3 research outputs found
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
Distance-based formulations for the position analysis of kinematic chains
This thesis addresses the kinematic analysis of mechanisms, in particular, the position
analysis of kinematic chains, or linkages, that is, mechanisms with rigid bodies (links)
interconnected by kinematic pairs (joints). This problem, of completely geometrical
nature, consists in finding the feasible assembly modes that a kinematic chain can adopt.
An assembly mode is a possible relative transformation between the links of a kinematic
chain. When an assignment of positions and orientations is made for all links with
respect to a given reference frame, an assembly mode is called a configuration. The
methods reported in the literature for solving the position analysis of kinematic chains
can be classified as graphical, analytical, or numerical.
The graphical approaches are mostly geometrical and designed to solve particular
problems. The analytical and numerical methods deal, in general, with kinematic chains
of any topology and translate the original geometric problem into a system of kinematic analysis of all the Assur kinematic chains resulting from replacing some of its revolute
joints by slider joints. Thus, it is concluded that the polynomials of all fully-parallel
planar robots can be derived directly from that of the widely known 3-RPR robot. In
addition to these results, this thesis also presents an efficient procedure, based on distance
and oriented area constraints, and geometrical arguments, to trace coupler curves
of pin-jointed Gr¨ubler kinematic chains. All these techniques and results together are
contributions to theoretical kinematics of mechanisms, robot kinematics, and distance
plane geometry.
equations that defines the location of each link based, mainly, on independent loop
equations. In the analytical approaches, the system of kinematic equations is reduced
to a polynomial, known as the characteristic polynomial of the linkage, using different
elimination methods —e.g., Gr¨obner bases or resultant techniques. In the numerical
approaches, the system of kinematic equations is solved using, for instance, polynomial
continuation or interval-based procedures.
In any case, the use of independent loop equations to solve the position analysis
of kinematic chains, almost a standard in kinematics of mechanisms, has seldom been
questioned despite the resulting system of kinematic equations becomes quite involved
even for simple linkages. Moreover, stating the position analysis of kinematic chains
directly in terms of poses, with or without using independent loop equations, introduces
two major disadvantages: arbitrary reference frames has to be included, and all formulas
involve translations and rotations simultaneously. This thesis departs from this standard
approach by, instead of directly computing Cartesian locations, expressing the original
position problem as a system of distance-based constraints that are then solved using
analytical and numerical procedures adapted to their particularities.
In favor of developing the basics and theory of the proposed approach, this thesis
focuses on the study of the most fundamental planar kinematic chains, namely, Baranov
trusses, Assur kinematic chains, and pin-jointed Gr¨ubler kinematic chains. The results
obtained have shown that the novel developed techniques are promising tools for the
position analysis of kinematic chains and related problems. For example, using these
techniques, the characteristic polynomials of most of the cataloged Baranov trusses can
be obtained without relying on variable eliminations or trigonometric substitutions and
using no other tools than elementary algebra. An outcome in clear contrast with the
complex variable eliminations require when independent loop equations are used to tackle
the problem.
The impact of the above result is actually greater because it is shown that the
characteristic polynomial of a Baranov truss, derived using the proposed distance-based
techniques, contains all the necessary and sufficient information for solving the positionEsta tesis aborda el problema de análisis de posición de cadenas cinemáticas, mecanismos con cuerpos rÃgidos (enlaces)
interconectados por pares cinemáticos (articulaciones). Este problema, de naturaleza geométrica, consiste en encontrar los
modos de ensamblaje factibles que una cadena cinemática puede adoptar. Un modo de ensamblaje es una transformación
relativa posible entre los enlaces de una cadena cinemática. Los métodos reportados en la literatura para la solución del análisis
de posición de cadenas cinemáticas se pueden clasificar como gráficos, analÃticos o numéricos.
Los enfoques gráficos son geométricos y se diseñan para resolver problemas particulares. Los métodos analÃticos y numéricos
tratan con cadenas cinemáticas de cualquier topologÃa y traducen el problema geométrico original en un sistema de ecuaciones
cinemáticas que define la ubicación de cada enlace, basado generalmente en ecuaciones de bucle independientes. En los
enfoques analÃticos, el sistema de ecuaciones cinemáticas se reduce a un polinomio, conocido como el polinomio caracterÃstico
de la cadena cinemática, utilizando diferentes métodos de eliminación. En los métodos numéricos, el sistema se resuelve
utilizando, por ejemplo, la continuación polinomial o procedimientos basados en intervalos.
En cualquier caso, el uso de ecuaciones de bucle independientes, un estándar en cinemática de mecanismos, rara vez ha sido
cuestionado a pesar de que el sistema resultante de ecuaciones es bastante complicado, incluso para cadenas simples. Por otra
parte, establecer el análisis de la posición de cadenas cinemáticas directamente en términos de poses, con o sin el uso de
ecuaciones de bucle independientes, presenta dos inconvenientes: sistemas de referencia arbitrarios deben ser introducidos, y
todas las fórmulas implican traslaciones y rotaciones de forma simultánea. Esta tesis se aparta de este enfoque estándar
expresando el problema de posición original como un sistema de restricciones basadas en distancias, en lugar de directamente
calcular posiciones cartesianas. Estas restricciones son posteriormente resueltas con procedimientos analÃticos y numéricos
adaptados a sus particularidades.
Con el propósito de desarrollar los conceptos básicos y la teorÃa del enfoque propuesto, esta tesis se centra en el estudio de las
cadenas cinemáticas planas más fundamentales, a saber, estructuras de Baranov, cadenas cinemáticas de Assur, y cadenas
cinemáticas de Grübler. Los resultados obtenidos han demostrado que las técnicas desarrolladas son herramientas
prometedoras para el análisis de posición de cadenas cinemáticas y problemas relacionados. Por ejemplo, usando dichas
técnicas, los polinomios caracterÃsticos de la mayorÃa de las estructuras de Baranov catalogadas se puede obtener sin realizar
eliminaciones de variables o sustituciones trigonométricas, y utilizando solo álgebra elemental. Un resultado en claro contraste
con las complejas eliminaciones de variables que se requieren cuando se utilizan ecuaciones de bucle independientes.
El impacto del resultado anterior es mayor porque se demuestra que el polinomio caracterÃstico de una estructura de Baranov,
derivado con las técnicas propuestas, contiene toda la información necesaria y suficiente para resolver el análisis de posición de
las cadenas cinemáticas de Assur que resultan de la sustitución de algunas de sus articulaciones de revolución por
articulaciones prismáticas. De esta forma, se concluye que los polinomios de todos los robots planares totalmente paralelos se
pueden derivar directamente del polinomio caracterÃstico del conocido robot 3-RPR. Adicionalmente, se presenta un
procedimiento eficaz, basado en restricciones de distancias y áreas orientadas, y argumentos geométricos, para trazar curvas
de acoplador de cadenas cinemáticas de Grübler. En conjunto, todas estas técnicas y resultados constituyen contribuciones a la
cinemática teórica de mecanismos, la cinemática de robots, y la geometrÃa plana de distancias.
Barcelona 13
Distance-based formulations for the position analysis of kinematic chains
Tesis presentada por Nicolás Rojas a través del programa de doctorado "Automà tica, Robòtica i Visió" y realizada en el Institut de Robòtica i Informà tica Industrial, CSIC-UPC.This thesis addresses the kinematic analysis of mechanisms, in particular, the position analysis of kinematic chains, or linkages, that is, mechanisms with rigid bodies (links) interconnected by kinematic pairs (joints). This problem, of completely geometrical nature, consists in finding the feasible assembly modes that a kinematic chain can adopt. An assembly mode is a possible relative transformation between the links of a kinematic chain. When an assignment of positions and orientations is made for all links with respect to a given reference frame, an assembly mode is called a configuration. The methods reported in the literature for solving the position analysis of kinematic chains can be classified as graphical, analytical, or numerical. The graphical approaches are mostly geometrical and designed to solve particular problems. The analytical and numerical methods deal, in general, with kinematic chains of any topology and translate the original geometric problem into a system of kinematic analysis of all the Assur kinematic chains resulting from replacing some of its revolute joints by slider joints. Thus, it is concluded that the polynomials of all fully-parallel planar robots can be derived directly from that of the widely known 3-RPR robot. In addition to these results, this thesis also presents an efficient procedure, based on distance and oriented area constraints, and geometrical arguments, to trace coupler curves of pin-jointed Gr¨ubler kinematic chains. All these techniques and results together are contributions to theoretical kinematics of mechanisms, robot kinematics, and distance plane geometry. equations that defines the location of each link based, mainly, on independent loop equations. In the analytical approaches, the system of kinematic equations is reduced to a polynomial, known as the characteristic polynomial of the linkage, using different elimination methods —e.g., Gr¨obner bases or resultant techniques. In the numerical approaches, the system of kinematic equations is solved using, for instance, polynomial continuation or interval-based procedures. In any case, the use of independent loop equations to solve the position analysis of kinematic chains, almost a standard in kinematics of mechanisms, has seldom been questioned despite the resulting system of kinematic equations becomes quite involved even for simple linkages. Moreover, stating the position analysis of kinematic chains directly in terms of poses, with or without using independent loop equations, introduces two major disadvantages: arbitrary reference frames has to be included, and all formulas involve translations and rotations simultaneously. This thesis departs from this standard approach by, instead of directly computing Cartesian locations, expressing the original position problem as a system of distance-based constraints that are then solved using analytical and numerical procedures adapted to their particularities.In favor of developing the basics and theory of the proposed approach, this thesis focuses on the study of the most fundamental planar kinematic chains, namely, Baranov trusses, Assur kinematic chains, and pin-jointed Gr¨ubler kinematic chains. The results obtained have shown that the novel developed techniques are promising tools for the position analysis of kinematic chains and related problems. For example, using these techniques, the characteristic polynomials of most of the cataloged Baranov trusses can be obtained without relying on variable eliminations or trigonometric substitutions and using no other tools than elementary algebra. An outcome in clear contrast with the complex variable eliminations require when independent loop equations are used to tackle the problem. The impact of the above result is actually greater because it is shown that the characteristic polynomial of a Baranov truss, derived using the proposed distance-based techniques, contains all the necessary and sufficient information for solving the positionMy doctoral studies and the research reported in this thesis have been partially developed under the activities of: The Catalonian Reference Network in Advanced Production Technologies (XaRTAP), and have been partially supported by: The Colombian Ministry of Communications and Colfuturo through the Information and Communications Technology (ICT) National Plan of Colombia,.Peer Reviewe