Spatial Reasoning and Intelligent Robotics
Our robotics research is based on a detailed computational theory of
an important body of human knowledge.
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 human spatial
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 knowledge that we call the Spatial
Semantic Hierarchy (SSH).
[sensorimotor <-> control] -> procedures -> topology -> geometry.
Each level of the hierarchy has its own ontology (the set of
objects and relations is uses for describing the world) and its own
set of inference and problem-solving methods. The objects, relations,
and assumptions required by each level are provided by those below it.
The structure of the Spatial Semantic Hierarchy provides robust
performance. 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. When metrical 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.
- 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 distinctive state is
defined as the local maximum found 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 distinctive state to the neighborhood of the next,
where hill-climbing can find a local maximum, reducing position
error and preventing its accumulation.
- At the procedural level of the hierarchy, the ontology
consists of views, which describe the sensory images at
distinctive states, and actions, which represent trajectories of
control laws allowing the robot to move from one view to another.
Sets of associations [V,A,V'] among views, actions, and resulting
views represent both declarative and imperative knowledge of routes or
- 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 procedural 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.
- At the geometrical 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 critical transition from continuous interaction to discrete symbols
is provided by the behavior of local control laws.
Within a neighborhood in the environment, if a ``distinctiveness
measure'' exists that provides an isolated local maximum for 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
The Spatial Semantic Hierarchy approach contrasts with more
traditional methods, which place geometrical sensor intepretation (the
most expensive and error-prone step) on 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'  subsumption architecture.
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:
These conditions have been adequate to support implementation of the
SSH mapping framework on several different simulated and physical
- The agent's sensory inputs must change continuously with its
actions, except at isolated discontinuities.
- 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.
- 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).
Progress and Extensions
The Spatial Semantic Hierarchy has supported several productive lines
Improved Formalization (in progress)
First, we are completing a new formalization of the SSH that clarifies
its dependence on the properties of the environment and the
sensorimotor system of the agent [Kuipers, manuscript]. Since the SSH
is an ontological hierarchy, different formalisms are required
for the different levels. The control level is formalized using the
continuous mathematics of control theory. The procedural and
topological levels are formalized in predicate logic. The geometrical
level is formalized in terms of state estimators such as Kalman
filters. The objects and relations defined at one level depend on
assumptions about the results of behaviors at the lower levels. This
formalization provides a rigorous description of a sophisticated and
important body of human knowledge.
Second, in order to provide the control level with the power and
flexibility it needs, we developed a method called heterogeneous
control, for creating complex control laws by composing simpler ones
and for proving properties of the resulting laws [Kuipers & Astrom
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 much clearer relationship to traditional
control theory, and methods for combining traditional and qualitative
analysis to prove properties of heterogeneous controllers. (This work
has led to an award under the NSF Intelligent Control initiative.)
Third, we have developed methods for extending the low-speed
quasi-static trajectory control laws used by the SSH control level to
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.
Hyper-Redundant Manipulator Planning
Fourth, we took an excursion into path planning for manipulators with
many degrees of freedom. This problem is intractable to
configuration-space methods, which are exponential in the number of
degrees of freedom. Using a continuous model of a many-jointed
manipulator, we developed an approximation-based path planning method
with complexity polynomial in the complexity of the workspace, and for
which the number of joints is a benefit (in the quality of the
approximation) rather than an exponential cost [Hayashi &
Kuipers, 1991, 1992].
Learning an Unknown Sensorimotor System
Fifth, we are developing methods for 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. We have developed a
lattice of learning methods based on modest, domain-independent
processing assumptions, that solve this problem for a significant
class of robots [Pierce & Kuipers, 1990; Pierce, 1991a, 1991b,
forthcoming doctoral dissertation]. With these methods, the robot's
cognitive map is much more thoroughly grounded in its own sensorimotor
experience, rather than being specified by a human programmer.
Finally, we have been experimenting with physical robot platforms to
embody the SSH approach to exploration and mapping. We have
demonstrated the ability to follow control laws and identify
distinctive places in the indoor office environment. The integration
of this control level foundation with the SSH procedural, topological,
and metrical levels is currently under way. We have also developed
the very substantial hardware and software infrastructure required for
physical robotics research [Lee, 1993; Kuipers, et al, 1993].
The bottom line is that we have demonstrated that the Spatial Semantic
Hierarchy provides a rigorous description of a foundational domain of
knowledge, and can be used to support and integrate a diverse body of
research which shares a focus on spatial knowledge.