Solving 3D geometric constraints for closed-loop assemblies

Abstract

In the design activity, part geometry is assembled to create an assembly model. The number of parts may range from a few tens to a few million and typically the relationship among them constructs closed-loops with under-constrained states. In this paper, a 3D constraint solving method is proposed for closed-loop assemblies with under-constrained states. The proposed constraint solving method determines assembly configurations by applying the following procedures: 1. Transform the geometric mating relations into the kinematic joint relations, 2. Convert the closed-chain kinematic assembly to an open kinematic assembly by removing a joint, 3. Compute an open kinematic configuration by solving the open kinematic problem and 4. Obtain the closed-loop kinematic configuration by pasting the 'cut' links of the open assembly. The cut and paste operations minimise the number of constraint variables that have to be solved simultaneously. Thus, it can maximise the efficiency and robustness of an assembly constraint solver. The proposed constraint solving method combines the simplicity of a sequential solving approach with the universality of a simultaneous solving approach.

An algorithm for CAD tolerancing integration: Generation of assembly configurations according to dimensional and geometrical tolerances

“…Kim et al defined a relationship between the joint alignments, which are defined by the mating constraints, and the assembly Degrees of Freedom (DoFs) [18,19]. In an assembly model, the mating joints are described by the 4×4 transformation matrix.…”

“…Indeed, the geometric modeler cannot find a solution for positioning components with required mating constraints. The above methods of assembly modeling solve three dimensional geometric constraints, which model the relationships between assembly parts, basing on the hypothesis of ideal-rigid parts [6,[17][18][19][20][21].…”

Relation between part position and kinematic freedom: An expository survey

“…Interactive specification of the mating relations for assembly modelling has been considered in [12,13] and used for kinematic simulations in [14]. Kim et al [15,16] developed an assembly modelling system based on a geometric constraint-solving method that takes well-constrained assembly mating relations (against, fits, and coplanar) and transforms them into T-matrices.…”

Generation of assembly models from kinematic constraints

“…To get the intended assembly configuration, we use a hybrid method that combines the simplicity of the sequential positioning approach with the universality of the numerical positioning approach [12]. This constraint-solving method can maximize the efficiency and the robustness of an assembly constraint solver.…”