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sequences called Higuchi fractal dimension (HFD). recordings and other biological signals [12,81]. Suppose, a given one-dimensional time series such

Figure 4 sequences called Higuchi fractal dimension (HFD). recordings and other biological signals [12,81]. Suppose, a given one-dimensional time series such

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The PDWs models are presented by two straight solid legs connected to a frictionless hinge [44-46]. The mass of upper body and the hip (MW, kg) is located at the hinge. Moreover, the mass of each leg (m, kg) is located at the end of the foot and is equal for both left and right legs [44]. Figure | illustrates a schematic view of PDW model. Fig. 1 a—d Simplest passive walking model at a typical step. b 6, y, M,m, 1, y and g are stance leg (thin line) angle, angle between swing and stance leg, hip mass, leg mass, length of leg, slope of ramp and acceleration due to gravity, respectively. This figure is reformed from [44,45] The nonlinear motion equation “Stride Function” (1) which is also known as Poincare map [47] consists of two coupled second-order differential equation in terms of stance leg angle 6 and swing angle ¢ as functions of time t. 6, 6 and @, @ are the angular velocity and acceleration of stance and swing angels, respectively. This nonlinear function has the capability to display several dynamics based on quantity and quality of its fixed points. According to this, a bounded area in the system’s state space forms that is called basin of attrac- tion.
The PDWs models are presented by two straight solid legs connected to a frictionless hinge [44-46]. The mass of upper body and the hip (MW, kg) is located at the hinge. Moreover, the mass of each leg (m, kg) is located at the end of the foot and is equal for both left and right legs [44]. Figure | illustrates a schematic view of PDW model. Fig. 1 a—d Simplest passive walking model at a typical step. b 6, y, M,m, 1, y and g are stance leg (thin line) angle, angle between swing and stance leg, hip mass, leg mass, length of leg, slope of ramp and acceleration due to gravity, respectively. This figure is reformed from [44,45] The nonlinear motion equation “Stride Function” (1) which is also known as Poincare map [47] consists of two coupled second-order differential equation in terms of stance leg angle 6 and swing angle ¢ as functions of time t. 6, 6 and @, @ are the angular velocity and acceleration of stance and swing angels, respectively. This nonlinear function has the capability to display several dynamics based on quantity and quality of its fixed points. According to this, a bounded area in the system’s state space forms that is called basin of attrac- tion.
a = 3.7 was chosen. Choosing other values of a shows less similarity to gait cyclic phase plane. When this condition is true, the new initial conditions for the next step must be set. In chaotic functions such as logistic map and Lorenz system, a strong principle between the consecutive points exists that binds these points to each other. These bindings are not independent and points are interre- lated. This is the sensible reason underlying using the chaotic functions in our model instead of using a ran- dom function. The relationship between each of the two consecutive points produced from a specific basin of attraction represents a rich information treasury that establishes a meaningful relationship between them [26]. For further analysis , instead of logistic map in model’s heel-strike collision rule, Lorenz system is used. Lorenz system is a significant mathematical model for representing chaotic behavior and was first introduced by Lorenz [71]. As illustrated in Eq. (7), this system consists of three coupled ordinary differential equa- tions with three different state variables. This model has three parameters (6 , e and a) which control responses and behavior of the system. As mentioned in [26], there are some latent ru es in unpredictable variations of states of equations which lead two wings in the strange attractor of Lorenz (6) needs one varia system. Heel-strike collision rule ble to control the chaotic behavior. As aresult, we arbitrary chose one of the state variables (x) from Lorenz system and set it as variable X’(n) in Eq. (6) after norma ization.
a = 3.7 was chosen. Choosing other values of a shows less similarity to gait cyclic phase plane. When this condition is true, the new initial conditions for the next step must be set. In chaotic functions such as logistic map and Lorenz system, a strong principle between the consecutive points exists that binds these points to each other. These bindings are not independent and points are interre- lated. This is the sensible reason underlying using the chaotic functions in our model instead of using a ran- dom function. The relationship between each of the two consecutive points produced from a specific basin of attraction represents a rich information treasury that establishes a meaningful relationship between them [26]. For further analysis , instead of logistic map in model’s heel-strike collision rule, Lorenz system is used. Lorenz system is a significant mathematical model for representing chaotic behavior and was first introduced by Lorenz [71]. As illustrated in Eq. (7), this system consists of three coupled ordinary differential equa- tions with three different state variables. This model has three parameters (6 , e and a) which control responses and behavior of the system. As mentioned in [26], there are some latent ru es in unpredictable variations of states of equations which lead two wings in the strange attractor of Lorenz (6) needs one varia system. Heel-strike collision rule ble to control the chaotic behavior. As aresult, we arbitrary chose one of the state variables (x) from Lorenz system and set it as variable X’(n) in Eq. (6) after norma ization.
Finally, Eq. (10) represents the modified version of for- mer transition rule in Eq. (8). When the transition is done and the initial values of
Finally, Eq. (10) represents the modified version of for- mer transition rule in Eq. (8). When the transition is done and the initial values of
able. However, the numerical analysis shows that the
able. However, the numerical analysis shows that the
the variation band for the swing leg angles is wider than the stance leg, the differences between peaks of steps in swing leg signal are significant. becomes unstable and finally falls. Thus, the main shortcoming of conventional models is the hypersen- sitivity to slope changes within a very limited range. Figures 12 and 13 highlight bifurcation diagram of the presented model with respect to slope of ramp y and initial value of swing angle, respectively. As can be seen in Fig. 12, by increasing the slope of walking surface from 0 to 0.523 radians (approximately 30°), the model continuously changes the number and posi- tion of its fixed points in state space in order to keep the stability. The response of the model in very low range of slopes is somehow periodic, but when the slope increases then the number of fixed points quickly increases and the model shows chaotic behavior. For slopes beyond 30°, the fixed points are not acceptable because of the assumptions in the basic walking model. Moreover, in ord er to further analyze the walking properties of the model and to compare them with the experimental data, t he extracted stride signal from the experimental data and the model are illustrated in Figs. 8 and 9, respective y. The comparison of these two figures shows that the model and subjects both have sharp stride-to-stride length variations, but the varia- tion of stride signal between two steps at model’s output are more considerable than the data of human walking group. slope increases then the number of fixed points quickly
the variation band for the swing leg angles is wider than the stance leg, the differences between peaks of steps in swing leg signal are significant. becomes unstable and finally falls. Thus, the main shortcoming of conventional models is the hypersen- sitivity to slope changes within a very limited range. Figures 12 and 13 highlight bifurcation diagram of the presented model with respect to slope of ramp y and initial value of swing angle, respectively. As can be seen in Fig. 12, by increasing the slope of walking surface from 0 to 0.523 radians (approximately 30°), the model continuously changes the number and posi- tion of its fixed points in state space in order to keep the stability. The response of the model in very low range of slopes is somehow periodic, but when the slope increases then the number of fixed points quickly increases and the model shows chaotic behavior. For slopes beyond 30°, the fixed points are not acceptable because of the assumptions in the basic walking model. Moreover, in ord er to further analyze the walking properties of the model and to compare them with the experimental data, t he extracted stride signal from the experimental data and the model are illustrated in Figs. 8 and 9, respective y. The comparison of these two figures shows that the model and subjects both have sharp stride-to-stride length variations, but the varia- tion of stride signal between two steps at model’s output are more considerable than the data of human walking group. slope increases then the number of fixed points quickly
Fig. 7 The signal of swing angles for 60 steps walking Fig. 6 The signal of stance angles for 60 steps walking
Fig. 7 The signal of swing angles for 60 steps walking Fig. 6 The signal of stance angles for 60 steps walking
Further analysis on the stride signal of the model is implemented using LLE. LLE is one of the well-known methods to validate chaotic behavior of the signal. We have studied this method on the extracted stride signals of both proposed models (containing logistic map and Lorenz systems) and also on the proposed PD model. The estimated LLE values for signals are illustrated in Table | shows that the mean LLE value of the outputs of the proposed model for both logistic and Lorenz sys- tems are nearly equal to LLE value of dataset. Also, the LLE of the proposed PD model and the stride data are nearly equal. As this table shows, the calculated LLE The estimated LLE values for signals are illustrated in
Further analysis on the stride signal of the model is implemented using LLE. LLE is one of the well-known methods to validate chaotic behavior of the signal. We have studied this method on the extracted stride signals of both proposed models (containing logistic map and Lorenz systems) and also on the proposed PD model. The estimated LLE values for signals are illustrated in Table | shows that the mean LLE value of the outputs of the proposed model for both logistic and Lorenz sys- tems are nearly equal to LLE value of dataset. Also, the LLE of the proposed PD model and the stride data are nearly equal. As this table shows, the calculated LLE The estimated LLE values for signals are illustrated in
Fig. 11 The period-doubling cascade route to chaos of simple walking model presented by Garcia [44]. No persistent walking wa: found at y much steeper than 0.019 radians for the PD group in the dataset is higher than the control group. The proposed model also shows this difference between the PD and the control groups. This results may confirm the [74-76] claims about an increase in variability of movements in PD patients. However, LLE alone is not enough to result in chaotic behavior and with further analysis like fractal dimension the result would be more reliable. The results of the study show that these signals are chaotic because their LLE value is positive and greater than zero. For the final analysis, the HFD analysis on both the model and the human stride signals in control and PD mode was carried out. HFD analysis is utilized to probe the similarities and differences of model and human stride signals in the basis of fractal and cybernetic sys- tems. The calculated HFDs are presented in a box- and-whisker plot, Fig. 14. As illustrated, the median HFD (MHED) of the proposed model in healthy gait
Fig. 11 The period-doubling cascade route to chaos of simple walking model presented by Garcia [44]. No persistent walking wa: found at y much steeper than 0.019 radians for the PD group in the dataset is higher than the control group. The proposed model also shows this difference between the PD and the control groups. This results may confirm the [74-76] claims about an increase in variability of movements in PD patients. However, LLE alone is not enough to result in chaotic behavior and with further analysis like fractal dimension the result would be more reliable. The results of the study show that these signals are chaotic because their LLE value is positive and greater than zero. For the final analysis, the HFD analysis on both the model and the human stride signals in control and PD mode was carried out. HFD analysis is utilized to probe the similarities and differences of model and human stride signals in the basis of fractal and cybernetic sys- tems. The calculated HFDs are presented in a box- and-whisker plot, Fig. 14. As illustrated, the median HFD (MHED) of the proposed model in healthy gait
Fig. 10 The fixed points of simplest PDW model which demon- strate the stable period-one dynamic when the y changes from 0.003 to 0.015
Fig. 10 The fixed points of simplest PDW model which demon- strate the stable period-one dynamic when the y changes from 0.003 to 0.015
Fig. 12 Bifurcation diagram of proposed model as a function of the slope y from 0 to 0.523 radians (~ 30°)
Fig. 12 Bifurcation diagram of proposed model as a function of the slope y from 0 to 0.523 radians (~ 30°)
Table 1 The estimated LLE for human stride length signal, pro- posed logistic model and proposed Lorenz model of gait
Table 1 The estimated LLE for human stride length signal, pro- posed logistic model and proposed Lorenz model of gait
Fig. 13 Bifurcation diagram of length differences between stride’s fixed points as a function of (0) from a range of 0- 0.523 radians
Fig. 13 Bifurcation diagram of length differences between stride’s fixed points as a function of (0) from a range of 0- 0.523 radians
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