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Machine Theory
Mechanism and Machine Theory 43 (2008) 271–284
ate/mechmt
Stiffness mapping pliant parallel mechanisms
in a serial arrangement
Hyun K. Jung a, Carl D. Crane III a,*, Rodney G. Roberts b
a Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, Florida 32611-6300, United States
b Florida State University, FAMU-FSU College of Engineering, Department of Electrical puter Engineering, United States
Received 6 February 2006; accepted 9 April 2007
Available online 21 June 2007
Abstract
This paper presents a stiffness mapping of a mechanism having pliant parallel mechanisms in a serial arrange-
ment. A derivative of the wrench connecting two moving bodies is derived and applied to obtain the stiffness matrix of the
mechanism. It is shown that the pliance of pliant parallel mechanisms that are serially arranged is
not the summation of pliances of the constituent mechanisms unless the external wrench applied to the mechanism
is zero. A numerical example is presented to support the result.
Ó 2007 Elsevier Ltd. All rights reserved.
Keywords: Compliance; Stiffness; Parallel mechanisms
1. Introduction
Compliant mechanisms can be considered as spatial springs having multiple degrees of freedom rather than
one as line springs do. For many robot applications which involve contact between the robot and its environ-
ment, compliant mechanisms can be essfully implemented pensate for the positional errors of the
robot system or to provide force control measures [1–3].
The concepts of twist and wrench from screw theory, which was introduced by Ball [4] are employed
throughout this paper to describe a small (or instantaneous) displacement of a rigid body and a force/torque
applied to a body [5]. A small twist applied to pliant mechanism generates a small change of the wrench
which pliant mechanism exerts on the environment and the stiffness matrix of the mechanism describes