Available online at wwwsciencedirectcom Procedia Engineering 34 (22 ) 28 223 9 th Conference of the International Sports Engineering Association (ISEA) Multi-body power analysis of kicking motion based on a double pendulum Hiroki Ozaki a Ken Ohta b sutomu Jinji a a Japan Institute of Sports Sciences 3-5- Nishigaoka Kita okyo 5-56 Japan b Keio Univercity 5322 Endo Fujisawa Kanagawa 252-882 Japan Accepted 5 March 22 Abstract o kick a ball with the maximum velocity the linear velocity of the kicking foot upon impact must be at the maximum he dynamical mechanism of the kicking motion must be clarified to better understand the mechanism to produce the maximum velocity of the kicking foot herefore 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 (ie centrifugal Coriolis and gravity) generate absorb and transfer energy in order to produce the maximum swing velocity of the leg 22 Published by Elsevier Ltd Open access under CC BY-NC-ND license Keywords: Football; multi-body power analysis; double pendulum; energy flow Introduction he 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 herefore 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 [] [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 he purpose of this study is to clarify the 877-758 22 Published by Elsevier Ltd doi:6/jproeng22438 Open access under CC BY-NC-ND license
Hiroki Ozaki et al / Procedia Engineering 34 ( 22 ) 28 223 29 mechanism used to produce the maximumm velocity of the foot using g mathematicall analysis based on a three-dimensional double pendulum model with a moving pivot 2 Double pendulum model of the kicking motion 2 he kinematics of the double pendulum We developed a nine degrees of freedom dynamic double pendulum model (Fig ) his 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 he firstt segment (thee thigh) is denoted as Link 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 herefore we defined thee knee joint ass a ball joint to adopt various form of kicks Symbols x g and xg2 are the positions of the center of mass in each segment he center of the hip joint of the kicking leg is x and that of the supporting leg is x L he center of the knee joint is x and center of the foot joint is x 2 Symbols el e l2 are unit vectors toward the normal lines in each of the segments Symbol e q is a cross product of e l 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 e l2 and the knee axis Finally e t and e t2 are calculated from e l l q and e l2 l q2 respectively Symbols m (m m 2 ) J ( J J 2 ) and l (l l 2 ) are the center of mass moments of inertia and lengths attached by Link and Link2 respectively Symbols l g and l g2 are the lengths from the proximal joint to x g and x g2 o simplify the analysis of the system the foot is not included in this system Fig Double pendulum model hen the acceleration vectors x g and x g 2 are given by x x g g2 = x + = x + 2 = x + l 2 l 2 l lge + ( lgel ) lg2e 2 + ( 2 lg2el 2) le + ( l e l ) + 2 l g2el 2 2 + ( 2 lg2el 2) () (2) (3) 22 kinemics of the double pendulum model he linear dynamics of each segment are m m (x g - g) = F - F2 2 (x g 2 - g) = 2 = F (4) (5)
22 Hiroki Ozaki et al / Procedia Engineering 34 ( 22 ) 28 223 where F and F 2 are respectively the force vectors acting on x and x herefore the joint force F 2 is the internal force the components of F 2 that have an effect on the acceleration of the proximal link can be described as F ( x g - ) (6) ( x - g)+ ( lel )+ ( ( lel ))+ ( 2 l g 2el 2) + ( 2 ( 2 l g 2el 2)) (7) 2 = 2 g = he rotational dynamics of each link is given by 3 Power for each links J + J = - 2 - lgel F + ( l - lg) el ( - F2 ) (8) J2 2 + 2 J22 = 2 - l g 2el 2 F2 (9) he kinetic energies of each link and 2 can be described as = / 2m x gx g +/ 2 J and 2 = / 2x g 2x g 2 +/ 22 J 22 Also the potential energies of each link U and U 2 areu = - m g xg U2 = - g xg2 he total kinetic energy and the potential energy of Link2 is given as E 2 = 2 + U2 () Also the power of Link2 can be described as follows: 2 = x g 2x g 2 + 2 J 22 - g x g 2 = F2 x + 2 2 E () he total kinetic energy and the potential energy of Link is E = + U (2) herefore the power of Link can be described as follows: E = m xgxg + J - m g xg = F (3) x - F2 x + - 2 4 Experiment en 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 he subjects performed at least three maximal-effort kicking trials toward a target ( m in front of the ball) wenty spherical reflective markers (8 mm in diameter) were used to identify player s key anatomical landmarks he motion of the reflective markers was recorded using a twelve-camera optoelectronic motion capture system (Vicon MXseries) at 5 Hz he analysis phase was defined as the time from the point at which the kicking foot left the floor (-2 s) to one frame before impact with the ball ( s) he data were smoothed by applying the bidirectional fourth-order Butterworth low-pass filter [3] he cutoff frequency was
Hiroki Ozaki et al / Procedia Engineering 34 ( 22 ) 28 223 22 calculated by subject 5 Results and discussion 5 Kinematic analysis Yu s method [4] In the following section we discuss the data that was collected from one x 2 (he velocity vector of x 2 ) can be divided between x (he velocity vector of x ) and 2 l 2e l 2 (the velocity due to shank rotation) and can describe as x 2 = x x + 2 l2el 2 Fig 2 a showss the results of x x 2 and 2 l2e l 2 he 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 Fig2 b shows the angular velocities of the thigh and shank e t expresses the thigh s angular velocity around e t andd also e t2 2 expresses the shank s angular velocity around e t2 he peak of thigh s angular velocity occurred after thee supporting leg (ie contralateral leg) landed On the other hand the peak of shank s angular velocity is observed after impact hese results suggest that it has the time lag of dynamics for the accelerating each link herefore we investigated the dynamical mechanism of kicking leg in the following section (a) kicking foot left l the floor supporting leg landedd impact (b) velocity(m/s) 35 8 3 6 x 2 (ankle velo ocity) 25 e t2 2 4 2 2 5 e 8 5 t x 6 (knee velo ocity) 4 528642864 2 2 2 ll 2e l 2 (velocity due to shank rotation) ime(s) 5 286428642 ime(s) Fig 2 (a) Comparison of components of ankle velocity; (b) comparison of angular velocity for the both segments angularvelocity(rad/s) 52 high 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 l ( - e g q ) F is one of the components of - l g e l F and it describes a moment around e t (Fig 3 a) Fig 3 b shows the change in the four moments around et that affect the rotation of the thigh In the first phase of the kicking motion e t is the main cause of the increased thigh rotation; however l g ( - e q )F gained after the supporting leg contacted the floor also contributes to the increase It was thoughtt that the increase in the l g ( - e q) F value was primarily caused by the impact force (F brake ) of the landing of the supporting leg (-66 s) being transferred to the pelvis
222 Hiroki Ozaki et al / Procedia Engineering 34 ( 22 ) 28 223 hen the acceleration of the thigh increased by l g ( - e q )F his method of acceleration that sudden stop is called the braking effect in this study uses a (a) (b) moment(nm) 2 5 e t l g (- e q ) F 5 ( l - l g ) (-e q ) F 2 5286428642 5 2 25 ime(s) - e t2 2 Fig 3 (a) Braking force of hip joint; (b) components of rotation torque in Link 53 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 maximumm swing velocity he internal force F 2 and x (whichh is the velocity vector of x ) can express F 2 = [ F q2 Ft 2 Fl 2] and x = [ x q2 x t 2 x l 2] Using t this equation E 2 can be described as E 2 = F 2 x + 22 = Ft 2 xt 2 + Fl 2xl 2 + t 2 l P 2 P F 2 2 + q x q 2 2 q2 P (4) (5) hen we observed that the main component of F 2 is the centrifugal force herefore by substituting Eq l 2 (7) in P F 2 ( e 2F2 )( l x 2 l l el 2x ) we obtain P F l x m x e l 2 2 l 2 = 2 l 2 m x e 2 l 2 l 2 { l 2P lin l 2 l l 2P L2a x - g}+ m x 2 l 2 l 2 l 2P L a { 2 lg 2e 2}+ x l 2el 2{ 2 e { lel}+ e x l 2el 2{ ( ( 2 lg 2el 2)} l 2 P L2c l 2P Lc l e l )}+ (6) Here l2 P l2 lin P l2 La P l2 Lc P L2 2a and l2 P L2c 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 l2 P lin and l2 P L the other hand l2 P Lc and l2 a show the low values compared with others On P L2c show higher values before impact Furthermore F2 x is the inner product of F 2 and x herefore when the subject places his knee at an angle of 9 degrees the energy from the thigh could be effectively transferred to the shank (Fig 4) In the experiment the subject kept his knee
Hiroki Ozaki et al / Procedia Engineering 34 ( 22 ) 28 223 223 angle close to 9 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 5 power(nm/s) 4 3 2 l2 P Lc l2 P La 2 l2 2 P L2c l2 P lin 286428642 2 ime(s) Fig 3 Sources of power for Link2 Fig 4 he 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 maximumm swing velocity of the leg he 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 thee 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 [] Zajac FE Neptune RR Kautz SA 22 Biomechanics and muscle coordinationn of human walking part I: introduction to concepts power transfer dynamics and simulations Gait & Posture Vol 6 No 3: 25 232 [2] Nunome H Asai Ikegami Y Sakurai S 22 hree-dimensional kinetic analysis of side-foot and instep soccer kicks Med Csi Sports Exerc 34 (2): 228 236 [3] Winter A D 99 Biomechanics and motor control of human movement Second edition John Wileyy & Sons New York pp 4 43 [4]Yu B Gabriel D Noble L An K N 999 Estimate of the Opitimum Cutoff Frequency for the Butterworth Low-Pass Digital Filter JOURNAL OF APPLIED BIOMECHANICS 5: 38 329 [5] Ohta K Ohgi Y Shibuya K 2 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)