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Machine Theory
Mechanism and Machine Theory 43 (2008) 297–309
ate/mechmt
Representations of multi-joint stiffness for prosthetic limb design
Chad E. English a, Donald L. Russell b,*
a Neptec Design Group Ltd., Ottawa, Ontario, Canada
b Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, Ontario, Canada
Received 7 April 2006; received in revised form 5 April 2007; accepted 6 April 2007
Available online 13 June 2007
Abstract
Human limbs have inherent spring-like properties that have been shown to be important for posture and movement
control. When an amputated limb is augmented with a powered prosthesis, the stiffness properties of the resulting
limb-bination are an important aspect of the performance of the system. Limb stiffness properties are typ-
ically represented as stiffness ellipses or isopotential ellipses at the hand. Such representations are useful visual tools for
studies of human limb mechanics and in the design and control of limb prostheses and robotics. However, these represen-
tations require the stiffness to be stable and symmetric. An often neglected term to the endpoint stiffness and the effects of
the controllers of powered prostheses can both result in unstable conditions of the limb. Understanding the state and
behaviour of the stiffness es difficult when the possibility of instability is introduced, particularly since the stiffness
can no longer be represented as an ellipse. The current work introduces an alternate stiffness representation based on the
Mohr’s circle used in stress analysis. Realistic examples demonstrate the benefits of this approach and an example shows
that this instability can occur in situations that are within the capabilities of the human limb.
Ó 2007 Elsevier Ltd. All rights reserved.
Keywords: Stiffness; Human limb; Geometric stiffness; Instability; Mohr’s circle; Prosthetic limb
1. Introduction
The dynamics of motion and contact forces in