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2444-8656
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Conventional Algorithms in Sports Training Based on Fractional Differential Equations

Published Online: 15 Jul 2022
Volume & Issue: AHEAD OF PRINT
Page range: -
Received: 17 Feb 2022
Accepted: 22 Apr 2022
Journal Details
License
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Introduction

At present, more and more young people join the basketball game. The spot-up shot is the foundation of modern basketball technology. Basketball shooting percentage is directly related to the initial speed and angle of the shot. We discussed the best shot angle from different angles [1]. The best shooting angle has always been a research hotspot in basketball theory. At present, there are many ideal basketball trajectory models in academia. Some scholars have used the variational method to study the problem of the best shooting angle. They discussed the shooting angle in detail from three angles: the best angle against angular deviation, the best angle against speed error, and the most effort-saving shot. They produce accurate quantitative results that are realistic. According to the ideal shooting trajectory model, some scholars have studied in detail the best position for the highest point of the shot. The author further gives the double 1/3 method of basketball shooting training under visual observation and improves the shooting percentage.

The no-resistance shooting trajectory model can analyze the best shot angle with a complex variational method. But the algorithm principle is complex, and the process is cumbersome. And its geometric meaning is the angle bisector. Therefore, this method is not conducive to guiding the actual shooting. The best release angle can also be estimated based on the best position for the estimated shot peak [2]. However, if the best shot angle is not directly given, it is also not conducive to guiding the actual shooting. Parabolic trajectory model with drag Although the functional equation for the best throw angle of the shot put can be derived from the principle of parabola, it is not the best throw angle for basketball. The process and result are quite complicated.

This paper establishes a model of shooting trajectory without resistance. We use the discriminant of the quadratic equation to derive the best shot angle simply and clearly. Its geometric meaning is a straight line between two points [3]. This helps guide the actual shot. We expand the model of shot trajectories with resistance based on Taylor. We perfectly unify the model of the shot trajectory with resistance and the model of the shot trajectory without resistance.

Shooting trajectory equation with resistance

Suppose the basketball is a particle and ignore air resistance. At this point, we can get the basketball equation and curve as: {x=υtcosθy=υtsinθ12gt2 \left\{ {\matrix{ {x = \upsilon t\,\cos \,\theta } \hfill \cr {y = \upsilon t\,\sin \,\theta - {1 \over 2}g{t^2}} \hfill \cr } } \right.

Where θ is the shooting angle. υ is the initial velocity of the shot. g is the acceleration of gravity. t is the basketball running time. The following formula is obtained by eliminating the time parameter t from formula (1). y=xsinθcosθgx22υ2cos2θ y = {{x\,\sin \,\theta } \over {\cos \,\theta }} - {{g{x^2}} \over {2{\upsilon ^2}{{\cos }^2}\,\theta }}

We first fix y, υ and find partial derivatives. The purpose is to find the best angle against angular deviation. xθ=xcos2θ+ysin2θgxυ212sin2θ {{\partial x} \over {\partial \theta }} = {{x\,\cos \,2\theta + y\,\sin \,2\theta } \over {{{gx} \over {{\upsilon ^2}}} - {1 \over 2}\sin \,2\theta }}

Then we get the best angle against angular deviation as follows: θ=12arctan(xy)+π2 \theta = {1 \over 2}\,\arctan \left( { - {x \over y}} \right) + {\pi \over 2}

But air resistance is unavoidable [4]. Assume that air resistance f is proportional to speed υ, f = . The relationship between the basketball position and the time parameter t can be obtained from Newton's second law: {x=mkυx0(1ekt/m)=mkυ0(1ekt/m)cosαy=mk(υ0sinα+mgk)(1ekt/m)mgtk \left\{ {\matrix{ {x = {m \over k}{\upsilon _{x0}}\left( {1 - {e^{ - kt/m}}} \right) = {m \over k}{\upsilon _0}\left( {1 - {e^{ - kt/m}}} \right)\cos \,\alpha } \hfill \cr {y = {m \over k}\left( {{\upsilon _0}\,\sin \,\alpha + {{mg} \over k}} \right)\,\left( {1 - {e^{ - kt/m}}} \right) - {{mgt} \over k}} \hfill \cr } } \right.

We can directly eliminate t from equation (5) to get: y=mk(υ0sinα+mgk)xkmυ0cosα+m2gk2ln(1xkmυ0cosα) y = {m \over k}\left( {{\upsilon _0}\,\sin \,\alpha + {{mg} \over k}} \right){{xk} \over {m{\upsilon _0}\,\cos \,\alpha }} + {{{m^2}g} \over {{k^2}}}\ln \left( {1 - {{xk} \over {m{\upsilon _0}\,\cos \,\alpha }}} \right)

This formula is too complicated and is not conducive to analyzing properties such as the optimum angle against angular deviation. Some scholars have simplified and approximated the above formula to find the shot angle when the shot put is thrown farthest. We expand ekt/m using series. The formula (5) is simplified to: {xυ0(tkt22m)cosαyυ0tsinαgt22(1+kυ0sinαmg) \left\{ {\matrix{ {x \approx {\upsilon _0}\left( {t - {{k{t^2}} \over {2m}}} \right)\cos \,\alpha } \hfill \cr {y \approx {\upsilon _0}t\,\sin \,\alpha - {{g{t^2}} \over 2}\,\left( {1 + {{k{\upsilon _0}\,\sin \,\alpha } \over {mg}}} \right)} \hfill \cr } } \right.

x, y is represented by a quadratic function of t. If t is eliminated directly, the expressions of x and y are still complicated [5]. This is also not conducive to further analysis of properties such as the optimal angle against angular deviation. This paper presents a new approximation method. This makes the expressions for x and y simpler. The algorithm is beneficial for further analyzing the optimal angle's properties against angle deviation. Generally, υx0g,km<<1,t<10s,a45°,kxmυ0cosα {\upsilon _{x0}}\sim g,{k \over m} < < 1,t < 10s,a\sim {45^\circ},{{kx} \over {m{\upsilon _0}\cos \alpha }} is a small amount. We expand ln(1kxmυ0cosα) \ln \left( {1 - {{kx} \over {m{\upsilon _0}\,\cos \,\alpha }}} \right) using series. If only the first two items are kept when expanding, then: y=xsinαcosαgx22υ2cos2α y = {{x\,\sin \,\alpha } \over {\cos \,\alpha }} - {{g{x^2}} \over {2{\upsilon ^2}\,{{\cos }^2}\alpha }}

This is the same relationship between x and y when air resistance is not considered [6]. Therefore, this article considers the expansion to retain the first three items: y=mk(υ0sinα+mgk)xkmυ0cosα+m2gk2(kxmυ0cosαk2x22m2υ02cos2αk3x33m3υ03cos3α)=xsinαcosαgx22υ2cos2αkgx33υ03cos3α \matrix{ {y = {m \over k}\left( {{\upsilon _0}\,\sin \,\alpha + {{mg} \over k}} \right){{xk} \over {m{\upsilon _0}\cos \alpha }} + {{{m^2}g} \over {{k^2}}}} \hfill \cr {\left( { - {{kx} \over {m{\upsilon _0}\,\cos \,\alpha }} - {{{k^2}{x^2}} \over {2{m^2}\upsilon _0^2\,{{\cos }^2}\alpha }} - {{{k^3}{x^3}} \over {3{m^3}\upsilon _0^3{{\cos }^3}\alpha }}} \right)} \hfill \cr { = {{x\,\sin \,\alpha } \over {\cos \,\alpha }} - {{g{x^2}} \over {2{\upsilon ^2}\,{{\cos }^2}\,\alpha }} - {{kg{x^3}} \over {3\upsilon _0^3{{\cos }^3}\,\alpha }}} \hfill \cr }

We add some terms without considering air resistance when there is air resistance. If the expansion is represented by an infinite series, the third term in equation (9) contains the resistance coefficient k. Hence the drag coefficient k = 0 when drag is not considered. By substituting it into the above formula, we get the same relationship between x and y when the resistance is not considered.

The most effortless shooting angle when there is no resistance

The model of the shot trajectory ignoring resistance is expressed as follows: {x=υtcosθy=υtsinθ12gt2+H1 \left\{ {\matrix{ {x = \upsilon t\,\cos \,\theta } \hfill \cr {y = \upsilon t\,\sin \,\theta - {1 \over 2}g{t^2} + H1} \hfill \cr } } \right.

Among them, H1 is the distance of the shot position from the ground. According to the discriminant of quadratic equation, this paper finds the most labor-saving shooting angle when there is no resistance. The specific method is as follows [7]. Assuming that the distance of the basket from the ground is H2, and the horizontal distance of the basket from the shot is S, then: {S=υtcosθH2=υtsinθ12gt2+H1 \left\{ {\matrix{ {S = \upsilon t\,\cos \,\theta } \hfill \cr {H2 = \upsilon t\,\sin \,\theta - {1 \over 2}g{t^2} + H1} \hfill \cr } } \right.

Eliminate time parameter t : H2=υsinθSυcosθ12gS2υ2cos2θ+H1 H2 = \upsilon \sin \,\theta {S \over {\upsilon \cos \,\theta }} - {1 \over 2}g{{{S^2}} \over {{\upsilon ^2}{{\cos }^2}\theta }} + H1 gx22υ2tan2θStanθ+gS22υ2H1+H2=0 {{g{x^2}} \over {2{\upsilon ^2}}}\,{\tan ^2}\theta - S\tan \theta + {{g{S^2}} \over {2{\upsilon ^2}}} - H1 + H2 = 0

We regard tan θ as a whole, then its discriminant: Δ=S24gS22υ2(gS22υ2H1+H2)0 \Delta = {S^2} - 4{{g{S^2}} \over {2{\upsilon ^2}}}\left( {{{g{S^2}} \over {2{\upsilon ^2}}} - H1 + H2} \right) \ge 0 4υ28g(H2H1)υ24g2S20 4{\upsilon ^2} - 8g\left( {H2 - H1} \right){\upsilon ^2} - 4{g^2}{S^2} \ge 0

So υ2g(H2H1)+g(H2H1)2+S2 {\upsilon ^2} \ge g\left( {H2 - H1} \right) + g\sqrt {{{\left( {H2 - H1} \right)}^2} + {S^2}} .

That is, the minimum square of the initial velocity is: g(H2H1)+g(H2H1)2+S2 g\left( {H2 - H1} \right) + g\sqrt {{{\left( {H2 - H1} \right)}^2} + {S^2}}

We can find out by substituting the initial velocity at this time into equation (13): tanθ=(H2H1)+(H2H1)2+S2S \tan \theta = {{\left( {H2 - H1} \right) + \sqrt {{{\left( {H2 - H1} \right)}^2} + {S^2}} } \over S}

This angle is the most labor-saving release angle [8]. The geometric meaning of this angle is as follows. As shown in Figure 1, O is the shot release point, and P is the center of the basket. PMOM,| OM | S,| PM |= H2 − H1, |PN|=(H2H1)2+S2 \left| {PN} \right| = \sqrt {{{\left( {H2 - H1} \right)}^2}+{S^2}} , ∠NOM = θ.

Figure 1

The most labor-saving corner (this article model)

It is demonstrated below that this angle is completely consistent with the angle θ=12arctan(xy)+π2 \theta = {1 \over 2}\arctan \left( { - {x \over y}} \right) + {\pi \over 2} in the literature.

The geometric meaning in the literature is shown in Figure 2. Know tanα=H2H1S=tan(2θ90) \tan \,\alpha = {{H2 - H1} \over S} = \tan \left( {2\theta - 90} \right) . Then tan(2θ)=SH2H1 \tan \left( {2\theta } \right) = {{ - S} \over {H2 - H1}} , tan(2θ)=2tanθ1tan2θ \tan \left( {2\theta } \right) = {{2\tan \,\theta } \over {1 - {{\tan }^2}\theta }} is known from the universal formula. Then SH2H1=2tanθ1tan2θ {{ - S} \over {H2 - H1}} = {{2\,\tan \,\theta } \over {1 - {{\tan }^2}\theta }} is S tan2 θ − 2(H2 − H1) tan θS = 0. tanθ=(H2H1)+(H2H1)2+S2S \tan \,\theta = {{\left( {H2 - H1} \right) + \sqrt {{{\left( {H2 - H1} \right)}^2} + {S^2}} } \over S} .

Figure 2

The most labor-saving corner (reference model)

The final result of the two methods is the same. In the literature, the complex variational method is used to obtain it. But this paper only uses the discriminant of the simple quadratic equation. In addition, the geometric meaning in the literature is the angle bisector and is not conducive to guiding the actual shooting. In this paper, the geometric meaning is a straight line. This method facilitates the actual shot aiming.

The most effortless shooting angle when there is resistance

The trajectory simulation equation with resistance is as follows y=xsinαcosαgx22υ2cos2αkgx33mυ3cos3α+H1 y = {{x\sin \alpha } \over {\cos \,\alpha }} - {{g{x^2}} \over {2{\upsilon ^2}{{\cos }^2}\,\alpha }} - {{kg{x^3}} \over {3m{\upsilon ^3}\,{{\cos }^3}\,\alpha }} + H1 H2=SsinαcosαgS22υ2cos2αkgS33mυ3cos3α+H1 H2 = {{S\,\sin \,\alpha } \over {\cos \,\alpha }} - {{g{S^2}} \over {2{\upsilon ^2}\,{{\cos }^2}\,\alpha }} - {{kg{S^3}} \over {3m{\upsilon ^3}\,{{\cos }^3}\,\alpha }} + H1

This formula is very complicated, and it is difficult to directly find the most labor-saving angle of the shot [9]. Therefore, this formula is approximately simplified in this paper. We substitute the most effortless release angle and release speed into υ cos α when there is no resistance, and the rest remain unchanged.

Assuming gS22υ2+kgS33mυ3cosα=δgS22υ2 {{g{S^2}} \over {2{\upsilon ^2}}} + {{kg{S^3}} \over {3m{\upsilon ^3}\,\cos \,\alpha }} = {{\delta g{S^2}} \over {2{\upsilon ^2}}} , this formula simplifies to (δgS22υ2)tan2αStanα+δgS22υ2H1+H2=0 \left( {{{\delta g{S^2}} \over {2{\upsilon ^2}}}} \right){\tan ^2}\,\alpha - S\,\tan \,\alpha + {{\delta g{S^2}} \over {2{\upsilon ^2}}} - H1 + H2 = 0 . According to the situation when there is no resistance, we regard tan α as a whole, and the discriminant is expressed as Δ=S24δgS22υ2(δgS22υ2H1+H2)0 \Delta = {S^2} - 4{{\delta g{S^2}} \over {2{\upsilon ^2}}}\left( {{{\delta g{S^2}} \over {2{\upsilon ^2}}} - H1 + H2} \right) \ge 0 , 4υ4 − 8δg (H2 − H1) υ2 − 4δ2 g2 S2 ≥ 0, υ2δg(H2H1)+δg(H2H1)2+S2 {\upsilon ^2} \ge \delta g\left( {H2 - H1} \right) + \delta g\sqrt {{{\left( {H2 - H1} \right)}^2} + {S^2}} .

The minimum square of the initial velocity is: δg(H2H1)+δg(H2H1)2+S2 \delta g\left( {H2 - H1} \right) + \delta g\sqrt {{{\left( {H2 - H1} \right)}^2} + {S^2}} .

We can obtain tan=(H2H1)+(H2H1)2+S2S \tan = {{\left( {H2 - H1} \right) + \sqrt {{{\left( {H2 - H1} \right)}^2} + {S^2}} } \over S} by substituting the initial velocity into the trajectory model with resistance. This angle is the most labor-saving release angle [10]. It can be seen from the results that the most effortless release angle with resistance is the same as when there is no resistance. The shot speed is a multiple of the shot speed when there is no resistance (Figure 3).

Figure 3

The most labor-saving corner with errors (the algorithm in this article)

Shooting trajectory simulation based on augmented reality

Although athletes training shots on the basketball court can increase the feel of the shot, they cannot see the trajectory of the basketball movement [11]. This paper develops an augmented reality-based shooting trajectory simulation system. This allows players to see the trajectory of their shots in real-time. The system software structure diagram is shown in Figure 4, and the system working diagram is shown in Figure 5.

Figure 4

Software structure diagram of shooting trajectory simulation system

Figure 5

Schematic diagram of system operation

The main purpose of the front module of the system is to obtain the shooting speed and shooting angle. The way to obtain it is to record the hand's position in real-time according to the magnetic tracker and the data glove [12]. The trajectory curve is fitted according to a certain algorithm. We end up with release speed and release angle.

The modules in the system are first based on the shot speed and shot angle obtained before the system. We render the trajectory of virtual basketball according to the previous illustrative model. Then it is judged whether the virtual basketball falls into the virtual basket.

The main purpose of the rear module of the system is based on the virtual basketball trajectory in front and whether it falls into the virtual basket. We give instructive advice to athletes.

Conclusion

The shooting trajectory can be approximated as a parabola. This paper first gives the most labor-saving angle of shot through theoretical analysis. The complex shooting trajectory is simplified by Taylor expansion when there is resistance. The algorithm proves that when the air resistance tends to 0, it is the same as the shooting trajectory without considering the air resistance. Secondly, through the discriminant of quadratic equation, the acquisition of the most labor-saving shooting angle without considering air resistance is given. We give its geometric meaning. The geometric meaning of the shooting angle given in this article is a straight line between two points. This is great for coaching the actual shot. Finally, this paper designs an augmented reality system. The system allows players to see the trajectory of the virtual basketball and the virtual basket in real-time. This effectively instructs the athlete to change his shooting speed and shooting angle in real-time.

Figure 1

The most labor-saving corner (this article model)
The most labor-saving corner (this article model)

Figure 2

The most labor-saving corner (reference model)
The most labor-saving corner (reference model)

Figure 3

The most labor-saving corner with errors (the algorithm in this article)
The most labor-saving corner with errors (the algorithm in this article)

Figure 4

Software structure diagram of shooting trajectory simulation system
Software structure diagram of shooting trajectory simulation system

Figure 5

Schematic diagram of system operation
Schematic diagram of system operation

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