Mathematical simulation analysis of optimal detection of shot-putters’ best path

With the successful bid for the 2008 Olympic Games, greatly improving the performance of various sports in my country has become the focus of attention of sports management departments, coaches and athletes. At the end of the 1980s, my country's shot-put athletes had achieved good results. However, the overall situation of the current level of shot-put in my country is in a downward trend, and there is still a certain gap compared with the world's first-class level. In order to narrow the gap, improve the performance of our shot-put athletes faster and win gold and silver medals in the 2008 Olympic Games, it is necessary to accurately establish a mathematical model of shot-put throwing. Using this mathematical model, coaches and athletes can analyse the best angle of shot throwing and obtain the functional relationship between the thrust size, the height of the shot, the speed of the shot and other parameters and the throwing distance so as to achieve the athlete's high performance through scientific training methods and improving them quickly.

In the past few decades, people have launched a lot of research on the mathematical model of shot-put. The establishment and use of these mathematical models have played a very good role in the development of the theoretical level of shot-put [1]. In these mathematical models, all numbers are accurate, that is, the mathematical model is based on precise and definite traditional mathematics. However, in the actual process, it is impossible to accurately obtain the value of the athlete's thrust applied to the shot-put and the height of the shot in the model, and the athlete's throwing state will not remain unchanged. All these will cause uncertainty in the model parameters. Obviously, the mathematical model based on certain parameters cannot accurately describe this uncertain shot-throwing process. Therefore, the optimal throwing mode and throwing distance calculated by these mathematical models will have corresponding errors. Based on the above analysis, in order to consider uncertain factors in the process of shot-putting, this paper introduces fuzzy mathematics and proposes a fuzzy optimization model for shot-putting. This model overcomes the shortcomings of the existing shot-putting model and can describe the fuzzy factors in the shot-putting process, thereby obtaining a more reasonable throwing mode and throwing distance [2].

As we all know, the shot throwing process can be roughly divided into the sliding phase and the force phase. (1) In the sliding phase, the shot will produce an initial speed v₀ with the human body, that is, the sliding speed. (2) In the exertion stage, the shot-put under the action of thrust F, following the linear motion of the human arm, obtains the speed v, that is, the shot speed θ. In the exertion stage, the relationship between each physical quantity is shown in Figure 1. Suppose that the shot height h, thrust F and shot angle 0 are not completely independent, then they satisfy the following functional relationship: (1) h(θ)=h1+l.sinθ−rshotF(θ)=F0−a.θ \matrix{ {h(\theta ) = {h_1} + l.\sin \theta - {r_{shot}}} \hfill \cr {F(\theta ) = {F_0} - a.\theta } \hfill \cr } In the formula, h₁ is the shoulder height of the human body; l is the arm length; r_shot is the radius of the shot put; F₀ is the horizontal thrust when the firing angle is 0° and a is the rate at which the thrust decreases as the firing angle increases (it is the rate of decline), which is related to the athlete's muscle strength and throwing technique. Different athletes have different values [3].

Fig. 1

On the basis of the above assumptions, according to the law of conservation of energy, the following equations are derived: (2) W=Flcos(φ−∂)=Fl[1−(mgF)2sin2(90∘+θ)]=12mv2−12mv02+mglsinθ W = Fl\cos (\varphi - \partial ) = Fl\left[ {1 - {{({{mg} \over F})}^2}\mathop {\sin }\nolimits^2 {{(90}^ \circ } + \theta )} \right] = {1 \over 2}m{v^2} - {1 \over 2}mv_0^2 + mgl\sin \theta where m is the quality of the shot. When the shot-put is released, the shot-put can be regarded as a projectile. Under the premise of ignoring the air resistance, the throwing distance s of the shot-put is given as follows: (3) s=v2sin2θ2g+(v2sin2θ2g)2+2hv2cos2θg s = {{{v^2}\sin 2\theta } \over {2g}} + \sqrt {{{\left( {{{{v^2}\sin 2\theta } \over {2g}}} \right)}^2} + 2h{{{v^2}{{\cos }^2}\theta } \over g}} In the sport of shot-putting, the goal pursued is to obtain the farthest throwing distance, even if s is the largest. Therefore, the mathematical model of the shot-putting process is as follows: (4) maxs=v2sin2θ2g+(v2sin2θ2g)2+2hv2cos2θg \max s = {{{v^2}\sin 2\theta } \over {2g}} + \sqrt {{{\left( {{{{v^2}\sin 2\theta } \over {2g}}} \right)}^2} + 2h{{{v^2}{{\cos }^2}\theta } \over g}} (5) s.t.Fl[1−(mgF)2sin2(90∘+θ)]=12mv2−12mv02+mglsinθh((θ)=h1+l.sinθ−rshot F(θ)=F0−a.θ 0≤θ≤90∘ F0=F0′, a=a′, h1=h1′, l=l′, v0=v0′ \matrix{ {\;\;\;\;\;\;\;\;s.t.Fl\left[ {1 - {{\left( {{{mg} \over F}} \right)}^2}\mathop {\sin }\nolimits^2 \left( {{{90}^ \circ } + \theta } \right)} \right] = {1 \over 2}m{v^2} - {1 \over 2}mv_0^2 + mgl\sin \theta } \hfill \cr {h((\theta ) = {h_1} + l.\sin \theta - {r_{shot}}} \hfill \cr {\;F(\theta ) = {F_0} - a.\theta } \hfill \cr {\;\;\;\;\;\;0 \le \theta \le {{90}^ \circ }} \hfill \cr {\;\;\;\;{F_0} = F_0^\prime,\quad a = {a^\prime},\quad {h_1} = h_1^\prime,\quad l = {l^\prime},\quad {v_0} = v_0^\prime} \hfill \cr } Among them, F0′,a′,h1′,l′,v0′ F_0^\prime,{a^\prime},h_1^\prime,{l^\prime},v_0^\prime are the athlete's horizontal thrust F₀, descent rate a, shoulder height h₀, arm length l and sliding speed v, respectively. Therefore, under the premise of knowing these anthropometric data and strength data, by solving the nonlinear optimisation problem (5), the best shot angle and the farthest throwing distance of the shot putter can be obtained [4].

In the mathematical model of shot throwing established above, all parameters are determined, and the relationship between equations and inequalities is clear. However, in the actual shot throwing process, there are various ambiguities, such as the measurement errors of data parameters such as F₀, a, h₀ and l and the fluctuation of the above parameters caused by the change of the athlete's throwing state. Therefore, a mathematical model based on certain and precise mathematical concepts cannot describe these ambiguities in shot-puts, and the best angle of shot-put is not really optimal. For the calculated throw, there will inevitably be a corresponding error between the distance and the actual situation [5].

In order to describe the fuzzy factors in the process of shot-putting and establish a more accurate mathematical model of shot-putting, this paper introduces fuzzy mathematics and proposes a fuzzy optimisation model of shot-putting. Its mathematical expression is as follows: (6) maxs=v2sin2θ2g+(v2sin2θ2g)2+2hv2cos2θg \max s = {{{v^2}\sin 2\theta } \over {2g}} + \sqrt {{{({{{v^2}\sin 2\theta } \over {2g}})}^2} + 2h{{{v^2}{{\cos }^2}\theta } \over g}} (7) s.t.Fl[1−(mgF)2sin2(900+θ)]=12mv2−12mv02+mglsinθ h(θ)=h1+l.sinθ−rshotF(θ)=F0−a.θ 0≤θ≤90∘ F0≈F0′, a≈a′, h1≈h1′, l≈l′, v0≈v0′ \matrix{ {\;\;\;\;\;\;\;\;s.t.Fl\left[ {1 - {{\left( {{{mg} \over F}} \right)}^2}\mathop {\sin }\nolimits^2 \left( {{{90}^0} + \theta } \right)} \right] = {1 \over 2}m{v^2} - {1 \over 2}mv_0^2 + mgl\sin \theta } \hfill \cr {\;h(\theta ) = {h_1} + l.\sin \theta - {r_{shot}}} \hfill \cr {F(\theta ) = {F_0} - a.\theta } \hfill \cr {\;\;\;\;\;0 \le \theta \le {{90}^ \circ }} \hfill \cr {\;\;\;\;{F_0} \approx F_0^\prime,\quad a \approx {a^\prime},\quad {h_1} \approx h_1^\prime,\quad l \approx {l^\prime},\quad {v_0} \approx v_0^\prime} \hfill \cr } Since the horizontal thrust, descent rate, shoulder height, arm length and sliding speed are all inaccurate quantities, in model (2), their corresponding equation constraints take the form of fuzzy constraints. The fuzzy optimisation model describes that under the condition that hard constraints (i.e. precise equality and inequality constraints) and soft constraints (i.e. fuzzy equality constraints) are satisfied as much as possible, the throwing distance as far as possible can be obtained. The model better captures the fuzzy characteristics of shot throwing and can get a more reasonable optimal throwing mode and throwing distance [6]. The following uses triangle membership to describe the fuzzy factors in the model. Taking horizontal thrust as an example, its value range may appear between [F0′−dF−, F0′, F0′+dF+] \left[ {F_0^\prime - {d_{F - }},\;F_0^\prime,\;F_0^\prime + {d_{F + }}} \right] and most likely to appear at F₀. The uncertainty of the horizontal thrust can be represented by the triangular fuzzy number F0=[F0′−dF−,F10,F0′+dF+] {F_0} = [F_0^\prime - {d_{F - }},{F^1}_0,F_0^\prime + {d_{F + }}] as shown in Figure 2. Its membership function (that is, the membership function of the fuzzy constraint F0≈F0′ {F_0} \approx F_0^\prime is given as follows: (8) μF={0,F′0+dF+≤F0F0′+dF−−F0dF+F0′≤F0≤F0′+dF+F0+F0′+dF−dF−dF−≤F0≤F0′0F0≤F0′−dF− {\mu _F} = \left\{ {\matrix{ {0,{F^\prime}0 + {d_{F + }} \le {F_0}} \hfill \cr {{{F_0^\prime + {d_{F - }} - {F_0}} \over {{d_{F + }}}}} \hfill \cr {F_0^\prime \le {F_0} \le F_0^\prime + {d_{F + }}} \hfill \cr {{{{F_0} + F_0^\prime + {d_F}_ - } \over {{d_{F - }}}}} \hfill \cr {{d_{F - }} \le {F_0} \le F_0^\prime} \hfill \cr {0{F_0} \le F_0^\prime - {d_{F - }}} \hfill \cr } } \right. Among them, d_F+, d_F− is the maximum range of positive or negative horizontal thrust that may fluctuate.

Fig. 2

It can be seen from Figure 2 that when F0=F0′ {F_0} = F_0^\prime , the membership function is 1, that is, the equality constraint is absolutely satisfied; when F0′<F0<F0′+dF+ F_0^\prime < {F_0} < F_0^\prime + {d_{F + }} or F0′−dF<F0<F0′ F_0^\prime - {d_F} < {F_0} < F_0^\prime , the membership function of the constraint is between (0, 1), and the degree of satisfaction of the equality constraint varies with the change changes and away from F. The farther the value, the lower the degree of satisfaction of the constraint; when F0′+dF<F0 F_0^\prime + {d_F} < {F_0} or F0≤F0′−dF− {F_0} \le F_0^\prime - {d_{F - }} , the membership function of the constraint is 0, indicating that the equality constraint is absolutely not satisfied. Similar to the horizontal thrust fuzzy number, the triangular fuzzy number is still used to characterise the fuzzy characteristics of descent rate, shoulder height, arm length and sliding speed, namely: (9) a˜=(a′−da−,a′,a′+da+)h˜1=(h′−dh−,h1′,h1′+dh+) l˜=(l′−dl−,l′,l′+dl+)v˜0=(v0′−dv−,v0′,v0′+dv+) \matrix{ {\,\,\,\widetilde a = \left\{ {{a^\prime} - {d_{a - }},{a^\prime},{a^\prime} + {d_{a + }}} \right)} \hfill \cr {{{\widetilde h}_1} = \left( {{h^\prime} - {d_{h - }},h_1^\prime,h_1^\prime + {d_{h + }}} \right)} \hfill \cr {\,\,\widetilde l = \left( {{l^\prime} - {d_{l - }},{l^\prime},{l^\prime} + {d_{l + }}} \right)} \hfill \cr {{{\widetilde v}_0} = \left( {v_0^\prime - {d_{v - }},v_0^\prime,v_0^\prime + {d_{v + }}} \right)} \hfill \cr } Then, the membership function μ_a, μ_h, μ_l, μ_v of their respective fuzzy constraints can be obtained. In the training of athletes, statistical data of horizontal thrust, descent rate, shoulder height, arm length and sliding speed can be obtained after multiple measurements. According to the frequency of the data, the respective membership functions can be determined [7].

Data measured by Maheras of college male shot-putters in 1995 as an example to illustrate the feasibility and effectiveness of the fuzzy optimisation model proposed in this paper are taken. After many measurements, the anthropometric data and strength data of a college male shot-put athlete are shown in Table 1. (Because the sliding speed has little effect on the shot throwing distance, Maheras did not measure the data, and it slipped during the following calculations. The speed is 2.5 ms, as shown in Figure 5). The Pulinix high-speed camera (frequency 120 Hz) was used to shoot 10 groups of athletes in the best throwing state of the two-dimensional film [11], through analysis to obtain the athlete's shooting angle, shooting speed, shooting height and throwing distance measurement data (Table 2).

Fig. 5

On the premise that the anthropometric data and strength data of the athletes are known, the fuzzy optimisation model proposed in this paper is used to obtain the fuzzy solution set of the best throwing state. From the measured values of anthropometric data and strength data (Table 1), their membership degrees can be obtained. In the triangle membership function, the values of the three key data are shown in Table 3. The fuzzy optimisation model is solved by the nonlinear programming method. The fuzzy solution set of the best shot angle, shot speed, shot height and throw distance is shown in Table 4. Corresponding to different throwing modes, the trajectory of the shot-put in the air is shown in Figure 4.

It can be seen from Table 4 that the athlete's throwing mode and throwing distance are also ambiguous. They are not given by a certain value but displayed by a set of values with known probability distribution. For example, the best shot angle varies between 31.94 and 33.36, where 31.94° is the most likely best shot angle. Corresponding to the throwing mode, the throwing distance will be between 13.39 m and 17.27 m, and the throwing distance of 15.21 m is most likely to occur [12]. This fuzzy representation of the throwing mode and throwing distance is basically consistent with the actual measured data of the throwing mode and throwing distance preferred by the athletes (Table 2), thus verifying the feasibility and effectiveness of the fuzzy optimisation model proposed in this paper.

In order to consider various fuzzy factors in the actual shot-putting process, this paper proposes a fuzzy optimisation model for shot-putting. It calculates the athlete's best throwing mode and the fuzzy solution set of the throwing distance based on the measured anthropometric data and the fuzzy characteristics of the strength data. It describes the athlete's shot throwing process more accurately than the non-fuzzy optimisation model. The training provided a more scientific theoretical basis. The idea of establishing a fuzzy optimisation model in this paper can also be applied to other sports and establish a fuzzy optimisation model of other sports considering fuzzy factors.