General order principle for multi-Bennett linkages

Order analysis for multi-Bennett linkages is a difficult topic in kinematics. Traditional methods fail to obtain the order of multi-Bennett linkages due to considering the special geometric distributions among joint axes. An order principle for multi-Bennett linkages is presented. For a summated multi-Bennett linkage, three procedures are included in the order principle. Firstly, a homogeneous screw equation is obtained by taking linear superposition operations and then the maximum order is determined according to linear dependency of all screws. Secondly, two theorems are employed to determine the maximum order, where the first is used to judge the linear independency of four-system screws and the second is fit for identifying the linear independency of five-system screws. Lastly, all possible cases in the order range are considered until the valid order is screened out. For a syncopated multi-Bennett linkage, an equivalent summated model is built and then the order analysis is the same as that of summated linkages. In order to verify the effectiveness of the presented order principle, the orders of summated 5R and 6R linkages as well as a syncopated 6R linkage are analyzed. The computed orders of the former two summated linkages are both 4 and the computed order of the last syncopated 6R linkage is 5. The results coincide with the prototype data. The advantage of the proposed principle is that it can get the correct order of a multi-Bennett linkage without solving the geometric conditions of joint axes and has wide application in variety of multi-Bennett linkages.