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

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 [2]. 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: (1) ξ=Jq˙ \xi = J\dot q

q ∈ R^N is the generalized coordinate of the system. J ∈ R^6×N is the Jacobian matrix of the calisthenics body. According to the geometric topology of the body of the aerobics, we can get: (2) 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\}

J_L ∈ R^3×N and J_A ∈ R^3×N 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. b_j−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 [3]. 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 J_L1, J_L2, J_L3, and J_L4, 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. q_i, i = 1, ⋯, 4 can be integrated and differentiated to obtain the joint angle q_i, i = 1, ⋯, 4 and joint angular acceleration q_i, i = 1, ⋯, 4.

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 = T − V. The system is shown in equation (3): (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 }

q_i 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. F_i 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): (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 ; τ_i 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. H_ij (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) . H_ij (j ≠ i) 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.

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 [4]. 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.

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 [5]. 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.

Figure 2

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 [6], only a reasonable threshold can produce a more accurate binary image. The threshold is generally expressed in the following form. (5) 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.

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 [7]. 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).

In this paper, a Gaussian mixture model is selected to achieve background subtraction based on the particularity of aerobics video images [8]. The principle is to use K Gaussian models to represent the features of each pixel in the aerobics video image.

3.4.2

Updating and matching models

Assuming that the pixel value of a new input frame of image is X_t, the formula for judging whether the pixel matches the Gaussian model is as follows: (7) | 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 [9]. We obtain the effect of background subtraction through the Gaussian model, as shown in Figure 3.

Figure 3

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).

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 [13]. 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.

Figure 6