The difference between stiffness and quasi-stiffness in the context of biomechanical modeling

Abstract

The ankle contributes the majority of mechanical power during walking and is a frequently studied joint in biomechanics. Specifically, researchers have extensively investigated the torque-angle relationship for the ankle during dynamic tasks, such as walking and running. The slope of this relationship has been termed the "quasi-stiffness." However, over time, researchers have begun to interchange the concepts of quasi-stiffness and stiffness. This is an especially important distinction as researchers currently begin to investigate the appropriate control systems for recently developed powered prosthetic legs. The quasi-stiffness and stiffness are distinct concepts in the context of powered joints, and are equivalent in the context of passive joints. The purpose of this paper is to demonstrate the difference between the stiffness and quasi-stiffness using a simple impedance-controlled inverted pendulum model and a more sophisticated biped walking model, each with the ability to modify the trajectory of an impedance controller's equilibrium angle position. In both cases, stiffness values are specified by the controller and the quasi-stiffness are shown during a single step. Both models have widely varying quasi-stiffness but each have a single stiffness value. Therefore, from this simple modeling approach, the differences and similarities between these two concepts are elucidated.

Fig.1

Model of the inverted pendulum shown with motor M, and pendulum weight W. Motor impedance control system shown with imposed controller stiffness represented as a physical spring.

Fig.2

The pendulum angle (black) and equilibrium angle (gray) are shown as a function of time (left) and the torque as a function of angle (right) for case 1 (A), case 2 (B) and case 3 (C).The quasi-stiffness values are given for each case and the mechanical stiffness is 10 Nm/rad/kg.

Fig.3

Diagram of biped model. The origin of the shank coordinate frame (dashed axes) is drawn above the ankle for clarity but is actually modeled to coincide with the ankle joint (located at the heel for simplicity).

Fig.4

Effective shape (dashed red curve) with a radius of curvature R_s. The center of rotation is a point P_s, which is constant in the shank coordinate frame (solid axes). Note that the relative ankle angle θ_ank is shown instead of the global leg angle.

Fig.5

a. Relative ankle angle trajectory over time. b. Ankle torque-angle plot (of the non-zero region) during simulated walking. Quasi-stiffness values (i.e., tangential slopes) are shown at points along the curve. Note the mechanical stiffness is 6 Nm/rad/kg.