Semantic Localization for IoT
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Non-Euclidean metric maps are useful when the standard Euclidean metric does
not really capture the interesting properties of a space. Consider a point x on the third
floor of a building and the point y directly below it on the second floor. Points x and
y are very close to each other in Euclidean space, but for the purposes of navigation,
this misrepresents reality. We can instead define a metric space with metric D, where
D(x, y) =
minimum length of a continuous path from x to y, if there is such a path
∞,
otherwise
This is easily shown to be a metric (or even an ultrametric, for some graphstructured metric spaces). If stairways and elevators are not navigable open space for
a particular robot, then this metric will yield D(x, y) = ∞, considerably more than
the Euclidean distance.
An inner product space (of which a Euclidean space is a common variety) introduces the notion of angles. Angles can facilitate special kinds of analysis like trilateration, and the use of trigonometric angle measurements to localize objects in
coordinate space.
As these examples illustrate, there are practical reasons to construct non-Euclidean
ontologies. However each of these mathematical ontologies has the property that any
map entity placed at a particular coordinate takes on all spatial relationships to other
map coordinates implied by the structure of the space. This is undesirable when only
a portion of those relations are positively known to be true and the rest are unknown.
A key advantage of relational ontologies is the expression of open ontologies, where
the absence of a relation does not imply its converse. This is analogous to ancient
maps that provided useful navigation information despite significant distortions in
the geometry and large gaps labeled “terra incognita”. Open ontologies translate
naturally into action plans that can deal with incomplete information.
This increased flexibility comes at the cost of a slightly more verbose vocabulary
for relations. Consider the containment map on the right hand side of Fig. 3. Because
this ontology is open, knowing that one place is not contained by another isn’t enough
to know they have no space in common. Another relation, “disjoint,” is necessary to
express that positive fact explicitly. We hope the reader can see a connection here
to intuitionisitic logic, in that for open ontologies it is not enough to know a spatial
relation is not not true to infer that it is true. Instead, relations must be constructively
built up from known facts.
2.4 Formalizing Open Ontologies
An open ontology A is a way of expressing partial knowledge about a spatial structure.
If we take the philosophical position that the unexpressed information in an open
ontology is fundamentally unknowabable, there is nothing to be done to increase
the amount of information represented in A. However, if we assume the missing
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Non-Euclidean metric maps are useful when the standard Euclidean metric does
not really capture the interesting properties of a space. Consider a point x on the third
floor of a building and the point y directly below it on the second floor. Points x and
y are very close to each other in Euclidean space, but for the purposes of navigation,
this misrepresents reality. We can instead define a metric space with metric D, where
D(x, y) =
minimum length of a continuous path from x to y, if there is such a path
∞,
otherwise
This is easily shown to be a metric (or even an ultrametric, for some graphstructured metric spaces). If stairways and elevators are not navigable open space for
a particular robot, then this metric will yield D(x, y) = ∞, considerably more than
the Euclidean distance.
An inner product space (of which a Euclidean space is a common variety) introduces the notion of angles. Angles can facilitate special kinds of analysis like trilateration, and the use of trigonometric angle measurements to localize objects in
coordinate space.
As these examples illustrate, there are practical reasons to construct non-Euclidean
ontologies. However each of these mathematical ontologies has the property that any
map entity placed at a particular coordinate takes on all spatial relationships to other
map coordinates implied by the structure of the space. This is undesirable when only
a portion of those relations are positively known to be true and the rest are unknown.
A key advantage of relational ontologies is the expression of open ontologies, where
the absence of a relation does not imply its converse. This is analogous to ancient
maps that provided useful navigation information despite significant distortions in
the geometry and large gaps labeled “terra incognita”. Open ontologies translate
naturally into action plans that can deal with incomplete information.
This increased flexibility comes at the cost of a slightly more verbose vocabulary
for relations. Consider the containment map on the right hand side of Fig. 3. Because
this ontology is open, knowing that one place is not contained by another isn’t enough
to know they have no space in common. Another relation, “disjoint,” is necessary to
express that positive fact explicitly. We hope the reader can see a connection here
to intuitionisitic logic, in that for open ontologies it is not enough to know a spatial
relation is not not true to infer that it is true. Instead, relations must be constructively
built up from known facts.
2.4 Formalizing Open Ontologies
An open ontology A is a way of expressing partial knowledge about a spatial structure.
If we take the philosophical position that the unexpressed information in an open
ontology is fundamentally unknowabable, there is nothing to be done to increase
the amount of information represented in A. However, if we assume the missing