From: AAAI Technical Report FS-94-03. Compilation copyright © 1994, AAAI (www.aaai.org). All rights reserved. An Ontological Hierarchy for Spatial Knowledge Benjamin Kuipers ComputerScience Department Universityof Texasat Austin Austin, Texas 78712 ku±pers@cs,at exas. edu Abstract communication paths exist, and how they fit together. An ontological hierarchy specifies what kinds of modules can exist, and what type of knowledge they can contain. A system architecture organizes inference steps, possibly drawn from more than one ontological level, into modules operating on the same data and at the same time-scale. The ontological hierarchy describes the "knowledge-level" structure of the system, and makes it possible to evaluate and refine theories that may be distributed amongthe modules of the system architecture. Knowledge in a complex domain may be organized into an ontological hierarchy, in which each level provides a different description of the world and a different set of feasible inferences. Wediscuss such a hierarchy for knowledgeof large-scale space: the Spatial Semantic Hierarchy (SSH). Wehave used the SSH as the basis for our research on cognitive maps and robot exploration for many years. Weillustrate by example how this knowledge-level hierarchy suggests a focused and productive set of research problems. 1 Ontological 2 The Spatial Semantic Hierarchy The cognitive map is the body of knowledge a human or robot has about its large-scale spatial environment. We have developed a computational theory of the human and robotic cognitive map [Kuipers, 1978, 1982; Kuipers and Levitt, 1988; Kuipers and Byun, 1988, 1991]. Our theory of the cognitive map is motivated by observations of humanspatial reasoning skills and the characteristic stages of child development [Lynch, 1960; Piaget & Inhelder, 1967; Hart & Moore, 1973]. These studies provide two fundamental insights. First, that a topological description of the environment is central to the cognitive map, and is logically prior to the metrical description. Second, that the spatial representation is grounded in the sensorimotor interaction between the agent and the environment. Our theory of the cognitive map is based on a hierarchy of representations for spatial knowledgethat we call the Spatial Semantic Hierarchy (SSH). Hierarchies A robotic system interacting with an environment represents knowledgeabout it, and performs inference about it, at multiple levels of description. Each level of the hierarchy has its ownontology (the set of objects and relations it uses for describing the world) and its ownset of inference and problem-solving methods. The objects, relations, and assumptions required by each level are provided by those belowit, by abstracting or reinterpreting information available at the lower level. The Spatial Semantic Hierarchy (SSH) is an ontological hierarchy for knowledgeof large-scale space. The SSHwas first described in [Kuipers & Levitt, 1988] as the result of analyzing the success of several early programsin qualitative robotics. Anontological hierarchy is a useful way to express a hypothesis about the structure of knowledgein a particular domain, for two reasons. First, each level of the hierarchy has a self-contained ontology (the set of describable object types and relations), so the theories associated with each level can be evaluated independently. Second, the objects and relations at each level are abstracted from lower levels, so the abstraction relations can be clearly stated and evaluated. As we shall see, this hierarchical structure allows the different levels to be based on very different bodies of mathematics, and therefore to have very different strengths and weaknesses. In the case of the Spatial Semantic Hierarchy, two of four levels have their foundations in continuous mathematics, while two have their foundations in logic. An ontological hierarchy is not a system architecture, although I believe that the Spatial Semantic Hierarchy helps explain the popularity and success of "three-level architectures" in recent generations of intelligent robotic systems [Kuipers & Byun, 1988; Bonasso, 1991; Connell, 1992; Gat, 1992]. A system architecture specifies what modules and [sensorimotor +-r control] --+ causality --r topology -~ geometry. The control level is based on the continuous mathematics of control theory; the causal and topological levels are based on logic; while the geometrical level is based on the theory of state estimators such as Kalmanfilters. 2.1 The Control Level At the control level of the hierarchy, the ontology is an egocentric sensorimotor one, without knowledge of fixed objects or places in an external environment. A locally distinctive state is defined as the local maximumfound by a hill-climbing control strategy, climbing the gradient of a selected sensory feature, or distinctiveness measure. Trajectory-following control laws take the robot from one locally distinctive state to the neighborhood of the next, where hill-climbing can find a local maximum,reducing position error and preventing its accumulation. The control level views the robot and its environment together as a dynamical system. The problem at the control level is to assemble a useful control law for the current 93 distinctive states: local optima of selected control laws, specific to each neighborhood. ¯ Metrical distance" direction shape 2.3 The Topological Level At the topological level of the hierarchy, the ontology consists of places, paths, and regions, with connectivity and containment relations. Relations among the distinctive states and trajectories defined by the control level, and among their summaries as views and actions at the causal level, are effectively described by the topological network. This network can be used to guide exploration of new environments and to solve new route-finding problems. Using the network representation, navigation is not dependent on the accuracy, or even the existence, of metrical knowledge of the environment. Within the topological level, the problem for the agent is to learn a graph describing the environment. There are a number of useful approaches to this exploration problem [Rivest &Schapire; Dudek, et all. Across levels, the abstraction problem is to define "places" that aggregate distinctive states defined at the causal level, implicitly distinguishing amongactions that change the agent’s location (take it from one place to another) and those that another aspect of the agent’s state (e.g. orientation) without changing its location. ¯ Topological places paths [ regions ¯ Causal views actions [ schemas connectivity order containment discrete space causality association discrete time ¯ Control sensor ] values magnitude gradient correlation ¯ Sensorimotor interaction with the environment. Figure 1: Ontology in the Spatial Semantic Hierarchy 2.4 The Metrical Level At the metrical level of the hierarchy, the ontology for places, paths, and sensory features is extended to include metrical properties such as distance, direction, shape, etc. Only at this level are sensory features considered to be transduced properties of the environment. Geometrical features are extracted from sensory input, and represented as annotations on the places and paths of the topological network. The metrical level describes places and paths, not as nodes and arcs in a graph, but as local 2-D and 1-D manifolds. The global structure of this set of manifold may simply be represented by their topological connections, and need not be integrated into a single global manifold. Inference at the topological level provides the most important objects (places and paths) about which metrical properties are asserted. However,other features, such as the walls of corridors, were treated implicitly as a source of exogenous signals at the control level, but are represented explicitly as objects with shape at the metrical level. neighborhood from locally-observable features and simple responses to those features. The agent must also be able to detect when it leaves the current neighborhood, so the current control law can be abandoned and a new one sought. In order to provide the control level with the power and flexibility it needs, we have developed a methodcalled heterogeneous control, for creating complex control laws by composing simpler ones and for proving properties of the resulting laws [Kuipers & AstrSm 1991, 1994]. This work has helped us understand the practical success of fuzzy control. Our approach provides an improved way to specify and apply such laws, with a muchclearer relationship to traditional control theory, and methods for combining traditional and qualitative analysis to prove properties of heterogeneous controllers. The Causal Level 2.2 At the causal level of the hierarchy, continuously changing values are abstracted to discrete states and transitions among states. The ontology for space consists of views, which describe the sensory images at locally distinctive states, and actions, which represent trajectories of control laws allowing the robot to movefrom one view to another. Sets of associations (V, A, W)amongviews, actions, and resulting views represent both declarative and imperative knowledgeof routes or action procedures. Within the causal level, the problem for the agent is to learn useful deterministic or stochastic models of the state-to-state transitions. Clearly, this is closely related to reinforcement learning. Across levels, the agent must be able to abstract from continuous states and trajectories at the control level to discrete states and transitions at the causal level. Our hypothesis in the SSHis that states at the causal level are 3 Performance Based on the SSH The structure of the Spatial Semantic Hierarchy prorides robust performance (figure 2). The control level definition of states and trajectories grounds the topological description of places and paths, which in turn supports navigation while more expensive sensor fusion methods accumulate metrical information. Whenmetrical information is available, it can be used to optimize travel plans or to disambiguate apparently identical places, but when it is absent navigation and exploration remain possible. The critical transition from continuous interaction to discrete symbols is provided by the behavior of local control laws. 94 (-) PS(Symm.Eq-S ’ P6(Symm-Eq-3) E5(Midline) IE6(Midline) F.,4 (Midline P4(Temp-Disc) PT(Temp-Disc) E23(Right-Wall) E3(L~t-W~I) I PS(Symm-Eq-2 ~-.,~E2 ( Left- Wall) ~,~ El(Left-Wall) ¯ P2(Symm-Eq-2) Er(L~t-w~l) P1 (Temp-Disc) (b) ,~ ~ ..~ ,e.-..~ p I I , ,~................... .-,".-; ~ : : , ~, [ ~,i ~1111111 . i ; ¯ ’! ¯ ¯ o- .’. .~ ~, ....:. ". . ~ :’ "i ~e~ I ":~e ~.... ~ : ’~ :¯ .e..... ¯ .o ¯ ., ¯ ~ .e .,m. ....¯ i i :" " .,,., "" .. .ir! ¯ ¯ I . .~.,.._...~..:.; ; " .: ,i ’q :oO ~ l "~’. ~,. ¯ .,*e,,’ I : ¯ : ;’ ~ ; :. ,~ .. ;¯ ~, . ¯ o ¯ "~. "’; . ¯ ¯ ’..~e , : ~’i " :,;¯ .. ¯ ’ ’ ¯ - "’’"?-" ii . ¯ ¯ "’ ~ " ’ Figure 2: Simulated NXrobot applies SSHexploration and mapping strategy. (a) The simulated NXrobot uses range-sensors explore and map an environment. The exploration and control strategies identify randomand systematic sensor errors, and thus provide robustness. (b) The topological map(fragment) identifies places and paths with the distinctiveness measures that define them(e.g., equidistance from nearby obstacles, discontinuoussensor changes), and represents their connectivity relations. (c) The metrical mapconsists annotations on each place and path, and can be relaxed into a global 2Dframe of reference. 95 Within a neighborhood in the environment, if a ~distinctiveness measure" exists that provides an isolated local maximumfor a hill-climbing control law, then the final state of the behavior resulting from following that control law can be considered a ~distinctive state", which is described by a view at the procedural level, and as a place at the topological level. Exploration and mapping can be viewed as search for suitable neighborhoods and distinctiveness measures, while accumulating their descriptions. The Spatial Semantic Hierarchy approach contrasts with more traditional methods, which place geometrical sensor intepretation (the most expensive and error-prone step) the critical path prior to creation of the topological map [Chatila and Laumond, 1985; Moravec and Elfes, 1985]. Our spatial representation hierarchy is consistent with levels 2 and 3 of Brooks’ [1986] subsumptionarchitecture. Research on reactive behavior languages [Brooks, 1990; Gat, 1991] can be viewed as exploring the nature of the control level. Our fundamental claim is that the Spatial Semantic Hierarchy describes the structure of an agent’s spatial knowledge, in a way that is relatively independent of its sensorimotor apparatus and the environment within which it moves. More specifically, an effective symbolic representation of an agent’s large-scale spatial environment can be built within the SSH approach, requiring only that the agent’s sensorimotor system and its environment satisfy a few conditions. Informally, these conditions are: 1. The agent’s sensory inputs must change continuously with its actions, except at isolated discontinuities. 2. It must be possible to define features (distinctiveness measures) over the sensory inputs with sufficiently steep gradients and sufficiently isolated local maxima to support hill-climbing and trajectory-following control laws within local regions. 3. The result of a trajectory-following control law must be a state from which a hill-climbing control law will reach a distinctive state (though occasional failures can be tolerated). These conditions have been adequate to support implementation of the SSH mapping framework on several different simulated and physical mobile robots. Figure 3 demonstrates a fragment of behavior of an l:tWI B12 robot, using a ring of 12 sonar sensors, as it follows control laws and identifies a distinctive place in the indoor office environment. The integration of this control level foundation with the SSHprocedural, topological, and metrical levels is currently under way. Ultimately, the robot’s learning process should not require programmer-supplied information about the properties of the robot’s sensors and effectors. Weare developing methodsfor learning the sensory features (i.e. distinctiveness measures) and control laws required to support the SSH framework, starting with an uninterpreted sensorimotor system gathering experience in an unknown environment. Wehave developed s lattice of learning methods based on modest, domain-independent processing assumptions, that solve this problem for a significant class of robots [Pierce ~ Kuipers, 1990, 1994; Pierce, 1991a, ¯ At the SSHmetrical level, how can metrical informstion be used to reduce topological ambiguity? Howis topological structure used to simplify the acquisition of local metrical manifolds? ¯ The Spatial Semantic Hierarchy is the motivating example for the more general concept of ontological hierarchies. Are there ontological hierarchies in other domains that are significantly different from the SSH? Or do the control, causal, topological, and metrical levels have useful analogs across many domains? The widespread use of spatial metaphors supports the latter position, but since they are not universal, there are presumably other ontological structures. ¯ Although information at the metrical level clearly makes a stronger ontological commitment than that at the topological level, there are many examples of metrical properties that are reliably, efficiently, and locally learnable from observations before more global topological properties become known. Therefore, most system architectures for a robot mapping system would learn local topological and metrical information together, followed by more global topological and metrical information. Should the ontological hierarchy be restructured, or is this purely an architectural issue? ¯ Assuming that an ontological hierarchy has a particular structure, what is the design space for system architectures based on that structure? © Figure 3: Spot, a physical robot, applying SSH control strategies. Spot movesalong a right wall, identifies a distinctive place, turns left, and begins to follow the next wall. This figure plots the position of the robot and the single most relevant sonar reading, on a mapof the corridor it is exploring. 1991b, forthcoming doctoral dissertation]. With these methods, the robot’s cognitive map is much more thoroughly grounded in its ownsensorimotor experience, rather than being specified by a human programmer. Another assumption built into many robot learning algorithms is that the robot’s position in the world, and hence its sensory input, changes only when the robot takes a deliberate action. This restriction to low-speed, quasistatic trajectory control laws is unnecessarily restrictive. Wewish the robot to build on the information in its cognitive map to acquire high-speed dynamic control laws for skilled motion along complex routes [Froom, 1991, 1992]. These methods use the incomplete spatial knowledge represented by the topological and metrical maps to formulate initial trajectory targets for locally-optimal dynamic control. With subsequent practice in the environment, the targets are incrementally refined to improve performance. This method strikes a balance between two unrealistic poles: globally optimal trajectory planning based on perfect knowledge, and overly conservative planning based only on visible obstacles. 4 References [1] R.A. Brooks. 1986. A robust layered control system for a mobile robot. IEEEJournal o] Robotics and Automation, v. RA-2,n. 1, pp. 14-23. [2] Raja Chatila and Jean-Paul Laumond.Position referenchag and consistent world modelingfor mobilerobots. IEEE International Conference on Robotics and Automation, 1985, pp. 138-170. [3] Jonathan Connell. 1992. SSS: a hybrid architecture applied to robot navigation. IEEEInternational Con]erence on Robotics and Automation, 1992. [4] Richard Froom. 1991. Acquiring effective knowledgeof environmental geometry for minimum-timecontrol of a mobile robot. Proceedingso.f the 1991 IEEEInternational Symposiumon Intelligent Control, Arlington, VA,August 13-15, 1991. [5] Richard Froom. 1992. Approximate maps for high-speed control of a mobile robot. In Mobile RobotsVII: Proceedings of SPIE- the International Society for Optical Engineering v. 1831, Boston, MA,November15-20 1992. [6] Erann Gat. 1991. ALFA:a language for programmingreactive robotic control systems. IEEEInternational Conference on Robotics and Automation. [7] Erann Gat. 1992. Integrating planning and reacting in a heterogenous asynchronousarchitecture for controlling real-world mobilerobots. Proceedingsof the National Con]erence on Artificial Intelligence (AAAI-9$), AAAI/MIT Press, 1992. [8] Hart, R. A., & Moore, G. T. The development of spatial cognition: A review. In R. M. Downs& D. Stea (Eds.), Image and Environment. Chicago: Aldine Publishing Company, 1973. Research Questions The benefit of an ontological hierarchy, like the benefit of a choice of system architecture, is that it focuses attention on certain technical questions which are hoped to be productive. ¯ At the SSHcontrol level, how can appropriate control laws be composed from simple elements responding to features idiosyncratic to each neighborhood of the environment? Howcan these control laws be incrementally improved with experience to achieve skilled performance? ¯ At the SSHcausal level, howshould probabilistic information be acquired and represented? At what point should a control level behavior be abstracted to a causal transition? ¯ At the SSH topological level, how should places be abstracted from states? When should locallyindistinguishable places be merged, and when should they be kept separate? 96 [9] Kuipers, B. J. 1978. Modeling spatial Science 2: 129-153. knowledge. Cognitive [113] B. Kuipers. 1982. The ’Map in the Head’ Metaphor. Environment and Behavior 14: 202-220. [11] Benjamin Kuipers & Karl Astr6m. 1991. The composition of heterogeneous control laws. In Proceedings of the American Control Conference, Boston, Massachusetts, June, 1991. [12] B. J. Kuipers and K. Astr6m. 1994. The composition and validation of heterogeneous control laws. Automatica 30(2), February 1994, to appear. [13] B. Kuipers and Y. T. Byun. 1988. A robust qualitative method for spatial learning in unknown environments. In Proceedings of the National Conference on Artificial Intelligence (AAAI-88). Los Altos, CA: Morgan Kaufman. [14] B. Kuipers & Y.-T. Byun. 1991. A robot exploration and mapping strategy based on a semantic hierarchy of spatial representations. Journal of Robotics and Autonomous Systems 8: 47-63. [15] B. Kuipers, R. Froom, W.-Y. Lee & D. Pierce. 1993. The semantic hierarchy in robot learning. In J. Connell and S. Mahadevan (Eds.), Robot Learning. Kluwer Academic Publishers, 1993, pages 141-170. [18] B. J. Kuipers and Tod Levitt. 1988. Navigation and mapping in large scale space. AI Magazine 9(2): 25-43. [17] Kevin Lynch. 1960. The Image of the City. Cambridge, MA: MIT Press. [18] H. Moravec & A. Elfes. 1985. High resolution maps from wide angle sonar. IEEE International Conference on Robotics and Automation, pp. 116-121. [19] Jean Piaget and Baerbel Inhelder. 1967. The Child’s Conception of Space. NewYork: Norton. (First published in French, 1948.) [20] D. M. Pierce. 1991. Learning turn and travel actions with an uninterpreted sensorimotor apparatus. In Proceedings of the 1991 IEEE International Conference on Robotics and Automation, Sacramento, CA. [21] Pierce, D.M. 1991. ~aming a set of primitive actions with an uninterpreted seusorimotor apparatus. In Birnbaum, L., and Collins, G. (eds.) Machine Learning: Proceedings of the Eighth International Workshop. San Mateo, CA: Morgan Kaufmann. [22] David M. Pierce and B. J. Kuipers. 1991. Learning hillclimbing functions as a strategy for generating behaviors in a mobile robot. In J.-A. Meyer and S. W. Wilson (Eds.), From Animals to Animats: Proceedings of the International Conference on Simulation of Adaptive Behavior, pp. 327-336. Cambridge, MA: MIT Press/Bradford Books, 1991. [23] D. Pierce & B. Kuipers. 1994. Learning to explore and build maps. Proceedings of the National Conference on Artificial Intelligence (AAAI-94), AAAI/MITPress, 1994. 97