Mobility Analysis of Limited-Degrees-of-Freedom Parallel Mechanisms in the Framework of Geometric Algebra

Abstract

Mobility is a basic property of a mechanism. The aim of mobility analysis is to determine the number of degrees-of-freedom (DOF) and the motion pattern of a mechanism. The existing methods for mobility analysis have some drawbacks when being applied to limited-DOF parallel mechanisms (PMs). Particularly, it is difficult to obtain a symbolic or closed-form expression of mobility and its geometric interpretations are not always straightforward. This paper presents a general method for mobility analysis of limited-DOF PMs in the framework of geometric algebra. The motion space and constraint space of each limb are expressed using geometric algebra. Then the mobility of the PM can be calculated based on the orthogonal complement relationship between the motion space and the constraint space. The detailed mobility analyses of a 3-RPS PM and a 3-RPC PM are presented. It is shown that this method can obtain a symbolic expression of mobility with straightforward geometric interpretations and is applicable to limited-DOF PMs with or without redundant constraints. Without solving complicated symbolic linear equations, this method also has computational advantages.

Higher-Order Kinematics Modeling of 3-RRS Parallel Mechanism Based on CGA

“…At the same time, Selig also proposed a method using an eight-dimensional algebra to build rigid body dynamics, and he claimed that inertias, velocities, and momenta all can be represented as the elements of that algebra and all the relationships between physical quantities could be given by the standard operations [38]. After that, attentions start to be paid by researchers in the robotic fields and amount of work has been done based on Clifford algebra [39,40], such as the forward and inverse position kinematics [41][42][43][44], singularity analysis [45,46], robot POC representation [47], type synthesis of parallel mechanisms [48], dynamics [49], freedom or constraint analysis [50][51][52], first-order kinematics and grub jobs [53]. As far as we know, the research of higher-order kinematics of parallel mechanisms based on CGA is relatively rare.…”

Geometric algebra approach to analyzing the singularity of six-DOF parallel mechanism

“…So, common constraint should be used in calculating the outer product only once. An algorithm (Li et al, 2016) is proposed to identifying the common constraints. For six-DOF parallel mechanisms 6-UPS or 6-PUS, if the wrenches acting on the moving platform are linearly independent, the motions of the moving platform vanish if all the prismatic pairs are locked.…”

Bézier motions with end-constraints on speed

“…Lemma 3.1. Suppose that P ∈ G 4 is a point in a body undergoing a Bézier motion S(t) of degree n given by equations (5) and (6). Then the point traces out the parametric curve…”

“…These ideas can be extended to describing and manipulating free-form motions. These have applications in such areas as robotics [2,3], cutter paths in manufacturing [4], mechanisms [5,6], neuroscience [7], and motion of spacecraft [8]. A free-form motion is where a body moves smoothly through space.…”

“…In specifying free-form motions, one approach is to deal separately with the (translational) motion of a reference point in the moving body and the (rotational) motion of the body with respect to it [20,21,22]. An alternative is to handle the motion of the body as a single variable transform [5,23,24].…”