impedance control for a golf swing robot to emulate different-arm

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C. C. Chen

Y. Inoue

K. Shibata

Department of Intelligent Mechanical SystemsEngineering,

Kochi University of Technology,Tosayamada-cho, Kochi-prefecture, 782-8502

Japan

Impedance Control for a GolfSwing Robot to EmulateDifferent-Arm-Mass GolfersVarious golfers can play different golf swing motions even if they hold the same golf club.This phenomenon casts light on the significance of the dynamic interaction between thegolfer’s arm and golf club. The dynamic interaction results in different swing motions,even if the robot has the same input torque of the shoulder joint as that of a golfer.Unfortunately, such influence has not been considered in the conventional control of agolf swing robot. An impedance control method is proposed for a golf swing robot toemulate different-arm-mass golfers in consideration of the dynamic interaction betweenhuman arm and golf club. Based on the Euler–Lagrange principle and assumed modestechnique, a mathematical model of golf swing considering the shaft bending flexibility isestablished to simulate the swing motions of different-arm-mass golfers. The impedancecontrol method is implemented to a prototype of golf swing robot composed of oneactuated joint and one passive joint. The comparison of the swing motions of the robotand different-arm-mass golfers is made and the results show that the proposed golf swingrobot with the impedance control method can emulate different-arm-mass golfers.�DOI: 10.1115/1.2837313�

Keywords: impedance control, mechanical impedance, golf swing robot, swing motion

IntroductionA large amount of research has been devoted to improve golf-

rs’ swing skills and golf club performance for decades. Amonghese studies, golf swing robots have formed a large body of lit-rature �1–5�. In their work, professional golfers’ swing motionsere expected to be emulated by robots and the evaluation of golf

lub performance was replaced by robots instead of golfers.Though much progress has been achieved in this area, there still

emains a long-standing challenge for a golf swing robot to accu-ately emulate the fast swing motions of professional golfers. Itas been noticed that conventional golf swing robots on the mar-et are usually controlled by the swing trajectory functions ofoints or of the club head directly measured from professionalolfers’ swings. The swing motions of these robots, unfortunately,re not completely the same as those of the advanced golfers, inhat they do not involve the dynamic interactions featured by dif-erent characteristics of human arms and golf clubs. Hunt and

iens �1� developed a parallel mechanism robotic testing machinehat captures the human golfer dynamics during golf club testing.uzuki and Inooka �2� proposed a new golf swing robot modelonsisting of one actuated joint and one passive joint. In theirodel, the robot like professional golfers was able to utilize the

nterference forces resulting from the dynamic features of indi-idual golf clubs on the arms, and the resulting optimal controlorques of the shoulder joint were obtained. Ming and Kajitani �3�ave a new motion planning method for this type of robot, usingifferent cost functions to gain the optimal control torques. Inheir work, the control input for the robot was the torque functionf the shoulder joint instead of the general ones such as the tra-ectory functions of joints or of club head. The change of theontrol input mainly results from the special dynamical character-stics of this new type of robot: The swing motion of the wrist

Contributed by the Dynamic Systems Division of ASME for publication in theOURNAL OF DYNAMIC, SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedune 8, 2006; final manuscript received July 20, 2007; published online February 29,008. Review conducted by Ranjan Mukherjee. Paper presented at The Engineering
f Sport 2006, The Third Asian Conference on Multibody Dynamics 2006.
ournal of Dynamic Systems, Measurement, and ControlCopyright © 20

aded 12 Jul 2010 to 131.183.72.12. Redistribution subject to ASME

joint is generated by the dynamic coupling drive of the shoulderjoint. This point was specifically explained in the work of Mingand Kajitani �3�. In their research, however, the difference be-tween the golfer’s arm and the robot’s arm in mass �or the mo-ment of inertia of the arm� was not considered. Therefore, if theoptimal control torque from their work is applied to other golferswho own different-mass arms, various swing motions would begained. In other words, the robots proposed by them can onlyemulate one kind of golfers who have the same arm mass as thatof the robot. The limitation of the golf swing robots promotes usto investigate a new control method to make the robots emulatemore general golfers.

In our study, an impedance control method based on velocityinstruction is proposed for a golf swing robot to emulate different-arm-mass golfers. A model of a golfer’s swing considering theshaft bending is given by using the Euler–Lagrange principle andassumed modes method. A prototype of golf swing robot with oneactuated joint and one passive joint is developed using the imped-ance control method. The comparison of swing motion is carriedout between golfers and the golf swing robot. The results demon-strate that our golf swing robot can simulate different-arm-massgolfers.

2 Dynamic Modeling of Golf SwingIn order to demonstrate the validity of the impedance control

method, the swing motions of different-arm-mass golfers shouldbe obtained so that the motions can be compared with those fromthe proposed golf swing robot using the impedance controlmethod. Therefore, a dynamic model has been developed to emu-late the swing motions of different-arm-mass golfers under thegiven input torques of the shoulder joint. The detailed derivationof the model is shown as follows.

2.1 Dynamic Model and Assumptions. The dynamic modelof golf swing is shown in Fig. 1. Here, the rotations of the armand golf club are assumed to occur in one plane during the down-swing and follow-through, and this plane is inclined with an angle� to the horizontal plane. The assumption of the planar movement
of the arm and golf club is well supported in the early work of
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ochran and Stobbs �6� and Jorgensen �7�. Therefore, the gravitycceleration vector in the swing plane can be expressed as g= �0g0�T, g0=g sin �. The arm and handgrip are considered as the

igid rods and the club head as a tip mass. The shaft is treated asn Euler–Bernoulli beam, in which the elastic modulus, density,nertia, and cross-sectional area are constant along the beamength. Two coordinate systems in the swing plane are introducedo describe the dynamics of the golf swing: a fixed referencerame XY and a rotational reference frame xy attached to the endf the handgrip where its x axis is along the undeformed configu-ation of the beam. The torque �1 is applied at the shoulder joint Oo drive the swing. The torque �2 is employed at the wrist joint so hold the golf club. Since the center of gravity of the club heads regarded as on the central axis of the shaft, the twisting of thehaft is neglected and the bending flexibility of the shaft in thewing plane is only considered. The bending displacement of thehaft is expressed as y�x , t� in the rotational coordinate system-xy. The rotational angle of the arm with the X axis is � and theotational angle of the club grip with the x axis is �.

2.2 Dynamic Equations of Motion. The Euler–Lagrange ap-roach is used to derive the dynamic equations of motion for aolfer’s swing. In the coordinate system O-XY, r2 and rR areefined as the position vectors of the centers of gravity of theandgrip and the club head, respectively; rp is the position vectorf a point p on the shaft; yp is the bending displacement of a pointon the shaft with respect to the coordinate system o-xy; J1 is theoment of inertia of the arm about the shoulder joint O; J2 and JR

re the moments of inertia of the handgrip and the club head,espectively; m1, m2, m3, and mR are the masses of the arm, theandgrip, the shaft, and the club head, respectively; a1, a2, and a3re the lengths of the arm, the handgrip, and the shaft, respec-ively; R is the radius of the club head; � is the mass per unitength of the shaft; E is Young’s modulus of the shaft material;nd I is the area moment of inertia of the shaft.

The following operators are defined:

��

�t�� and ��

�

�x��

he total kinetic energy of the system is given by

T = T1 + T2 + T3 + T4 �1�

here T1, T2, T3, and T4 are the kinetic energies associated withhe arm, the handgrip, the shaft, and the club head, respectively.hey are

T1 =1

2J1�2 �2�

T2 =1

J2�� + ��2 +1

m2r2Tr2 �3�

ig. 1 Dynamic model of golf swing. The Z and z axes areerpendicular to the swing plane.

2 2

21007-2 / Vol. 130, MARCH 2008


T3 =1

2��

0

a3

rpTrpdx �4�

T4 =1

2mRrR

TrR +1

2JR�� + � + y��x=a3

�2 �5�

The total potential energy of this system can be written as

U = U1 + U2 + U3 + U4 �6�

where U1, U2, and U4 are the potential energies resulting from thegravitational forces of the arm, the handgrip, and the club head,respectively, and U3 is the potential energy of the shaft.

U1 = − m1gTr1 �7�

U2 = − m2gTr2 �8�

U3 = U31 + U32 �9�

where U31= 12EI�0

a3�yp��2dx and U32=−�gT�0

a3rpdx. U31 and U32are the potential energies due to the elastic deformation and gravi-tational force of the shaft, respectively.

U4 = − mRgTrR �10�

Therefore, the Lagrangian L of the system can be obtained as

L = T − U �11�

The internal structural damping in the golf shaft should also beconsidered. By using Rayleigh’s dissipation function, the dissipa-tion energy for the golf shaft is written as

ED = i=1

m1

2diqi

2 �12�

where di and qi are the damping coefficient and the mode ampli-tude associated with the ith mode of the shaft bending vibration,respectively.

According to the assumed modes technique of Theodore andGhosal �8�, a finite-dimensional model of the shaft bending dis-placement is written as

y�x,t� = i=1

m

�i�x�qi�t� �13�

where �i�x� and qi�t� are the ith assumed mode eigenfunction andith time-varying mode amplitude, respectively. Since the golfshaft is modeled as an Euler–Bernoulli beam with uniform densityand constant flexural rigidity �EI�, it satisfies the following partialdifferential equation:

EI�4y�x,t�

�x4 + ��2y�x,t�

�t2 = 0 �14�

We can obtain the general solution of Eq. �14�

qi�t� = exp�jwit� �15�

where wi is the ith natural angular frequency.Furthermore, �i�x� can be expressed as

�i�x� = C1i sin��ix� + C2i cos��ix� + C3i sinh��ix� + C4i cosh��ix��16�

where �i4=wi

2� /EI.We consider the golf club as a cantilever that has a tip mass; the

following four expressions associated with the boundary condi-tions can be obtained �9�:

�y�x=0 = 0 �17�

�y��x=0 = 0 �18�

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EI�y��x=a3 = − JR�y��x=a3 �19�

EI�y��x=a3 = mR�y�x=a3 �20�

rom these boundary conditions, the following results are given.

C1i = − C3i and C2i = − C4i �21�

A11 A12

A21 A22�C1i

C2i� = 0 �22�

he ith natural angular frequency wi can be obtained by solvinghe eigenvalue problem of the matrix equation �22� and the coef-cients C1i and C2i are chosen by normalizing the mode eigen-unctions �i�x� such that

��0

a3

�i2�x�dx = m3, i = 1,2, . . . ,m �23�

he impedance of the wrist joint is also considered in our simu-ation. The wrist impedance function is written as

� f = c� �24�

here c is the viscous damping constant of the wrist joint and � fs the wrist retarding torque due to the impedance of the wristoint.

Since professional golfers such as Jones �10� turned the wristoint freely and felt that the golf club freewheeled at the lattertage of the downswing, the value of c used here is relativelymall compared to that from Milne and Davis �11�. Here, it isssumed that the value of c is equivalent to that of the golf swingobot. On the basis of the Euler–Lagrange equation

�

�t� �L

�Qi −

�L

�Qi+

�ED

�Qi

= f i �25�

ith the Lagrangian L, the dissipation energy of the shaft ED, theeneralized coordinates Qi, and the corresponding generalizedorces f i, the dynamic equations of motion of the golf swing cane obtained. Since the amplitudes of the lower modes of the bend-ng vibration of the golf club are significantly larger than those ofhe higher ones, m is simplified to 2 in this paper. As a result, thequations of motion of the golf swing can be written as

B�� + h��,�� + K� + G�� + D� = � �26�

here

B = �B11 B12 B13 B14

B12 B22 B23 B24

B13 B23 B33 0

B14 B24 0 B44

�, � = ��

�

q1

q2

�, h = �h1

h2

h3

h4

� ,

K = �0 0 0 0

0 0 0 0

0 0 K33 K34

0 0 K34 K44

�, G = �G1

G2

G3

G4

� ,

D = �0 0 0 0

0 c 0 0

0 0 d1 0

0 0 0 d2

�, � = ��1

�2

0

0�

, K, and D are the inertia, stiffness, and damping matrices; h ishe vector of Coriolis and centrifugal forces; G is the gravityector and � is the input vector; q1 and q2 are the first and secondime-varying mode amplitudes of the shaft bending vibration, re-
pectively; c is the wrist viscous coefficient; and d1 and d2 are the
ournal of Dynamic Systems, Measurement, and Control


damping coefficients of the first and second modes of the bendingvibration of the shaft, respectively.

3 Impedance Control DesignThe dynamic equation of a mechanical system is always ex-

pressed as

Mx + Cx + Kx = F �27�

where F is the external force and M, C, and K are denoted as theinertia, viscosity, and stiffness, respectively. These parameterswere called mechanical impedance in the work of Hogan �12�. Inthis paper, the golfer’s arm as a mechanical system is investigated.Since it has been found that a golfer’s hands move in a circular arcabout the shoulder joint during the golf swing, the golfer’s arm isassumed to be a rigid body and the stretch reflex of the armmuscle is neglected. The fact that the swing motions obtainedfrom the numerical simulation using the rigid arm link�6,7,11,13–15� agree well with those from the swing photographsof professional golfers have assured the above assumption. There-fore, the dynamic equation of the golfer’s arm can be given as

Mx = F �28�where the viscous damping and stiffness of the golfer’s arm areignored and the moment of inertia of the arm about the shoulderjoint M is defined as the mechanical impedance. The virtual sys-tem representing the dynamic model of a golfer’s arm and therobot system expressing the dynamic model of a robot’s arm areshown in Figs. 2 and 3, respectively.

The equations of motion of the virtual and robot systems arewritten as Eqs. �29� and �30�.

J1h�h = �1h + FhL1h + fgh��h� + Nh �29�

J1r�r = �1r + FrL1r + fgr��r� + Nr �30�

where the subscripts h and r denote the golfer and robot, respec-tively; fg��=−Fgl1 cos �; Fg is the gravitational force of thearm; F is the reaction force from the club to arm, and the direction

Fig. 2 Virtual system

Fig. 3 Robot system

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s perpendicular to the arm; and N is the reaction torque from thelub to the arm. Here, we assume L1h=L1r.

In our control method, the dynamic parameters J1h and J1r inqs. �29� and �30� are defined as the mechanical impedance. With

he various arm masses for the golfer and robot, the mechanicalmpedances J1h and J1r are varied. Consequently, the swing mo-ion of the robot is not the same as that of the golfer, even if thenput torques of the shoulder joints are equal. In order to realizehe dynamic swing motion of the virtual system, the followingontrol algorithm is proposed for the robot.

According to the Euler method, angular acceleration of the arman approximate the following expressions:

�hn =

�hn − �h

n−1

�t, n = 1, . . . ,M �31�

Table 1 Parameter

Parameter

Mass of armMoment of inertia of arm about shoulder Joint OLength of armMass of handgripMoment of inertia of handgripLength of handgripMass per unit length of shaftLength of shaftArea moment of inertia of shaftYoung’s modulus of shaft materialMass of club headMoment of inertia of club headRadius of club headWrist viscous damping coefficientDamping coefficient of first mode of clubDamping coefficient of second mode of club

Fig. 4 Configuration of the control system

Fig. 5 Prototype of golf swing robot

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�rn =

�rn − �r

n−1

�t, n = 1, . . . ,M �32�

where �t is the sampling time, M is an integer, and �n and �n arethe angular velocity and acceleration in the nth sampling period.

Substituting Eq. �31� into Eq. �29�, and Eq. �32� into Eq. �30�,and after some manipulations, Eqs. �33� and �34� are obtained.

�hn = �h

n−1 +�t

J1h��1h

n−1 + Fhn−1L1h + fgh��h

n−1� + Nhn−1� �33�

�rn = �r

n−1 +�t

J1r��1r

n−1 + Frn−1L1r + fgr��r

n−1� + Nrn−1� �34�

In the impedance control method, the arm angular velocity ofthe golfer is regarded as the control input reference for the robot.As shown in Eq. �33�, the reaction force Fh

n−1 and reaction torqueNh

n−1 should be known in advance, if we expect to acquire thecontrol input reference �h

n. Since the PI control in the velocityloop for the robot is used to assure that the arm angular velocity ofthe robot is equivalent to that of the golfer, �h= �r, the motions ofthe same golf club used by the golfer and robot are the same.Therefore, the reaction force and reaction torque from the samegolf club to the arms of the golfer and robot are equal, that is,Fh

n−1=Frn−1 and Nh

n−1=Nrn−1. Note that the reaction force Fr

n−1 andreaction torque Nr

n−1 can be obtained by a six-dimensional forcesensor and a one-dimensional force sensor installed on the robot’sarm, respectively. Substituting the above results into Eq. �33�, thecontrol input reference �h

n for the golf swing robot can be calcu-lated from the following equation:

�hn = �h

n−1 +�t

J1h��1h

n−1 + Frn−1L1h + fgh��r

n−1� + Nrn−1� �35�

where the input torque of the shoulder joint for a golfer �1hn−1 can

be given by the operator of the robot.The whole configuration of the control system is described in

Fig. 4.

4 Simulation, Experiment, and ResultsA fourth-order Runge–Kutta integration method at intervals of

1.010−4 s was used to drive the golfer swing model and theswing motions of different-arm-mass golfers were obtained.

The experiment was implemented by a prototype of golf swingrobot composed of an actuated joint driven by a direct drive motor�NSK MEGATORQUE� and a passive joint with a mechanicalstopper. The stopper carries out the wrist cock action. A mechani-cal brake is used to stop the golf club after impact. An absolute

f the robot system

Symbol Value

m1 3.779 kgJ1 0.082 kg m2

a1 0.3 mm2 0.603 kgJ2 6.4010−4 kg m2

a2 0.108 m� 0.242 kg /ma3 0.5 mI 6.7510−11 m4

E 7.001010 N /m2

mR 0.06 kgJR 6.8910−6 kg m2

R 0.007 mc 0.033 N m s / radd1 0.269 N s /md2 0.490 N s /m

s o



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esolver �feedback signal of 51,200 pulse/rev� and an incrementalncoder �9000 pulse/rev� situated at the actuated and passiveoints are utilized to measure the arm and club rotational angles,espectively. A one-dimensional force sensor �KYOWA LCN-A-00N� and a six-dimensional force sensor �NITTA IFS-7M25A50-I40� are adopted to obtain the reaction torque Nr andeaction force Fr from the club to the arm, respectively. A flexibleolid beam made of aluminum is used to replace the shaft, and its clamped at the grip. The natural frequencies and vibration modehapes of the golf club are obtained by numerical calculation andxperimental modal analysis. The photograph of the robot systems shown in Fig. 5. Table 1 gives the parameters of the robotystem. The experimental control system is indicated schemati-ally in Fig. 6. A personal computer is used to implement thempedance control program and its sampling rate is 1 kHz. Bysing the velocity control mode of motor driver, the DD motor isontrolled by the velocity �voltage� reference from the D/A con-erter. Here, JR3 is a DSP-based signal receiver for the six-imensional force sensor.

The obtainment of the reaction torque Nr is shown in Fig. 7.ased on the figure, the reaction torque Nr can be calculated by

Nr = − Fnb �36�

here Fn is the force measured from the one-dimensional forceensor and b is the distance between the force contact point of theensor and the wrist joint of the robot.

It is noted that not only the force measured from the six-imensional force sensor is needed to calculate the reaction forcer but also the inertia force of the sensor f t caused by the arm

ig. 6 Schematic diagram of the experimental control system

Fig. 7 Obtainment of the reaction torque Nr

Fig. 8 Force analysis of the handgrip

ournal of Dynamic Systems, Measurement, and Control


angular acceleration �r should be considered because the sensor isregarded as a part of handgrip. The specific force analysis of thehandgrip is shown in Fig. 8.

According to Fig. 8, the following equations are obtained:

Fr = f t + fs �37�

f t = ms�rL1r �38�

Fr = − Fr �39�

where ms is the sensor mass; Fr is the reaction force from the armto the club; and fs is the force obtained from the sensor, and itsdirection is perpendicular to the arm, as shown in Fig. 9. The forcefs can be calculated by

fs = − TX sin�� + TY cos�� 40�

where TX and TY are the forces measured from the six-dimensional force sensor, and they are perpendicular to eachother.

Therefore, the reaction force Fr from the club to the arm isobtained as

Fr = − ms�rL1r + TX sin�� − TY cos�� 41�

In Eq. �41�, the arm angular acceleration �r should be known inorder to calculate the reaction force Fr. Here, a constant-coefficient Kalman filter is used to obtain the arm angular velocityto reject undesirable position measurement noise, and then �r isacquired by the filtered velocity using the difference method.

Lampsa �14� thought that a pause usually occurs at the momentwhen a golfer completes his backswing and is just about to beginhis downswing, indicating that the angular velocities of the armand club are equal to zero at the start of the downswing. There-fore, in this paper, it is assumed that the swing commences with

�=0, �=0 and also the bending of the shaft at the start of thedownswing is ignored. The posture values of the arm and club at

Fig. 9 Obtainment of the force fs whose direction is perpen-dicular to the arm

Fig. 10 Torque of shoulder joint for different-arm-mass golfers

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he start of the downswing are taken with �=120 deg, �=90 deg, which were regarded as the general initial configuration

or professional golfers in the work of Sprigings and Mackenzie15�. The simulation and experiment are terminated when the clubead hits the golf ball, and the ball is placed at the center of theidth of golfer’s stance �the same golf ball position is shown inigs. 7 and 8 of Ref. �16��, that is, the test result of up to impact

s all that is of concern. The gravity effect is neglected because ofhe horizontal positioning of the direct drive motor. The golfwing robot is defined as R, and three golfers, labeled by H1, H2,nd H3, own the same arm length as that of the robot but withifferent arm masses, 7 kg, 5 kg, and 3 kg, respectively. It haseen noted that there are many kinds of torque input functions ofhe shoulder joint applied in the previous research work of golfwing, and here these functions are referred as �1 employed to thehoulder Joint O. Jorgensen �7� thought that the shoulder inputorque was constant during the swing. Milne and Davis �11� used

ramp as the torque function and Suzuki and Inooka �2� set theorque function as a trapezoid. Here, the trapezoid-shaped torquesFig. 10� are employed at the shoulder joint for the golfers. Theorques are linearly increased with the rise time of 0.12 s and thenp to the corresponding maximum values, which are maintained.05 s, and finally are terminated at the time of 0.29 s. Here, theaximum torque values for H1, H2, and H3 are 25 N m, 20 N m,

nd 17 N m, respectively.Figure 11 illustrates a simulation comparison of the swing mo-

ions of different-arm-mass golfers. It should be noted that thewing motions of H1, H2, and H3 plotted in Fig. 11 are generatedy the model developed in Sec. 2. From Fig. 11, it is evident thathe arm rotational motions of these different-arm-mass golfers areot the same and the club rotational motions of H3 are differentrom those of H1 and H2. However, we note that the club rota-ional motions of H1 are almost the same as those of H2 even if

Fig. 11 Comparison of the swing motions of the different-angle, „c… arm angular velocity, and „d… club angular velocit

he input torques of shoulder joint for them are different. It clearly

21007-6 / Vol. 130, MARCH 2008


demonstrates that dissimilar torque profiles of shoulder joint cangenerate very similar looking club rotational motions fordifferent-arm-mass golfers. Figures 12–14 show the results of theerrors of the swing motions by the proposed golf swing robot toemulate different-arm-mass golfers. From Figs. 12�a�, 12�c�,13�a�, 13�c�, 14�a�, and 14�c�, it can be seen that the motion errorsof the arm of the robot are very small. As for the swing motions ofthe club, it is clear that no substantial differences are found be-tween the robot and different-arm-mass golfers from Figs. 12�b�,12�d�, 13�b�, 13�d�, 14�b�, and 14�d�. It is noteworthy to mentionthat the club swing motion differences between the robot andgolfers are clearly larger than the arm swing motion differences.We note that there are two forces retarding the club during thedownswing: one is the wrist retarding force due to the mechanicalimpedance of the wrist joint; the other is the air retarding forcedue to the high-speed rotational motion of the club. As for obtain-ing the club swing motions of different-arm-mass golfers in simu-lation, the air retarding force of the club was neglected due to thecomplex dynamical modeling and practical measurement diffi-culty for this air retarding force despite that it works in the ex-periment for the robot. Therefore, the club swing motion differ-ences between the robot and golfers become relatively large ascompared with the arm swing motion differences.

5 ConclusionsAn impedance control method based on the velocity instruction

was proposed to control a golf swing robot. By this method, theinterference forces from the club to the arm were considered, andthe influence of different human arms on the golf swing motionwas also involved.

Simulation of the swing motions of the different-arm-massgolfers was carried out by a mathematical model of golf swing

-mass golfers: „a… arm rotational angle, „b… club rotational
army
considering the shaft bending flexibility. This model was derived



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Fig. 12 The errors in the swing motions of the 7 kg arm-mass golfer „H1…: „a… error in arm rotationalangle, „b… error in club rotational angle, „c… error in arm angular velocity, and „d… error in club angular
velocity
Fig. 13 The errors in the swing motions of the 5 kg arm-mass golfer „H2…: „a… error in arm rotationalangle, „b… error in club rotational angle, „c… error in arm angular velocity, and „d… error in club angular
velocity
ournal of Dynamic Systems, Measurement, and Control MARCH 2008, Vol. 130 / 021007-7

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y the Euler–Lagrange principle and assumed modes method. Arototype of golf swing robot consisting of one actuated joint andne passive joint was developed to verify the impedance controlethod. The simulational and experimental results showed that

he robot with the impedance control method could emulate thewing motions of different-arm-mass golfers.

eferences�1� Hunt, E. A., and Wiens, G. J., 2003, “Design and Analysis of a Parallel Mecha-

nism for the Golf Industry,” Proceedings of Tenth International Congress onSound and Vibration, Stockholm, Sweden, Jul. 7–10, p. 8.

�2� Suzuki, S., and Inooka, H., 1998, “A New Golf-Swing Robot Model UtilizingShaft Elasticity,” J. Sound Vib., 217�1�, pp. 17–31.

�3� Ming, A., and Kajitani, M., 2000, “Motion Planning for a New Golf SwingRobot,” Proceedings of 2000 IEEE International Conference on System, Manand Cybernetics, Nashville, pp. 3282–3287.

�4� Ming, A., and Kajitani, M., 2003, “A New Golf Swing Robot to SimulateHuman Skill-Accuracy Improvement of Swing Motion by Learning Control,”Mechatronics, 13�8–9�, pp. 809–823.

�5� Hoshino, Y., Kobayashi, Y., and Yamada, G., 2005, “Vibration Control Usinga State Observer That Considers Disturbance of a Golf Swing Robot,” JSME

Fig. 14 The errors in the swing motions of the 3angle, „b… error in club rotational angle, „c… error ivelocity

Int. J., Ser. C, 48�1�, pp. 60–69.

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�6� Cochran, A., and Stobbs, J., 1999, Search for the Perfect Swing, TriumphBooks, Chicago.

�7� Jorgensen, T., 1994, The Physics of Golf, AIP Press, New York.�8� Theodore, R. J., and Ghosal, A., 1995, “Comparison of the Assumed Modes

and Finite Element Models for Flexible Multi-Link Manipulators,” Int. J. Ro-bot. Res., 14�2�, pp. 91–111.

�9� De Luca, A., and Siciliano, B., 1991, “Closed-Form Dynamic Model of PlanarMultilink Lightweight Robots,” IEEE Trans. Syst. Man Cybern., 21�4�, pp.826–839.

�10� Jones, R., 1966, Bobby Jones on Golf, Doubleday, New York.�11� Milne, R. D., and Davis, J. P., 1992, “The Role of the Shaft in the Golf

Swing,” J. Biomech., 25�9�, pp. 975–983.�12� Hogan, N., 1984, “Adaptive Control of Mechanical Impedance by Coactiva-

tion of Antagonistmuscles,” IEEE Trans. Autom. Control, AC-29�8�, pp. 681–690.

�13� Williams, D., 1967, “The Dynamics of the Golf Swing,” Q. J. Mech. Appl.Math., 20�2�, pp. 247–264.

�14� Lampsa, M. A., 1975, “Maximizing Distance of the Golf Drive: An OptimalControl Study,” ASME J. Dyn. Syst., Meas., Control, 97�4�, pp. 362–367.

�15� Sprigings, E. J., and Mackenzie, S. J., 2002, “Examining the Delayed Releasein the Golf Swing Using Computer Simulation,” Sports Eng., 5�1�, pp. 23–32.

�16� Jorgensen, T., 1970, “On the Dynamics of the Swing of a Golf Club,” Am. J.Phys., 38�5�, pp. 644–651.

arm-mass golfer „H3…: „a… error in arm rotationalrm angular velocity, and „d… error in club angular

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impedance control for a golf swing robot to emulate different-arm

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