Linear and nonlinear stiffness and friction in biological rhythmic movements

Abstract

Biological rhythmic movements can be viewed as instances of self-sustained oscillators. Auto-oscillatory phenomena must involve a nonlinear friction function, and usually involve a nonlinear elastic function. With respect to rhythmic movements, the question is: What kinds of nonlinear friction and elastic functions are involved? The nonlinear friction functions of the kind identified by Rayleigh (involving terms such as theta3) and van der Pol (involving terms such as theta2theta), and the nonlinear elastic functions identified by Duffing (involving terms such as theta3), constitute elementary nonlinear components for the assembling of self-sustained oscillators, Recently, additional elementary nonlinear friction and stiffness functions expressed, respectively, through terms such as theta2theta3 and thetatheta2, and a methodology for evaluating the contribution of the elementary components to any given cyclic activity have been identified. The methodology uses a quantification of the continuous deviation of oscillatory motion from ideal (harmonic) motion. Multiple regression of this quantity on the elementary linear and nonlinear terms reveals the individual contribution of each term to the oscillator's non-harmonic behavior. In the present article the methodology was applied to the data from three experiments in which human subjects produced pendular rhythmic movements under manipulations of rotational inertia (experiment 1), rotational inertia and frequency (experiment 2), and rotational inertia and amplitude (experiment 3). The analysis revealed that the pendular oscillators assembled in the three experiments were compositionally rich, braiding linear and nonlinear friction and elastic functions in a manner that depended on the nature of the task.

Measuring Multi-Joint Stiffness during Single Movements: Numerical Validation of a Novel Time-Frequency Approach

“…Non-linear approximations to characterize limb mechanics often include a cubic stiffness term in addition to linear stiffness and damping terms [78], [79], [80], [81]. Maintaining constant stiffness and damping parameters and as in (52), we included a cubic stiffness term so that in the matrix version of (17) .…”

Influence of stimulus velocity profile on rhythmic visuomotor coordination.

“…(Beek, Rikkert, & van Wieringen, 1996;Beek, Schmidt, Morris, Sim, & Turvey, 1995;Kay, Kelso, Saltzman, & Schöner, 1987;Mottet & Bootsma, 1999;Roerdink et al, 2008). For example, Mottet and Bootsma (1999) have shown that continuous deviations from ideal (harmonic) motion occur in the kinematics of goal-directed aiming movements in a reciprocal Fitts' task and that these nonlinearities change as a function of the distance and precision constraints of the task.…”

Modeling inter-human movement coordination: synchronization governs joint task dynamics

“…While also transients of a pair of coupled oscillators can be reconstructed from simulated data, the method is rather sensitive to noise and requires extensive observation of transient regimes to yield stable results, since the whole statespace region is reconstructed. Inspired by the numerous variations of coupled oscillators models of rhythmic limb movements, Beek et al [1995a] systematically analyze how different components such as linear and nonlinear elastic and friction terms contribute to the composition of rhythmic movement. Jirsa and Kelso [2005] show in their work on dynamical movement models how the attractor landscape in its state space can be formed to reproduce a variety of both discrete and rhythmic movement behaviors, using their so-called excitators.…”