Geometric constraint subsets and subgraphs in the analysis of assemblies and mechanisms

Geometric Reasoning ability is central to many applications in CAD/CAM/CAPP environments. An increasing demand exists for Geometric Reasoning systems which evaluate the feasibility of virtual scenes specified by geometric relations. Thus, the Geometric Constraint Satisfaction or Scene Feasibility (GCS/SF) problem consists of a basic scenario containing geometric entities, whose context is used to propose constraining relations among still undefined entities. If the constraint specification is consistent, the answer of the problem is one of finitely or infinitely many solution scenarios satisfying the prescribed constraints. Otherwise, a diagnostic of inconsistency is expected. The three main approaches used for this problem are numerical, procedural or operational and mathematical. Numerical and procedural approaches answer only part of the problem, and are not complete in the sense that a failure to provide an answer does not preclude the existence of one. The mathematical approach previously presented by the authors describes the problem using a set of polynomial equations. The common roots to this set of polynomials characterizes the solution space for such a problem. That work presents the use of Groebner basis techniques for verifying the consistency of the constraints. It also integrates subgroups of the Special Euclidean Group of Displacements SE(3) in the problem formulation to exploit the structure implied by geometric relations. Although theoretically sound, these techniques require large amounts of computing resources. This work proposes Divide-and-Conquer techniques applied to local GCS/SF subproblems to identify strongly constrained clusters of geometric entities. The identification and preprocessing of these clusters generally reduces the effort required in solving the overall problem. Cluster identification can be related to identifying short cycles in the Spatial Constraint graph for the GCS/SF problem. Their preprocessing uses the aforementioned Algebraic Geometry and Group theoretical techniques on the local GCS/SF problems that correspond to these cycles. Besides improving the efficiency of the solution approach, the Divide-and-Conquer techniques capture the physical essence of the problem. This is illustrated by applying the discussed techniques to the analysis of the degrees of freedom of mechanisms.MSC: 68U07La habilidad del Razonamiento Geométrico es central a muchas aplicaciones de CAD/CAM/CAPP (Computer Aided Design, Manufacturing and Process Planning). Existe una demanda creciente de sistemas de Razonamiento Geométrico que evalúen la factibilidad de escenas virtuales, especificados por relaciones geométricas. Por lo tanto, el problema de Satisfacción de Restricciones Geométricas o de Factibilidad de Escena (GCS/SF) consta de un escenario básico conteniendo entidades geométricas, cuyo contexto es usado para proponer relaciones de restricción entre entidades aún indefinidas. Si la especificación de las restricciones es consistente, la respuesta al problema es uno del finito o infinito número de escenarios solución que satisfacen las restricciones propuestas. De otra forma, un diagnóstico de inconsistencia es esperado. Las tres principales estrategias usadas para este problema son: numérica, procedimental y matemática. Las soluciones numérica y procedimental resuelven solo parte del problema, y no son completas en el sentido de que una ausencia de respuesta no significa la ausencia de ella. La aproximación matemática previamente presentada por los autores describe el problema usando una serie de ecuaciones polinómicas. Las raíces comunes a este conjunto de polinomios caracterizan el espacio solución para el problema. Ese trabajo presenta el uso de técnicas con Bases de Groebner para verificar la consistencia de las restricciones. Ella también integra los subgrupos del grupo especial Euclídeo de desplazamientos SE(3) en la formulación del problema para explotar la estructura implicada por las relaciones geométricas. Aunque teóricamente sólidas, estas técnicas requieren grandes cantidades de recursos computacionales. Este trabajo propone técnicas de Dividir y Conquistar aplicadas a subproblemas GCS/SF locales para identificar conjuntos de entidades geométricas fuertemente restringidas entre sí. La identificación y pre-procesamiento de dichos conjuntos locales, generalmente reduce el esfuerzo requerido para resolver el problema completo. La identificación de dichos sub-problemas locales está relacionada con la identificación de ciclos cortos en el grafo de Restricciones Geométricas del problema GCS/SF. Su preprocesamiento usa las ya mencionadas técnicas de Geometría Algebraica y Grupos en los problemas locales que corresponden a dichos ciclos. Además de mejorar la eficiencia de la solución, las técnicas de Dividir y Conquistar capturan la esencia física del problema. Esto es ilustrado por medio de su aplicación al análisis de grados de libertad de mecanismos.MSC: 68U0

On Crossley's contribution to the development of graph based algorithms for the analysis of mechanisms and gear trains

This paper celebrates a particular branch of Crossley's early work dedicated to Mechanism Science, which deals with a rigorous introduction of Graph Theory to the study of some fundamental and intrinsic properties of kinematic chains and mechanisms. Although such idea gave its main outcome in Type and Number Synthesis (which has been much better and extensively described in another paper of the present special issue) some other intriguing side effects appeared, later in Mechanism Science, which yielded several results, and are still in the center of research and industrial world interest, such as, to name but a few, the automatic generation of the equations governing kinematic, static force and dynamic analysis of mechanisms and geared trains, the power flow analysis, the computation of the efficiency and, finally, the never fully explored structure-to-function mapping, which the present contribution points out to be still a challenge in the field

Geometric constraint subsets and subgraphs in the analysis of assemblies and mechanisms

Geometric Reasoning ability is central to many applications in CAD/CAM/CAPP environments -- An increasing demand exists for Geometric Reasoning systems which evaluate the feasibility of virtual scenes specified by geometric relations -- Thus, the Geometric Constraint Satisfaction or Scene Feasibility (GCS/SF) problem consists of a basic scenario containing geometric entities, whose context is used to propose constraining relations among still undefined entities -- If the constraint specification is consistent, the answer of the problem is one of finitely or infinitely many solution scenarios satisfying the prescribed constraints -- Otherwise, a diagnostic of inconsistency is expected -- The three main approaches used for this problem are numerical, procedural or operational and mathematical -- Numerical and procedural approaches answer only part of the problem, and are not complete in the sense that a failure to provide an answer does not preclude the existence of one -- The mathematical approach previously presented by the authors describes the problem using a set of polynomial equations -- The common roots to this set of polynomials characterizes the solution space for such a problem -- That work presents the use of Groebner basis techniques for verifying the consistency of the constraints -- It also integrates subgroups of the Special Euclidean Group of Displacements SE(3) in the problem formulation to exploit the structure implied by geometric relations -- Although theoretically sound, these techniques require large amounts of computing resources -- This work proposes Divide-and-Conquer techniques applied to local GCS/SF subproblems to identify strongly constrained clusters of geometric entities -- The identification and preprocessing of these clusters generally reduces the effort required in solving the overall problem -- Cluster identification can be related to identifying short cycles in the Spatial Con straint graph for the GCS/SF problem -- Their preprocessing uses the aforementioned Algebraic Geometry and Group theoretical techniques on the local GCS/SF problems that correspond to these cycles -- Besides improving theefficiency of the solution approach, the Divide-and-Conquer techniques capture the physical essence of the problem -- This is illustrated by applying the discussed techniques to the analysis of the degrees of freedom of mechanism

Computational kinematics of multibody systems: Two formulations for a modular approach based on natural coordinates

Multibody systems can be divided into an ordered set of kinematically determined modules, known as structural groups, in order to compute their kinematics more efficiently. In this work a procedure for the kinematic analysis of any kind of structural group is introduced, and two different methods for their solution in natural coordinates are presented: the time derivative (TD) and the third-order tensor (3OT) approaches. Moreover, the newly derived methods are compared in terms of efficiency with a global formulation, consisting in solving the kinematics of the multibody system as a whole using dense and sparse solvers. Two scalable case studies have been considered: a 2D four-bar linkage and a 3D slider-crank mechanism with an increasing number of constraint equations. The results show that the TD approach performs better in all cases with speed ups in a range of 27 to 61 times faster in 2D, and of 2.3 to 3.7 times faster in 3D with respect to the global sparse solution

Configraphics:

This dissertation reports a PhD research on mathematical-computational models, methods, and techniques for analysis, synthesis, and evaluation of spatial configurations in architecture and urban design. Spatial configuration is a technical term that refers to the particular way in which a set of spaces are connected to one another as a network. Spatial configuration affects safety, security, and efficiency of functioning of complex buildings by facilitating certain patterns of movement and/or impeding other patterns. In cities and suburban built environments, spatial configuration affects accessibilities and influences travel behavioural patterns, e.g. choosing walking and cycling for short trips instead of travelling by cars. As such, spatial configuration effectively influences the social, economic, and environmental functioning of cities and complex buildings, by conducting human movement patterns. In this research, graph theory is used to mathematically model spatial configurations in order to provide intuitive ways of studying and designing spatial arrangements for architects and urban designers. The methods and tools presented in this dissertation are applicable in: arranging spatial layouts based on configuration graphs, e.g. by using bubble diagrams to ensure certain spatial requirements and qualities in complex buildings; and analysing the potential effects of decisions on the likely spatial performance of buildings and on mobility patterns in built environments for systematic comparison of designs or plans, e.g. as to their aptitude for pedestrians and cyclists. The dissertation reports two parallel tracks of work on architectural and urban configurations. The core concept of the architectural configuration track is the ‘bubble diagram’ and the core concept of the urban configuration track is the ‘easiest paths’ for walking and cycling. Walking and cycling have been chosen as the foci of this theme as they involve active physical, cognitive, and social encounter of people with built environments, all of which are influenced by spatial configuration. The methodologies presented in this dissertation have been implemented in design toolkits and made publicly available as freeware applications

Equivariant Cosheaves and Finite Group Representations in Graphic Statics

This work extends the theory of reciprocal diagrams in graphic statics to frameworks that are invariant under finite group actions by utilizing the homology and representation theory of cellular cosheaves, recent tools from applied algebraic topology. By introducing the structure of an equivariant cellular cosheaf, we prove that pairs of self-stresses and reciprocal diagrams of symmetric frameworks are classified by the irreducible representations of the underlying group. We further derive the symmetry-aligned Euler characteristics of a finite dimensional equivariant chain complex, which for the force cosheaf yields a new formulation of the symmetry-adapted Maxwell counting rule for detecting symmetric self-stresses and kinematic degrees of freedom in frameworks. A freely available program is used to implement the relevant cosheaf homologies and illustrate the theory with examples.Comment: 29 pages, 9 figures, for code see https://github.com/zcooperband/EquivariantGraphicStatic

Evaluating the boundary and covering degree of planar Minkowski sums and other geometrical convolutions

AbstractAlgorithms are developed, based on topological principles, to evaluate the boundary and “internal structure” of the Minkowski sum of two planar curves. A graph isotopic to the envelope curve is constructed by computing its characteristic points. The edges of this graph are in one-to-one correspondence with a set of monotone envelope segments. A simple formula allows a degree to be assigned to each face defined by the graph, indicating the number of times its points are covered by the Minkowski sum. The boundary can then be identified with the set of edges that separate faces of zero and non-zero degree, and the boundary segments corresponding to these edges can be approximated to any desired geometrical accuracy. For applications that require only the Minkowski sum boundary, the algorithm minimizes geometrical computations on the “internal” envelope edges, that do not contribute to the final boundary. In other applications, this internal structure is of interest, and the algorithm provides comprehensive information on the covering degree for different regions within the Minkowski sum. Extensions of the algorithm to the computation of Minkowski sums in R3, and other forms of geometrical convolution, are briefly discussed