New Schoenflies-Motion Manipulator Implementing Isosceles Triangle and Delassus Parallelogram

Schoenflies (X) motion is a 4D displacement Lie group including a spatial translation and any rotation whose axis is parallel to a given direction. Delassus parallelogram has four parallel screw (H) pairs with related pitches and the isosceles triangle is a special HHHP. After merging these two chains, an HHH-//-HHH generator of 2-DoF translation along a right helicoid is derived. It produces a 2-DoF motion mathematically modeled by a 2D submanifold of a 4D group of X-motion. Because of the product closure in an X-group, the 4-DoF generator with HHH-//-HHH loop serving as a subchain is revealed by adding two H pairs with axes parallel to fixed H axes. Parallel arrangement of two generators of the same X motion results in a new Schoenflies-motion manipulator with hybrid topology for 4-DoF pick-and-place operations. Four fixed H pairs (two double Hs) can actuate this manipulator and the two coaxial Hs must have distinct pitches. In addition, the possible design choices of special architectures are introduced for practical applications. Computer simulations of the new parallel manipulator with Schoenflies motion verify the effectiveness.

New Mechanical Generators of Schoenflies Motion Implementing Bennett Linkages

Based on the Bennett 4R chain, we construct a rotating loop by fixing one R axis to the frame and the fixed R becomes a coaxial double R pair. The R pair opposite to the fixed double R is replaced by a spherical S pair which can be equivalent to a (RRR) open chain with non-coplanar intersecting axes. In the (RRR) sub-chain, we choose special axes and derive R|- R|(R(RRR)R chain moving with 2 DoFs. That moving R becomes a coaxial double R with the addition of another rigid body and the obtained chain with hybrid topology generates a 3-dof motion, which is mathematically modeled by a 3D submanifold of a 4D group of X motions. Because of the product closure in an X-motion group, adding an H pair with any pitch and an axis parallel to the fixed R axis leads to a mechanical generator of a 4D X-motion group. Then, parallel arrangement of two generators of the same X motion gives a new parallel generator of X motion, which can be actuated by four fixed R pairs; the two Hs must have distinct pitches. A special design with four collinear actuated axes is revealed too.

ON SEMIGROUPS WHOSE IDEMPOTENT-GENERATED SUBSEMIGROUP IS APERIODIC

We show that if S is a finite semigroup with aperiodic idempotent-generated subsemigroup and H is a pseudovariety of groups, then the sequence of iterated H-kernels of S stops at the idempotent-generated subsemigroup if and only if each subgroup of S belongs to the wreath product closure of H. Applications are given to Mal'cev products.