1. bookAHEAD OF PRINT
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2444-8656
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01 Jan 2016
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Football Offense Training Strategy Based on Fractional Differential Mathematical Modeling

Published Online: 15 Jul 2022
Volume & Issue: AHEAD OF PRINT
Page range: -
Received: 27 Apr 2022
Accepted: 28 Jun 2022
Journal Details
License
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Introduction

The optimal tracking control of the falling ball trajectory in football training is the basis for improving athletes' skills. Generally, the academic circle adopts the precision map trajectory tracking control method. At the same time, we combine the fuzzy control law to track the optimal drop point trajectory. This can improve the optimal drop ball positioning and dynamic follow-up control capabilities [1]. The optimal trajectory tracking control method of football training has received significant attention.

The optimal ball trajectory tracking methods for football training mainly include fuzzy tracking control method, SLAM tracking control method, adaptive dynamic following control method, and so on. Scholars established the constraint reference model of trajectory following control and combined template matching and grid matching methods to realize trajectory following control. In this way, the tracking control ability of the ball drop point trajectory in football training can be improved. However, the adaptive performance of the above methods for tracking the trajectory of the falling ball in football training is not good, and the time cost is too high. Therefore, this paper proposes a tracking method for the trajectory of the falling ball in football training based on fractional differential equations. We construct a constrained parameter model for trajectory tracking control of the falling ball in football training [2]. At the same time, we use fractional differential equations for optimal trajectory optimization control. At the same time, this paper adopts the multi-parameter dynamic following method to realize the trajectory tracking of the falling ball in football training. Finally, this paper conducts an experimental test analysis and draws a valid conclusion.

Trajectory tracking control object model and constraint parameter analysis
Trajectory tracking control object model

In this paper, the end position reference model is used to carry out the spatial planning and design of the football training drop point trajectory. The two-degree-of-freedom model is used for the spatial planning of the trajectory of the ball drop point in football training. We first use the attitude sensor to sample the flight trajectory information and mechanical information of the football training ball [3]. We construct the spatial planning model of the football training sphere's flight trajectory (Fig. 1).

Figure 1

The spatial planning model of the flight trajectory of the football training sphere

We construct the kinematics model of the sphere based on the spatial position distribution model of the trajectory of the football training drop point. We use the end position reference model for the spatial planning and design of the football training drop point trajectory. Suppose the GPS to be converted is the point A(φ A; λA. We use the carrier phase difference technology RTK to correct the position error offset of the trajectory endpoint. This paper d(t) = d1(t) + d2(t) is defined in the mathematical model of the feedback regulation of the ball drop point track in football training [4]. We extend the trajectory tracking model of the football training drop point to an integer order system. At the same time, we use the proportional-differential controller to track the trajectory of the ball drop point to obtain the spatial coordinate distribution of the trajectory tracking control (Fig. 2).

Figure 2

Spatial coordinate distribution of trajectory tracking of ball drop points in football training

We get adaptive learning weights in the plane Cartesian coordinate system: Y(s)/R(s)=GC(s)G0(s)eτs1GC(s)+G0(s) Y\left(s \right)/R\left(s \right) = {{{G_C}\left(s \right) - {G_0}\left(s \right){e^{- \tau s}}} \over {1 - {G_C}\left(s \right) + {G_0}\left(s \right)}}

Where x(t) = [x1(t), x2(t), ⋯, xt(t)]T is the spatial position state vector of the football training drop point. d1(t) and d2(t) represent the stable and time-delayed solutions of the Kalman filter, respectively. Therefore, the controlled object model and control constraint parameter model of the trajectory tracking of the ball drop point in football training is constructed [5]. The trajectory tracking adjustment is carried out in combination with the adjustment method of the flight trajectory parameters of the ball drop point in football training. This can improve the ball drop point's trajectory tracking and control ability in football training.

Constraint parameter adjustment of trajectory tracking control

We construct a flight kinematics model for the trajectory tracking of the ball in football training based on the control as mentioned above constraints of the trajectory tracking of the ball drop point [6]. We construct a linear control matrix of dynamic motion primitives for tracking the ball's trajectory in football training under a single degree of freedom. In this paper, the football training drop point trajectory track is divided by the segmented trajectory learning method [7]. We determine the primitive points of the abscissa of point A on both sides of the meridian. In this way, we get the optimization process of the trajectory of the football training drop point: Bl(A)=minβ0{ω(β)+ω(ATβ)} {B_l}\left(A \right) = \mathop {\min}\limits_{\beta \ne 0} \left\{{\omega \left(\beta \right) + \omega \left({{A^T}\,\beta} \right)} \right\}

Where AT represents the transpose of matrix A. The primitive function of the trajectory tracking of the ball drop point in football training is recorded as: k1=maxtI01|G(t,s)a(s)|ds {k_1} = \mathop {\max}\limits_{t \in I} \int_0^1 {\left| {G\left({t,\,s} \right)a\left(s \right)} \right|ds} k2=maxtI01|Gs(s,τ)a(τ)|dτ {k_2} = \mathop {\max}\limits_{t \in I} \int_0^1 {\left| {{G^{'}}_s\left({s,\,\tau} \right)a\left(\tau \right)} \right|d\tau} k=maxtI{k1,k2} k = \mathop {\max}\limits_{t \in I} \left\{{{k_1},{k_2}} \right\} d=max{2A1,4A2} d = \max \left\{{2{A_1},4\,{A_2}} \right\}

We construct a constrained parameter model for trajectory tracking control of the falling ball in football training. In this paper, the SLAM positioning method of a high-precision map is used to establish the grid structure model of the trajectory tracking of the falling ball. We analyze the latitude value of the distance between points A, C relative to the earth's radius. At this point, we build a two-degree-of-freedom model for tracking the football training drop point [8]. The geodesic distance between points A, C satisfies the following: A=[fx1fx2gx1gx2]=[r12x1N1x1σ1x1N2x2σ2x2N1r2σ2x1N1] A = \left[ {\matrix{{{f_{{x_1}}}} & {{f_{{x_2}}}} \cr {{g_{{x_1}}}} & {{g_{{x_2}}}} \cr}} \right] = \left[ {\matrix{{{r_1}{{2{x_1}} \over {{N_1}}}} & {- {{{x_1}{\sigma _1}{x_1}} \over {{N_2}}}} \cr {- {{{x_2}{\sigma _2}{x_2}} \over {{N_1}}}} & {{r_2}{{{\sigma _2}{x_1}} \over {{N_1}}}} \cr}} \right]

Tracking optimization of the ball drop point
Construction of kinematics model

In this paper, the control model of the trajectory tracking of the ball drop point in football training is divided into a deterministic model and an uncertain model. We call them x˙(t)=Ax(t)+Bu(t) \dot x\left(t \right) = Ax\left(t \right) + Bu\left(t \right) and uc(t) = Kxc(t), respectively. At this point, we get the closed-loop control system for trajectory tracking in the positive kinematic space: x˙(t)=Ax(t)+BHx(tds(t)da(t)) \dot x\left(t \right) = Ax\left(t \right) + BHx\left({t - {d_s}\left(t \right) - {d_a}\left(t \right)} \right)

In this paper, the proportional-integral control is used for trajectory correction control to obtain the boundary conditions: dx2dt=r2x2(1σ2x1N1x2N2) {{d{x_2}} \over {dt}} = {r_2}{x_2}\left({1 - {\sigma _2}{{{x_1}} \over {{N_1}}} - {{{x_2}} \over {{N_2}}}} \right)

This paper uses a section of RTK-GPS data from the campus playground for steady-state tracking control. At this point, we obtain the sufficient conditions for the flight stability of the trajectory of the football training drop point: φ(x1,x2)=1x1N1σ1x2N2 \varphi \left({{x_1},{x_2}} \right) = 1 - {{{x_1}} \over {{N_1}}} - {\sigma _1}{{{x_2}} \over {{N_2}}}

When p > 2, it is determined that the tracking error of the dynamic primitive trajectory in the range of the earth ellipsoid satisfies: limx[sup|X(t)y(t)|p]=0 \mathop {\lim}\limits_{x \to \infty} \left[ {\sup {{\left| {X\left(t \right) - y\left(t \right)} \right|}^p}} \right] = 0

We combine the Lyapunov stability principle for coordinate transformation. At this point, we set the Lyapunov function to be V=12σTσ V = {1 \over 2}{\sigma ^T}\sigma . At this point, we determine the distinct components of each GPS point on the trajectory. V˙=σTσ˙=σTCn{f(X,t)x1dnp(t)(n)+Cn1k=1n1Ck[e(k)p(t)(k)]}+σTCnb(X,t)u+σTCn[Δf(X,t)+d(t)]σTCn{f(X,t)x1dnp(t)(n)+Cn1k=1n1Ck[e(k)p(t)(k)]}+σTCnb(X,t)u+σTCnΔf(X,t)+d(t) \matrix{{\dot V = {\sigma ^T}\dot \sigma = {\sigma ^T}{C_n}\left\{{f\left({X,t} \right) - x_{1d}^n - p{{\left(t \right)}^{\left(n \right)}} + C_n^{- 1}\sum\limits_{k = 1}^{n - 1} {{C_k}\left[ {{e^{\left(k \right)}} - p{{\left(t \right)}^{\left(k \right)}}} \right]}} \right\} +} \hfill \cr {{\sigma ^T}{C_n}b\left({X,t} \right)u + {\sigma ^T}{C_n}\left[ {\Delta f\left({X,t} \right) + d\left(t \right)} \right] \le} \hfill \cr {{\sigma ^T}{C_n}\left\{{f\left({X,t} \right) - x_{1d}^n - p{{\left(t \right)}^{\left(n \right)}} + C_n^{- 1}\sum\limits_{k = 1}^{n - 1} {{C_k}\left[ {{e^{\left(k \right)}} - p{{\left(t \right)}^{\left(k \right)}}} \right]}} \right\} +} \hfill \cr {{\sigma ^T}{C_n}b\left({X,t} \right)u + \left\| {{\sigma ^T}{C_n}} \right\|\,\left\| {\Delta f\left({X,t} \right) + d\left(t \right)} \right\|} \hfill \cr}

Thus, a kinematic model for determining each GPS point on the trajectory is constructed. We get a time lag of τs = τc + τCSA. This paper uses the standard Kalman filter for trajectory tracking control and kinematic modeling.

Optimization of control law for trajectory tracking of ball drop point in football training

In this paper, the trajectory motion of the ball drop point in football training is regarded as a two-dimensional motion in the plane [9]. The football training sphere enters the initial value r = 1, kjr=kj k_j^r = {k_j} , bjr=bj b_j^r = {b_j} , (j = 1, 2, ⋯, m), X = ∅, Y = ∅, Kr=j{kjr}, {K^r} = \bigcup\limits_j {\left\{{k_j^r} \right\},} , Br=j{bjr} {B^r} = \bigcup\limits_j {\left\{{b_j^r} \right\}} in continuous flight. At this point, we obtain the output inertial feature quantity after the latitude and longitude of the motion trajectory is sequentially converted to the plane rectangular coordinate system. Thanks σTCnCnTσ=CnTσ2 {\sigma ^T}{C_n}C_n^T\sigma = {\left\| {C_n^T\sigma} \right\|^2} . So we get the Lyapunov functional under the optimal trajectory tracking configuration from the latitude and longitude transformation: V˙CnTσ{Δf(X,t)+d(t)[F(X,t)+D(t)]}KCnTσ=CnTσ{Δf(X,t)F(X,t)+[d(t)D(t)]}KCnTσKCnTσ<0(|σ|0) \matrix{{\dot V \le \left\| {C_n^T\sigma} \right\|\left\{{\left\| {\Delta f\left({X,t} \right) + d\left(t \right)} \right\| - \left[ {F\left({X,t} \right) + D\left(t \right)} \right]} \right\} - K\left\| {C_n^T\sigma} \right\| =} \hfill \cr {\left\| {C_n^T\sigma} \right\|\left\{{\left\| {\Delta f\left({X,t} \right)} \right\| - F\left({X,t} \right) + \left[ {\left\| {d\left(t \right)} \right\| - D\left(t \right)} \right]} \right\} - K\left\| {C_n^T\sigma} \right\| \le} \hfill \cr {- K\left\| {C_n^T\sigma} \right\| < 0\left({\left| \sigma \right| \ne 0} \right)} \hfill \cr}

In this paper, the optimal landing point XΛ k + 1 is optimally estimated based on the principle of Lyapunov stability. In Bernoulli space, this paper combines nonlinear hyperbolic differential equations to track the trajectory of football training falling balls. At this point we get the tracked optimized model: (u,tu)|t=0=(u0,u1)Hxsc×Hxsc1 \left({u,{\partial _t}u} \right)\left| {_{t = 0} = \left({{u_0},{u_1}} \right) \in H_x^{{s_c}} \times H_x^{{s_c} - 1}} \right.

Where u: I × IRdIR is the Gaussian white noise during state transition, and we perform adaptive parameter adjustment for the trajectory tracking of the ball drop point in football training. In this paper, the dynamic parameter model of the trajectory of the football landing point is constructed: Ψ(h1,0)=Ψ+h1K(Z1+Z2+Z3)1KT+h2L(Z2+Z3)1LT<0 \Psi \left({{h_1},0} \right) = \Psi + {h_1}K{\left({{Z_1} + {Z_2} + {Z_3}} \right)^{- 1}}{K^T} + {h_2}L{\left({{Z_2} + {Z_3}} \right)^{- 1}}{L^T} < 0

In football training, the low-altitude landing point of the sphere satisfies the spatial distribution law of automatic optimization of inertial navigation. Suppose x* is a limit point in the optimal solution set (xk} of saturation control.

Simulation experiment and result analysis

The experiment was designed using Matlab7. We realize the electronic sensor for collecting the flight attitude parameters of the football training sphere. In this paper, the ARM5.0 video acquisition instrument is used to capture the time-lapse of the football training drop point [10]. The time interval of data collection is 0.12s. The sampling frequency is 200KHz. The interference signal-to-noise ratio is -20dB. The minimum ground clearance is 3cm. The operating frequency is 1.2GHz. Set L = 500mm, D1 = 10mm. According to the above simulation environment and parameter settings, the trajectory tracking control of the ball drop point in football training is carried out. At this time, we get the tracking result of the ball drop point trajectory according to the football training, as shown in Figure 3. It is found that this method has better adaptive performance in tracking the trajectory of the falling ball in football training. The trajectory tracking has high precision and strong control ability. The error of trajectory tracking can be converged to a minimum quickly. The football trajectory simulation model improves the tracking control ability of the landing point trajectory through automatic adjustment.

Figure 3

Tracking of the ball drop point in football training

Conclusion

This paper proposes a tracking method for the trajectory of the falling ball in football training based on fractional differential equations. At the same time, we construct a constrained parameter model for trajectory tracking control of the ball drop point in football training. The grid structure model of the trajectory tracking of the falling ball is established using the high-precision map's SLAM positioning method. The optimal trajectory optimization control is carried out by using fractional differential equations. We use the multi-parameter dynamic following method to track the trajectory of the falling ball in football training. The research shows that this method has better adaptive performance in tracking the trajectory of the ball drop point in football training. The trajectory tracking has high precision and strong control ability. The model improves the football training effect.

Figure 1

The spatial planning model of the flight trajectory of the football training sphere
The spatial planning model of the flight trajectory of the football training sphere

Figure 2

Spatial coordinate distribution of trajectory tracking of ball drop points in football training
Spatial coordinate distribution of trajectory tracking of ball drop points in football training

Figure 3

Tracking of the ball drop point in football training
Tracking of the ball drop point in football training

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