Modern Robot Mapping TechniquesA broad mini-survey in outline form!

Kristopher R. BeeversDepartment of Computer ScienceRensselaer Polytechnic Institute

beevek@cs.rpi.edu

January 12, 2006

Abstract

Traditionally, two distinct approaches have dominated the robot mapping literature:topological mapping and geometric mapping. A topological map is an abstract encod-ing of the structural characteristics of an environment. Often, topological maps representthe environment as a set of “distinctive places” (e.g., rooms), connected by sequences ofrobot behaviors (e.g., hall-following). A geometric map, on the other hand, is a represen-tation of the precise geometric characteristics of the environment, much like a floorplan.Typical approaches to acquiring geometric maps rely heavily on probabilistic techniquesdue to inherent uncertainties in geometric measurements. Recently, probabilistic meth-ods for topological mapping have also begun to appear, and the mapping literature hasmoved in some sense toward a confluence of topological and geometric techniques inwhich the strengths of both approaches can be exploited. This mini-survey examinesthe convergence of topological and geometric mapping techniques by highlighting sev-eral important works in the literature that signify the trend.

1 IntroductionRobot mapping is an exceedingly well-studied problem, but it is not yet “solved” and isthe subject of large amounts of ongoing research. The basic goal of robot mapping is toacquire (usually autonomously) some representation of the environment in which a robotis situated. The particular type of map depends on a number of factors, including thepurpose of the map (e.g., for planning, navigation, or use by people), the characteristics ofthe robot (e.g., sensing and computation capabilities and the robot’s motion and perceptualmodels), the type of environment (e.g., indoor or outdoor, static or dynamic) and so on.These many factors have led to the development of several mapping paradigms. Twoof these paradigms, topological mapping and geometric mapping, dominate the modernrobot mapping literature.

Topological maps offer a qualitative representation of the environment. In general,topological maps represent distinctive places that can be reliably recognized by a robot,along with the connectivity of these places via paths that can be reliably followed by arobot. In this way, the global structure (or topology) of the environment is encoded. How-ever, topological maps need not be embedded in a single global metrical frame of reference— the robot can know “where it is” in terms of the structure of the environment with-out knowing “where it is” to a high degree of global metrical accuracy. This property oftopological maps makes them especially attractive for use with robots with limited sensing

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or noisy odometry or for use in very large environments where taking globally consistentmeasurements is difficult. It is also useful from a planning and navigation standpoint:topological maps are typically represented as graphs, and finding paths from one distinc-tive place to another is generally as simple as a straightforward graph search operation.

Geometrical maps are instead concerned with producing a globally accurate, geometri-cally precise representation of the environment. Such maps can be embedded directly into ametrical space (typically the Euclidean plane in the case of two-dimensional mapping) andoften are suitable for visual inspection in that they yield a “picture” of the environment thatcan be interpreted similarly to, say, a floor plan or road map. Geometrical maps are com-monly represented either as a set of embedded landmarks, or as a probabilistic grid (knownas an occupancy grid) where the intensity of each grid cell represents the likelihood thatthe corresponding region in the environment is unoccupied. The majority of modern robotmapping literature has focused on the problem of building geometrical maps using SLAM(simultaneous localization and mapping) techniques that employ probabilistic inference torecover the map from the robot’s motions and sensor measurements. Recent progress hasyielded algorithms for SLAM that are capable of building geometrically accurate maps inlarge environments in real time. Often, geometrical maps (particularly occupancy maps)are suitable for planning using grid-based techniques.

Recent work on both topological and geometrical mapping techniques has begun to ex-hibit some “cross-pollination” of ideas. In particular, ideas from the geometrical mappingliterature are being applied to topological maps. In some cases, hybrid mapping tech-niques have been developed that acquire global topological maps consisting of smaller,“local” geometrical maps at distinctive places (Tomatis et al., 2002). Perhaps more inter-estingly, geometry has been incorporated directly into algorithms for building topologicalmaps, as in (Kuipers, 2000) and (Shatkay and Kaelbling, 2002). Inference techniques thatwere first exploited by the SLAM community have been applied to the problem of inferringthe structure of an environment (Ranganathan et al., 2005). In some cases, ideas flow in theother direction. For example, in (Stachniss et al., 2005), the authors choose from a discreteset of actions to take (typically a characteristic of the topological approach) while buildinga geometrical map.

In this “mini-survey,” we highlight a representative set of papers from the modernmapping literature along with a few of the seminal papers that inspired modern tech-niques. Some of these papers present purely topological or purely geometrical mappingtechniques. Others combine ideas from both paradigms. We believe these papers sup-port the hypothesis that a convergence is beginning to appear in the literature: maps thataddress issues of both topology and geometry — and that exploit both structure and ge-ometry — can be acquired accurately and are more useful in practice than maps of eitherindividual type.

The mini-survey presents reasonably detailed summaries of the main ideas in each pa-per and attempts to point out contributions, details of the approach, and potential issueswith each technique. We begin with a set of five “primary papers” that introduce the topo-logical and geometric paradigms through modern techniques and show how ideas havebeen applied across paradigms. We then continue with twelve “secondary papers” thatgive more in-depth background and further emphasize the convergence of topological andgeometric mapping.

2 Primary papersWe begin with five recent papers. Kuipers (2000) and Thrun et al. (2004) are representativepapers from the topological mapping and SLAM literature, respectively. Ranganathan et al.

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(2005) shows how a Bayesian inference technique from the SLAM literature can be appliedtoward building topological maps. Shatkay and Kaelbling (2002) describes a different typeof topological mapping algorithm based on HMMs and shows how geometrical informationfrom odometry measurements can be incorporated into the approach. Finally, Stachnisset al. (2005) presents an autonomous exploration method for use in geometrical mappingthat considers a discrete set of actions to be taken, much like topological approaches.

2.1 Kuipers (2000)• Main contribution: a complete description of the Spatial Semantic Hierarchy (SSH),

which is presented as being both a model of the human cognitive map and a methodfor robot exploration and mapping that is primarily topological in nature

• Breaks down mapping beyond metrical and topological into a system of hierarchicallayers with different knowledge representations:

1. sensory: continuous sensing

2. control: continuous behavior

3. causal: discrete states/actions4. topological: hypothesized places/paths to account for observed states/actions

5. metrical: merging of local geometrical information using topological knowledge

• Topological map is necessary because “distinctive places” are not truly unique —they do not have unique sensing signatures

• Control:

– behaviors that take the robot between locally distinctive states

– (1) leave a distinctive state; (2) execute behavior to reach the vicinity of anotherdistinctive state; (3) hill-climb until at the actual state

– if local maps are available, localization in the local map can replace hill-climbing

– trajectory-following control laws, e.g. WF/HF, keep two out of three of therobot’s degrees of freedom fixed

• Causal:

– abstracts sequences of control laws into schemas: 〈V, A, V ′〉, where A is an actionand V, V ′ are sensory images (views) at distinctive states

– consider two primary types of actions:

∗ turns leave the robot at the same place∗ travels take the agent between places

– a routine is a sequence of schemas, e.g. a path through distinctive states

– Question: what is the difference between the causal and topological layers ofSSH?

∗ causal does not deal with the problem of revisitation/registration, which isresolved in the topological layer

• Topological:

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– create structural relations by abduction1, “positing the minimal additional set ofplaces, paths, and regions required to explain the sequence of observed viewsand actions.”

– defines topological relations between views, places, paths, etc. in a FOL frame-work

– loop-closing done using physical exploration (“rehearsal procedure”) to checkneighbors of places hypothesized to be the same

• Metrical:

– goal: map that specifies robot configuration and locations of environment fea-tures in a single global reference frame

– “patchwork mapping:” local maps with separate frames of reference– not really all that different from SLAM, where “local maps” are, e.g., scans at each

time step, or features extracted over a window of steps

• SSH is capable of building maps opportunistically although place ambiguity increasesbecause loop-closing actions can’t be performed

• SSH supports qualitative mapping using limited sensing since it is still possible toimplement effective control laws given sensing limitations

• “The rehearsal procedure for resolving topological ambiguity is effective but ad hoc,and should perhaps be replaced by a more principled POMDP-based strategy”

• “A state s is localizable if, starting from s, the agent can reliably reach any distinctivestate in the topological map.” Question: can we precisely define the class of localiz-able environments?

• “A state s is reachable if, starting at any distinctive state in the environment, thereis a reliable method for the agent to travel to s . . . Clearly, every reachable state islocalizable, but not necessarily vice versa.” Question: Is this really true? I’m unclearon it.

2.2 Thrun et al. (2004)• Main contribution: FastSLAM 1.0 and 2.0, particle filtering algorithms for SLAM that

estimate trajectories (not just poses) and maps using a particle filtering approach.FastSLAM 2.0 uses an improved proposal distribution that takes into account boththe robot’s motion model and its measurement model to improve particle diversityby sampling in areas of high likelihood. Proof of convergence of FastSLAM 2.0 in thesimplified case of linear Gaussian SLAM.

• SLAM:

– The usual landmark based SLAM problem (Bayes filter):

p(st, Θ|zt, ut, nt) = ηp(zt|st, θnt , nt)

∫dst−1 p(st|st−1, ut)p(st−1, Θ|zt−1, ut−1, nt−1)

∗ when both measurement and motion models are linear and Gaussian: Kalmanfilter; nonlinear Gaussian: EKF approximates using Taylor expansion

1Dictionary.com: The process of inference to the best explanation. “Abduction” is sometimes used to meanjust the generation of hypotheses to explain observations or conclusions.

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– Filter for calculating posterior over trajectories and maps:

p(st, Θ|nt, zt, ut) = ηp(zt|st, θnt , nt)p(st|st−1, ut)p(st−1, Θ|nt−1, zt−1, ut−1)

Note that the pose at time t − 1, st−1, is not integrated out.

• Approach:

– Bayesian filtering using a Rao-Blackwellized particle filter:∗ particles represent the distribution of robot trajectories; enables nonlinear

motion models since particle filters can represent multimodal distributions∗ every particle has its own trajectory and map∗ EKFs to represent the parameters of each landmark in the maps (requires

linearization of the measurement model)– Exploit a structural property of SLAM: correlations between map features arise

only through robot pose uncertainty; given the path, errors in feature estimatesare independent. By representing the trajectory distribution using particles andconditioning on the particles, the map can be factored into small filters for eachfeature:

p(st, Θ|nt, zt, ut) = p(st|nt, zt, ut)N

∏n=1

p(θn|st, nt, zt)

– Data association:∗ per-particle, enabling multiple hypotheses: only need a few particles to find

the correct data association∗ maximum likelihood (ML) or data association sampling (DAS)∗ use a threshold minimum value of likelihood for initializing a new feature

instead of associating with one in the map– Negative information: track log-odds of existence for features based on whether

they are sensed when in range– FastSLAM 2.0: proposal distribution takes measurement into consideration

∗ FastSLAM 1.0: sample poses based on control, then use measurement to re-sample; more resampling when control accuracy much less than sensingaccuracy

∗ FastSLAM 2.0 approach samples more in areas of high likelihood so less re-sampling occurs and higher particle diversity is maintained

– Basic algorithm:1. sample new poses for each particle based on motion command and model

(and measurement and sensing model for FastSLAM 2.0)2. observe a landmark and do data association for each particle3. update estimate for associated landmark in the maps of each particle4. compute importance factor and resample particles

– An extension: considering multiple features simultaneously during data asso-ciation (no landmark can be observed in multiple locations at the same time —decreases errors)

• Big assumptions:

– measurement model is nonlinear, Gaussian

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– motion model is nonlinear, Gaussian

– FastSLAM 2.0: linear Gaussian approximation of proposal distribution in orderto do sampling; however, this is done on a per-particle basis so the PF still rep-resents multimodal distributions

• Primary benefits of FastSLAM:

– considers multiple data association hypotheses (different particles)

– computational: O(M log N) instead of O(N2) for EKF

• Question: is it true that the number of particles needed depends on the length of thetrajectory (since we’re estimating a posterior over trajectories)? [Yes.]

– Related important question: how many particles are necessary to produce acorrect trajectory/map with high probability? Or even just to close a single loopwith high probability? “It is unknown at this time whether the number of par-ticles necessary to achieve a given accuracy is polynomial or exponential in thesize of the loop.”

– Related problem: lack of “long-range correlations” when traversing large loopsdue to loss of particle diversity near the beginning of the loop

• Convergence:

– FastSLAM 2.0 converges for linear Gaussian motion and measurement models(LG-SLAM) to the correct map in expectation with one particle as the number ofobservations → ∞ if the location of one feature is known a priori

– Proof: robot pose error only shrinks in expectation; thus pose error eventuallyis smaller than largest feature error; in turn, the largest feature error shrinks

– LG-SLAM: robot in a Cartesian space with noise-free compass and sensors thatmeasure distances along coordinate axes

– for LG-SLAM, number of particles required for convergence is independent of the sizeof the map

– no results for more realistic SLAM problems

– no results describing convergence speed

• Implementation:

– naı̈ve implementation: O(MN) updates

– organize map as balanced binary tree, share unmodified features between parti-cles with a common ancestor: O(M log N) updates with known data association

– with unknown data association: O(M log N) if we assume the number of in-range features is independent of map size

• Results using a Brownian motion model equivalent to those with real odometry;Question: is this just due to the use of “easy” data sets?

• Question: it seems like the main goal (which the experiments are mainly gearedtoward) is to extract good trajectories; recovering a “correct map” is less important?

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2.3 Ranganathan et al. (2005)• Main contribution: the idea of computing the posterior distribution over the space

of topologies given odometry measurements from traveling between landmarks andsensing signatures from the landmarks themselves.

• Claims that the perceptual aliasing problem is hard by considering it as a balls andbins problem (set partitioning):

– each measurement (ball) is assigned to some bin (landmark)– the set of all possible assignments of measurements to landmarks is the space of

all topologies– Question: isn’t this a dramatic oversimplification? Don’t we have significant

problem-specific knowledge such that the space is smaller? For example, for twolandmarks each with several measurements, switching a single measurementbetween the landmarks may not change the topology — just the metric details.At least, this shows that the space of topologies is not exactly one-to-one withthe space of set partitions.

– the space of topologies (if it is taken to be equivalent to the space of set parti-tions) is exponentially large in the size of the environment; this leads to the needfor a sampling based representation.

• Important idea: “an ideal mapping algorithm, capable of producing an accurate mapin any environment using just the available measurements, may not exist. Instead,the mapping algorithm should be able to reason about and flag any uncertainty itmight have about the maps it is generating.”

• Big advantages:

– systematic approach to solving the loop closing (perceptual aliasing) problem intopological maps

– multi-hypothesis decision theoretic approach: can keep around several topolo-gies until one emerges as the best (note: this paper doesn’t discuss the problemof choosing a particular topology, just finding the posterior over all topologies)

– topology estimates can be obtained given very little information

• Some assumptions:

– different types of observations (e.g., appearance and odometry) are independent– measurements taken at different landmarks are independent (i.e. appearance

information at two landmarks is independent)– known priors over landmark locations in the environment, measurements (appearance)

in the environment, and topologies. The latter is assumed to be well-approximatedby the “Classical Occupancy Distribution” from Urn/Ball-bin models, which inturn requires a prior on the total number of landmarks in the environment.

– distribution assumptions: proposal is multivariate Gaussian, Gaussian mea-surement noise, σ2

jk ∼ IG, etc.

• Problems:

– the algorithm appears to be offline: optimization is performed for a fixed set ofmeasurements and it is unclear how the posterior can be efficiently recomputedafter the acquisition of new data.

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– proposal distibution: uniform over two simple actions (split a landmark ormerge two landmarks); Question: what is an ideal proposal distribution? whypick landmarks for splitting or merging uniformly rather than according to,e.g., number of assigned measurements or uncertainty? why is it a good thingthat the proposal doesn’t incorporate any domain knowledge?

– unclear how the algorithm scales as the number of landmarks increases (thelargest experiment considers only 12 landmarks, computation time is not dis-cussed, parameters such as sample count are not given)

– inference is sensitive to proposal distribution parameters: fundamental issuein combining two irreconcilable measures of “goodness”: continuous pdf fromodometry information and discrete preference for fewer nodes in the map.

– convergence properties?

2.4 Shatkay and Kaelbling (2002)You will come to a place where the streets are not marked.Some windows are lighted but mostly they’re darked.A place you could sprain both your elbow and chin!Do you dare to stay out? Do you dare to go in?. . .And if you go in, should you turn left or right. . .or right-and-three-quarters? or, maybe, not quite?. . .Simple it’s not, I’m afraid you will find,for a mind-maker-upper to make up his mind.

Oh, the Places You’ll Go, Dr. Seuss.

• Main contribution: a formal framework for incorporating odometric informationand geometrical constraints into HMM/POMDP topological environment models

• A good description of the environment modeling problem: “A robot moving in theenvironment suffers from two main limitations: its noisy sensors prevent it from con-fidently knowing where it is, while its noisy effectors preventing from knowing withcertainty where its actions will take it. We concentrate here on structured environ-ments, which can in turn be characterized by two main properties: such environ-ments consist of vast uneventful and uninteresting areas, and are interspersed withrelatively few interesting positions or situations.”

• Probabilistic topological map: states represent interesting configurations, edges rep-resent actions leading between states; pdfs over transitions and possible observationsat each situation model noisy effectors and sensors

• First work to learn probabilistic topological models directly without making use ofhuman-provided or metrical models for bootstrapping

– states are more general than just geometrical locations: may include other infor-mation, e.g. battery level, etc.

– Main improvement over previous HMM/POMDP mapping approaches: use odom-etry to help learn the environment structure

• “When learning a model that captures interaction, the agent acquiring the model isthe one who is also using it. Hence, the noisy sensors and actuators specific to theagent are reflected in the model. A different model is likely to be needed by differentagents.”

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• Previous work:

– Kuipers and Byun (1991); Kuipers (2000): deterministic topological maps; rely onability to either uniquely identify states or use the ad-hoc rehearsal procedureto differentiate

– “diktiometric” maps: topological maps with metric relations between nodes

– POMDP models: Koenig and Simmons (1996): use an initial human-providedmap; represent halls as chains of one-meter segments, yielding models with sizeproportional to that of the environment

– learning correct HMMs is hard; instead, learn an HMM where only weak guaranteesexist about the goodness of the model, but the learning procedure is tuned inorder to obtain practically good results

• HMMs vs. POMDPs:

– a POMDP requires the robot to make decisions regarding its next action at everystate; an HMM allows only one action to be executed from each state

– HMM approach extends to POMDPs: learn an HMM for each possible action.Question: aren’t there an exponential number of such HMMs to learn?

• Major restrictive assumptions:

– the number of states in the model (size of the map) is known a priori∗ this essentially means that all loops are closed!

– static environments (“there is an inherent ‘true’ odometric relation between theposition of every two states in the world”)

– noise is Gaussian (von Mises ≡ circular Gaussian for rotational error)

• Approach:

– “augmented” HMM: states, observation vectors for each state (sensing signa-tures), transition probabilities, observation probabilities, mean/variance of odo-metric relations between each pair of states

– geometrical constraints: requires R (odometric relations) to be geometricallyconsistent (embedded in 2D): d(a, a) = 0, d(a, b) = −d(b, a) (anti-symmetry),d(a, c) = d(a, b) + d(b, c) (additivity)

∗ full constraints reduce quality of learned models for large environments;can remove additivity constraint for these environments

– two models:

1. global coordinate system: assume rectilinearity, no cumulative rotationalerror

2. state-relative coordinate systems: each state has its own frame; requiresmapping of headings onto an additive linear space in order to enforce addi-tivity constraint; unproven that this preserves convergence properties of learningHMMs

– learning algorithm:

∗ modification of Baum-Welch, an EM algorithm:1. E-step: compute state occupation and state transition probabilities given

data and current model

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2. M-step: find a new model that maximizes likelihood of data given themodel and probabilities

∗ finds local maxima; use multiple initial model guesses and learning runs totry and find the best

∗ in addition to normal HMM learning, incorporate odometric informationand geometrical constraints· Problem: constraints introduce interdependencies between optimal mean

and variance estimates· resolving using “lag-behind” updates: calculate new estimate for mean

using old variance, then use new mean to update variance

– initial model

∗ use a clustering algorithm to extract the set of states, e.g., k-means (underperpendicularity constraints), “tag-based initialization” (bucketing of odo-metric readings; when cumulative rotational error exists)

∗ Big problem: all of the loop-closing work is done during initialization incompletely ad-hoc fashion! After initialization, the number of states is fixed.

∗ after initialization, we just learn the transition probabilities and geometricrelationships (i.e. we learn connectivity, given the set of states)

∗ there are some approaches (“cross-validation,” “regularization,” heuristics)for selecting model complexity; Question: to what extent are these generallearning methods vs. tools tailored for solving the loop-closing problem?

• Results:

– using odometry speeds up convergence and gives more accurate maps thanlearning an HMM considering only sensing signatures

– computational performance unclear

2.5 Stachniss et al. (2005)• Main contribution: an exploration strategy for SLAM that selects an action to take

based on the expected utility of the action. Expected utility is a value encoding thetradeoff between the cost of an action and the “expected information gain” it pro-vides.

• SLAM itself is done using occupancy grids, scan matching, and a Rao-Blackwellizedparticle filter approach.

• In choosing an action, a robot should take into account both the uncertainty in itsmap and the uncertainty in its trajectory in order to satisfy the dual objectives of:

1. closing loops

2. exploring new territory

• Approach:

– Entropy of the RBPF system is the sum of two terms:

1. entropy of the posterior over the trajectories of the robot2. sum of the entropies of the maps for each trajectory weighted by the likeli-

hood of the trajectory

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Thus minimizing uncertainty requires reducing the uncertainty of each map aswell as the uncertainty of the trajectories.∗ Entropy of an occupancy grid map: sum of entropies of all cells∗ Entropy of the trajectory posterior is hard to compute; approximate it as the

average entropy of the pose posteriors over time:

H(p(x1:t|dt)) ≈1t

t

∑t′=1

H(p(xt′ |dt))

Problem with this: entropies of pose posteriors are not independent, par-ticularly when the robot revisits a place. Thus, also average over placescovered by the robot.

– Pick an action: Given a set of actions, compute the expected information gainE[I(at)] for each, along with the cost, and choose the action that maximizes

E[U(at)] = E[I(at)] − α · cost(at)

where information gain is the difference in total system entropy before and afterthe action occurs.

∗ Requires integration over all possible measurement sequences for the givenaction

∗ Approximated using ray-casting of lasers in the map of each particle∗ For computational reasons, only done with one particle trajectory for each

action, selected randomly with probability proportional to particle weight.Question: is this better than using the expected trajectory?

∗ Idea: perhaps sampling a small set of locations on the action trajectory andonly performing ray-casting at these locations is sufficient

– Exploration is complete when no action has positive expected utility

– Generating actions:

∗ exploration actions: for each frontier between explored/unexplored areas,generate an action leading robot along shortest path to the frontier

∗ revisiting actions: move the robot back along previously taken path∗ loop-closing actions: move the robot forward to a hypothesized loop-closing

opportunity

• Claim: this approach “can be regarded as an approximation of the POMDP [approachto choosing an exploration strategy] with a one-step lookahead” — a benefit sincethe POMDP approach is to compute the expected optimal strategy w.r.t. belief over allpossible states of the world, which quickly becomes intractable.

3 Secondary papersWe now highlight several more papers. We begin by building some background by ex-amining important precursors of modern techniques in topological mapping (Chatila andLaumond, 1985; Kuipers and Byun, 1991) and geometrical mapping (Cox and Leonard,1994; Fox et al., 1999; Konolige, 1997; Murphy, 2000; Pagac et al., 1998; Smith et al., 1990;Thrun et al., 1998). We then introduce several other interesting mapping papers from bothparadigms (Choset and Nagatani, 2001; Leonard et al., 2002; Lu and Milios, 1997).

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3.1 BackgroundThe following papers have been particularly influential and are responsible for many ofthe ideas that led to modern algorithms for topological mapping and SLAM.

3.1.1 Chatila and Laumond (1985)

• Main contribution: one of the first attempts to deal with issues of consistency anduncertainty in robot mapping; also introduces the idea of extracting a topologicalmap from a geometrical map, and goes even further, extracting from the topologicalmap a “semantic model” that is useful in planning.

• Tiered environment models:

1. Geometrical: deduced directly from sensor data; polygonal model of the envi-ronment in an absolute frame

2. Topological: extracted from the geometrical model based on cellular decomposi-tions of the geometrical map; the topological map is itself hierarchical (e.g., room,storey, building)

3. Semantic: symbolic model for use in decision making; in fact, this informationcan be incorporated directly into the topological model (which is the approach takenin the paper)

• Gaussian error models for sensing and odometry

• Basic approach: same as SLAM (match prediction with perception); developed priorto modern SLAM techniques

• Global correction: updating the most recent pose requires correction of the entiretrajectory. Question: this is basically an iterative EM process; in this paper, is just oneiteration going on? The “fading” model is a little unclear and ad-hoc.

3.1.2 Kuipers and Byun (1991)

• Main contribution: an early (less developed) version of the SSH (Kuipers, 2000).Includes the control, topological, and geometric levels along with the rehearsal pro-cedure for closing loops.

• Use location-specific control algorithms that define distinctive places and paths, linkedto form a topological map

• Less differentiation than more recent work on the SSH: sensory level is incorporatedinto the control level, and causal level is part of the topological level

• Using low-level control policies makes navigation and localization independent ofdetailed geometric information

• Distinctive places have sensing signatures that help to identify them (not necessarilyuniquely)

• Distinctive paths: follow midline of corridor, follow wall, etc.; points on the interiorof a large space have no qualitatively distinctive characteristics so the robot does notenter the interior

• Rehearsal procedure: same as for SSH (Kuipers, 2000)

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– in this paper, the rehearsal procedure is “not called recursively” and topologicalmatching is done at a fixed radius (one path)

– rehearsal procedure eliminates the problem of cumulative position error in “suf-ficiently well-behaved [non-self-similar] environments”

– can be used to relocalize the robot in a previously-acquired map

• Dynamic environments:

– require more sophisticated control strategies, e.g. to avoid obstacles, etc.

– requires recognition of fixed versus dynamic portions of the environment

3.1.3 Cox and Leonard (1994)

• Main contribution: a mapping approach in which uncertainty of the values of mea-surements is decoupled from uncertainty in the origins of measurements (data as-sociation uncertainty). Models the data association uncertainty using an hypothe-sis tree where branches represent different possible assignments of measurements tofeatures. This allows the postponement of data association decisions until more in-formation is available. It also allows for mapping in dynamic environments since itis easy to represent false alarm and feature deletion hypotheses.

• Based on ideas from the target tracking literature; mapping is equivalent to targettracking where the targets are features in the environment

• “The Kalman filter is a powerful tool for dealing with noise, but is of little help inmodeling the uncertainty in the origin of sensor measurements.”

• Recognizes that conditioned on the trajectory, measurement errors between featuresare uncorrelated (the observation that makes FastSLAM possible)

– this paper assumes the trajectory is precisely known so that each feature has itsown Kalman filter

– joining the stochastic map (Smith et al., 1990) with the approaches of this paperleads to serious compuational problems because of correlations

• “A global hypothesis is one possible world model of the robot’s environment to-gether with an associated probability of the hypothesis given all past and currentmeasurements. An hypothesis or world model is a collection of tracks that parti-tion the measurement data into sets, each set representing measurements assumed tooriginate from the same geometric feature.”

• Approach:

– each hypothesis is a different set of assignments of measurements to features– thus, each hypothesis predicts a different set of sensor measurements at each

time step

– upon taking new measurements, each hypothesis can be propagated by gener-ating children corresponding to possible interpretations of the data: (1) belongsto a previously known feature; (2) new feature; (3) false alarm; or (4) deletion ofa feature (from too few observations when predicted)

– the probability of each hypothesis can be computed and used in, e.g., pruning

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– in implementation, tracks with common measurements can share the measure-ments; similar to the log M data structure for FastSLAM 2.0

– Important note: this is very much the same as the core ideas behind FastSLAM.In fact, the techniques presented here are essentially a deterministic version ofFastSLAM, plus some extra heuristics for pruning.

– pruning: the number of hypotheses is exponential in the number of measure-ments; need to do pruning of unlikely hypotheses:∗ N-scan-back pruning algorithm: assumes that any ambiguity at iteration k

has been resolved by iteration k + N∗ if a hypothesis at time k has q children, at time N the sum of the probabilities

of the leaf nodes is calculated for each of the q branches; the branch withhighest probability is kept

∗ this is essentially an irrevocable decision based on looking ahead N steps∗ more pruning using a beam search technique, e.g. that keeps at most 100 hy-

potheses

• Assumes that false alarms and new features are Poisson distributed; assumes Gaus-sian measurement model

3.1.4 Fox et al. (1999)

• Main contribution: a Markov localization technique that uses a 3D grid to representa probability density over the space of robot configurations in an environment. Alsoincludes filtering techniques for discarding measurements due to dynamic obstaclesso the Markov assumption remains valid in dynamic environments, and introducesways of reducing computation so that localization can be performed in real time.

• Benefits of Markov localization with a grid representation:

– accurate global position estimation– relocalization in case of localization failure (kidnapped robot)– more accurate than techniques which assume a Gaussian posterior (unimodal

vs. multimodal) or which use topological representations (which are coarse-grained)

– using a grid enables incorporation of raw sensory input — no features need beextracted

– able to deal with dynamic environments using a filtering technique to use onlymeasurements due to known objects

• Approach:

– represent the location posterior as a fine-grained discretization of the state space(x, y, θ)

– Markov assumption (static environment): if the position of the robot is known,future measurements are independent of past measurements (and vice versa)∗ Note: this is really the same as the FastSLAM insight applied in a different

way∗ clearly violated if the environment is dynamic (e.g. contains people)

– divide updates into ones due to sensor measurements and ones due to odometrymeasurements; updates follow the familiar Bayes’ rule derivation

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– initialization: reflects location prior, e.g. uniform (unknown position) or Gaus-sian (approximately known)

– motion model: Gaussian; measurement model: incorporates possiblity of ob-struction by unknown obstacles

– in dynamic environments, to maintain Markov assumption, filter sensor mea-surements to remove “corrupted” data generated by non-permanent obstacles(e.g. people)

∗ entropy filter: exclude sensor measurements that increase entropy of theposterior (include only measurements that reflect current belief); Problem:if robot’s belief is wrong, this fails

∗ distance filter: remove measurements that are with high probability shorterthan expected; Note: computes expected value and thus doesn’t suffer fromthe same issue as the entropy filter

– grid representation: too computationally expensive∗ sensor model precomputation: given a map, precompute for each grid cell

the expected sensor measurements∗ selective update: only do the full Bayesian update for cells that have belief

greater than some small threshold

• Important advantage: works well in incomplete maps (treats unmapped objects asdynamic ones and ignores them)

• Note: this is a precursor to the Monte Carlo localization techniques introduced bythe same authors

3.1.5 Konolige (1997)

• Main contribution: A sensor model for sensors that experience specular reflection(e.g. SONAR, RADAR) to be used in updating occupancy grid maps.

• Previous models have two important problems:

1. specular reflections are not modeled; instead, an averaging assumption is madethat specular reflections affect all readings equally (i.e. the uncertainty of allreadings is increased to deal with specular reflections)

2. sensor readings from the same pose may not be independent — although thetarget occupies only one cell, traditional methods increase the probability for allcells on the arc; Question: I don’t really understand the justification for theseprobabilities being conditionally dependent (page 361)

• Approach:

– address the above two issues as follows:

1. use information that helps to determine if a sensor reading is the result ofa specular reflection to discount the data from the reading (weight updatesby their chance of not being specular); e.g., if the free space of a readingintersects a high-confidence obstacle, the reading is likely to be specular

2. at each cell, store a list of (binned) poses from which the cell has been ob-served; discard readings from poses already in the list

– assumes Gaussian error model for diffuse readings, and a static environment

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– performs a log-likelihood based update rather than a standard Bayesian update;Question: the paper says this is simpler; why?

• “The main insight is that it is possible to discount redundant and specular readingsin a local fashion by keeping track of the readings that impinge on a given cell”

3.1.6 Murphy (2000)

• Main contribution: formulation of the mapping problem in a Bayesian inferenceframework in which the map is treated as a random variable. The first to notice thatwhen conditioned on trajectory, the map posterior can be factored (Rao-Blackwellization)— i.e., the origin of the techniques used for FastSLAM (Thrun et al., 2004). Also com-pares the Bayesian approach to the EM approach where the map is treated as a fixedparameter and finds that the Bayesian method produces significantly better results.

• Uses a simplified model of the environment: a grid where each cell has a label (sensedby the robot’s noisy sensors). The robot takes actions (which experience errors) fromeach grid cell.

• Dynamic environments are included in the basic formulation but it doesn’t appearthat they are actually considered in the solution

• Bayesian inference is exponential; instead use particle filtering (see the discussionof (Thrun et al., 2004))

• Rao-Blackwellization:

– particle filtering is generally inefficient in high-dimensional spaces

– here it is tractable since if the trajectory is known, then the posterior on the mapwould be factored; thus, we can sample the trajectory and marginalize out themap analytically

– in fact we only need to store the most recent pose

• “The optimal proposal distribution (the one which minimizes the variance of the im-portance weights) takes the most recent evidence into account. Question: is thisbasically the idea of the improved proposal distribution as used in FastSLAM 2.0?

• How many particles?

– depends on motion/measurement models and the structure of the environment

– we are sampling trajectories, so the number of hypotheses (and thus the numberof particles needed) grows exponentially with time

– number of particles depends on the length of the longest cycle in the environ-ment since closing a loop kills off most particles

– using active learning can help in picking the number of particles

• Choose actions based (essentially) on “information gain.” Question: origins of theideas in (Stachniss et al., 2005)?

• Also mentions alternative approximate inference techniques: Factorial HMM, dy-namic Bayes nets

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3.1.7 Pagac et al. (1998)

• Main contribution: develops a more accurate model of the SONAR sensor for usein building occupancy grid maps, and applies Dempster-Shafer theory in an eviden-tial approach to updating the map with new information, instead of the Bayesianapproach.

• Shortcomings with Bayesian approach:

– for cell values with high uncertainty, “one cannot deduce whether simply notenough information has been received, or whether the information received wascontradictory” — information that could be useful in planning and estimatingthe quality of the posteriors

– requires use of a sensor model that isn’t well supported by the actual character-istics of SONAR (probability of empty cells decreases as a function of range); thisleads to “messy” maps without clearly defined obstacle boundaries

– requires knowledge of priors

• Approach:

– taking a SONAR measurement gives evidence about the cells intersected by theSONAR cone

– Dempster-Shafer basic probability assignment:

∗ mE = 0 (no evidence that the cells are empty) for cells on the arc at the rangereported by the sensor

∗ mF = 1/n (uncertain knowledge of which cells are occupied) on the arc∗ mE = ρ (constant) for cells within the cone (no evidence that they are occu-

pied but no knowledge of which are unoccupied)∗ mF = 0 for cells within the cone (no evidence that they are occupied)∗ m{E,F} = 1 − (mE + mF) (ignorance — lack of knowledge about the occu-

pancy of the cell due to lack of evidence one way or the other)

– initialization: every cell has mE = 0, mF = 0, m{E,F} = 1 (complete ignorance)

– with this approach, long range readings have low probability assignments

3.1.8 Smith et al. (1990)

• Main contribution: the seminal SLAM paper in which the use of the Kalman filterand EKF was proposed for maintaining probabilistic estimates of the locations of arobot and objects in the environment.

• “the explicit estimation of uncertaint spatial information makes it possible to decidein advance whether proposed operations are likely to fail because of accumulated un-certainty, and whether proposed sensor information will be sufficient to reduce theuncertainty to tolerable limits”

• Approach:

– estimate uncertain spatial relationships as probability distributions over the spa-tial variables; in practice, represent the first two moments (mean and covariance)

– system state vector: robot pose and object parameters

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– this paper does not “assume that the uncertain spatial relationships are de-scribed by normal distributions” — it estimates the mean and variance of ar-bitrary distributions; Problem: the EKF approach still doesn’t deal well withmultimodal distributions (which are often encountered in SLAM)

– approximate nonlinear spatial relationships by Taylor expansion and using theEKF

– operations for new measurements: merging (combining the estimates from twomeasurements of the same quantity), compounding (using, e.g. ij and jk to de-rive ik)

– can handle dynamic environments where a continuous dynamics model is knownfor the objects that move (easiest when only the robot moves)

– can incorporate geometrical constraints (e.g. rectilinearity) to update the map

• Weaknesses:

– model assumptions: angular errors are small (necessary because of lineariza-tion), estimating one or two moments of the distribution is sufficient

– data association is not considered

– computational issues

• Estimated values need not be spatial relationships; they might instead be forces, ve-locities, etc.

3.1.9 Thrun et al. (1998)

• Main contribution: an expectation maximization algorithm for landmark-based ge-ometric mapping in a global frame based on posing the problem in a constrainedprobabilistic maximum likelihood formulation. Optimizes both forwards and back-wards in time (modifying the estimate of the robot’s trajectory) using a Baum-Welchvariant to find the best estimate of the robot’s path and the landmark map.

• Assumes the robot correctly recognizes (non-unique) landmarks; in this paper, land-marks are hand-specified

• Motion model assumes triangular error distributions for translational and rotationalerror

• Probabilities represented using uniform grids

• Uniform prior over maps

• Finding the most likely map is equivalent to maximizing a function of the data, theperceptual model, and the motion model

– hard since it involves a search in the space of maps (high-dimensional), whereevaluation of each map requires integration over many variables

– instead use Baum-Welch (EM) to find a local maximum:

∗ E-step: localization assuming a fixed map∗ M-step: mapping assuming a fixed trajectory

• Shows that some distributions are multimodal, so Kalman filters are not appropriate

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• “Principle topology of the environment was already known after two iterations ofthe Baum-Welch algorithm; after approximately four iterations, the location of thelandmarks were consistently known with high certainty”

• Extends easily to multiple robots with unknown relative locations: initialize the beliefin location of the second robot to be uniform (rather than Dirac)

3.2 More modern techniquesThese papers exhibit interesting alternatives to techniques presented earlier for topologi-cal (Choset and Nagatani, 2001) and geometrical (Leonard et al., 2002; Lu and Milios, 1997)mapping.

3.2.1 Choset and Nagatani (2001)

• Main contribution: a topological approach to mapping based on tracing the GVGof a (planar) environment using simple control laws. Loops are closed based on thetopology of the GVG and the geometric lengths of GVG edges.

• Since the robot localizes itself topologically it never needs to determine its coordi-nates in a global frame

• GVG in the plane: one-dimensional set of points equidistant to two obstacles

– useful for sensor-based implementation since it is defined in terms of a localdistance (minimum distance to the nearest obstacle)

– a type of roadmap since any point in an environment can be reached by (1)moving to the nearest GVG edge; (2) tracing the GVG to the point closest to thegoal; and (3) leaving the GVG to reach the goal

• Exploration:

– (1) get on the GVG; (2) trace GVG edge; (3) determine meet point location;(4) recursively explore branches emanating from meet points; (5) terminate allbranches are explored

– stable control law is derived for tracing a GVG edge

• Topological localization:

– three approaches: sensing signatures, backtracking through the known map,graph matching

– all approaches have problems with close-together meet points since they can bemistaken for each other

– graph matching: upon possibly revisiting a meet point, pick what would be analready-explored edge and traverse/measure it, comparing the edge and thenext endpoint; if they match, repeat until disambiguation occurs or until onlyone possible meet point remains

• Problems:

– assumes sufficient sensing range (solved using SGVD)

– in self-similar environments, can use a hierarchical approach: (1) topologicallocalization; (2) feature-based localization; (3) dead reckoning

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– weak meet points: features in the environment that are on the boundary of beingsignificant enough to warrant a meet point; due to sensor noise, they sometimesgenerate meet points and sometimes do not

∗ reduced GVG: just delete all meet points that have adjacent boundary nodes;doesn’t completely solve the problem (two closeby meet points can still ex-ist)

• Future work: incorporate probabilistic approaches to deal with meet points that ap-pear and disappear; also deals with dynamic environments

3.2.2 Leonard et al. (2002)

• Main contribution: a modification of the traditional EKF-based SLAM algorithm thatadds the robot’s recent pose history to the state vector. In this way, feature initial-ization and data association decisions can be delayed until more data is available.Delayed initialization is consistent because correlations between current and previ-ous poses are maintained and used in the initialization.

• In some circumstances the data from a single sensor scan is insufficient to initialize anew feature (e.g., with sparse sensing not enough data is available to initialize a linefeature)

• Approach: delayed stochastic mapping: store the measurements from multiple consec-utive poses along with the correlations between the poses, and use this to initializenew features instead

– this also enables composite features comprised of multiple simple features (e.g.by joining lines and points to form polygons)

– measurements can be applied “asynchronously” in batches

∗ use a few “seed” measurements to initialize a feature, and use the rest thatassociate with that feature to perform a standard EKF update

∗ alternative: use non-linear least squares performed on many measurements;yields a covariance matrix suitable for incorporation into the map (Note:this is what we do)

– Note: this paper stores a fixed size pose history (40 states); “with a fixed windowsize, the process of adding past states is similar to a fixed-lag Kalman smoother”

– Problem: computing Jacobians is hard since they depend on the pose history;can do this numerically

– this approach extends to feature initialization with data from multiple robots;Problem: only if the relative poses of the robots are known

– vulnerable to large odometric errors

3.2.3 Lu and Milios (1997)

• Main contribution: an optimization algorithm based on considering relationshipsbetween robot poses (obtained via odometry and scan matching) in order to findglobally consistent values for the poses and create consistent, aligned maps.

• Most mapping algorithms are incremental: each frame is aligned to the map and thenmerged with the map; this may lead to inconsistency

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• Approach:

– maintain local frames (poses) for all measurements and relationships betweenthem (obtained from odometry or scan matching)

– use a maximum likelihood procedure to perform an energy minimization whereeach relation is a constraint modeled as a spring with minimum energy whenthe spring is at the measured distance of the relation

– this approach considers all spatial relations between poses simultaneously– Note: this approach is very similar to the idea of maintaining local maps and

finding a consistent sequence among them

– Note: for simple cases or compounding (two sequential relations) or merging(two parallel relations) this approach is equivalent to the Kalman filter

– can be extended to a sequential algorithm that maintains the current best esti-mate of the map

• Advantages:

– scan matching avoids the problem of extracting and matching features

– doesn’t require a priori covariance information about relations — can be derivedfrom the ML estimation procedure

• Problems:

– assumes that all measurements are independent, that noise is zero mean Gaus-sian, and that point correspondences from scan matching are correct

– sequential algorithm: requires inversion of a matrix whose size is proportionalto the length of the trajectory

∗ Strategy for reducing the number of variables: check all pairs of poses andpick the one where the poses are most strongly correlated; then fix the rela-tion between the poses and remove one pose from the state vector (it can beretrieved from the fixed relation and the other pose)

∗ Another strategy: decompose the full network into small components (i.e. lo-cal maps) and do full estimation only within local maps

– robot must stop to collect range scans; continuous scanning produces too muchdata that has to instead be sampled

4 ConclusionWe have presented summaries of a number of papers from the robot mapping literature.Some of the papers take purely topological approaches to mapping, and others purely geo-metrical approaches. Some papers apply ideas from both paradigms. It is our belief that themapping literature is beginning to exhibit convergence between these two markedly differ-ent approaches to mapping. Maps that explicitly encode both topological and geometricalaspects of the environment are useful in many respects: for simple and accurate automaticplanning and navigation, for visual inspection by people, and for use with robots whosesensing and computational abilities are limited or noisy. The continued “cross-pollination”between topological and geometrical mapping techniques will lead to efficient and accu-rate autonomously generated maps for use by both robots and humans.

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