1Lecture 20:The Spatial Semantic HierarchyCS 344R/393R: RoboticsBenjamin KuipersWhat is a Map?• A map is a model of an environment thathelps an agent plan and take action.• A topological map is useful for travelplanning.• A metrical map is useful for inferringdirections and distances.• Both must be learned from observations.2Scale of Space• Small-scale space is within the agent’sperceptual surround.– “visual space” or “perceptual space”• Large-scale space has structure that mustbe integrated from the agent’s observationsgathered over time and travel.– the “cognitive map”Local Metrical Mapping Works• In small-scale space, modern SLAM methodswork extremely well with lasers.– Great progress with visual SLAM.Large-scalespaceLocal SLAMSmall-scalespaceTopologicalMappingMetricalMapping3Global Metrical Mapping Is Hard• Within a single global frame of reference overlarge-scale space, errors accumulate.– Sufficiently large loops are always a problem.Cumulative errorsScalabilityLarge-scalespaceLocal SLAMSmall-scalespaceTopologicalMappingMetricalMappingProblem: Closing Large LoopsRaw Odometry SLAM Corrected Odometry4Local matching can find false,but locally optimal, loop closuresTopological Mapping• Describe large-scale space in terms of– Places (with local frames of reference)– Paths (with ordered sequences of places)– Regions (with sets of places and paths)• Paths can serve as boundaries• Handles many practical planning problems,even without a metrical map5The Spatial Semantic HierarchyA hierarchy of ontologies.• Control: select control laws to movereliably among distinctive states.• Causal: actions such as turn and travel linkstates, which have sensory views.• Topological: places, paths, and regionslinked by connectivity, order, containment.• Metrical: frames of reference, distance,direction, shape.The Basic SSH• Strengths– The robustness of commonsense knowledgecomes from having multiple, different,coordinated representations for knowledge.– Makes few assumptions about sensors,effectors, or the environment.• Weaknesses– Hill-climbing to distinctive states is awkward,and seems like unnecessary physical motion.– What if we really want a global metrical map?– What if we really know about our sensors?6Solution: The Hybrid SSH• Local metrical maps– Metrical SLAM methods work well locally.– Localization substitutes for hill-climbing• Global topological maps– Represent structural hypotheses explicitly.• Global metrical map– Build on the skeleton of the topological mapIdentify the Local Topology• Identify the local decision structure of eachplace neighborhood.– Travel experience as graph explorationLarge-scalespaceLocal decisionstructureLocal SLAMSmall-scalespaceTopologicalMappingMetricalMapping7Build the Global Topological Map• Decide when and how loops are closed– When does the next place match a previous place?• Build a tree of all possible topologiesGlobaltopological mapLarge-scalespaceLocal decisionstructureLocal SLAMSmall-scalespaceTopologicalMappingMetricalMappingSearching the Tree ofAll Possible Maps• The tree is guaranteed to contain the truemap– All consistent maps are created.– Only inconsistent ones are deleted.• Select the best consistent map for planning.– Remember the tree.– The current best map could be refuted.8Axioms for Map Structure• These axioms can rule out possible maps.– Logically inconsistent, hence impossible– Outside the set of permissible maps• Causal: predict results of actions• Topological: order relations on paths• Boundary: paths divide the world• Metrical: triangle inequalityThe Topological Map is a Graphof Places and Paths• The topological map is a bipartite graph:– Nodes = Places ∪ Paths– Edges = relations: on(place,path)• Each path has a 1-D direction dir ∈ {+,−}• An order relation, order(path,a,b,dir), forthe places on each path.• Each directed path is a boundary, describingplaces as on its right and its left.9Deeper Topological Inference• Each map has richertopological concepts andrelations:– A place has a circularorder of directed paths– Boundary relations holdbetween path & places– Useful for route planning• Refute maps that violatethe topological axiomsThe Topological MapLinks Local Place Maps10Roadmap• Local metrical maps– Given local maps of each place…• Global topological maps– Given a single best structural hypothesis …• Global metrical map– Displacement along each travel segment– Global layout of places– All robot poses in the global frame of referenceGlobal Metrical Map• Use the topological map as a skeleton.– Lay out places in a single global frame of reference.– Fill in the details from local places and segments.Globaltopological mapGlobal metricalmapLarge-scalespaceLocal decisionstructureLocal SLAMSmall-scalespaceTopologicalMappingMetricalMapping11Given the Topological Map …• The loop-closing problem is solved.– The topological map specifies which loopsclose, and where.• Each place has an accurate local metricalmap in its own local frame of reference.• Continuous behavior divides into segmentsat distinctive place neighborhoods• The global metrical map combinesinformation from separate local maps.The Global Metrical Map:Factoring the Problem• Displacements: the pose of each place in theframe of reference of its predecessor.• Layout: the pose of each place in the globalframe of reference.• Robot poses: the robot pose at each timestep inthe global frame of reference.• Global map: range sensor endpoints startingfrom known robot poses.12Estimating Displacements• Use incremental SLAM to estimate posexi+1,0 in the frame of reference of mi.• Localize to get xi+1,0 in frame mi+1.• Derive displacement λi between the twoplace poses.Estimating Place Layout• Local displacementspropagate to globalplace layout.– Loop-closings areespecially helpful.• Relaxation searchconverges quickly toa maximumlikelihood layout.13Estimating Robot Poses• Given a max likelihoodplace layout• and the trajectory of robotposes• define a fixed anchor poseeach time the trajectorypasses through a placeneighborhood• interpolate poses in eachsegment, using correctedodometry.Global SLAM with new poses• The pose distribution is ahighly accurate proposaldistribution.• Treat it as providingcorrected odometry.• Now do SLAM in theglobal frame of