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
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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
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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
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96
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