Application of Nonlinear Dynamics to Human Knee Movement on Plane & Inclined Treadmill Daniela Tarnita – University of Craiova, Romania Marius Georgescu – University of Craiova, Romania Dan Tarnita - University of Medicine and Pharmacy Craiova, Romania
Introduction
Introduction The objective of this study was to quantify and investigate nonlinear motion of the human knee joint on plane and inclined treadmill, using nonlinear dynamics stability analysis. The largest Lyapunov exponent (LLE) and correlation dimension where calculated as chaotic measures from the experimental time series of the flexion- extension angle of human knee joint.
Introduction Nonlinear Dynamics
State Space Reconstruction ◦ One method to reconstruct the state space is to generate the so-called delay coordinates vectors ◦ The space constructed using the vector ◦ x n = {s(t 0 +nT s ); s(t 0 +nT s +T);...; s(t 0 +nT s +(d E -1)T} ◦ is called the reconstructed space, where the integer d E is the embedding dimension. ◦ It is assumed that the geometry and the dynamics of the trajectory obtained using the vectors x n are the same as the geometry and the dynamics of the trajectory in the actual phase space of the system ◦ The embedding dimension d E must be large enough so that the reconstructed orbit does not overlap with itself. ◦ The dynamics in the reconstructed state space is equivalent to the original dynamics, so an attractor in the reconstructed state space has the same invariants, such as Lyapunov exponents and correlation dimension
Correlation Dimension ◦ An attractor’s dimension is a measure of its geometric structure ◦ The correlation dimension, d F, is one of the most used measures of the fractal dimension, and one of the chaotic characteristics which allows to define the dimension of an attractor and shows the existence of the self-similarity ◦ The correlation dimension, C 2 (1), represents the probability that the distance between two arbitrary points x i and x j of the reconstructed space will be 1. d F C (l) l 2 ◦ The correlation dimension is given by the saturation value of the slopes of the curves for an increasing embedding dimension. ◦ If there is no saturation of the slopes, and they keep increasing with the increasing of the embedding dimension, then the system is stochastic.
Embedding Dimension ◦ The embedding is a mapping from one dimensional space to a m-dimensional space and is based on the principle that all the variables of a dynamical system influence one another ◦ One of the most used method for measuring the minimal embedding dimension it is called the false nearest neighbor (FNN) method ◦ The main idea of the method is to unfold the observed orbits from self overlap arising from the projection of an attractor of a dynamical system on a lower dimensional space ◦ If two nearest neighbors are nearest neighbors in the d i dimension, but they are not in the d i+1 dimension they are called false nearest neighbors ◦ The value where the percentage of false nearest neighbors approaches zero is chosen as d E value
Lyapunov Exponents Lyapunov exponents are used to distinguish the chaotic and non-chaotic behavior because they exhibit the rate of divergence or convergence of the nearby trajectories from each other in state space, providing a measure of the system sensitivity to its initial conditions For a 3-dimensional state space there will be an exponent for each dimension: ◦ all negative exponents will indicate the presence of a fixed point; ◦ one zero and the other negative indicate a limit cycle; ◦ one positive indicate a chaotic attractor ; If the system has more than one positive Lyapunov exponent, the magnitude of the LLE indicates the maximum amount of instability in any direction in the attractor The value of the LLE is expressed in bits of information/second and it is the main exponent that quantifies the exponential divergence of the neighboring trajectories in the reconstructed state space and reflects the degree of chaos in the system.
Introduction Nonlinear Dynamics Experimental study
Experimental study ◦ The experimental method which allows obtaining the kinematic parameters diagrams for the human knee joint uses a data acquisition system based on electrogonimeters; Block schema of the acquisition data system based on electrogoniometer. Subject with data acquisition system
Anthropometric data Healthy subjects - Mean values and standard deviations of anthropometric data Subject Age Weight Height Leg length Hip – knee Knee – ankle [years] [kg] [cm] [cm] length [cm] length [cm] Average 31.57 77 174.71 85.29 45.28 40 St. Dev. 4.27 6.63 4.27 6.5 4.75 1.91 Patients group - Mean values and standard deviations of anthropometric data Patient Age Weight Height Leg length Hip – knee Knee – ankle [years] [kg] [cm] [cm] length [cm] length [cm] Average 55.7 81 172.27 82.35 43.15 39.2 St. Dev. 3.86 8.16 8.14 3.72 2.85 1.21
Introduction Nonlinear Dynamics Experimental study Results
Knee flexion-extension angles ◦ The angular amplitudes of human knee flexion-extension during the gait on the treadmill were obtained for each test as data files. ◦ In the data analysis the beginning region and the end region of the time series were cut off in order to remove the transient data. Diagram of the knee flexion-extension angles F 1 [deg] for Test 1 and F 2 [deg] for Test 2 in respect with time [s] for subject 1.
Phase plane ◦ Phase plane portraits are used to characterize the kinematics of the system when it attained this equilibrium. ◦ The plots traced for plane treadmill show less divergence in their trajectories, while the trajectories obtained for inclined treadmill at Phase plane plots for both plane TM and for inclined TM for Subject 1 10 o are confined within a tighter space. ◦ The phase plane curves are more compact for walking on TM 0 than for walking on TM 10. ◦ For all patients, the right knee is the pathological one and the left knee is the Phase plane plots for both knees of Patient 1 normal one.
Phase plane ◦ Comparing the four phase-plane graphics, it can be seen that for healthy subject the cycles curves of both tests are more compact and their spread is decreasing, while the plots traced for patient show more divergence in their trajectories. ◦ So at normal speed the amplitude of consecutive steps tend to be constant, while in the case of patient’s kneees the amplitude varies. ◦ It can be seen that the phase planes are almost concentric curves, inside is the curve corresponding to the osteoarthritic knee, and outside are the curves corresponding to the healthy subject.
Correlation Dimension ◦ Using Chaos Data Analyzer (CDA) software, we calculated Mean Correlation Dimension for each time series obtained in the both experimental tests. Graphics of correlation dimension values in respect with ◦ As we can see, the correlation dimensions embedding dimension D for each subject; were increasing, the higher the dimension D becomes and, also, they increase in the case of inclined treadmill comparing with plane treadmill walking. Graphics of correlation dimension in respect with all subjects for each D (Test1)
Mean Correlation Dimension WALKING ON PLANE TREADMILL WALKING ON INCLINED TREADMILL D=3 D=4 D=5 D=6 D=7 D=8 D=9 D=10 D=3 D=4 D=5 D=6 D=7 D=8 D=9 D=10 Sub.1 2.043 2.178 2.219 2.43 2.515 2.560 2.596 2.765 Sub.1 2.134 2.297 2.338 2.459 2.547 2.593 2.619 2.788 Sub.2 1.968 1.983 2.056 2.069 2.175 2.156 2.232 2.305 Sub.2 2.055 2.277 2.405 2.601 2.646 2.918 3.023 3.024 Sub.3 1.957 1.973 2.023 2.056 2.240 2.352 2.398 2.461 Sub.3 1.981 2.105 2.159 2.394 2.353 2.408 2.602 2.625 Sub.4 2.048 2.218 2.299 2.454 2.590 2.657 2.645 2.752 Sub.4 2.186 2.402 2.503 2.805 2.782 2.819 2.903 3.093 Sub.5 1.966 2.009 2.043 2.162 2.152 2.245 2.262 2.283 Sub.5 1.986 2.123 2.323 2.398 2.417 2.378 2.396 2.542 Sub.6 1.995 2.138 2.319 2.457 2.546 2.619 2.646 2.848 Sub.6 2.111 2.29 2.397 2.479 2.572 2.765 2.783 2.909 Sub.7 1.961 2.102 2.278 2.405 2.500 2.572 2.618 2.837 Sub.7 1.996 2.138 2.284 2.434 2.529 2.589 2.658 2.737
Largest Lyapunov Exponent HEALTHY SUBJECTS PATIENTS We can see that the LLE values are smaller The LLE values are bigger for the normal knees on the plane treadmill than on the inclined of the patients and much bigger for treadmill for healthy subjects. osteoarthritic knees. One of the explanations could be that the Larger values of LLEs obtained for normal effects of the inclined plane on the knees of patients are explained by the variability are more pronounced than in the influence of the osteoarthritic knees pain and situation of plane walking. instability. Sub1 Sub2 Sub3 Sub4 Sub5 Sub6 Sub7 Sub1 Sub2 Sub3 LLE LLE Left 0.132 0.098 0.108 TM 0 0.047 0.078 0.049 0.057 0.042 0.058 0.055 Right 0.163 0.131 0.157 TM 10 0.048 0.082 0.056 0.06 0.047 0.058 0.06
Largest Lyapunov Exponent
Introduction Nonlinear Dynamics Experimental study ` Results Conclusions
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