=Paper=
{{Paper
|id=Vol-2969/paper78-RobOntics
|storemode=property
|title=Qualitative Spatial Ontologies for Robot Dynamics
|pdfUrl=https://ceur-ws.org/Vol-2969/paper78-RobOntics.pdf
|volume=Vol-2969
|authors=Jona Thai,Michael Grüninger
|dblpUrl=https://dblp.org/rec/conf/jowo/ThaiG21
}}
==Qualitative Spatial Ontologies for Robot Dynamics==
,
Qualitative Spatial Ontologies for Robot Dynamics
Jona Thai1 , Michael Grüninger1
1 Department of Mechanical and Industrial Engineering, University of Toronto, Ontario, Canada M5S 3G8
Abstract
Following Smith and Gasser’s work on embodied cognition, one can consider a robot as an intelligent
agent that interacts with its external environment through sensorimotor activities, such as touching, lifting,
standing, sitting, and walking. In this paper we explore the ontologies that are required to represent and
reason about robot dynamics. We propose new ontologies for robotic components and poses, including a
new nonclassical mereotopology for touch contact. The design of the ontologies is driven by semantic
parsing of natural language instructions (e.g. “Lift the box that is beside the chair and place it on the
table"), through which we identify the spatial and mereotopological relations among a robot’s components
and the external world, as well as the activities that the robot can perform to change these relationships.
Keywords
spatial ontologies, mereotopology, robotics, semantic parsing
1. Introduction
Many within the field of artificial intelligence(AI) research are familiar with the "Monkey &
Banana Problem" [1] - a famous toy problem aimed at solving and optimizing the best sequence
of actions for a monkey to obtain bananas suspended from a ceiling, given a chair and a stick.
Although the optimal solution is straightforward to find, what is the background knowledge and
context necessary to execute it?
It is typically assumed that the monkey is aware of how to navigate and use tools such as
sticks. To instill the same level of spatial awareness, it is common to turn to the quantitative
precision of Euclidean geometry. However, a perfect understanding of Euclidean geometry does
not necessarily translate to a perfect understanding of verbal instruction - an extension of the
symbol-grounding problem [2] in embodied cognition [3]. This is somewhat consistent with the
embodiment hypothesis, which descrives intelligence as an emergent phenomena of sensorimotor
activity between an agent and its environment [4].Even simple natural-language instructions,
such as "stand up" or "raise your arms" do not have a straightforward translation in logic. Perhaps
one could argue that it is a problem easily solved through manual programming, but what comes
of interpreting prepositions or state-of verbs? "Rinsing off a mug" or "Placing a block behind
another" do not directly map to an equation or first-order preposition, and current semantic parsing
infrastructure is ill-equipped to perform such operations with rigor, precision and reproducibility.
[5][6].
RobOntics 2021: 2nd International Workshop on Ontologies for Autonomous Robotics, held at JOWO 2021: Episode
VII The Bolzano Summer of Knowledge, September 11–18, 2021
© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
This paper aims to highlight an ontological approach to supplement current work in robotic-
natural-language-processing and spatial understanding. After presenting a set of motivating
scenarios in robot dynamics, we introduce a set of new ontologies for qualitative spatial relations
that are needed to formalize the logical sentences that correspond to natural language instructions
within the scenarios.
2. Motivating Scenarios
We return to the "Monkey & Banana Problem" (or perhaps the "Robot & Banana Problem") in
guiding our ontology design. Below are common poses we deem necessary to not only perform
the activity of grabbing bananas, but are also indicative of basic self-awareness and qualitative
spatial understanding.
Although simple to understand from a human perspective, these poses are not trivial. Mereotopo-
logically speaking, all the poses in the motivating scenarios above are equivalent – in each case,
the parthood and connection relations between objects are the same. However, their embedding
within the physical space and intended function differ based on assumptions. For instance, Moti-
vating Scenario 1 & 3 are unique in that they are independent of external environment. Regardless
of the presence of a floor, gravity or ceiling, it should be achievable with spatial self-awareness
alone.
1. Stand Up
a) Feet are completely touching the ground.
b) Other limbs and body parts are not touching any external object.
Figure 1: Stand Up
2. Arms Raised Over Head
a) Feet are completely touching the ground
b) Hands are above head.
c) Shoulders and upper arms are next to head.
3. Right Hand Touching Top of Head
a) Feet are completely touching the ground
Figure 2: Arms Raised Over Head
b) Only the right hand is touching the head.
c) No other body part is touching an external object.
Figure 3: Right Hand Touching Top of Head
4. Left Leg Lift
a) Right foot is completely touching the ground.
b) Left foot is not touching the ground, or any other external object
c) No other body part, other than the right foot is touching an external object.
3. Ontologies for Spatial Relations in the Motivating Scenarios
We recognize two core components to qualitative spatial reasoning - self awareness and interacting
with external objects. An ontology for self-awareness would provide the necessary axioms to
define models as described in Motivating Scenarios 1-4. However, we recognize that Motivating
Scenarios 1-4 may not be feasible on certain body anatomies. For example, BB8 from the Star
Wars franchise would find it impossible to lift its left leg (Motivating Scenario 4), due to its lack
of such a limb.
Figure 4: Left Leg Lift
Hence, for the sake of a holistic solution, we also define three large classes of anatomies as a
reference. The ontologies for self-awareness and interacting with external objects are relative to
these anatomies. Besides effectively constraining the set of feasible models, this approach also
uniquely allows the robot to have two frames of reference – one of itself, and possibly one of a
human, which could aid in the semantic parsing of human instruction.
3.1. Robot Anatomy
At its core, all anatomies and skeletons can be represented as a connected induced subgraph
structure [7]. This is due to the inherent convexity of its mereotopology - for example, the sum of
upper limb, joint, lower limb is different from upper limb, lower limb, joint. Convexity is the
concept of importance of order in sums. Moreover, it allows us to represent hiearchical relations,
which are particularly important in the context of Zoomorphic and Anthropomorphic Anatomies.
To represent differences in direction, such as "right" or "left", which tend to be innate and not
relative to an external compass, we utilize the concept of "half-rays" from Hilbert’s Foundations
of Geometry [8]. We also utilize the between relation, where we interpret between(a,b,c) as a is
between b and c. We translate Hilbert’s axioms for half rays below:
∀A∀A′ ∀O∀B∀abody_part(A) ∧ body_part(A′ ) ∧ body_part(O) ∧ body(a) ∧ part(A, a)
∧part(A′ a) ∧ part(O, a) ∧ between(O, A, B) ∧ ¬between(O, A, A′ )
⊃ same_side(A, A′ ) ∧ ¬same_side(A, B) ∧ ¬same_side(A′ , B) (1)
3.1.1. General Anatomy
The basic building blocks for a skeleton are fixed, immovable parts(bones) and movable parts(joints).
Joints dictate possible movements of bones – in other words, if possible movements are models
of axioms, joints constrain the number of feasible models. Hence, the weakest anatomy structure
contains only three classes - joint(x) , bone(y) and limb(z)(a limb can be any combination of bone
and joint).
3.1.2. Zoomorphic Anatomy
The second class of anatomy we define is zoomorphic/quadri-pedal. It builds upon the previously
defined General Anatomy, with the added restriction of the existence of a maximal element within
the hierarchy of joints, limbs and bones. This maximal element is the head(h).
3.1.3. Anthropomorphic Anatomy
Following the trend above, the Anthropomorphic Anatomy builds upon the Zoomorphic and
General Anatomy. The Anthropomorphic Anatomy not only has a single unique maximal element-
head(h), but also has two minimal elements - foot(f).
3.2. Ontology for Self-Awareness
Representations of space, and their use in qualitative spatial reasoning, are widely recognized as
key aspects in commonsense reasoning, with applications ranging from biology to geography.
The predominant approach to spatial representation within the applied ontology community has
used mereotopologies, which combine topological (expressing connectedness) with mereological
(expressing parthood) relations. A variety of first-order mereotopological ontologies have been
proposed, the most widespread being the Region Connection Calculus (RCC) [9], the ontology
RT [10], and the ontologies introduced by Casati and Varzi [11].
All of these approaches implicitly propose that there is a single connection relation. The
weakest classical mereotopology, Tmt , has a signature that consists of two primitive binary
relations, parthood (P) and connection (C). The axioms of the theory (Axioms 2 to 7) state that
connection is a reflexive and symmetric relation, while parthood is a reflexive, transitive, and
anti-symmetric relation. In addition, if one individual is connected to another, then the first one is
also connected to any individual which the second is part of.
C(x, x). (2)
C(x, y) ⊃ C(y, x). (3)
P(x, x). (4)
P(x, y) ∧ P(y, x) ⊃ (x = y). (5)
P(x, y) ∧ P(y, z) ⊃ P(x, z). (6)
P(y, z) ∧C(x, y) ⊃ C(x, z). (7)
However, the problems encountered in the motivating scenarios presented earlier in this paper
lead to the proposal of topological pluralism – there are multiple distinct connection relations
with different axiomatizations. Consider the relation touches(x, y) that formalizes the relation
that appears in the following natural language sentences:
My feet are touching the floor.
My hands are touching each other.
Touch your head.
Touch is a connection relation distinct from the connection relation between components and it
also has an axiomatization different from the connection relation in classical mereotopologies. In
particular, touches is not reflexive – a hand cannot touch itself, and a body only touches itself
if there are two disjoint proper parts of the body that touch each other. Similarly, a hand (as a
proper part of the body) only touches the entire body if there is another disjoint part of the body
that touches the hand e.g. two hands can touch each other and in this sense each hand is touching
the body.
(∀x, y) touches(x, y) ⊃ physical_body(x) ∧ physical_body(y) (8)
(∀x, y) touches(x, y) ⊃ touches(y, x) (9)
(∀x, y, z) touches(x, y) ∧ component_o f (y, z) ⊃ touches(x, z) (10)
touches is a subproperty of physical connection:
(∀x, y, z) touches(x, y)wedgeppart(y, z) ⊃ touches(x, z) (11)
(∀x) touches(x, x) ⊃ (∃y, z) ppart(y, x) ∧ ppart(z, x) ∧ ¬overlaps(y, z) ∧ touches(y, z) (12)
(∀x, y) touches(x, y) ∧ ppart(x, y) ⊃ (∃z) ppart(z, y) ∧ ¬overlaps(x, z) ∧ touches(x, z) (13)
3.2.1. Revisiting the Motivating Scenarios
With an established ontology signature and anatomy, we can now revisit our motivating scenarios
and redefine each instruction in first order logic. Each bipedal robot pose can be defined by a set
of conditions on the spatial relations between the robot’s body parts, as we can see in the next
three axiomatization of poses. Note that these are not instruction axioms, rather they are a way of
finding out what spatial relations are needed to understand each pose e.g. arms raised over head,
1. Arms Raised Over Head
a) Feet are completely touching the ground
∀x∀y f oot(x) ∧ f oot(y) ∧ x ̸= y ⊃ ∃z f loor(z) ∧ touches(x, z) ∧ touches(y, z) (14)
b) Hands are above head.
∀x∀y∀h hand(x) ∧ hand(y) ∧ head(h) ∧ x ̸= y ⊃ above(x, h) ∧ above(y, h) (15)
c) Shoulders and upper arms are next to head.
∀x∀y∀h shoulders(x) ∧ upper_arms(y) ∧ head(h) ∧ above(y, x) ⊃ ad jacent_to(x, h) ∧ ad jacent_to(y, h)
(16)
2. Right Hand Touching Top of Head
a) Feet are completely touching the ground
∀x∀y f oot(x) ∧ f oot(y) ∧ f loor(z) ∧ x ̸= y ⊃ ∃z f loor(z) ∧ touches(x, z) ∧ touches(y, z)
(17)
b) Only the right hand is touching the head.
∀l∀o body_part(l) ∧ body_part(o) ∧ touches(l, o) ⊃ right_hand(l) ∧ head(o)
(18)
c) No other body part is touching an external object.
∀l∀o body_part(l) ∧ ob ject(o) ∧ touches(l, o) ⊃ ( f oot(l) ∧ f loor(o)) ∨ (right_hand(l) ∧ head(o))
(19)
3. Left Leg Lift
a) Right foot is completely touching the floor.
∀l∀o body_part(l) ∧ ob ject(o) ∧ touches(l, o) ⊃ right_ f oot(l) ∧ f loor(o) (20)
b) Left foot is not touching the ground, or any other external object
c) No other body part, other than the right foot is touching an external object.
∀l∀o body_part(l) ∧ ob ject(o) ∧ touches(l, o) ⊃ f oot(l) ∧ right(l) ∧ f loor(o)
(21)
Both 4(b) and 4(c) are represented by axiom 23.
3.3. Ontology for Interacting with External Objects
In a typical role-playing game tutorial (RPG), you first figure out how to move your avatar before
interacting with the world within the game. Similarly, the Ontology for Interacting with External
Objects builds upon the Ontology for Self-Awareness.
Specifically, the Ontology for Interacting with External Objects differentiates the touch relation
used for touching/grasping from the connection relation used to describe the mereotopology
between a joint and a bone. This is also a reason why Euclidean geometry is simultaneously too
strong, yet insufficient to represent physical relations - it does not account for cases of topological
pluralism (as demonstrated).
If a robot is standing, then one of its feet touches the floor:
(∀x) standing(x) ⊃ (∃y) component_o f (y, x) ∧ f oot(y) ∧ touches(y, Floor) (22)
In fact, both of its feet are probably touching the ground, if it is bipedal:
(∀x)standing(x) ⊃ ∀x∀y∃z f oot(x) f oot(y) f loor(z)∧x ̸= y ⊃ touches(x, z)∧touches(y, z) (23)
Since it is only described as standing up, it is assumed that the robot is not touching any other
external object:
(∀x) standing(x) ⊃ ∀l∀o body_part(l) ob ject(o) ∧ touches(l, o) ⊃ f oot(l) ∧ f loor(o) (24)
If a robot is sitting, then its torso is touching a chair and its feet are touching the floor:
(∀x) sitting(x) ⊃ (∃y, z, u) component_o f (y, x) ∧ torso(y)
∧chair(u) ∧ touches(y, u) ∧ component_o f (z, x) ∧ f oot(z) ∧ touches(z, Floor) (25)
In the walk activity, one foot is always touching the ground:
(∀o, x) occurrence_o f (o, walk(x)) ⊃ ((∀s) subactivity_occurrence(s, o)
⊃ (∃y) componentO f (y, x) ∧ f oot(y) ∧ prior(tc(y, Floor), s) (26)
4. Methodology
Embodied question answering [12], [13] provides an interesting platform for identifying new
ontologies for qualitative spatial relations that formalize the semantic properties of the robot and
its environment [14]. In order to answer such questions, the agent must first intelligently navigate
to explore the environment, gather necessary visual information, and then answer the question.
The long-term goal is to build intelligent agents that can perceive their environment (through
vision and other sensors), communicate (i.e., hold a natural language dialog grounded in the
environment), and act. In this approach, a key capability is for intelligent robots to understand
natural language instructions ([15], [16], [17], [18], [19]) This includes the problem of parsing
natural language commands to actions and control structures that can be readily implemented
in a robot execution system. [20] and the ability for robots to interact with human partners in
following spoken instructions [21].
4.1. From Instructions to Process Descriptions
Semantic parsing maps natural language sentences to first-order logic formulae ([22], [23]). The
ontology is correct and complete with respect to the natural language corpus iff any conclusion
that the answer to a question about the natural language sentences is mapped to a logical formula
that is entailed by the ontology.
We are therefore interested in using the ontology to represent the intended semantics of the
terms that appear in the corpus.
Instructions are mapped to process descriptions with the PSL Ontology, which are logical
formulae representing activities and the constraints on their occurrences. A process description
for an atomic activity contains constraints that arise from the following two questions:
• Under what conditions does an atomic activity occur?
• How do occurrences of atomic activities change fluents?
Classes of complex activities are defined with respect to the following two questions:
• What is the relationship between the occurrence of the complex activity and occurrences of
its subactivities?
• Under what conditions does a complex activity occur?
An activity may have subactivities that do not occur; the only constraint is that any
subactivity occurrence must correspond to a subtree of the activity tree that characterizes
the occurrence of the activity.
Within the PSL Ontology, the notion of state is represented by reified fluents. Intuitively, a
change in state is captured by fluents that are either achieved or falsified by an activity occurrence.
The prior relation is used to specify the fluents that are intuitively true prior to an activity
occurrence and the holds relation specifies the fluents that are intuitively true after an activity
occurrence. Furthermore, a fluent can only be changed by the occurrence of activities. Thus, if
some fluent holds after an activity occurrence, but after an activity occurrence later along the
branch it is false, then an activity must occur at some point between that changes the fluent. This
also leads to the requirement that the fluents holding after an activity occurrence will be the same
fluents that are prior to any successor occurrence, since there cannot be an activity occurring
between them.
The design of the ontologies is driven by semantic parsing in two ways. In the first approach,
stative verbs and participles of dynamic verbs are mapped to relations between entities within
the signature of the ontology. For example, sitting, standing, touching are all present participles
of the dynamic verbs sit, stand, touch, respectively, Each of the first three are represented by a
relationship between an entity and its environment, while each of the last three are represented by
an activity that can possibly change the relationship. This leads to a series of domain ontologies
for new mereotopologies and spatial relations. Each domain ontology is translated into a domain
state ontology. Relations in the domain ontology are mapped to fluents in the domain state
ontology. Given the domain state ontology, we axiomatize the domain process ontology. Classes
of activities in the domain process ontology are associated with dynamic verbs within the phrase
map for the semantic parser.
In the converse direction, we start with the dynamic verbs, which are mapped to activities
within the signature of the ontology. We identify the fluents that are changed when the process
corresponding to the verb occurs. Axiomatize the domain state ontologies that contain the fluents
in their signature Identify the domain ontologies for these domain state ontologies. Axiomatize
the domain process ontologies. The process corresponding to the verb will either be an atomic
activity in one of the domain process ontologies or it is a complex activity that is composed of
activities in the domain process ontologies. The fluents which are changed by the activities within
the domain process ontology are associated with stative verbs and participles of dynamic verbs
within the phrase map for the semantic parser.
One technique for ontology validation is to demonstrate that these two directions indeed
converge on the same domain state and process ontologies. For example, the activities that change
the standing(x) fluent include stand(x), sit(x), walk(x).
The domain process methodology guarantees that we have a complete classification of activities
that change a given set of fluents. In turn, the set of fluents is a complete characterization of the
possible states of the world given the set of domain ontologies.
Figure 5: Two-dimensional cross-section
Figure 6: One-dimensional cross-section
4.2. Hilbert’s Foundations of Geometry
In Section 1, we argued the case for qualitative spatial reasoning; the primary reason is the lack of
a clear mapping between equation and natural language statement. The secondary reason is that
Euclidean geometry (the underlying framework of quantitative reasoning) is both too weak and
too strong to account for the nuances of spatial relationships between active agents and external
objects. The development of a qualitative spatial reasoning framework is to combat both of the
above issues.
A tertiary problem we seek to solve is that of undecideability. Other mereotopologies for qualita-
tive spatial reasoning, such as the region-connection-calculus (RCC8) and classical extensional
mereology are not decideable. Hence, concepts of parthood and connection are also undecideable.
However, we have found a potential workaround through Hilbert’s geometry (which is decideable,
due to the fact that it is interpretable by real arithmetic, which is decideable).
Our ontology is faithfully interpretable in Hilbert’s geometry. This was a solution devised to
combat the problem of not being able to discern poses mereotopologically. By mapping each
skeleton(anatomy) to a polygon under Hilbert’s geometry, we can utilize Hilbert’s theories of
connection and order to discern direction and contact without relying on an external surface or
compass. Hilbert’s geometry is traditionally interpreted as second-order due to the definition of a
line of a set of segments – where a "set" is mathematically a second order concept. However, we
propose an alternative interpretation of Hilbert’s Geometry – as a mereology of lines. Not only is
it applicable to describe different parthood relations, it is allows us to represent planes (2D) and
polygons(3D) as one dimensional lines. For example, Figure 5 can be represented by Figure 6, as
taken from Hilbert’s Foundations in Geometry [8]. In Figure 5, the line A-A’ is on a different
side as the line A-B relative to the plane a. All of this information is consolidated within Figure 6,
and describable with the between relation.
5. Conclusion
We began this paper with the goal of building the case for embodied AI and design the ontological
foundations necessary for robot understanding of natural language instruction. This is to build a
logically consistent, modular framework for performing qualitative physical reasoning. However,
there were two immediate challenges to this endeavor. There was the problem of differentiating
different directions and poses, independent of external environment, despite the fact that they
are mereotopologically equivalent. There was also the issue of differentiating the weakness and
strength of different connection relations (e.g. touching or being fused together). Hence, we
developed a general methodology based on Hilbert’s Geometry and the Process Specification
Language(PSL). We then applied this methodology to design an ontology for Robot Anatomy, as
well as an Ontology for Self-Awareness and an Ontology for Touch/Contact relations, which are
relative to the robot’s type and specification of anatomy. We then used these ontologies to define
axioms to describe poses related to sub-problems of the "Monkey and Banana Problem".
From a research program perspective, we will follow up this paper by further applying our
established methodology to discover new domain and process ontologies related to additional
natural language instructions in a robot context. From an ontology perspective, we will continue
fine-tuning other aspects of spatial awareness. For example, defining concepts of floor and ceiling,
and how they relate to each other e.g. the relationship between the first and second floor, or
between the ground floor ceiling and the top of the building.
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