key: cord-0058848-zlwvau2c
authors: González, Ramón E. R.; Collazos-Morales, Carlos; Galdino, João P.; Figueiredo, P. H.; Lombana, Juan; Moreno, Yésica; Segura, Sara M.; Ruiz, Iván; Ospina, Juan P.; Cárdenas, César A.; Meléndez-Pertuz, Farid; Ariza-Colpas, Paola
title: Hybrid Model for the Analysis of Human Gait: A Non-linear Approach
date: 2020-08-24
journal: Computational Science and Its Applications - ICCSA 2020
DOI: 10.1007/978-3-030-58799-4_16
sha: 5b89abf7b52d9219f1e5742d14d499e29ce9f53e
doc_id: 58848
cord_uid: zlwvau2c

In this work, a generalization of the study of the human gait was made from already existent models in the literature, like models of Keller and Kockshenev. In this hybrid model, a strategy of metabolic energy minimization is combined in a race process, with a non-linear description of the movement of the mass center’s libration, trying to reproduce the behavior of the walk-run transition. The results of the experimental data, for different speed regimes, indicate that the perimeter of the trajectory of the mass center is a relevant quantity in the quantification of this dynamic. An experimental procedure was put into practice in collaboration with the research group in Biomedical Engineering, Basic Sciences and Laboratories of the Manuela Beltrán University in Bogotá, Colombia.

Modern biomechanics arose with the studies of Archivald V. Hill on the transformation of heat from the mechanical work of muscular contractions [1] . The correlation of thermodynamic parameters with the heat transformed by the muscles led him to win the Nobel Prize in Physiology and Medicine in 1922. With the discovery of Hill, the links between the macroscopic biological systems and the universal characteristics of Physics were tightened.

The complexity of human locomotion comes from the fact that there are diverse and varied interactions between the body and the environment. A simplification of these interactions could be that the chemical potential energy originating from the muscles and the elastic potential energy of the tendons and type of the muscular elasticity end up transforming into work and heat [2] . The cyclic contractions in the active muscles give rise to reaction forces of the soil along the lower extremities. The force resulting from the gravitational action and the strength of resistance accelerates and decelerates the mass center of the body. Studies concerning the mechanical efficiency of locomotion [3] show that for different constant speeds, the rapidity in which energy is transformed, determined by oxygen consumption and by external work, is different for walking and running, the energy cost is lower in the first. When we deal with bipedal movement in particular, nonlinear effects are often observed, but most of the approximate models for this movement ignore such effects. These models can be reliable for primary studies but end up being insufficient in relation to the complexity of the movement, losing the naturalness and clinical accuracy of the locomotion.

A reasonable approximation, when describing the process of gait, is to suppose a cyclic pattern of corporal movements that are repeated at each step [4] . The process of normal gait is one in which the erect body in movement is consistent with one leg and then on the other. This process has two phases, the support phase, which begins when the leg is in contact with the ground and lasts until this contact is lost. This phase represents 60% of the entire cycle of the March. The remaining 40% are the balance or balancing phase, since the leg loses contact with the ground until they come into contact again.

In the process of walking there are two very important basic aspects, the continuous reaction forces of the ground that sustain and provide the body with torque and the periodic movements of the feet from one position of support to the next, in the direction of movement. The reaction force of the soil depends intimately on the speed of travel [5] . The entire process is controlled by the neuromuscular system, which is why it is a complex process, which means that locomotion is a system in which changes in small components result in significant changes.

People walk naturally in a way that energy consumption is optimized [4] . The metabolic rate, the variation of energy per time of physical activity, is generally measured indirectly by the amount of oxygen consumed during bodily activity [6] . In order to minimize energy dissipation, the neuromuscular system selects the patient. Deviations from this normal gait pattern result in increased energy consumption and limit locomotion [7] .

In 1974, J.B. Keller [8] proposed a model based only on variational calculus and elementary dynamics to study the running of extraordinary human performance. In this model, it is assumed that the reaction force of the soil affects the amount E(t) of power per unit mass from muscle that store N_2 of the food and the consumption of O_2 of the individual. The reactions that occur in the body of the individual use these chemical reserves to provide power for locomotion. For prolonged periods of activity, the individual's biology supplies a r rate of energy supply obtained from respiration and circulation. Keller determined the theoretical relationship between the shortest time T, based on physiological parameters, in which a distance D can be traversed (1) .

In this calculation, it was considered that the propulsive force of the sprinter cannot exceed a certain maximum value (2) .

The constant c is a physiological constant and has units of time inverse. So that there is an optimized solution, according to the results of Keller's work, for the sprinter in starting a finite number of velocity curves described according to

Where the constant C is arbitrary and is determined by the initial velocity. The important cases in Keller's work are: an acceleration curve during the interval in which the runner exerts the maximum momentum; the constant velocity curve from the moment the runner reaches its maximum speed and the deceleration curve that begins when E t ð Þ 0. These three curves are combined ensuring continuity throughout the run and that the area under the speed curve must be maximum. A summary of the above can be seen in (4) and in Figs. 1 and 2. 

When analyzing competitions between professional athletes and world record data. Keller estimated parameters for a run with optimal or next optimization strategy. These values, corresponding to physiological variables are the following (5). 

The parameters T C and E 0 are the critical time (the minimum for the distance D) and the initial power per unit mass.

For the study of the oscillations that the center of mass experience in a locomotion at constant speed it is important to observe the movement of the center of mass in a sagittal plane. Kokshenev, in his 2004 paper [9] , used this plane as a reference to study the movement of the center of mass during human walking at constant speed. In this model, an inertial reference system is considered moving as a virtual mass center defined by the displacement vector R 0 ! t ð Þ. The conditions established in the model for displacement are: x 0 t ð Þ ¼ Vt, where V is a constant and y 0 t ð Þ ¼ H is the average height of the center of mass in relation to the ground, where the origin of the inertial ¼ 0. In the transition, the curve presents a decline in speed referring to the athlete's energy limit.

coordinate system is. In this way, the displacement vector relative to the movement of the center of mass of the human body with the virtual center of mass is

is the displacement vector of the center of mass of the human body in relation to the same referential of R 0 ! t ð Þ.

In the model, a driving force DF ! t ð Þ is defined, which is the neuromuscular capacity to exercise work, which is derived from observations of small oscillations close to the weight support of the body. In (6), we can see this force related to the ground reaction forceF t ð Þ and the force of gravity.

The driving force, as well as the velocity and displacement of the center of mass must respect the condition of cyclometry that preserves the amount of movement of the system. Applying the Lagrangean formalism, and considering harmonic and non-harmonic solutions for the movement of the center of mass, Kokshenev found the force that characterizes the forced oscillatory movement of the center of mass around the point

The first term of (6) is the force that describes the free movement of the center of masses as a superposition of linear oscillations. With the increase of the speed, the anharmonic effects become important, being necessary the introduction of a potential resulting from the expansion in a Taylor series of the elastic potential of the Hamiltonian. The second term then results from the gradient of this expanded potential up to the order of the anharmonic effects. Result of all this, the components of the force given by (6) are presented in (8) and (9).

The coefficients Dl 0 v ð Þ; Dh 0 v ð Þ; Dl 1 v ð Þ and Dh 1 v ð Þ are harmonic and anharmonic amplitudes respectively, whose values correspond to experimental data and x 0 v ð Þ corresponds to the frequency of a cycle on the step cycle.

Finally, the introduction of a locomotive resistive force DF res

ð Þ represents the coefficient of friction, results in the following functions for the respective positions in a steady state with x 0 t ) 1 and in a low resistance approach (10) and (11) .

These equations are the solution of the following equation of motion (12) .

The force DF 1 ! t ð Þ is given by (7) and (8) previous. The results of Kokshenev show a closed orbit given by a hypocycloid Dr 1 \Dr 0 ( H ð Þ , around a fixed point and clockwise. It is assumed, in the work of Kokshenev, that this orbit is described, for walking, as a characteristic ellipse, with

, horizontal and vertical, respectively. Given the conditions of the model, the center of mass moves with constant speed V at a certain height H and rotates along a hypocycloid circumscribed by a "flattened" or "shrunken" ellipse of eccentricity e þ e À ð Þ given by the following expression (13) .

The study was approved by the Ethics Committees of the Manuela Beltrán University. Written informed consent was obtained by the patients.

Experimental data were collected in a space of approximately 16 m 2 , where there is a track formed by four platforms of force and six motion detection cameras around the platforms, see in Fig. 3 .

The cameras used are part of the data acquisition system for BTS GAITLAB motion analysis. These optoelectronic cameras measure the displacement, with an accuracy of ±10 −7 m, of body segments in a time interval of AE10 À2 s [10] . The experimental data to validate the models are extracted from.

In this study used only the markers of the hip of the Davis protocol distributed in different segments and corporal regions of the volunteer (Fig. 4) . For each range of speed (around 1.03 m/s, 1.81 m/s and 3.2 m/s), In the study was included both female and male volunteers (5 females and 6 males).

In literature, we can find several references in relation to the position of the center of mass of the human body. For Miralles [11] the center of mass lies behind the lumbar vertebra L5. Yet for Dufour and Pillu [12] it is located before the sacral vertebra S 2 . We suggest in this work that the center of mass would be placed just between these two vertebrae without using reaction forces. The Smart TRACKER and Smart Analyzer was used, a program with which we are able see the displacement along the track and to capture the position of each of the markers placed on the volunteer. A simple model was created that virtually simulates the markers and their connection (Fig. 5) .

With the Smart TRACKER and Smart ANALYZER, the displacements of the real markers coupled with the volunteer was interpolated, guaranteeing the continuity of the information throughout the capture of the data. With the defined function, a virtual point situated at the midpoint of sacrum and anterior superior iliac spine (left and right) was created. On this point all the clinical analysis corresponding to the March was done.

In order to find results for different walking speed using the Kokshenev model for nonlinear running, we used the experimental data conceived in the experiments described above for comparison with important aspects of the model. In our model we use (12) , deduced in the Kokshenev model, which, from a simpler mechanical analog, represents oscillations in the two-dimensional plane under the action of a viscous resistant force and the reaction force of the soil acting on favor of the movement, turning the system a two-dimensional pendulum damped and forced.

The diversity of the velocities was issued by (4). The physiological parameters used in optimization by Keller were adopted (5) . We noted that, by varying these parameters and implementing them in the equation for speed, there was a dependence between the terms related to the transitions between the different speed regimes. At time t 1 , which separates the velocity regime with exponential growth of the constant velocity regime, the following behavior is obtained (14) .

From this, a maximum value is induced for the ratio F/c from which we can find t 1 . In this way, the relationship between the parameters F and c is established as the maximum limit for the value of the speed reached (maximum speed). From the previous Eq. (14) we can see that the maximum speed that the system can reach is equal to the limit in which time tends to infinity in the following relation (15) .

Making the maximum speed can be expressed according to (16) .

The minimum speed, on the other hand, refers to the case in which the system works for long periods of time. At the time t 2 when the physiological wear begins, the velocity transition occurs and this time we can obtain it from (4), as follows (17) [woodside].

In order to maintain the coherence of the function from t_2 to T, in case that v is less than the root of the term r/c, then we will have a divergence on the function that characterizes this regime transition. We can then interpret the following as the minimum speed (18) .

For the values of the physiological parameters cited, the maximum time T that an individual can reach, at the end of the slope curve due to physiological wear is (19) .

In this equation, we can see that values of r ( V 2 c there will be a value of T while for r ) V 2 c the equation diverges, where T is in that case inaccessible. The connection between the parameters r and T, as well as the minimum speed allows us to find different values of r respecting the maximum and minimum values of the speed. In (12) , the terms k 0 and c are functions of velocity. The form as

varies was obtained by Kokshenev [9] using the results of experimental data reported in [3] . It was defined that x 0 v ð Þ is a linear function of velocity, as follows (20).

With this result, the coefficient of friction per unit of mass

m was found as follows (21).

Introducing Keller's optimal speed and substituting the dependence with speed for the dependence with time, for small oscillations we can affirm that (22).

x 0 t ð Þ % 4:94 þ 4:02 1 À e Àv t

In the numerical solution (12) , using (20) and (21) e with v t ð Þ being Keller's optimal velocity, we note that it grows rapidly for the constant velocity value and (12) is quickly damped by the growth of x 1 v ð Þ. The model presented here never comes to contemplate the physiological wear due to the quadratic growth of x 1 v ð Þ and its mathematical complexity.

The physiological parameter chosen to determine the minimum velocity was r. For each value of r, the last oscillation of the x and y positions was recorded and parametric curves were constructed. These curves vary between the maximum time t max of the occurrence of the movement and the difference between that time and an R term dependent on the angular velocity x 0 in t max as follows (23).

The registration time of a parametric curve is (24):

With the equations already defined, the behaviors of the components x t ð Þ e and y t ð Þ for the trajectory of the center of masses were determined. Three velocity values were chosen: v = 1.03 m/s, 1.81 m/s and 3.20 m/s, referring to those obtained in collaboration with the Research Group in Biomedical Engineering, Basic Sciences and Laboratories of the Manuela Beltrán University, in Bogotá, Colombia. The final period of oscillation and comparison with a period taken from the real data was plotted from the model. We chose the data closest to an average of a characteristic behavior of force, for the experimental data. The characteristic behavior of the normalized force of the actual data is related to the average behavior of the data obtained with all the volunteers and platforms. In this way, we selected the data of the individual MF7P1, female, on the P 1 platform, one of the four used to capture the FRS, see Figs. 6 and 7.

We see in Fig. 6 , for the first two speed, that the oscillations have the same phase, both in time and in amplitude. Yet for the speed of 3.2 m/s, we see a phase shift between the curves of approximately p/2 and the values of the amplitude referring to the real data are in a proportion four times greater than the amplitude of the model.

In Fig. 7 , the same time interval was plotted. It is observed, to 1.03 m/s a point of maximum similar in both graphs, of the model and of the real data. For the intermediate speed, 1.81 m/s we see extremes of inverted phases and for 3.20 m/s, although the amplitude obtained from the model is three times lower than that obtained from the actual data, the phases of the oscillations are quite similar.

From the model, parametric curves for x(t) and y(t) were obtained by varying the value of the physiological constant r, reported by Keller (5) from 0.2 m 2 /s 3 to 12 m 2 / s 3 . It was observed that there is a perimeter for the trajectory, associated with each translation speed and that the velocity value that maximizes the perimeter is v = 1.38 m/s. Figure 8 illustrates the behavior of the perimeter of the trajectories of the center of masses for different translation speed. At speeds up to v = 1.38 m/s, the perimeter of the trajectories increases and higher speed it decreases. For each of the curves in Fig. 8 , the perimeter was calculated, corresponding to the trajectory of the center of mass for each velocity. A graph of the perimeter as a function of speed was constructed where the maximum point was easily identified, corresponding to the velocity v = 1.38 m/s. The perimeters corresponding to experimental data found in the literature were also calculated [13] and trajectories of the center of mass obtained from the experimental data, in collaboration with the Research Group on Biomedical Engineering, Basic Sciences and Laboratories of Manuela Beltrán University, in Bogotá, Colombia. These last data are referring to a certain running regime, with a definite velocity of the center of mass of the volunteer. With this data obtained an average of the trajectories of the center of mass of each volunteer and with such means a closed parametric curve was generated, for which its characteristic parameter was calculated. The perimeter of the path of just one individual was likewise calculated, which approximated the result obtained by the average in each speed regime.

In the graph of Fig. 9 , we can see all the information regarding perimeters as a function of speed. From experimental results reported by Kokshenev, it was possible to find the perimeter corresponding to each plotted eccentricity, as a function of the speed and they were represented using black squares. With red squares the average values of the perimeters corresponding to the experimental data were represented in each speed regime. The blue points are referring to real data from the literature [14] and the green points are the perimeters of the trajectories of a single individual, MPF5P1, which is adapted to the values obtained for the average of the parametric trajectories. The dashed red line marks the speed considered by Kokshenev as the transition point between the March and the race, this speed is v = 1.73 m/s. The dashed black line represents the maximum perimeter velocity v = 1.38 m/s. This speed represents, for our model, what would be the transition between the running and running regimes. It is reasonable to think that close to this point of maximum of the gait is located, said in some way, normal for a healthy adult [15] . This walking speed is the most stable for the mass center of healthy adults.

The model proposed presents peculiar behavior in relation to the perimeter of the trajectory of the center of masses as a function of the translation speed. Trying to solve the model for variable speeds, in the way proposed by Keller, has ineffective implications, since the translation speed has a very short duration in the regime in which the speed grows exponentially, observed in Eq. (4) and lasts a long time in the stationary speed regime, before reaching physiological wear.

In fact, for practical purpose, physiological wear is unattainable for the simulation time that is generated.

Kokshenev work with the hypothesis that the eccentricity varies depending on the speed. In this work we chose to study the perimeter depending on the speed as an approach to obtain the results. This approach is plausible due to the relationship between the eccentricity and the amplitude of the curves. The visibility between the perimeter and the speed was another point for the use of this approach. A result of this relationship is the maximum point found, for v = 1.38 m/s. This value coincides with one of the most accepted values in the literature for the "normal" speed of a healthy adult [16] [17] [18] .

For the topic addressed in this paper there are various methods and models in the literature [10] , on the other hand, the model studied here, despite being a simplification of effects of other natures, is acceptable because, using only non-linear mechanics, the results obtained result in a good approximation of reality.

The objective of this work was to approach non-linear effects in biomechanics. Using models already known from the literature, such as Kokshenev and Keller, an association of these models was achieved in order to obtain results from a model with more general characteristics. An experimental procedure was adopted in collaboration with the research group in Biomedical Engineering, Basic Sciences and Laboratories of the Manuela Beltrán University in Bogotá, Colombia. These experiments generated data that, finally, were compared with the created model [16] [17] [18] .

The results of the proposed model for low speeds, works quite well, from v = 1.0 m/s to approximately v = 1.38 m/s, for which the perimeter of the center of mass calculated from the model coincides or results fairly close to the experimental perimeter. As the speed increases, deviations are observed more and more accentuated. It is also observed that the trajectory is more accentuated for the critical speed v = 1.38 m/s and decreases both, with the increase and with the decrease in speed.

It is noticeable that the presented model manages to approximate the experimental results for low speed of the March, in spite of the model using physiological parameters that optimize the gait. Deviations at high speed are hypothetically associated with noise from the central pattern generator (PCG), a biological neural network responsible for locomotion that produces a rhythmic pattern in the absence of sensory responses or descendants that carry specific temporal information [19, 20] .

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