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Research Papers

# Quotient Kinematics Machines: Concept, Analysis, and Synthesis

[+] Author and Article Information
Yuanqing Wu1

Research Institute of Robotics, School of Mechanical Engineering, Shanghai Jiaotong University, 800 Dongchuan Rd. 200240, Shanghai, P.R. Chinatroy.woo@gmail.com

Hong Wang2

Control and Mechatronics Department, HIT Shenzhen Graduate School, Shenzhen, P.R. Chinahit.wang.zc@gmail.com

Zexiang Li3

Professor Department of ECE, Hong Kong University of Science and Technology, Hong Kong S.A.R., P.R. Chinaeezxli@ust.hk

We use SO(3) to denote both the the 3-dimensional rotation subgroup of SE(3) and the special orthogonal group of $R3$.

Sim+ (3) denotes the group of all proper similarity transform of $R3:Sim+(3)≜{[λI3×3001]·g∈R4×4|λ∈R++,g∈SE(3)}$

1

Corresponding author. Present address: Department of ECE, Hong Kong University of Science and Technology, Hong Kong S.A.R., P.R.C.

2

Present address: Department of ECE, Hong Kong University of Science and Technology, Hong Kong S.A.R., P. R. C.

3

Present address: Control and Mechatronics Department, HIT Shenzhen Graduate School, Shenzhen, P. R. C.

J. Mechanisms Robotics 3(4), 041004 (Sep 26, 2011) (11 pages) doi:10.1115/1.4004891 History: Received August 05, 2010; Revised May 24, 2011; Published September 26, 2011; Online September 26, 2011

## Abstract

This paper presents a geometric analysis and synthesis theory for quotient kinematics machines (QKMs). Given a desired motion type described by a subgroup G of the special Euclidean group SE(3), QKM refers to a left-and-right hand system that realizes G through the coordinated motion of two mechanism modules, one synthesizing a subgroup H of G, and the other a complement of H in G, denoted by G/H. In the past, QKMs were often categorized into hybrid kinematics machines (HKMs) and were treated on a case-by-case basis. We show that QKMs do have a unique and well-defined kinematic structure that permits a unified and systematic treatment of their synthesis and design. We also study the properties of G/H as a novel motion type for parallel kinematics machine (PKM) synthesis. Another contribution of the paper is to model five-axis machines by SE(3)/R(o,z) (where R(o,z) represents the spindle symmetry) and give a complete classification of five-axis QKMs using the same geometric framework.

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## Figures

Figure 1

(a) Delta robot with ℜℜPaℜ subchains; (b) A HKM with equivalent SKM (ℜPaPaℜ) representation.

Figure 2

(a) A T2 (z) PKM with two TT subchains; (b) a U* (z) PKM with two ℜℜPaℜ subchains and one UU subchain.

Figure 3

T(3) QKM with M1  = T(z); M2  = T2 (z) and U* , both realized by a PKM

Figure 4

X(z) QKM with M1  = T2 (z) and M2  = C(o,z)

Figure 5

DS technology Echospeed-F-HT machine center, consisting of a 1T2R Sprint Z3 PKM head and a 2Txy-table generating 5-axis machining motion (Courtesy of DS Technology)

Figure 6

A classification of Lie subgroups of SE(3). The upper part of each box denotes the Lie subalgebra of the corresponding Lie subgroup in its normal form, and the lower part denotes the conjugacy class. Enclosed in the parenthesis is a generic member of the conjugacy class.

Figure 7

A QKM consists of two motion modules M1 and M2 acting in unison. Their relative rigid displacement is given by g1-1·g2∈M1-1·M2.

Figure 8

Two T2 (z) QKMs: (a) T2 (z)(T(x),T(y)) ; (b) T2(z)(T(x),Pa(z,y))

Figure 9

A collection of PL(z)(R(p,z),M2) QKMs

Figure 10

(a) Schematic of Z3 Head; (b) projected configuration space of Z3 Head: θ = 0; (c) projected configuration space of Z3 Head: θ=π2

Figure 11

Schematic of a {X(z)/T2 (z)} PKM, with one ℜℜPaℜ subchain, and two US subchains

Figure 12

Configuration of a 5-axis machine with spindle symmetry: g1  ∼ g2 iff g1-1g2∈R(o,z)

Figure 13

(a) the Tricept PKM [36], with one UT subchain and three UTS subchains. It has a motion type of U(-hz,x,y)T(z)∈{SE(3)/S(o)}; (b) a five-axis QKM having a T(3) module with three cascading linear axes, and a U(o,x,y) pan-and-tilt table.

Figure 14

(a) A 3-TℜℜU PKM, having the motion type of T(z)U(o,x,y) [23] (Courtesy of Q.C. Li); (b) Schematic of the inverted Exechon [37], a SE(3)/PL(z) PKM with two ℜTU subchains and one ℜTS subchain

Figure 15

(a) Omni wrist [38], a SE(3)/X(z) PKM with four identical UU subchains; (b) Heckett SKM400 3-axis machine tool [39], with one ℜPaPa subchain and three UTS subchains. It has a motion type of R(-hy,z)T2(x)∈{X(z)/R(o,z)}.

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