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Procedia Engineering 34 (2012) 218 - 223

9th Conference of the International Sports Engineering Association (ISEA)

Multi-body power analysis of kicking motion based on a

double pendulum

Hiroki Ozakia, Ken Ohtab, Tsutomu Jinjia

aJapan Institute of Sports Sciences, 3-15-1 Nishigaoka, Kita, Tokyo, 115-0056 Japan bKeio Univercity, 5322 Endo, Fujisawa, Kanagawa, 252-0882 Japan

Accepted 05 March 2012

Abstract

To kick a ball with the maximum velocity, the linear velocity of the kicking foot upon impact must be at the maximum. The dynamical mechanism of the kicking motion must be clarified to better understand the mechanism to produce the maximum velocity of the kicking foot. Therefore, the aim of this study was to clarify the mechanism that produces the maximum foot velocity using mathematical analysis based on a three-dimensional double pendulum model with a moving pivot. We investigated how the non-muscular forces of three components (i.e. centrifugal, Coriolis and gravity) generate, absorb, and transfer energy in order to produce the maximum swing velocity of the leg.

© 2012 Published by Elsevier Ltd.

Keywords: Football; multi-body power analysis; double pendulum; energy flow

1. Introduction

The ball velocity of an instep kick depends on various factors including the foot mass of the kicking leg, the rigidity of the kicking foot, the impact point, and the linear velocity of the swing. Among these factors, one of the most important is to generate the maximum ball velocity is the linear velocity of kicking foot's swing. Therefore, the kicker must use the dynamics of the kicking leg efficiently to generate as much kinetic energy as possible and transfer it to gain foot swing velocity. Considerable research on kinetic analysis and energy flow of the swing motion has been reported [1] [2]. However, the biomechanical mechanism by which the mechanical energy flows through the limb segments to the ball is not well explained. For this reason, a free-body power analysis of the entire limb was used to analyze the mechanical energy flow using a double pendulum model. The purpose of this study is to clarify the

1877-7058 © 2012 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.04.038

mechanism used to produce the maximum velocity of the foot using mathematical analysis based on a three-dimensional double pendulum model with a moving pivot.

2. Double pendulum model of the kicking motion

2.1. The kinematics of the double pendulum

We developed a nine degrees of freedom, dynamic double pendulum model (Fig. 1). This double pendulum model consists of two segments: the first segment freely suspended from a point in 3D space and the second suspended from the end of the first segment. The first segment (the thigh) is denoted as Linkl and the second (the shank) as Link2. It is known that the knee joint of human being rotates internally or externally in a flexed position. Therefore, we defined the knee joint as a ball joint to adopt various form of kicks. Symbols xg1 and xg2 are the positions of the center of mass in each segment. The center of the hip joint of the kicking leg is x0 and that of the supporting leg is xL0. The center of the knee joint is x1 and center of the foot joint is x2. Symbols e11, e12 are unit vectors toward the normal lines in each of the segments. Symbol eq1 is a cross product of en and a vector from the knee medial to the knee lateral (this vector was named the knee axis). Symbol eq2 is also a cross product of el2 and the knee axis. Finally, et1 and et2 are calculated from en * lq1 and el2 x iq2, respectively. Symbols m (m1, m2), J ( J1, J2), and l (l1, l2) are the center of mass, moments of inertia and lengths attached by Linkl and Link2, respectively. Symbols lg1 and lg2 are the lengths from the proximal joint to xg1 and xg2. To simplify the analysis of the system, the foot is not included in this system.

Fig. 1. Double pendulum model

Then, the acceleration vectors .¿gi and xg2 are given by

= Xo + àh xlgie,i+ mi xlgien). Xg2 = Xi + o>2 xlge2 + fflzx(ffli xlg2e,2),

= Xo + mi xlieli + ^i x(^i xlieli) +¿2 x^e^ + ^2 x(^2 x^e^).

(1) (2) (3)

2.2. kinemics of the double pendulum model

The linear dynamics of each segment are

mi( Xgi -g) = Fi- F2,

m2(Xg2 -g) = F2,

where F1 and F2 are, respectively, the force vectors acting on x0 and x1. Therefore, the joint force F2 is the internal force, the components of F2 that have an effect on the acceleration of the proximal link can be described as

F2= m2( Xg 2- g), (6)

= m2(x -g)+m2(tai x/ie,i)+m2(mi xhen)) + m2 (<02 * lg2ei2) + m2(^2 x(^2 * lg2^2)). (7)

The rotational dynamics of each link is given by

Jitoi + ^i * JiWi= Ti - T2 - lgien x Fi +(li - lgi )en x(- F2), (3)

J2CO2 + ®2*J2W2 = T2 -lg2el2 *F2. (9)

3. Power for each links

The kinetic energies of each link Ti and T2 can be described as Ti = i / 2mixTixxgi + i / 2m? JiWi and T2 =i / 2m2xg,2xcg2 +i / 2mlJ1w1. Also, the potential energies of each link Ui and U2 are Ui =- migTxgi,

U2 =- m2 gT xg 2.

The total kinetic energy and the potential energy of Link2 is given as

E2 = T2+U2. (10)

Also, the power of Link2 can be described as follows:

E2 = m2-»¿T2xxg2 + «>2T J2W2 - m2gTxxg2 = F2-xi + r2T^2. (11)

The total kinetic energy and the potential energy of Linki is

Ei= Ti+Ui. (12)

Therefore, the power of Linki can be described as follows:

i?i = mixCgTicCgi + iOiT Ji^i - migTcCgi = FiTxo - F2cCi + zf®i - rtm . (U)

4. Experiment

Ten professional male futsal players participated in this study (members from five national teams were included). Each subject preferred to kick the ball using his right leg. The subjects performed at least three maximal-effort kicking trials toward a target (i0 m in front of the ball). Twenty spherical reflective markers (8 mm in diameter) were used to identify player's key anatomical landmarks. The motion of the reflective markers was recorded using a twelve-camera optoelectronic motion capture system (Vicon MXseries) at 500 Hz. The analysis phase was defined as the time from the point at which the kicking foot left the floor (-0.2 s) to one frame before impact with the ball (0.0 s). The data were smoothed by applying the bidirectional fourth-order Butterworth low-pass filter [3]. The cutoff frequency was

calculated by Yu's method [4]. In the following section, we discuss the data that was collected from one subject.

5. Results and discussion

5.1. Kinematic analysis

X2 (The velocity vector of x2) can be divided between Xi (The velocity vector of xj) and Wi xhen (the velocity due to shank rotation) and can describe asXi = Xi + ®ixlie,2. Fig. 2 a shows the results ofXi, X2 anda^xhe,2. The horizontal axis represents time of kicking motion. It shows that the ankle velocity depends on the knee velocity. However, the velocity due to shank rotation suddenly increases after the supporting foot landed and exceeds the knee velocity upon impact. It indicates that the rotation of the shank is important to produce a maximum ankle velocity. Fig.2 b shows the angular velocities of the thigh and shank. et1Tm1 expresses the thigh's angular velocity around et1 and also, et2Tm2 expresses the shank's angular velocity around et2. The peak of thigh's angular velocity occurred after the supporting leg (i.e. contralateral leg) landed. On the other hand, the peak of shank's angular velocity is observed after impact. These results suggest that it has the time lag of dynamics for the accelerating each link. Therefore, we investigated the dynamical mechanism of kicking leg in the following section.

kicking foot left the floor supporting leg landed impact

110 H ■s 8H

(U 6 ->

4 42 0

| X1 | (knee velocity) >

| W2 x l2e,21 (velocity due to shank rotation)

ra-10 --15

1.18 -0.16 -0.14 -0.12 -O.1O,;0?08 -0.06 -0.04 -0.02 0.00 Time (s)

-0.20 -0.18 -0.16 -0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 Time (s)

Fig. 2. (a) Comparison of components of ankle velocity; (b) comparison of angular velocity for the both segments

5.2. Thigh acceleration

On the right side of Eq. (8), the second and third terms describe the effect of the acceleration of the thigh in the swing direction. Moreover, lg1(-eis one of the components of -lg1e„xF1 and it describes a moment around en (Fig. 3 a). Fig. 3 b shows the change in the four moments around en that affect the rotation of the thigh. In the first phase of the kicking motion, et1T r1 is the main cause of the increased thigh rotation; however, l^ (- eT )F gained after the supporting leg contacted the floor, also contributes to the increase. It was thought that the increase in the l^ (- eT )F value was primarily caused by the impact force (Fbrake) of the landing of the supporting leg (-0.06 s) being transferred to the pelvis.

Then the acceleration of the thigh increased by lsl( - eh)F . This method of acceleration that uses a sudden stop is called the "braking effect" in this study.

Fig. 3. (a) Braking force of hip joint; (b) components of rotation torque in Linkl

5.3. Shank acceleration

Using the multi-body power analysis [5], we discuss how the non-muscular forces act to increase energy in order to produce the maximum swing velocity. The internal force F2 and ¿1 (which is the velocity vector of x1) can express F2 = [Fq2,Fn,F,2]T andx =\1xq2,lx,2,1xi2]T. Using this equation, E can be described as

E2= F2T ¿1+ riot, (14)

= Fhxt2 + FnXl2 + F2Xq2j + TTW2 .

V Y ( 15)

12 P 12p q2p

Then, we observed that the main component of F2is the centrifugal force. Therefore, by substituting Eq. (7) in '2P = F2x,2 = (e,2F2)(ei2.¿1), we obtain

12 P = Fi2 x, 2,

= «21 xnen{x0 -g} + «21 xnen{d)1 xAen} + M21 xx,2en{m1 x(®1 x,^,1)} +

V_° V V__^_V__^ __-___>

12V I2p 12 p

pin PL1a PL1c

M21 x,2el2{m2 x 42e,2}+ W21 XX,2e;T2{ffl2 X^ x le2e,2)}.

12 P 12P

PL2a PL2c

Here, 12Pun, nPL1a, aPL1c, 12PL2a and ,2PL2c are the components of linear acceleration, angular acceleration of the thigh, centrifugal acceleration of the thigh, angular acceleration of the shank, and centrifugal acceleration of the shank respectively. Fig. 3 shows the change in these power components that have an effect on the energy of the shank. Symbols 12PUn and ,2PL1a show the low values compared with others. On the other hand, ,2PL1c and ,2PL2c show higher values before impact. Furthermore, FTx, is the inner product of F2 and ¿1 . Therefore, when the subject places his knee at an angle of 90 degrees, the energy from the thigh could be effectively transferred to the shank (Fig. 4). In the experiment, the subject kept his knee

angle close to 90 degrees at the peak of the thigh's angular acceleration. In other words, the shank was accelerated by the energy produced by the thigh and effectively transferred to the shank by the internal force of the action. Moreover, the knee extension torque was the main contributor to the increase in swing velocity after the supporting leg landed.

N20 -0.18 -0.16 -0.14--0.12 -0.^-0.08 -0.06 -0.04 -0.02"0,00 Time (s)

Fig. 3. Sources of power for Link2

Fig. 4. The idealized posture for an effective energy transfer

Conclusion

We investigated how the non-muscular forces generate, absorb, and transfer energy in order to produce the maximum swing velocity of the leg. The dominant force to accelerate the thigh was the muscle force generated by the hip extension torque. Following this energy production, the braking effect contributed to the increase after the supporting leg landed. On the other hand, the shank was accelerated by the muscle force generated from the knee extension torque at approximately the same time as the braking effect. Finally, the non-muscular forces generated by the thigh action contributed to increase the ankle velocity. It was thought that the subject controlled their motion of kicking leg skilfully to transfer the energy effectively.

References

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concepts, power transfer, dynamics and simulations. Gait & Posture Vol. 16, No. 3: 215-232.

[2] Nunome, H., Asai, T., Ikegami, Y., Sakurai, S., 2002. Three-dimensional kinetic analysis of side-foot and instep soccer kicks.

Med Csi Sports Exerc 34 (12): 2028-2036.

[3] Winter, A D., 1990. Biomechanics and motor control of human movement. Second edition. John Wiley & Sons, New York, pp.

41-43.

[4]Yu, B., Gabriel, D., Noble, L., An, K. N., 1999. Estimate of the Opitimum Cutoff Frequency for the Butterworth Low-Pass Digital Filter. JOURNAL OF APPLIED BIOMECHANICS 15: 318-329.

[5] Ohta, K., Ohgi, Y., Shibuya, K., 2011. Multi-body power analysis of golf swing based on a double pendulum with moving pivot.

Procedia Engineering: 9th Conference of the International Sports Engineering Association (ISEA).