PhD defence: Closed Linkages with Six Revolute Joints
In this dissertation, we explore spatial overconstrained closed linkages with six revolute joints and a single-degree-of-freedom (6R linkages). The first 6R linkage was invented by Pierre Frédéric Sarrus in 1853. In the literature, a lot of 6R linkages were found by numerous methods. New 6R linkages are still being found by new methods, too. But the answer on the classification question of 6R linkages is open. The aim of this dissertation is to try to fill the gap. We will use two new methods: bond theory and factorization of motion polynomial to analyze 6R linkages. These two methods, which were invented by my supervisor Josef Schicho and his collaborators, are based on algebraic geometry and computer algebra. In the first part, we will recall bond theory. Simultaneously, we give the genus bound for mobile 6R linkages. Using this new theory, we introduce a new technique for deriving equational conditions on the Denavit-Hartenberg parameters of 6R linkages that are necessary for movability. Several new families of 6R linkages are derived by this new technique. In the second part, we will recall the method of factorization of motion polynomials. There are cases where the factorization does not exist. But, even in these cases, we can do some reduction to the cases where the factorization does exist. Using the factorization method and bond theory, we construct several new 6R linkages. In the third part, we will give the sub classification of 6R linkages that have three equal pairs of opposite rotation angles (angle-symmetric 6R linkages). In the classification, there are three families. Two families are known and one family is new. This new 6R linkage has an additional parallel property, namely, three parallel pairs of joints. We also give the classification of the parallel 6R linkages. The new angle-symmetric family appears in both classifications. In addition, we find two other types. One has the translation property: three rotational axes can be obtained by a single translation from other three axes. The other one is a special case of the known family of angle-symmetric 6R linkages. In the final part, we will give an overview of results and open questions on the classification of 6R linkages.