Convex Polygonal Hull for a Pair of Irregular Objects

Introduction. Optimization placement problems are NP-hard. In most cases related to cutting and packing problems, heuristic approaches are used. The development of analytical methods for mathematical modeling of the problems is of paramount important for expanding the class of placement problems that can be solved optimally using state of the art NLP-solvers. The problem of placing two irregular two-dimensional objects in a convex polygonal region of the minimum size, which is a convex polygonal hull of the minimum area or perimeter, is considered. Continuous rotations and translations of non-overlapping objects are allowed. To solve the problem of optimal compaction of a pair of objects, two algorithms are proposed. The first is a sequentially search for local extrema on all feasible subdomains using a solution tree. The second algorithm searches for a locally optimal extremum on a single subdomain using a "good" feasible starting point. Purpose of the paper. Show how to construct a minimal convex polygonal hull for two continuously moving irregular objects bounded by circular arcs and line segments. Results. A mathematical model is constructed in the form of a nonlinear programming problem using the phi-function technique. Two algorithms are proposed for solving the problem of placing a pair of objects in order to minimize the area and perimeter of the enclosing polygonal area. The results of computational experiments are presented. Conclusions. The construction of a minimal convex polygonal hull for a pair of two-dimensional objects having an arbitrary spatial shape and allowing continuous rotations and translations makes it possible to speed up the process of finding feasible solutions for the problem of placing a large number of objects with complex geometry. Keywords: convex polygonal hull, irregular objects, phi-function technique, nonlinear optimization.

Type Synthesis of 3-RSR Equivalent 2R1T Parallel Mechanisms

Parallel mechanisms (PMs) with two rotational and one translational (2R1T) degrees of freedom (DOFs) have attracted much attention these years. The 2R1T PMs can be divided into various categories due to different motion patterns, such as the UP equivalent PMs, the RPR equivalent PMs, the PU equivalent PMs and the 3-PPS equivalent PMs. In this paper, the 2R1T PMs have the same motion characteristics with the 3-RSR PM are studied and synthesized. This kind of PMs can be called as 3-RSR equivalent 2R1T PMs. The 3-RSR equivalent 2R1T PMs can realize both continuous rotations about fixed axes and continuous translation along fixed directions. The constraint and motion characteristics of the 3-RSR equivalent 2R1T PMs are analyzed. The design of the branches for the 3-RSR equivalent 2R1T PMs is dealt with using the screw theory and the subchains. A group of novel 3-RSR equivalent 2R1T PMs are obtained.