where A is the anatomical constrain

where A is the anatomical constraint coefficient matrix of the generalised velocity vector.
Substituting equations (2) and (3) after differentiating with respect to time into equation (1), the equation of
motion for the system can be obtained, which is written by the following matrix form expression as follows:
= + ++ e ext act v V A F A T AV A G F Ta G (4)
where AFe, ATa, and AG are the coefficient matrices of the external force vector Fext, the active joint torque vector
Tact, and the gravitational acceleration vector G, and Av V denotes the motion dependent term.
Integrating equation (4) with respect to time, one can obtain dynamic equation relating the generalized velocity
vector to the terms such as external force, active joint torques, motion dependent and gravitational terms.
Furthermore, by extracting evaluation variables using a selecting matrix, one can quantify contribution of the terms
to the generation of target variables.
2.3. Modelling of elastic characteristics of racket shaft
The elastic characteristics of the racket shaft were represented by using the restoring torques exerted about the
individual virtual joints. The restoring torques were calculated as the product of the angular displacements of
divided shaft segments and the stiffness of the rotational spring settled at the virtual joints. The joint torque term in
equation (6) can be divided into the human active joint torque term and the racket shaft restoring torque term as
follows:
AT A T AT Ta act = + TH H TS S (5)
where ATH and ATS are the coefficient matrices of the active joint torque vector TH and of the shaft restoring torque
vector TS.
The shaft restoring torques can be expressed as follows:
T KX S S = (6)
where KS is the coefficient matrix including stiffness matrix for shaft joints, and X is the vector calculated by
integration of the generalized velocity vector V.
Substituting equations (5) and (6) into equation (4) yields the equation of motion for the racket-human system
expressed as follows:
Sekiya Koike and Tomohiro Hashiguchi / Procedia Engineering 72 ( 2014 ) 496 – 501 499
= + ++ + v e ext V A T A K X AV A G A F TH H TS S G F (7)
where A T TH H represents the accelerations of individual segments caused by joint torques, the second term,
A KX TS S represents the accelerations caused by shaft deformation, A Vv represents the accelerations caused by the
motion-dependent torques, e.g. centrifugal force, Coriolis force, and gyro moment, A GG represents the
accelerations caused by gravity, and A FFe ext represents the accelerations caused by shoulder joint forces. By
integrating equation (7) with respect to time and by extracting racket face velocity from the generalized velocity
vector V, one can obtain the contribution of individual terms to the generation of the racket head speed vspd,fc as
follows:
T T TT T
spd,fc fc fc fc fc fc e ext TH H TS S G F = + ++ + e S A T e S A K X e S AV e S A G e S A F vv v ³ ³ ³³ ³ v v v v dt dt dt dt dt (8)
where S denotes a selective matrix, and ev,fc is the unit vector of racket head velocity vector. The five terms on the
right side of equation (8) represent the active joint torque term, shaft restoring torque term, motion dependent term,
gravitational term, and external force term.
2.4. Modelling of racket shaft deformation
The shaft deformation was described as the angular displacement at individual virtual joints. The deformation
curve of racket shaft was approximated by using a 2nd order polynomial function fitted from measured coordinate
data obtained under the smash and drive shot conditions in order to calculate the angular displacements of each
virtual joint. The coordinate data were captured with a motion capture system (VICON-MX, VICON Motion
Systems Inc.). Marker position data were recorded at 500Hz. The translational velocity of each shaft segment’s
centre of gravity (CG) and the angular velocity of each segment were calculated by using the approximate function.
2.5. Converting algorithm of motion dependent term into other terms
The previous studies regarding to the speed generation mechanism in the baseball pitching motion have shown
that the motion dependent term play a crucial role to generate the hand speed prior to the ball release. Since the
racket head is accelerated to about 50 m·s-1 before the impact in order to obtain large shuttlecock speed in the
badminton smash motion, the motion dependent term must be a great contributor to the generation of racket head
speed in the smash motion. Therefore, we convert it into the terms such as active joint torque term, shaft restoring
torque term, gravitational term and external force term.
The equation of motion for system was discretized as follows:
() () () () V k k kk = + ȁV V A V (9a)
ȁV = A F A T A T AG F TH H TS S G e ext + ++ (9b)
The generalised acceleration vector was expressed by difference approximation shown as
( ) V V ( ) () 1
V
+ − = Δ
 k k
k
t
(10)
Combining equations (9a) and (10) yields a recurrence formula for the generalised velocity vector V as follows:
V ( ) () () () () k t k k k, k t k + = + = +Δ 1 ǻ ȁ Ȍ VV V V V Ȍ E A ( ) (11)
Equations (9b) and (11) provide us the information about the contribution of input terms (e.g., individual axial
torques at racket-side upper limb joints, external force exerted on the racket-side shoulder joint, and shaft restoring
torque) except MDT to the generation of speed of racket face segment.