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Machine Theory
Mechanism and Machine Theory 43 (2008) 1175–1185
ate/mechmt
A simple method to calculate mobility with Jacobian
Dong-Chao Yang *, Jing Xiong, Xiang-Dong Yang
Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, China
Received 14 January 2007; received in revised form 4 August 2007; accepted 6 August 2007
Available online 21 September 2007
Abstract
Although many mobility formulae have been set in the last 150 years, unfortunately, some of them are not fit for many
classical mechanisms and some are indigestible. This paper gives a simple method to calculate the mobility of all kinds of
parallel mechanisms only with Jacobian matrixes. It can position of degrees-of-freedom and output speeds of
moving platform in passing. Furthermore, this method can be used to determine the existence of inactive joints and equiv-
alent serial chain plicated parallel chain.
Ó 2007 Elsevier Ltd. All rights reserved.
Keywords: Mobility; Spatiality; Degree-of-freedom; Parallel mechanisms
1. Introduction
Many mobility formulae have been set in the last 150 years, for example, the formulae based on theory of
linear transformation [1], screw theory [2–5], Lie group and Lie algebra [6–9], theory of topological features
[10]. Gogu studied 35 famous mobility formulae in [11], and he grouped formulae into two categories: (a)
approaches for mobility calculation based on setting up the kinematic constraint equations and their rank cal-
culation for a given position of the mechanism with specific joint location, (b) formulae for a quick calculation
of mobility without need to develop the set of constraint equations.
The approaches for mobility calculation based on setting up the kinematic constraint equations and their
rank calculation are valid without exception. The major drawback of these approaches is that the mobility
cannot be determined quickly without setting up the kinemati