Movement on Plane & Inclined Treadmill Daniela Tarnita - - PowerPoint PPT Presentation

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

• One method to reconstruct the state space is to generate the so-called delay coordinates

vectors

• The space constructed using the vector
• xn = {s(t0+nTs); s(t0+nTs+T);...; s(t0+nTs+(dE -1)T}
• is called the reconstructed space, where the integer dE is the embedding dimension.
• It is assumed that the geometry and the dynamics of the trajectory obtained using the

vectors xn are the same as the geometry and the dynamics of the trajectory in the actual phase space of the system

• The embedding dimension dE must be large enough so that the reconstructed orbit does not
• verlap 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

State Space Reconstruction

Correlation Dimension

• An attractor’s dimension is a measure of its geometric structure
• The correlation dimension, dF, is one of the most used measures of the fractal dimension, and
• ne of the chaotic characteristics which allows to define the dimension of an attractor and

shows the existence of the self-similarity

• The correlation dimension, C2(1), represents the probability that the distance between two

arbitrary points xi and xj of the reconstructed space will be 1.

• 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.

F

d 2

C (l) l 

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 di dimension, but they are not in the di+1 dimension they

are called false nearest neighbors

• The value where the percentage of false nearest neighbors approaches zero is chosen as dE 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

Subject Age [years] Weight [kg] Height [cm] Leg length [cm] Hip–knee length [cm] Knee–ankle 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 Patient Age [years] Weight [kg] Height [cm] Leg length [cm] Hip–knee length [cm] Knee–ankle 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

Healthy subjects - Mean values and standard deviations of anthropometric data Patients group - Mean values and standard deviations of anthropometric data

Anthropometric data

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
• rder to remove the transient data.

Diagram of the knee flexion-extension angles F1 [deg] for Test 1 and F2 [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 10o 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 normal one.

Phase plane plots for both plane TM and for inclined TM for Subject 1 Phase plane plots for both knees of Patient 1

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

• utside 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.

• As we can see, the correlation dimensions

were increasing, the higher the dimension D becomes and, also, they increase in the case

• f inclined treadmill comparing with plane

treadmill walking.

Graphics of correlation dimension in respect with all subjects for each D (Test1) Graphics of correlation dimension values in respect with embedding dimension D for each subject;

Mean Correlation Dimension

WALKING ON PLANE TREADMILL

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.2 1.968 1.983 2.056 2.069 2.175 2.156 2.232 2.305 Sub.3 1.957 1.973 2.023 2.056 2.240 2.352 2.398 2.461 Sub.4 2.048 2.218 2.299 2.454 2.590 2.657 2.645 2.752 Sub.5 1.966 2.009 2.043 2.162 2.152 2.245 2.262 2.283 Sub.6 1.995 2.138 2.319 2.457 2.546 2.619 2.646 2.848 Sub.7 1.961 2.102 2.278 2.405 2.500 2.572 2.618 2.837

WALKING ON INCLINED TREADMILL

D=3 D=4 D=5 D=6 D=7 D=8 D=9 D=10 Sub.1 2.134 2.297 2.338 2.459 2.547 2.593 2.619 2.788 Sub.2 2.055 2.277 2.405 2.601 2.646 2.918 3.023 3.024 Sub.3 1.981 2.105 2.159 2.394 2.353 2.408 2.602 2.625 Sub.4 2.186 2.402 2.503 2.805 2.782 2.819 2.903 3.093 Sub.5 1.986 2.123 2.323 2.398 2.417 2.378 2.396 2.542 Sub.6 2.111 2.29 2.397 2.479 2.572 2.765 2.783 2.909 Sub.7 1.996 2.138 2.284 2.434 2.529 2.589 2.658 2.737

Largest Lyapunov Exponent

HEALTHY SUBJECTS

LLE Sub1 Sub2 Sub3 Sub4 Sub5 Sub6 Sub7 TM 0 0.047 0.078 0.049 0.057 0.042 0.058 0.055 TM 10 0.048 0.082 0.056 0.06 0.047 0.058 0.06

PATIENTS

LLE Sub1 Sub2 Sub3 Left 0.132 0.098 0.108 Right 0.163 0.131 0.157

We can see that the LLE values are smaller

• n the plane treadmill than on the inclined

treadmill for healthy subjects. One of the explanations could be that the effects of the inclined plane on the variability are more pronounced than in the situation of plane walking. The LLE values are bigger for the normal knees

• f

the patients and much bigger for

• steoarthritic knees.

Larger values of LLEs obtained for normal knees of patients are explained by the influence of the osteoarthritic knees pain and instability.

Largest Lyapunov Exponent

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Introduction Nonlinear Dynamics Experimental study Results Conclusions

Conclusions 1/2

• The purpose of this study was to investigate the spatio-temporal characteristics, the

biomechanics of chaotic characteristics of movements of the knee joint

• The LLE obtained for each test of human knee joint were positive values which suggest that

human knee motions show chaotic characteristics.

• The mean LLE values for healthy human knee joints ranged from 0.042 to 0.082, while for the
• steoarthritic knees of patients LLE values ranged from 0.131 to 0.163.
• Larger values of LLEs obtained for patients are associated with more divergence and increase of

knee flexion variability, while smaller values obtained by healthy group for plane treadmill walking and inclined treadmill walking reflect a local stability, less divergence and variability, less sensitivity to perturbations and higher resistance to stride-to-stride variability.

Conclusions 2/2

• The flexion-extension movement of the human knee during the locomotion presents interest for

the kinematic and dynamic study of the bipedal locomotion applied to humanoid robots.

• The results obtained can be used as a reference for the normal knee joint movement for further

studies of abnormal movement

• The results can show the potential of the Lyapunov exponents used to measure the balance

control of human walking, an important aspect in the development of bipedal walking machines.

• The results of this study can be used in the medical field to develop new artificial devices that

could reproduce the motion of the lower limb, in robotics for humanoid robots which movements are inspired by human movement or as educational robot in order to study the complex process of locomotion.

Introduction Nonlinear Dynamics Experimental study Results Conclusions Future work

Future work

We will continue studies and we will do the same type of analysis:

• With a larger sample size
• With more type of tests
• walking on different degrees of inclination both ascent and descent
• walking up and down on a large number of stairs steps
• walking on different environments
• Considering many other variables
• influence of age , weight, and height;
• Influence of different degrees of osteoarthritis;
• Simultaneous analysis of the three lower limb joints (hip, knee, ankle)

Thank you!