Josef Schicho

In a foregoing publication, the authors studied pentapods with mobility 2, where neither all the platform anchor points nor all the base anchor points are located on a line. It turned out that the given classification is incomplete. This article is devoted to the discussion of the missing cases resulting in additional solutions already known to Duporcq.

In a foregoing publication the authors studied pentapods with mobility 2, where neither all platform anchor points nor all base anchor points are located on a line. It turned out that the given classification is incomplete. This addendum is devoted to the discussion of the missing cases resulting in additional solutions already known to Duporcq.

We use the recently introduced factorization theory of motion polynomials over the dual quaternions and cubic interpolation on quadrics for the synthesis of closed kinematic loops with six revolute joints that visit four prescribed poses. The resulting 6R linkages are special in the sense that the relative motions of all links are rational. They exhibit certain elegant properties such as symmetry in the rotation angles and, at least in theory, full-cycle mobility. Our synthesis approach admits either no solution or two one-parametric families of solutions. We suggest strategies for picking good solutions from these families. They ensure a fair coupler motion and optimize other linkage characteristics such as total rotation angle or linkage size. A comprehensive synthesis example is provided.

In this paper, we give a full classification of all pentapods with mobility 2, where neither all platform anchor points nor all base anchor points are located on a line. Therefore, this paper solves the famous Borel–Bricard problem for two-dimensional motions beside the excluded case of five collinear points with spherical trajectories. But even for this special case, we present three new types as a side-result. Based on our study of pentapods, we also give a complete list of all nonarchitecturally singular hexapods with two-dimensional self-motions.

We show how to use formal desingularizations (defined earlier by the first author) in order to compute the global sections (also called adjoints) of twisted pluricanonical sheaves. These sections define maps that play an important role in the birational classification of schemes.