The ultimate goal of upper limb prosthesis development is to duplicate the behaviour of the natural arm. To duplicate the natural behaviour we must first understand it. The current work investigates the behaviour of stiffness in a two-link manipulator based on a musculo-skeletal model of the human arm. The focus of the work is on the development of an above-elbow upper limb prosthesis but the results are relevant to robotic manipulators, haptic devices, human movement studies, and any linkage mechanism that interacts with external forces.

Derivation of the limb stiffness as seen at the hand shows two distinct types of stiffness:​ the endpoint spring stiffness that is dependent on actuator stiffness (such as muscles) and the geometric stiffness that is solely dependent on the changing geometry and is often neglected in stiffness studies. A key accomplishment of the current work is the decomposition of the endpoint spring stiffness into a sum of projections of antagonistic actuator stiffnesses, each contributing to the endpoint stiffness in a unique way. Limitations, techniques, and desired properties for the control of endpoint stiffness are investigated. One consequence of these limitations is that natural limb behaviour cannot be reproduced in an artificial manipulator without replacing the stiffness effects of bi-articular muscles in the upper arm.

The geometric stiffness is investigated and shown to be significant at relatively low contact forces, potentially causing instability with forces on the order of 50N (10 lb) for typical joint stiffness ranges seen in human limb studies. The geometric stiffness has predictable behaviour based on the direction of endpoint force. The current work characterizes this behaviour in terms of stiffness size, shape, and orientation with respect to the direction of applied force. External forces toward the shoulder tend to decrease the endpoint stiffness and potentially destabilize the arm. Forces away from the shoulder increase the stiffness and further stabilize it. Methods for using and compensating for the geometric stiffness are discussed.

We develop several analytical tools to understand stiffness. Mohr's circle is adapted to represent a general stiffness, including stiffnesses that are non-conservative and unstable. Traditional stiffness representations do not allow such cases. We also develop a figure of merit for determining when the geometric stiffness is negligible.