CONE-CUTTING: A VARIANT REPRESENTATION OF PIVOT IN SIMPLEX

This study presents a variant representation of pivot in simplex, which performs cone-cutting on a cone C in dual space to match the pivot performed on a basis B, while the edge-vectors of C are indicated by the row vectors of the feature matrix F = B-1 in the simplex table. Under this representation, we can see the dual cone C of basis B through the feature matrix F directly, and we can perform pivot motivated by the monitor viewing toward the dual space. As an example, a constraint plane in the dual space is delete-able for the optimal searching if it does not pass through the dual optimal point, while such a plane corresponds to a variable being not in the optimal basis. Motivated by the cone-cutting's vision, a variable-sifting algorithm is presented in Sec. 3, which marks those variables corresponding to delete-able planes into a list to forbid them enter pivot and put zero to their components in the final solution.

A Novel Method on Singularity Analysis in Parallel Manipulators

A parallel manipulator is a closed loop mechanism in which a moving platform is connected to the base by at least two serial kinematic chains. The main problem engaged in these mechanisms, is their restricted working space as a result of singularities. In order to tackle these problems, many methods have been introduced by scholars. However, most of the mentioned methods are too much time consuming and need a great amount of computations. They also in most cases do not provide a good insight to the existence of singularity for the designer. In this paper a novel approach is introduced and utilized to identify singularities in parallel manipulators. By applying the new method, one could get a better understanding of geometrical interpretation of singularities in parallel mechanisms. Here we have introduced the Constraint Plane Method (CPM) and some of its applications in parallel mechanisms. The main technique used here, is based on Ceva Theorem.

Analysis of Singularities of a 3DOF Parallel Manipulator Based on a Novel Geometrical Method

In this article singular points of a parallel manipulator are obtained based on a novel geometrical method. Here we introduce the constrained plain method (CPM) and some of its application in parallel mechanism. Given the definition of constraint plane (CP) and infinite constraint plane (ICP) the dependency conditions of constraints is achieved with the use of a new theorem based on the Ceva geometrical theorem. The direction of angular velocity of a body is achieved by having three ICPs with the use of another theorem. Finally, with the use of the above two novel theorems singularities of the 3UPF_PU mechanism are obtained. It should be emphasized that this method is completely geometrical, involving no complex or massive calculations. In the previous methods based on the Grassmann Geometry, the mechanism needs to be statically analyzed at first, so that the Inverse Jacobian Matrix is achieved, and then the Plucker-Vector is derived. This task is somewhat inconvenient and in the end there are plenty of conditions remained to be pondered in order to obtain the singularity conditions, while the novel method introduced here, involves no tiring calculations neither the analysis of numerous conditions and yields the answer quickly.