1Lecture 19:Topological MappingCS 344R/393R: RoboticsBenjamin KuipersExploration DefinesImportant Places and Paths2Abstract the Exploration Patternto the Topological MapThe Topological Map• The topological map is the set of places andedges linking them.• A place is a decision point among edges.– It has a local topology: radial order among edges.– It has a local geometry: directions of edges.• An edge links two places.– An edge has a control law for travel along it.3Scale 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”Two Approaches toDistinctive States and Places• Hill-climb to a distinctive state– Makes very weak assumptions about sensors– Voronoi graph: points equidistant from nearbyobstacles• Localize in place neighborhood– Requires local metrical map of neighborhood– Use Voronoi graph to define local topology4What is a Place?• In small-scale space:– A place is a region.– It’s a neighborhood where the agent canreliably localize itself completely.– It’s bounded by gateways, which connect topath segments for travel to other places.• In large-scale space:– A place is a decision point.– It’s a graph node connected to other places,representing a 0-D location.Topological Mapping Overview• Build local perceptual maps of placeneighborhoods, each a small-scale space.• Build local topology descriptions of thecomplete qualitative structure of each placeneighborhood.• Build the global topological mapabductively, using:– completeness of the local topology description,– pose in local topology to serve as a “view”.5Local Place Neighborhood Map• For each place neighborhood, build a smalllocal metrical map, with its own frame ofreference.– Use it for “virtual range sensing” when specularreflection makes sonar sensors unreliable.• Put the origin at a central point, and storedirections of outgoing edges.– Store the local map as an attribute of the place.A Scrolling Metrical Map• During travel, maintain a limited-rangemetrical map of immediate surroundings.– Keep robot pose (x,y) in the center cell.– Robot’s orientation θ can vary in the map.– Robot pose is high resolution, not map cell.• Scroll the map as the robot moves.– Shift in (x,y) only, not in orientation.– Shift only by integral numbers of cells, toprevent information loss.• Cells that fall off the edge are lost.6Gateways• A gateway is a transition between a travelaction and a place neighborhood– i.e., between a trajectory-following control lawand a local perceptual map.– Transitions can be inbound or outbound.– Detected from local properties of theenvironment and the conditions on the controllaw.7Detect and Describe a PlaceIdentify Constrictions8Define GatewaysDefine Local Path Fragments9Local Topology Description• The small-scale star is a circular order ofpath fragments, gateways, and control laws.DeadEnd(gw2,in)PF2-Midline(gw5,out)PF3-DeadEnd(gw3,in)PF4-Midline(gw4,out) & (gw1,in)PF1-Midline(gw3,out)PF4+DeadEnd(gw5,in)PF3+Midline(gw2,out)PF2+Midline(gw1,out) & (gw4,in)PF1+Local Topology Description• The large-scale star describes the placewith distinctive states and directed paths.EndpointPa4, −ds8Pa3, −ds7EndpointPa2, −ds6Pa1, −ds5Pa4, +ds4EndpointPa3, +ds3Pa2, +ds2Pa1, +ds1Pa1Pa2Pa3Pa410Turn Actions• A Turn action may follow a trajectorythrough the local place neighborhood.Pa1Pa2Pa3Pa4in large-scale spacein small-scale spaceTopology fromLocal Metrical MapsPa1Pa2Pa3Pa411Places and Gateways• The agent can localize reliably anywhere in aplace neighborhood.– Gateways act as distinctive states– state = (place, gateway, orientation)• Actions move the agent deterministically,from one state to another,.– Travel: from outbound gateway at one placeneighborhood to inbound gateway at another– Turn: from inbound to outbound gateway at aplace neighborhood• Every 〈q, Turn, q′ 〉 at a place is known.Does a place abstraction always exist?• Not in truly pathological environments– open ocean or with pathological sensors– video snow• Conjecture: Yes, with sufficiently richsensors in a sufficiently rich environment.– office environments– campus/urban indoor/outdoor environments12Build the Global Topological Map• Define a tree of all possible topological mapsconsistent with exploration experience.– They are the leaves of this tree.• For each new action+observation– If the map predicts the observation, OK.– If it contradicts the observation, prune it.– Otherwise, branch on maps with new edges:• All possible loop-closing hypotheses• One hypothesis of a brand-new place– Identify the current best map.Building the Tree of Maps12345613Tree of Maps (1)1Tree of Maps (2)1214Tree of Maps (3)123Tree of Maps (4)123415Tree of Maps (5)12345Tree of Maps (6)12345616Find the Current Best Map• The tree is guaranteed to contain the true map– All consistent maps are created.– Only inconsistent ones are deleted.• Rank the consistent maps by simplicity andlikelihood.– Each map is a loop-closing hypothesis.– The true map is often simpler than the others.• Use the current best map for planning.– Remember the tree.– The current best map could be refuted.The Topological MapLinks Local Place Maps17Bizarre Map Hypotheses Ruled Out ByTopology, Planarity, & ProbabilityResult: Single correct topological map hypothesisNext• The Hybrid Spatial Semantic Hierarchy• Building the global metrical map– Using the topological map as a