On complex fixture. I. Mathematical descriptions

Presented is the complex fixture, as a mathematical model, for the purpose of designing large-scale fixture, planning force assembly, synthesizing mechanism tolerance and analyzing contact restraint. In contrast to a simplex fixture, a complex fixture is not necessarily accessible and deterministic. In order to recognize what is an effective fixel, and what is an active fixel for a complex fixture, a set of fundamental equations is developed, such as locating equation (1), rigidity constraint (2), clamping equation (3), and compliance model (4) etc. A complex fixture is regarded as a rigid fixture system (RFS), if it can be algebraically described by (1), (2), and (3). A complex fixture is an elastic fixture system (EFS), if it can be algebraically described by (1), (3), and (4). By exploiting equations (3) and (2), we may construct different geometrical primitives both in wrench space and in twist space to characterize a complex fixture. Based on the duality of (3) and (2), any RSF can be represented by a pair of linear programs: the primal P and the dual D