Solving 3d Geometric Constraints for Closed-loop Assemblies

In the design activity, part geometry is assembled together to create an assembly model. The number of parts can range from a few tens to a few millions and typically the relationships among them construct a closed-loop with under-constrained states in a constraint graph. In this paper, we propose an efficient and robust 3D constraint solving method for determining part positions in closed-loop assemblies with under-constrained states. The proposed constraint solving process consists of two steps: 1) cut operation and 2) paste operation. In the cut operation, the closed-loop constraint graph is first converted into an open-kinematic chain graph by removing a ‘cut’ constraint and determining kinematic joints from assembly constraints. Then, an initial kinematic configuration satisfying all the assembly constraints of the open-chain graph is computed incrementally. In the paste operation, the final assembly configuration of the closed-loop assembly is determined by pasting the ‘cut’ constraint and by simultaneously reconfiguring the initial kinematic chain that satisfies the ‘cut’ constraint. The proposed constraint solving method combines the simplicity of an incremental solving approach with the universality of a simultaneous solving approach. Thus, the proposed approach can maximize the efficiency and robustness of an assembly constraint solver.