Solving 3D geometric constraints for closed-loop assemblies

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.