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# Optimization System of Strength and Flexibility Training in Aerobics Course Based on Lagrangian Mathematical Equation

###### Accettato: 22 Jun 2022
Dettagli della rivista
Formato
Rivista
eISSN
2444-8656
Prima pubblicazione
01 Jan 2016
Frequenza di pubblicazione
2 volte all'anno
Lingue
Inglese
Introduction

In recent years, computer image processing capabilities have been continuously improved. The development of computer vision is also maturing. Computer vision application to action recognition in aerobics video images has multiple implications. It can accurately and comprehensively recognize the movement posture of the aerobics in the video. This makes it easy to evaluate it objectively to make suggestions for improvement for the aerobics. Action recognition has gone through a long development process. Gesture recognition, body motion recognition, etc., belong to the low-level human motion analysis. The essence of action recognition in aerobics video images is motion recognition. It belongs to the advanced stage of human motion analysis . The purpose is to analyze, judge, or extract the motion in the video. It can create personalized teaching, training, and virtual simulation systems. Action recognition in aerobics video images is more complex. Our simple limb localization algorithm does not accurately reflect the movements in the aerobics video images. So we need to use mathematical models to help us solve the above problems. The Lagrangian mathematical model can help us establish a model of the body movement characteristics of aerobics athletes.

Modeling and Simulation of Aerobics by Lagrange Method
Inverse kinematics model

Suppose we know the trajectories of the end segments of the aerobics in the workspace. At this point, we solve the kinematic parameters of each joint of the aerobics, which is the inverse kinematics problem . The inverse kinematics problem can be regarded as a nonlinear mapping from the workspace to the joint space. The movement speed ξ of the athlete's hand is a 6-dimensional vector. ξ. There is the following relationship with the N-dimensional vector $q˙$ \dot q that describes the speed of each joint of the human body of aerobics: $ξ=Jq˙$ \xi = J\dot q

qRN is the generalized coordinate of the system. JRN is the Jacobian matrix of the calisthenics body. According to the geometric topology of the body of the aerobics, we can get: $J=[ JL JA ]TJLj=bj−1×rJAj=bj−1 }$ \left. {\matrix{ {J = {{\left[ {{J_L}\,{J_A}} \right]}^T}} \hfill \cr {{J_{Lj}} = {b_{j - 1}} \times r} \hfill \cr {{J_{Aj}} = {b_{j - 1}}} \hfill \cr } } \right\}

JLRN and JARN are the linear velocity and angular velocity transfer matrices, respectively. j represents the j column vector of the matrix. r is the representation of the end segment position vector in the base coordinate system. bj−1 is the unit vector of the j − 1 joint axis. If the spatial velocity of the end segment is known, then the joint velocity $q˙$ \dot q can be obtained by solving equation (1). We integrate and differentiate them to get the joint angle and joint angular acceleration.

This paper simplifies the human body of aerobics in EVA into four segments: trunk, upper arm, forearm, and hand. The aerobics system has four rotating joints of hip, shoulder, elbow, and wrist rotating joints. The corresponding joint angular displacements are q1, q2, q3, and q4, respectively. r is the radius of the hand motion trajectory. We simplify the problem to one with four segments. It is a conventional open-chain system with 4 degrees of freedom and two redundancy. Each segment is connected by a plane rotation joint. We establish a right-handed coordinate system . The spatial poses of the aerobics’ shoulders, elbows, wrist joints, and hands can be calculated according to the coordinate transformation. We can obtain the expressions of JL1, JL2, JL3, and JL4, according to formula (2) and the calculated spatial pose.

In this way, the Jacobian matrix of this system can be obtained. The joint angular velocity can be obtained by solving equation (1) when the hand's is known. qi, i = 1, ⋯, 4 can be integrated and differentiated to obtain the joint angle qi, i = 1, ⋯, 4 and joint angular acceleration qi, i = 1, ⋯, 4.

Inverse dynamics model

The Lagrangian function L is defined as the difference between the kinetic energy T and the potential energy V of the system. That is, the Lagrangian equation of the L = TV. The system is shown in equation (3): $ddt[ ∂L∂q˙i ]−∂L∂qi=Fii=1,2,⋯,n$ \matrix{ {{d \over {dt}}\left[ {{{\partial L} \over {\partial {{\dot q}_i}}}} \right] - {{\partial L} \over {\partial {q_i}}} = {F_i}} \hfill & {i = 1,\,2,\, \cdots ,\,n} \hfill \cr }

qi is the generalized coordinate of the system. $q˙i$ {\dot q_i} is the derivative of generalized coordinates concerning time. We call it generalized velocity. Fi is the generalized force or generalized moment acting on the system at a joint i to drive the rigid body. The 4-segment Lagrange dynamic equation of the simplified aerobics body we deduced is shown in Eq. (4): $∑j=14Hijq¨j+∑j=14∑k=14hijkq˙jq˙k=τii=1,⋯,4$ \matrix{ {\sum\limits_{j = 1}^4 {{H_{ij}}{{\ddot q}_j} + \sum\limits_{j = 1}^4 {\sum\limits_{k = 1}^4 {{h_{ijk}}{{\dot q}_j}{{\dot q}_k} = {\tau _i}} } } } \hfill & {i = 1, \cdots ,4} \hfill \cr }

$hijk=∂Hij∂qk−12∂Hjk∂qi;τi$ {h_{ijk}} = {{\partial {H_{ij}}} \over {\partial {q_k}}} - {1 \over 2}{{\partial {H_{jk}}} \over {\partial {q_i}}}; \tau_i ; τi represents the joint moment, the first term on the left side represents the moment of inertia, and the second term represents the centripetal force and the Coriolis force. Hij (j = i) is called the effective inertia of joint i. The acceleration of joint i will generate an inertial force on joint i equal to $Hijq¨i(j=i)$ {H_{ij}}{\ddot q_i}\left( {j = i} \right) . Hij (ji) is called the coupled inertia between joints i and j. The $q¨j$ {\ddot q_j} of joint j will generate an inertial force on joint i equal to $Hijq¨j$ {H_{ij}}{\ddot q_j} . The $hijkq˙j2(k=j)$ {h_{ijk}}\dot q_j^2\left( {k = j} \right) term is the centripetal force on joint i produced by the velocity $q˙j$ {\dot q_j} of joint i. The $hijkq˙jq˙k(k≠j)$ {h_{ijk}}{\dot q_j}{\dot q_k}\left( {k \ne j} \right) term is the Coriolis force acting on joint i caused by $q˙j$ {\dot q_j} and $q˙k$ {\dot q_k} of joints i and k.

Aerobics video image action preprocessing technology
Convert video image to grayscale image

The images in the aerobics videos are mostly in color. If we feed the image directly into the computer vision system, it will increase the amount of information in the image input. This will increase the number of subsequent operations . The principle of grayscale processing of color images is to reduce the dimensionality of the three-dimensional channels of the color space RGB. We make it into one dimension. The grayscale process of color images is shown in Figure 1.

There are many methods for grayscale transformation of images, such as single-component method, maximum value method, global mapping method, etc. Different methods apply to different processing objects. Our use of an inappropriate grayscale transformation method will likely result in an unsatisfactory grayscale effect . Therefore, this paper adopts a hybrid grayscale processing method based on image fusion. We use Plass weighting to calculate the local transformation sum of each channel input of R, G, and B. We use the global contrast weighted mapping method to establish the objective function through elements such as gray value and color distance difference. Then this paper obtains the grayscale result by minimizing the objective function. In this paper, this hybrid method can preserve the image's global structural information and local contrast information to the greatest extent. This paper uses this method to grayscale the images in the aerobics video. The effect is shown in Figure 2.

Thresholding video images

The essence of image thresholding is to segment aerobics images. Its purpose is to compress the data volume of color images and simplify the image analysis steps. It divides the set of pixels according to the grayscale set. Each subset of this set corresponds to an individual aerobics move. Determining an appropriate threshold is critical when we threshold aerobics video images. Because it relates to where the pixels in the image belong , only a reasonable threshold can produce a more accurate binary image. The threshold is generally expressed in the following form. $T=T[ x,y,f(x,y),p(x,y) ]$ T = T\left[ {x,y,f\left( {x,y} \right),p\left( {x,y} \right)} \right]

f(x, y) is the gray value at the pixel point (x, y); p(x, y) is the gray gradient function of the point. We can get the binarized image through this formula.

Segmentation of video images

We segment aerobics video images to change the aerobics image representation. This makes the image easier to understand and analyze by computer vision systems. This allows each pixel in the image to have its unique label. The image finally forms a multi-pixel collection . After thresholding the aerobics action image, we need to separate the dancer's motion area from the whole scene to obtain the binary image. This requires segmenting the dancer's motion area, and after determining the appropriate threshold, the edge of the dancer's human silhouette can be found. Assuming that the current moment is t, the frame detection value of the video image at the moment t is p(x, y, t). The dimension we get at this point is M × N. Then the binary image we get at this time is A(x, y, t).

Background subtraction and noise reduction for video images

In this paper, a Gaussian mixture model is selected to achieve background subtraction based on the particularity of aerobics video images . The principle is to use K Gaussian models to represent the features of each pixel in the aerobics video image.

Building the model

Assuming that the value of the pixel at time t is Xt, its probability formula is as follows: $P(Xi)=∑i=1Kωi,t η(Xt,μi,t,δi,t)$ P\left( {{X_i}} \right) = \sum\nolimits_{i = 1}^K {{\omega _{i,t}}\,\eta \left( {{X_t},{\mu _{i,t}},{\delta _{i,t}}} \right)}

ωi,j represents the weight of the i Gaussian distribution at time t. η(Xt, μi,t, δi,t) represents its corresponding probability density function. μi,t represents the corresponding mean. δi,t is the period variance.

Updating and matching models

Assuming that the pixel value of a new input frame of image is Xt, the formula for judging whether the pixel matches the Gaussian model is as follows: $| Xt−μi,t−1 |≤2.5 δi,t−1$ \left| {{X_t} - {\mu _{i,t - 1}}} \right| \le 2.5\,{\delta _{i,t - 1}}

If this formula condition can be satisfied, it can be judged that the pixel point matches the model. It is the background point, and if it cannot be satisfied, it can be judged that it does not match the model. It is the fore spot . We obtain the effect of background subtraction through the Gaussian model, as shown in Figure 3.

Noise reduction

It can be seen from the background removal effect diagram of the Gaussian model that there is noise around the foreground, which will cause interference pulses for the computer vision system to recognize the edge of the target image. Therefore, we can achieve the purpose of protecting the edge of the dancer's body image through the noise reduction operation. In this paper, the median filter method is used to reduce noise. First, we move the template in the aerobics video image until the center of the template coincides with the center pixel in the aerobics video during the movement . This pixel can be used as the center of the window. We use this center to build windows of different shapes, such as squares and circles. Second, we read the grayscale values of all pixels in the window under the template. Then we sort the grayscale values. We are generally arranged in order from smallest to largest. Finally, we calculate the median value of the gray value of the pixel arrangement and use it as the pixel's gray value at the center point of the window. The calculation formula of the output pixel gray value after median filter noise reduction follows. $g(x,y)=median{ f(x−i,y−j) }(i,j)∈W$ g\left( {x,y} \right) = median\left\{ {f\left( {x - i,\,y - j} \right)} \right\}\left( {i,j} \right) \in W

W stands for template window. g(x, y) refers to the output pixel gray value. f(xi, yj) refers to the input pixel gray value.

Action pose extraction and joint modeling from aerobics video images
Recognition of action features by pose feature extraction
Classification of dancer action recognition features

There is a big difference between the aerobics movements and the daily movements of an ordinary person. When selecting the target area for background recognition, we must master the dancer's whole body movement information to identify its movements accurately. Dancer's action recognition can be divided into several categories: Static features. The main form of expression is the size, color, body contour, depth, etc., of the dancer's human target . We can derive the current basic shape of the dancer through the outline features dynamic features. It is mainly manifested in the dancer's movement, speed, direction, and trajectory. These can reflect the movement path of the dancer. These feature recognition can calculate the movement direction characteristics of the dancer. This creates the conditions for modeling. The spatiotemporal features are mainly represented as shapes, points of interest, etc. Defining features include the scene where the dancer is located, surrounding objects, pose, etc.

Dancer pose feature extraction

When we use the pose feature extraction method, we can use the pose estimation sensor. It can determine the direction of the dancer's movement . In this way, the dancer's joint coordinates area can be obtained. The shadowing and influence of the dancer's clothing and other factors on the dancer's movement can be eliminated. The dancer pose feature map is shown in Figure 4.

Using Kinect for dancer joint recognition modeling

The Kinect method regards the human body as a coordinate axis composed of 25 joint point coordinates. We use these joint points to establish the dancer's human skeleton structure to obtain the dancer's human skeleton model (Figure 5).

The joints of dancers are mainly distributed in the limbs. The head, neck, spine, and shoulder center have an articulation point. The most concentrated distribution of joint points is located in the upper limbs. The left upper limb has joints such as the left shoulder, left elbow, left wrist, and left finger. The right upper limb has joints such as the right shoulder, right elbow, right wrist, and right fingers . The left lower limb has joints such as the left hip, left knee, left ankle, and left foot. The right lower extremity has critical points: the right hip, right knee, right ankle, and right foot. The principle is to accurately record the movement of each joint point in the process of dancers doing various movements. In this way, each dancer's movement can be accurately identified to output the correct dancer's motion skeleton. The dancer motion skeleton model can significantly improve dancer movements’ recognition accuracy and efficiency by computer vision systems. The whole identification process is shown in Figure 6.

Conclusion

Aerobics video image recognition should consider the influence of aerobics background, clothing, etc., on action recognition. At the same time, we also need to consider the problem of occlusion and self-occlusion in the aerobics’ movements. We adopt an action recognition technology that can accurately and completely record and reflect the aerobics’ movement information. In this way, the aerobics body's static and action information is obtained. This paper proposes a specific identification method based on the existing research. These methods fit the dancer's movement characteristics. The algorithm proposed in this paper is of great value for video action analysis of aerobics and other applications.

Leitao, A., Margotti, F., & Svaiter, B. F. Range-relaxed criteria for choosing the Lagrange multipliers in the Levenberg–Marquardt method. IMA Journal of Numerical Analysis., 2021;41(4): 2962–2989 LeitaoA. MargottiF. SvaiterB. F. Range-relaxed criteria for choosing the Lagrange multipliers in the Levenberg–Marquardt method IMA Journal of Numerical Analysis 2021 41 4 2962 2989 10.1093/imanum/draa050 Search in Google Scholar

Wang, J., & Qu, H. Analysis of regression prediction model of competitive sports based on SVM and artificial intelligence. Journal of Intelligent & Fuzzy Systems., 2020; 39(4): 5859–5869 WangJ. QuH. Analysis of regression prediction model of competitive sports based on SVM and artificial intelligence Journal of Intelligent & Fuzzy Systems 2020 39 4 5859 5869 10.3233/JIFS-189061 Search in Google Scholar

Rahaman, H., Hasan, M. K., Ali, A., & Alam, M. S. Implicit methods for numerical solution of singular initial value problems. Applied Mathematics and Nonlinear Sciences., 2021; 6(1): 1–8 RahamanH. HasanM. K. AliA. AlamM. S. Implicit methods for numerical solution of singular initial value problems Applied Mathematics and Nonlinear Sciences 2021 6 1 1 8 10.2478/amns.2020.2.00001 Search in Google Scholar

Demir, F., Akbulut, Y., & Tasci, B. An effective and robust machine learning approach for automated human posture detection from IoTs module. Selcuk University Journal of Engineering Sciences., 2021; 20(3): 84–88 DemirF. AkbulutY. TasciB. An effective and robust machine learning approach for automated human posture detection from IoTs module Selcuk University Journal of Engineering Sciences 2021 20 3 84 88 Search in Google Scholar

El-Borhamy, M., & Mosalam, N. On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic. Applied Mathematics and Nonlinear Sciences., 2020; 5(1): 93–108 El-BorhamyM. MosalamN. On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic Applied Mathematics and Nonlinear Sciences 2020 5 1 93 108 10.2478/amns.2020.1.00010 Search in Google Scholar

Kuchta, M., Laurino, F., Mardal, K. A., & Zunino, P. Analysis and approximation of mixed-dimensional PDEs on 3D-1D domains coupled with Lagrange multipliers. SIAM Journal on Numerical Analysis., 2021;59(1): 558–582 KuchtaM. LaurinoF. MardalK. A. ZuninoP. Analysis and approximation of mixed-dimensional PDEs on 3D-1D domains coupled with Lagrange multipliers SIAM Journal on Numerical Analysis 2021 59 1 558 582 10.1137/20M1329664 Search in Google Scholar

Lemos, N. A., & Moriconi, M. On the consistency of the Lagrange multiplier method in classical mechanics. American Journal of Physics., 2021;89(8): 776–782 LemosN. A. MoriconiM. On the consistency of the Lagrange multiplier method in classical mechanics American Journal of Physics 2021 89 8 776 782 10.1119/10.0004135 Search in Google Scholar

Wei, R., Cao, J., & Huang, C. Lagrange exponential stability of quaternion-valued memristive neural networks with time delays. Mathematical Methods in the Applied Sciences., 2020; 43(12): 7269–7291 WeiR. CaoJ. HuangC. Lagrange exponential stability of quaternion-valued memristive neural networks with time delays Mathematical Methods in the Applied Sciences 2020 43 12 7269 7291 10.1002/mma.6463 Search in Google Scholar

van Dung, N., & Thi Le Hang, V. Solution to Kim-Rassias's question on stability of generalized Euler-Lagrange quadratic functional equations in quasi-Banach spaces. Mathematical Methods in the Applied Sciences., 2020;43(5): 2709–2720 van DungN. Thi Le HangV. Solution to Kim-Rassias's question on stability of generalized Euler-Lagrange quadratic functional equations in quasi-Banach spaces Mathematical Methods in the Applied Sciences 2020 43 5 2709 2720 10.1002/mma.6077 Search in Google Scholar

Srivastava, H. M., Saad, K. M., Gómez-Aguilar, J. F., & Almadiy, A. A. Some new mathematical models of the fractional-order system of human immune against IAV infection. Mathematical Biosciences and Engineering., 2020; 17(5): 4942–4969 SrivastavaH. M. SaadK. M. Gómez-AguilarJ. F. AlmadiyA. A. Some new mathematical models of the fractional-order system of human immune against IAV infection Mathematical Biosciences and Engineering 2020 17 5 4942 4969 10.3934/mbe.202026833120535 Search in Google Scholar

Zhang, X., Zhao, W., & Wan, D. A Hybrid Volume-of-Fluid/Euler-Lagrange Method for Vertical Plunging Jet Flows. International Journal of Offshore and Polar Engineering., 2022; 32(01): 31–38 ZhangX. ZhaoW. WanD. A Hybrid Volume-of-Fluid/Euler-Lagrange Method for Vertical Plunging Jet Flows International Journal of Offshore and Polar Engineering 2022 32 01 31 38 10.17736/ijope.2022.jc838 Search in Google Scholar

Roy, S., Baldi, S., Li, P., & Sankaranarayanan, V. N. Artificial-delay adaptive control for underactuated Euler–Lagrange robotics. IEEE/ASME Transactions on Mechatronics., 2021; 26(6): 3064–3075 RoyS. BaldiS. LiP. SankaranarayananV. N. Artificial-delay adaptive control for underactuated Euler–Lagrange robotics IEEE/ASME Transactions on Mechatronics 2021 26 6 3064 3075 10.1109/TMECH.2021.3052068 Search in Google Scholar

Zhang, X., Wang, J., & Wan, D. Euler–Lagrange study of bubble drag reduction in turbulent channel flow and boundary layer flow. Physics of Fluids., 2020; 32(2): 027101 ZhangX. WangJ. WanD. Euler–Lagrange study of bubble drag reduction in turbulent channel flow and boundary layer flow Physics of Fluids 2020 32 2 027101 10.1063/1.5141608 Search in Google Scholar

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