Student(s):
Tim Dunn and Ignas Satkauskas
Dates of Involvement:
Fall 2009 - Present
Faculty Advisor(s):
Alaa Ahmed
Graduate Mentor:
Krissy Snyder
Using Ideas of Dynamical Stability to Explore Gait
Introduction
We will be using ideas from non-linear dynamics to study differences in gait under various imposed conditions ideas originally proposed in [1,2]. We will take motion capture data of human movement, then analyze the collected data to compare stability across conditions. For the state-space analysis we will utilize Floquet theory to measure overall stability. To track stability of individual markers and segments known to be important in gait, we will first embed time-delayed copies in state-space, and then use measures developed from the idea of Lyapunov exponents.
Floquet Theory: A little background
Floquet analysis allows measuring the orbital stability of a system, describing how well a system responds to small perturbations across periodic cycles. Floquet theory makes the assumption that the system is purely periodic with a limit cycle embedded in the system. While the existence of a true limit cycle in the either walk or running gait is still not fully resolved current gait analysis allows for the assumption that it does. In order to accommodate this assumption the motion captured gait data is normalized to 101 Poincaré samples (0 % - 100%) for each data set.
Poincaré sections are used to reduce the two dimensional problem into a simpler 1 dimensional model. Finding the Floquet multipliers for the system is done by setting a point X on the limit cycle for some function F which satisfies
Fitting a fixed point in the map function into equation 1 gives
(2) 
Looking at what happens between points 
and 
near 
shows the evolution of the system as
Now exploiting a Taylor expansion around the series
(4) 
Letting 
equation 4 becomes
(5) 
Using equation 5, 
becomes the Floquet multiplier 
and generalizing equation 5 we find
(6) 
Looking at equation 6 we see that when
meaning that as the fixed point is stable as the points 
approach the fixed point. It follows that the trajectories of these points about the limit cycle correspond to the 1D subspace perpendicular to the flow of the system, yielding the Poincaré section. Taking the average of the 101 normalized Poincaré sections produces the fixed points of the Poincaré section.
The Floquet multipliers FM can be further characterized in exponential form using eigenvalues as
(7) 
Again showing that 
produces stable orbital systems for 
.
Lyapunov Exponent Theory: Analytical and Practical
The individual marker and segment data will, once embedded in state space, be analyzed using ideas from Lyapunov Theory. We will begin with both an analytical and practical derivation of the exponent proper.
Analytical Derivation
Lyapunov exponent is used to measure the sensitivity of the dynamical systems to the initial conditions. Consider 1-D map:
Now take somewhat perturbed initial condition
After n iterations we have
Taylor expanding about initial condition gives us
After canceling same terms and taking absolute value on both sides we get
Using chain rule we get
Now we are assuming that initial perturbation either grows or decays exponentially. Hence, from equation (5) we get
Taking natural logarithm on both sides and multiplying by n we arrive at
Using equation (6) and rewriting natural logarithm of the product as the sum of natural logarithms we get
Now we are ready to define Lyapunov exponent as a limit of equation (9)
If Lyapunov exponent is less than zero then initial perturbation decays exponentially and the dynamical system is not sensitive to initial conditions. On the other hand, if Lyapunov exponent is greater that zero then initial perturbation grows exponentially and the dynamical system proves to be very sensitive to initial conditions.
Practical Calculation
In practical experiments, we do not have explicit iteration function, and hence, can not calculate Lyapunov exponent using definition in equation (10). In practice, we must use finitely many experimental data points to calculate Lyapunov exponent. We start with two nearby initial conditions, 
and 
. Each time step n we measure changed value 
of initial perturbation 
. As in analytical section, we assume that perturbation either grows or decays exponentially, and hence, we use formula:
Taking natural logarithm on both sides of equation (11) we arrive at
Hence, if we plot 
versus n we should get a straight line that has a slope equal to Lyapunov exponent.
What we will be doing
Past work has used pure Lyapunov exponents, but this has at times proven to be problematic for experimental data. Therefore, one of our tasks will be to find or create a more appropriate quantity.
About Tim Dunn and Ignas Satkauskas:
Tim Dunn is a return student to CU currently concentrating on degrees in both Applied Mathematics and Computer Science. In addition he is finishing off a degree in Molecular, Cellular and Developmental Biology. His interests are in nonlinear dynamics and high performance scientific computing.
Ignas Satkauskas is a senior Mathematics major. He is currently taking classes in pure and applied mathematics, works as a grader, and, obviously, participates in MTCP research on gait. Next fall, he is planning to go to graduate school in Applied Mathematics in order to extend his knowledge, do research, and play a little part in education of this fascinating field.
References:
J.B. Dingwell, and J.P. Cusumano. Nonlinear time series analysis of normal and pathological human walking. Chaos: An Interdisciplinary Journal of Nonlinear Science, 10:848-863, 2000.
Y. Hurmuzlu, and C. Basdogan. On the measurement of dynamic stability in human locomotion. Medicine & Science in Sports & Exercise, 25(10):1158-1164, 1993.
