An automatic approach for identification of natural reciprocal screw systems of serial kinematic chains based on the invariance properties matrix

This paper describes an automatic procedure for identifying a set of (6 − f) natural constraint wrenches, termed natural reciprocal screws, for f < 6 degrees of freedom serial kinematic chains featuring special geometric properties. By a natural constraint wrench we mean a constraint wrench that can be directly related to pairs of geometric elements point, line and plane within the chain, irrespective of the values the joint variables may attain. While in the general case such natural constraint wrenches do not exist, for many practical applications the lines of action of the constraint wrenches can be predicted a priori, leading to advantages over the numerical solution. This paper regards such R/P serial chains with corresponding special geometries. The approach is based on the concept of the invariance properties matrix (IPM), which was proposed in the past to find the constraint conditions for single-loop spatial mechanisms allowing for closed-form solutions. The method is further developed here to identify the invariant geometric properties associated with the natural constraint wrenches. Several typical lower-mobility serial chains are taken as examples to demonstrate the effectiveness of the approach.