Abstract
The orthogonal group On is defined as the group of all invertible (n×n)matrices A whose
transposed matrix AT is the inverse of A. The special orthogonal group SOn consists of
all orthogonal (n × n)matrices with a determinant of 1. It represents rotations around
the origin in Rn. The special Euclidean group SEn comprises all pairs (R, t), where R is
a rotation in SOn and t is a translation vector in Rn. An element of SEn can be used to
represent the pose of an object. These three subgroups of GLn are smooth manifolds.
Factor graphs are bipartite graphs with variable nodes and factor nodes and define the
factorization of a function. They can encode geometrical relations among certain objects.
Additionally, a factor graph can be equipped with a probabilistic structure.
A retraction is a mapping from the tangent bundle TM of a smooth manifold M to the
manifoldM that satisfies certain properties, such as the local rigidity condition. By utilizing
the exponential map for matrices, retractions can be defined on SOn and SEn. Retractions
allow simple implementations of iterative optimization techniques on manifolds.
In the following specific application scenario, the relative positions of partially movable
components are estimated. Considering a truck as a composite object composed of simpler
components, such as its wheels, leads to a representation of the truck as a factor graph.
Variable nodes in the factor graph represent different parts of the truck, while factor nodes
represent the relative poses of these parts to each other. Introducing a sensor observing
specific parts of the truck expands the factor graph by adding variable nodes for the sensor
at each time step and factor nodes for the observations. Equipping factor nodes with
probability densities enables the computation of the maximum a posteriori estimate of
some state X given observations Z by maximizing the joint probability function p(X,Z)
through optimization on manifolds. This approach provides estimates for the configuration
of the truck and the pose of the sensor. Implementation and testing of this pose estimation
method for composite objects can be achieved using the Python package GTSAM.
transposed matrix AT is the inverse of A. The special orthogonal group SOn consists of
all orthogonal (n × n)matrices with a determinant of 1. It represents rotations around
the origin in Rn. The special Euclidean group SEn comprises all pairs (R, t), where R is
a rotation in SOn and t is a translation vector in Rn. An element of SEn can be used to
represent the pose of an object. These three subgroups of GLn are smooth manifolds.
Factor graphs are bipartite graphs with variable nodes and factor nodes and define the
factorization of a function. They can encode geometrical relations among certain objects.
Additionally, a factor graph can be equipped with a probabilistic structure.
A retraction is a mapping from the tangent bundle TM of a smooth manifold M to the
manifoldM that satisfies certain properties, such as the local rigidity condition. By utilizing
the exponential map for matrices, retractions can be defined on SOn and SEn. Retractions
allow simple implementations of iterative optimization techniques on manifolds.
In the following specific application scenario, the relative positions of partially movable
components are estimated. Considering a truck as a composite object composed of simpler
components, such as its wheels, leads to a representation of the truck as a factor graph.
Variable nodes in the factor graph represent different parts of the truck, while factor nodes
represent the relative poses of these parts to each other. Introducing a sensor observing
specific parts of the truck expands the factor graph by adding variable nodes for the sensor
at each time step and factor nodes for the observations. Equipping factor nodes with
probability densities enables the computation of the maximum a posteriori estimate of
some state X given observations Z by maximizing the joint probability function p(X,Z)
through optimization on manifolds. This approach provides estimates for the configuration
of the truck and the pose of the sensor. Implementation and testing of this pose estimation
method for composite objects can be achieved using the Python package GTSAM.
Originalsprache  Englisch 

Qualifikation  Master of Science 
Gradverleihende Hochschule 

Betreuer/in / Berater/in 

Datum der Bewilligung  12 Dez. 2023 
Publikationsstatus  Veröffentlicht  31 Dez. 2012 
Research Field
 Assistive and Autonomous Systems