Kinematic Redundancy Analysis for (2$n$+1)R Circular Manipulators

The kinematic analysis of redundant serial manipulators with
$2n+1$
revolute joints (integer
$n \geq 3$
), which we call circular manipulators, is presented in this article. The structure of the kinematic chain of circular manipulators has special properties that can be seen in the Denavit–Hartenberg parameters: all orthogonal distances are zero, all even-numbered offsets are zeros, but odd-numbered offsets are not. Typical manipulators that fulfill these properties are redundant 7R serial chains (
$n=3$
) that mimic the human arm, e.g., the lightweight robot arm KUKA LBR iiwa. This 7R circular manipulator has self-motion as rotation around an axis that goes through two fixed points for a fixed pose. First, radical reparametrization is presented based on the swivel angle of the closed-form inverse kinematics solution for the 7R circular manipulator. Second, for a six-dimensional task, the inverse kinematics solution for redundant serial manipulators with
$2n+1$
revolute joints (
$n\geq 3$
) is reparametrized by the swivel angle and other
$2n-6$
rotation parameters. From a geometric point of view, for a circular manipulator with
$2n+1$
revolute joints, one can have
${n(n-1)}/{2}$
choices of such circular rotations. Third, we conjecture numerical kinematic singularities for circular manipulators in a recursive formula, confirming
$n=5,6,7$
.