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and
Machine Theory
Mechanism and Machine Theory 43 (2008) 1112–1128
ate/mechmt
Unification and simplification of velocity/acceleration
of limited-dof parallel manipulators with linear active legs
Yi Lu *,BoHu
Robotics Research Center, College of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, PR China
Received 31 July 2006; received in revised form 15 August 2007; accepted 21 August 2007
Available online 24 October 2007
Abstract
A unified and simple approach is proposed for solving inverse/forward velocity and acceleration of the limited-dof
PKMs (parallel kinematic machines) with linear active legs. First, a general PKM with n active legs is constructed and
some formulae for solving inverse displacement are derived. Second, the poses of the constrained wrench are determined
and the formulae for solving auxiliary Jacobian/Hessian matrices and statics are derived. Third, some formulae are derived
for solving the general velocity/acceleration, the 6 · 6 Jacobian matrix and the 6 · 6 · 6 Hessian matrix. Finally, the solv-
ing procedure is illustrated by applying it to a 3-SPR, a 4SPS + SPR and a 3-UPU PKMs.
Ó 2007 Elsevier Ltd. All rights reserved.
Keywords: Parallel manipulator; Kinematics; Jacobian matrix; Hessian matrix
1. Introduction
PKMs (parallel kinematic machines) have drawn continuous interest in both academia and industries in
recent years [1–8]. To unify and simplify their velocity/acceleration is greatly significant for analyzing kine-
matic/dynamic characteristics, determining singularity, and simplifying calculation of PKMs [6–8]. In the
aspect of kinematics, Huang et al. [1] studied the first/second-order kinematic influence coefficient matrices
which are proved to be Jacobian/Hessian matrices later. Based on Huang’s influence coefficient matrices,
Fang and Huang [6] solved the velocity/acceleration of a 3-RPS PKM. By means of screw theory, Rico
et al. [9] solved the accelerat