Tools for Analysis of Dynamical Systems
A variety of different tools have been used to study biodynamic systems. The most common are measures of kinematic variability. System stability has been quantified using stability diffusion analysis and maximum Lyapunov exponents. Finally, a new approach to locate boundaries in the dynamical system is presented.
Kinematic Variability
Kinematic variability(KV) parameters measure the amount of variability in the system over the total time series. Common parameters are the root mean squared (rms) value of movement in the anterior-posterior (AP) direction, rms value in the medial-lateral direction, or the rms value in the radial direction. These measures may be applied to either the center of mass (COM) or the center of pressure (COP). Other common measures are the 95% ellipse area and the mean path velocity.
Stability
The stability of a dynamic system is often quantified using KV, however, KV is not a true measure of system stability. System stability is better quantified using stability diffusion analysis or maximum Lyapunov exponents. In both of these methods, trajectories are followed for a finite time, and the stability of the system is determined by averaging the results over the time series.
Lagrangian Coherent Structures
Lagrangian Coherent Sturctures are defined as state space boundaries in the finite time Lyapunov exponent (FTLE) field. These can be used to locate the boundaries that separate regions of state space in which qualitatively different behavior is exhibited. The approach uses similar methods to calculate the FTLE to those used to determine stability in the paragraph above. However, unlike previous approaches that average the FTLE over the entire time series, in this method an FTLE field is generated based on the initial location of each pair of trajectories in state space. This results in an entire field of data rather than a single scalar value for the system.
Inverse Dynamics Analysis
Inverse Dynamic Analysis is a tool that can be used to estimate the net internal joint forces and moments that connot be measured without using invasive procedures.