software of a deployed robotic system for mapping\nsubterranean spaces such as abandoned mines. Subsidence of abandoned mines poses a\nmajor problem for society, as do ground water contaminations, mine \u00a3res, and so on. Most\nabandoned mines are inaccessible to people, but some are accessible to robots. Autonomy\nis a key requirement for robots operating in such environments, due to a lack of wireless\ncommunication technology for subterranean spaces.\nOur vehicle, shown in Figure 1 (see [1] for a detailed hardware description) is equipped\nwith two actuated laser range \u00a3nders. When exploring and mapping unknown mines, it\nalternates short phases of motion guided by 2D range scans, with phases in which the\nvehicle rests to acquire 3D range scans. An analysis of the 3D scans leads to a path that\nis then executed, using rapidly acquired 2D scans to determine the robot\u2019s motion relative\nto the 3D map. If no such path is found a high-level control module adjusts the motion\ndirection accordingly.\nAcquiring consistent large-scale maps without external geo-referencing through GPS is\nlargely considered an open research issue. Our approach relies on ef\u00a3cient statistical tech-\nniques for generating such maps in real-time. At the lowest level, we employ a fast scan\nmatching algorithm for registering successive scans, thereby recovering robot odometry.\nGroups of scans are then converted into local maps, using Markov random \u00a3eld repre-\nsentations (MRFs) to characterize the residual path uncertainty. Loop closure is attained\nby adding constraints into those MRFs, based on a maximum likelihood (ML) estimator.\nHowever, the brittleness of the ML approach is overcome by a \u201clazy\u201d data association\nmechanism that can undo and redo past associations so as to maximize the overall map\nconsistency.\nTo navigate, local 3D scans are mapped into 2 1\n2 D terrain maps, by analyzing surface gradi-\nents and vertical clearance in the 3D scans. The result is subsequently transformed into cost\n\n\fFigure 1: The Groundhog robot\nis\na 1,500 pound custom-built vehicle\nequipped with onboard computing, laser\nrange sensing, gas and sinkage sensors,\nand video recording equipment. Its pur-\npose is to explore and map abandoned\nmines.\n\nfunctions expressed in the robot\u2019s three-dimensional con\u00a3guration space, by convolving\nthe 2 1\n2 D terrain maps with kernels that describe the robot\u2019s footprints in different orienta-\ntions. Fast A* planning is then employed in con\u00a3guration space to generate paths executed\nthrough PD control.\nThe system has been tested in a number of mines. Some of the results reported here\nwere obtained via manual control in mines accessible to people. Others involved fully\nautonomous exploration, for which our robot operated fully self-guided for several hours\nbeyond the reach of radio communication.\n\n2D Mapping\n\n2\n2.1 Generating Locally Consistent Maps\nAs in [6, 9], we apply an incremental scan matching technique for registering scans, ac-\nquired using a forward-pointed laser range \u00a3nder while the vehicle is in motion. This\nalgorithm aligns scans by iteratively identifying nearby points in pairs of consecutive range\nscans, and then calculating the relative displacement and orientation of these scans by min-\nimizing the quadratic distance of these pairs of points [2]. This approach leads to the\nrecovery of two quantities: locally consistent maps and an estimate of the robot\u2019s motion.\nIt is well-understood [3, 6], however, that local scan matching is incapable of achieving\nglobally consistent maps. This is because of the residual error in scan matching, which\naccumulates over time. The limitation is apparent in the map shown in Figure 2a, which is\nthe result of applying local scan matching in a mine that is approximately 250 meters wide.\nOur approach addresses this problem by explicitly representing the uncertainty in the map\nand the path using a Markov random \u00a3eld (MRF) [11]. More speci\u00a3cally, the data acquired\nthrough every \u00a3ve meters of consecutive robot motion is mapped into a local map [3].\nFigure 3a shows such a local map. The absolute location of orientation of the k-th map will\nbe denoted by \u00bbk = ( xk yk (cid:181)k )T ; here x and y are the Cartesian coordinates and (cid:181) is\nthe orientation. From the scan matcher, we can retrieve relative displacement information\nof the form \u2013k;k\u00a11 = ( \u00a2xk;k\u00a11 \u00a2yk;k\u00a11 \u00a2(cid:181)k;k\u00a11 )T which, if scan matching was error-\nfree, would enable us to recover absolute information via the following recursion (under\nthe boundary condition \u00bb0 = (0; 0; 0)T )\n\n\u00bbk = f (\u00bbk\u00a11; \u2013k;k\u00a11) = \u02c6 xk\u00a11 + \u00a2xk;k\u00a11 cos (cid:181)k;k\u00a11 + \u00a2yk;k\u00a11 sin (cid:181)k\u00a11\n\nyk\u00a11 \u00a1 \u00a2xk;k\u00a11 sin (cid:181)k;k\u00a11 + \u00a2yk;k\u00a11 cos (cid:181)k\u00a11\n\n(cid:181)k\u00a11 + \u00a2(cid:181)k;k\u00a11\n\n!(1)\n\nHowever, scan matching is not without errors. To account for those errors, our approach\ngeneralizes this recursion into a Markov random \u00a3eld (MRF), in which each variable \u00a5 =\n\u00bb1; \u00bb2; : : : is a (three-dimensional) node. This MRF is de\u00a3ned through the potentials:\n\n`(\u00bbk; \u00bbk\u00a11) = exp \u00a1 1\n\n2 (\u00bbk \u00a1 f (\u00bbk\u00a11; \u2013k;k\u00a11))T Rk;k\u00a11(\u00bbk \u00a1 f (\u00bbk\u00a11; \u2013k;k\u00a11)) (2)\nHere Rk;k\u00a11 is the inverse covariance of the uncertainty associated with the transition\n\u2013k;k\u00a11. Since the MRF is a linear chain without cycles, the mode of this MRF is the solution\nto the recursion de\u00a3ned in (1). Figure 3b shows the MRF for the data collected in the\n\n\f(a)\n\n(b)\n\nFigure 2: Mine map with incremental ML scan matching (left) and using our lazy data association\napproach (right). The map is approximately 250 meters wide.\n\nBruceton Research Mine, over a distance of more than a mile. We note this representation\ngeneralizes the one in [11], who represent posteriors by a local bank of Kalman \u00a3lters.\n\n2.2 Enforcing Global Consistency\nThe key advantage of the MRF representation is that it encompasses the residual uncer-\ntainty in local scan matching. This enables us to alter the shape of the map in accordance\nwith global consistency constraints. These constraints are obtained by matching local maps\nacquired at different points in time (e.g., when closing a large cycle). In particular, if the\nk-th map overlaps with some map j acquired at an earlier point in time, our approach lo-\ncalizes the robot relative to this map using once again local scan matching. As a result, it\nrecovers a relative constraint `(\u00bbk; \u00bbj) between the coordinates of non-adjacent maps \u00bbk\nand \u00bbj. This constraint is of the same form as the local constraints in (2), hence is repre-\nsented by a potential. For any \u00a3xed set of such potentials ' = f`(\u00bb k; \u00bbj)g, the resulting\nMRF is described through the following negative log-likelihood function\n\n\u00a1 log p(\u00a5) = const: + 1\n\n2Xk;j\n\n(\u00bbk \u00a1 f (\u00bbj; \u2013k;j))T Rk;j (\u00bbk \u00a1 f (\u00bbj; \u2013k;j))\n\n(3)\n\nwhere \u00a5 = \u00bb1; \u00bb2; : : : is the set of all map poses, and f is de\u00a3ned in (1).\nUnfortunately, the resulting MRF is not a linear chain any longer.\nInstead, it contains\ncycles. The variables \u00a5 = \u00bb1; \u00bb2; : : : can be recovered using any of the standard inference\nalgorithms for inference on graphs with cycles, such as the popular loopy belief propagation\nalgorithm and related techniques [5, 14, 17]. Our approach solves this problem by matrix\ninversion. In particular, we linearize the function f using a Taylor expansion:\n\nf (\u00bbj; \u2013k;j) \u2026 f ( \u201e\u00bbj) + Fk;j(\u00bbj \u00a1 \u201e\u00bbj)\n\n(4)\n\nwhere \u201e\u00bbj denotes a momentary estimate of the variables \u00bbj (e.g., the solution of the\nrecursion (1) without the additional data association constraints). The matrix Fk;j =\n\n\f(a)\n\n(b)\n\nFigure 3: (a) Example of a local map.\n(b) The Markov random \u00a3eld: Each\nnode is the center of a local map, ac-\nquired when traversing the Bruceton Re-\nsearch Mine near Pittsburgh, PA.\n\nr\u00bbj f ( \u201e\u00bbj; \u2013k;j) is the Jacobean of f (\u00bbj; \u2013k;j) at \u201e\u00bbj:\n\nFk;jx = 0\n@\n\n1\n0\n0\n\n0 \u00a1\u00a2xk;j sin \u201e(cid:181)k + \u00a2yk;j cos \u201e(cid:181)k\n1 \u00a1\u00a2xk;j cos \u201e(cid:181)k \u00a1 \u00a2yk;j sin \u201e(cid:181)k\n0\n\n1\n\nThe resulting negative log-likelihood is given by\n\n1\nA\n\n(5)\n\n\u00a1 log p(\u00a5) \u2026 const: + 1\n\n2Xk;j\n\n(\u00bbk \u00a1 f ( \u201e\u00bbj) \u00a1 Fk;j(\u00bbj \u00a1 \u201e\u00bbj))T (cid:190)\u00a11\n\nk;j (\u00bbk \u00a1 f ( \u201e\u00bbj) \u00a1 Fk;j(\u00bbj \u00a1 \u201e\u00bbj))\n\nis quadratic in the variables \u00a5 of the form const: + (A\u00a5 \u00a1 a)T R (A\u00a5 \u00a1 a), where A is a\ndiagonal matrix, a is a vector, and R is a sparse matrix that is non-zero for all elements j; k\nin the set of potentials. The minimum of this function is attained at (AT RA)\u00a11AT Ra. This\nsolution requires the inversion of a sparse matrix. Empirically, we \u00a3nd that this inversion\ncan be performed very ef\u00a3ciently using an inversion algorithm described in [15]; it only\nrequires a few seconds for matrices composed of hundreds of local map positions (and it\nappears to be numerically more stable than the solution in [11, 6]). Iterative application of\nthis linearized optimization quickly converges to the mode of the MRF, which is the set of\nlocations and orientations \u00a5. However, we conjecture that recent advances on inference in\nloopy graphs can further increase the ef\u00a3ciency of our approach.\n\n2.3 Lazy Data Association Search\nUnfortunately, the approach described thus far leads only to a consistent map when the\nadditional constraints `(\u00bbk; \u00bbj) obtained after loop closure are correct. These constraints\namount to a maximum likelihood solution for the challenging data association problem\nthat arises when closing a loop. When loops are large, this ML solution might be wrong\u2014a\nproblem that has been the source of an entire literature on SLAM (simultaneous localization\nand mapping) algorithms. Figure 4a depicts such a situation, obtained when operating our\nvehicle in a large abandoned mine.\nThe current best algorithms apply proactive particle \u00a3lter (PF) techniques to solve this\nproblem [4, 8, 12, 13]. PF techniques sample from the path posterior. When closing a loop,\nrandom variations in these samples lead to different loop closures. As long as the correct\nsuch closure is in the set of surviving particle \u00a3lters, the correct map can be recovered.\nIn the context of our present system, this approach suffers from two disadvantages: it is\ncomputationally expensive due to its proactive nature, and it provides no mechanism for\nrecovery should the correct loop closure not be represented in the particle set.\nOur approach overcomes both of these limitations. When closing a loop, it always picks the\nmost likely data association. However, it also provides a mechanism to undo and redo past\ndata association decisions. The exact data association algorithm involves a step that moni-\ntors the likelihood of the most recent sensor measurement given the map. If this likelihood\nfalls below a threshold, data association constraints are recursively undone and replaced\nby other constraints of decreasing likelihood (including the possibility of not generating a\nconstraint at all). The search terminates if the likelihood of the most recent measurement\n\n\f(a)\n\n(b)\n\ncon\u00a4ict\n\n''\u2026\n\n(cid:190)\n\nstart\n\nmap after adjustment\n\nFigure 4: Example of our lazy data association technique: When closing a large loop, the robot\n\u00a3rst erroneously assumes the existence of a second, parallel hallway. However, this model leads to\na gross inconsistency as the robot encounters a corridor at a right angle. At this point, our approach\nrecursively searches for improved data association decisions, arriving at the map shown on the right.\n\nexceeds the threshold [7]. In practice, the threshold test works well, since global inconsis-\ntencies tend to induce gross inconsistencies in the robot\u2019s measurements at some point in\ntime.\nThe algorithm is illustrated in Figure 4. The left panel shows the ML association after\ntraversing a large loop inside a mine: At \u00a3rst, it appears that the existence of two adjacent\ncorridors is more likely than a single one, according to the estimated robot motion. How-\never, as the robot approaches a turn, a noticeable inconsistency is detected. Inconsistencies\nare found by monitoring the measurement likelihood, using a threshold for triggering an\nexception. As a result, our data association mechanism recursively removes past data asso-\nciation constraints back to the most recent loop closure, and then \u201ctries\u201d the second most\nlikely hypothesis. The result of this backtracking step is shown in the right panel of Fig-\nure 4. The backtracking requires a fraction of a second, and with high likelihood leads to\na globally consistent map and, as a side-effect, to an improved estimate of the map coordi-\nnates \u00a5. Figure 2b shows a proto-typical corrected map, which is globally consistent.\n\n3 Autonomous Navigation\n2D maps are suf\u00a3cient for localizing robots inside mines; however, they are insuf\u00a3cient to\nnavigate a robot due to the rugged nature of abandoned mines. Our approach to navigation\nis based on 3D maps, acquired in periodic intervals while the vehicle suspends motion to\nscan its environment. A typical 3D scan is shown in Figure 5a; others are shown in Figure 7.\n\n3.1\n\n2 1\n2 D Terrain Maps\n\nIn a \u00a3rst processing step, the robot projects local 3D maps onto 2 1\n2 D terrain maps, such as\nthe one shown in Figure 5b. The gray-level in this map illustrates the degree at which the\nmap is traversable: the brighter a 2D location, the better suited it is for navigation.\nThe terrain map is obtained by analyzing all measurements hx; y; zi in the 3D scan\n(where z is the vertical dimension). For each rectangular surface region fxmin; xmaxg \u00a3\nfymin; ymaxg, it identi\u00a3es the minimum z-value, denoted z. It then searches for the largest\nz value in this region whose distance to z does not exceed the vehicle height (plus a safety\nmargin); this value will be called \u201ez. The difference \u201ez \u00a1 z is the navigational coef\u00a3cient:\nit loosely corresponds to the ruggedness of the terrain under the height of the robot. If no\nmeasurement is available for the target region fxmin; xmaxg \u00a3 fymin; ymaxg, the region is\nmarked as unknown. For safety reasons, multiple regions fxmin; xmaxg \u00a3 fymin; ymaxg\noverlap when building the terrain map. The terrain map is subsequently convolved with\na narrow radial kernel that serves as a repellent potential \u00a3eld, to keep the robot clear of\nobstacles.\n\n3.2 Con\u00a3guration Space Maps\nThe terrain map is used to construct a collection of maps that describe the robot\u2019s con\u00a3g-\nuration space, or C-space [10]. The C-space is the three-dimensional space of poses that\n\n\f(a)\n\n(b)\n\n(c)\n\nFigure 5: (a) A local 3D model of the mine corridor, obtained by a scanning laser range \u00a3nder. (b)\n2 D terrain map extracted from this 3D snapshot: the brighter a location, the\nThe corresponding 2 1\neasier it is to navigate. (c) Kernels for generating directional C-space maps from the 2 1\n2 D terrain\nmap. The two black bars in each kernel correspond to the vehicle\u2019s tires. Planning in these C-space\nmaps ensures that the terrain under the tires is maximally navigable.\n\nthe vehicle can assume; it comprises the x-y location along with the vehicle\u2019s orientation\n(cid:181). The C-space maps are obtained by convolving the terrain map with oriented kernels that\ndescribe the robot\u2019s footprint. Figure 5c shows some of these kernels: Most value is placed\nin the wheel area of the vehicle, with only a small portion assigned to the area in between,\nwhere the vehicle\u2019s clearance is approximately 30 centimeters. The intuition of using such\na kernel is as follows: Abandoned mines often possess railroad tracks, and while it is per-\nfectly acceptable to navigate with a track between the wheels, traversing or riding these\ntracks causes unnecessary damage to the tires and will increase the energy consumption.\nThe result of this transformation is a collection of C-space maps, each of which applies to\na different vehicle orientation.\n\n3.3 Corridor Following\nFinally, A* search is employed in C-space to determine a path to an unexplored area. The\nA* search is initiated with an array of goal points, which places the highest value at loca-\ntions at maximum distance straight down a mine corridor. This approach \u00a3nds the best path\nto traverse, and then executes it using a PD controller.\nIf no such path can be found even within a short range (2.5 meters), the robot decides\nthat the hallway is not navigable and initiates a high-level decision to turn around. This\ntechnique has been suf\u00a3cient for our autonomous exploration runs thus far (which involved\nstraight hallway exploration), but it does not yet provide a viable solution for exploring\nmultiple hallways connected by intersections (see [16] for recent work on this topic).\n\n4 Results\nThe approach was tested in multiple experiments, some of which were remotely operated\nwhile in others the robot operated autonomously, outside the reach of radio communication.\nOn October 27, 2002, Groundhog was driven under manual control into the Florence Mine\nnear Burgettstown, PA. Figure 6b shows a picture of the tethered and remotely controlled\nvehicle inside this mine, which has not been entered by people for many decades.\nIts\npartially \u00a4ooded nature prevented an entry into the mine for more than approximately 40\nmeters. Maps acquired in this mine are shown in Figure 9.\nOn May 30, 2003, Groundhog successfully explored an abandoned mine using the fully\nautonomous mode. The mine, known as the Mathies Mine near Pittsburgh, is part of a\nlarge mine system near Courtney, PA. Existing maps for this mine are highly inaccurate,\nand the conditions inside the mine were unknown to us. Figure 6a shows the robot as it\nenters the mine, and Figure 7a depicts a typical 3D scan acquired in the entrance area.\n\n\f(a)\n\n(b)\n\nFigure 6: (a) The vehicle as it enters the Mathies Mine on May 30, 2003. It autonomously descended\n308 meters into the mine before making the correct decision to turn around due to a blockage inside\nthe mine. (b) The vehicle, as it negotiates acidic mud under manual remote control approximately 30\nmeters into the Florence Mine near Burgettstown, PA.\n\n(a)\n\n(b)\n\nFigure 7: 3D local maps: (a) a typical corridor map that is highly navigable. (b) a map of a broken\nceiling bar that renders the corridor segment unnavigable. This obstacle was encountered 308 meters\ninto the abandoned Mathies Mine.\n\nFigure 8: Fraction of the 2D mine map of the Mathies Mine, autonomously explored by the Ground-\nhog vehicle. Also shown is the path of the robot and the locations at which it chose to take 3D scans.\nThe protruding obstacle shows up as a small dot-like obstacle in the 2D map.\n\n(a)\n\n(b)\n\n(c)\n\nFigure 9: (a) A small 2D map acquired by Groundhog in the Florence Mine near Burgettstown, PA.\nThis remotely-controlled mission was aborted when the robot\u2019s computer was \u00a4ooded by water and\nmud in the mine. (b) View of a local 3D map of the ceiling. (c) Image acquired by Groundhog inside\nthe Mathies Mine (a dry mine).\n\n\fAfter successfully descending 308 meters into the Mathies Mine, negotiating some rough\nterrain along the way, the robot encountered a broken ceiling beam that draped diagonally\nacross the robot\u2019s path. The corresponding 3D scan is shown in Figure 7b: it shows rubble\non the ground, along with the ceiling bar and two ceiling cables dragged down by the bar.\nThe robot\u2019s A* motion planner failed to identify a navigable path, and the robot made the\nappropriate decision to retreat. Figure 8 shows the corresponding 2D map; the entire map\nis 308 meters long, but here we only show the \u00a3nal section, along with the path and the\nlocation at which the robot stop to take a 3D scan. An image acquired in this mine is\ndepicted in Figure 9c.\n\n5 Conclusion\nWe have described the software architecture of a deployed system for robotic mine map-\nping. The most important algorithmic innovations of our approach are new, lazy techniques\nfor data association, and a fast technique for navigating rugged terrain. The system has\nbeen tested under extreme conditions, and generated accurate maps of abandoned mines\ninaccessible to people.\n\nAcknowledgements\nWe acknowledge the contributions of the students of the class 16865 Mobile Robot Development\nat CMU who helped build Groundhog. We also acknowledge the assistance provided by Bruceton\nResearch Mine (Paul Stefko), MSHA, PA-DEP, Workhorse Technologies, and the various people\nin the mining industry who supported this work. Finally, we also gratefully acknowledge \u00a3nancial\nsupport by DARPA\u2019s MARS program.\n\nReferences\n[1] C. Baker, Z. Omohundro, S. Thayer, W. Whittaker, M. Montemerlo, and S. Thrun. A case study\n\nin robotic mapping of abandoned mines. FSR-03.\n\n[2] P. Besl and N. McKay. A method for registration of 3d shapes. PAMI 14(2), 1992.\n[3] M. Bosse, P. Newman, M. Soika, W. Feiten, J. Leonard, and S. Teller. An atlas framework for\n\n[4] A. Eliazar and R. Parr. DP-SLAM: Fast, robust simultaneous localization and mapping without\n\nscalable mapping. ICRA-03.\n\npredetermined landmarks. IJCAI-03.\n\n[5] Anshul Gupta, George Karypis, and Vipin Kumar. Highly scalable parallel algorithms for sparse\n\nmatrix factorization. Trans. Parallel and Distrib. Systems, 8(5), 1997.\n\n[6] J.-S. Gutmann and K. Konolige. Incremental mapping of large cyclic environments. CIRA-00.\n[7] D. H\u00a8ahnel, W. Burgard, B. Wegbreit, and S. Thrun. Towards lazy data association in SLAM.\n\n11th International Symposium of Robotics Research, Sienna, 2003.\n\n[8] D. H\u00a8ahnel, D. Fox, W. Burgard, and S. Thrun. A highly ef\u00a3cient FastSLAM algorithm for gen-\nerating cyclic maps of large-scale environments from raw laser range measurements. Submitted\nto IROS-03.\n\n[9] D. H\u00a8ahnel, D. Schulz, and W. Burgard. Map building with mobile robots in populated environ-\n\n[10] J.-C. Latombe. Robot Motion Planning. Kluwer, 1991.\n[11] F. Lu and E. Milios. Globally consistent range scan alignment for environment mapping. Au-\n\nments. IROS-02.\n\ntonomous Robots: 4, 1997.\n\n[12] M. Montemerlo, S. Thrun, D. Koller, and B. Wegbreit. FastSLAM 2.0: An improved particle\n\u00a3ltering algorithm for simultaneous localization and mapping that provably converges. IJCAI-\n03.\n\n[13] K. Murphy. Bayesian map learning in dynamic environments. NIPS-99.\n[14] K.P. Murphy, Y. Weiss, and M.I. Jordan. Loopy belief propagation for approximate inference:\n\nAn empirical study. UAI-99\n\n1988.\n\n[15] W. H. Press. Numerical recipes in C: the art of scienti\u00a3c computing. Cambridge Univ. Press,\n\n[16] R. Simmons, D. Apfelbaum, W. Burgard, M. Fox, D. an Moors, S. Thrun, and H. Younes.\n\nCoordination for multi-robot exploration and mapping. AAAI-00.\n\n[17] M. J. Wainwright. Stochastic processes on graphs with cycles: geometric and variational ap-\n\nproaches. PhD thesis, MIT, 2002.\n\n\f", "award": [], "sourceid": 2458, "authors": [{"given_name": "David", "family_name": "Ferguson", "institution": null}, {"given_name": "Aaron", "family_name": "Morris", "institution": null}, {"given_name": "Dirk", "family_name": "H\u00e4hnel", "institution": null}, {"given_name": "Christopher", "family_name": "Baker", "institution": null}, {"given_name": "Zachary", "family_name": "Omohundro", "institution": null}, {"given_name": "Carlos", "family_name": "Reverte", "institution": null}, {"given_name": "Scott", "family_name": "Thayer", "institution": null}, {"given_name": "Charles", "family_name": "Whittaker", "institution": null}, {"given_name": "William", "family_name": "Whittaker", "institution": null}, {"given_name": "Wolfram", "family_name": "Burgard", "institution": null}, {"given_name": "Sebastian", "family_name": "Thrun", "institution": null}]}