**Abstract** | In kinematics, the classifification of spatial paradoxical closed linkages with six revolute joints and a single degree-of-freedom (6R linkages) is still an open problem. In this talk, we will see how to use two algebraic methods, the bond theory and the factorization of motion polynomials, to solve some problems for 6R linkages and further kinematic problems. Using bond theory, we give a sharp bound for the genus of confifiguration curves of mobile 6R linkages. In the meantime, 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 from this new technique. The linkages with prismatic joints (P-joints) or helical joints (H-joints) are also considered in the framework of bond theory. For factorization of motion polynomials, we focus on the motion polynomials which admit a factorization. Algorithms for computing a factorization for such motion polynomials are developed. We use the factorization of motion polynomials for constructing single loop linkages and producing new Kempe linkages (multiple loops), which can follow a prescribed motion in 2D or 3D. In the end, we will give an overview of results and open questions on this topic – the algebra of motions in space. |