Study Of Modeling And Solving Methods Of Geometric Constraints System

Geometric constraints solving is a kernel technology of CAD systems. The development of a geometric constraint solver involves some key technologies such as modeling, decomposition, constraint maintenance and solving. The model of geometric constraints system is firstly studied adopting Euler parameter and basic constraints to express geometric constraints and geometric elements. After analyzing the decoupled property of orientation and position constraints, the basic bodies are summarized into ball, box and ball-box body to express spatial geometric elements which forms a particular hierarchy structure of geometric constraints model. A geometric constraints system is expressed with a directed geometric constraint graph which describes the relation among the basic geometric elements.On the basis of the united expressions of geometric constraints and elements, some solving methods are advanced in this thesis to improve solving efficiency as well. For the decoupled case of orientation constraints and position constraints, a new algorithm based on spherical geometry and spherical four-linkage mechanism to solve orientation constraints is presented. The approach maps orientation constraints in 3D Euclidean space to spherical surface and reasons them with translation, rotation and rigid transformation on spherical surface. The combination of orientation constraints can be categorized into two cases: operable and un-operable. The former can be solved efficiently by introducing simple spherical surface reasoning. The latter is solved by employing spherical surface four-linkage mechanism. This method can be applied for under-constrained and well-constrained system; in addition, can detect redundant constraints for over-constrained system. Also, analytical solution to position constraints is investigated in this thesis. The basic position constraints are mapped to translation spaces which are expressed by parametric equations. The mixed method of the incremental analytical intersections of translation spaces and numerical interation method is employed to obtain the solutions satisfying design intend. This approach keeps the independency of basic constraints expression, which is beneficial to modeling and manage constraints of geometric constraint system.In 2D well-constrained geometric constraints system, there exsit rigid strongly connected components assembly patterns. The constructible merge patterns of rigid strongly connected components assembly patterns is analyzed and distinguished in this thesis. A geometric incremental construction algorithm is presented to solve the constructible merge patterns. After constructing the sub-configurations of constructible merge patterns, free space of rigid components can be worked out. The unsatisfactory constraint can be constructed in free space. The proposed method is of practical benefit to geometric constraints solving.However, the solution of non-constructible flexible merge patterns is a key issue for geometric constraints system and can not be solved by the above methods. An equivalent iteration method to solving flexible merge patterns is advanced. In the proposed approach, the strongly coupled connected components are broken by introducing equivalent constraints and tearing part constraints, and a solving sequence, only including basic geometric elements, is obtained. If given values of equivalent constraints variables, every vertex of the solving sequence has closed solution. On the basis of properties and structure of equivalent constraint, equivalent iteration Jacobian matrix can be worked out linearly if the vertexes of the solving sequence have real solutions; otherwise, local iteration is performed. The solution of non-constructible flexible merge patterns can be accomplished by equivalent constraints variables iteration, rather than undertaking simultaneously numeric iteration.Before solving constraints, the redundant constraints must be eliminated because singularity is an important factor that affects directly solving efficiency and ability of a geometric constraint solver. So, a novel perturbation on tangent-plane algorithm to effectively differentiate between redundant and embranchment singular constraints is presented. The presented algorithm obtains redundant constraint set by analyzing singular Jacobian matrix with Gaussian Elimination Method. Singular Jacobian matrix is separated into independent configuration space and singular space and constraint system variables are divided into independent configuration space variables and singular space variables. Set a perturbing value for singular space variables, the increment of independent configuration space variables can be worked out by the relation of the two kinds of variables. Recalculating the rank of Jacobian matrix, embranchment singularity and redundant singularity can be differentiated in terms of the change of the rank deficit that the constraint is a redundant singular constraint if the rank deficit unchanged, otherwise an embranchment singular constraint. Some examples illustrate the feasibility and validity of the achievement of the researches above.