Patent Publication Number: US-9403077-B2

Title: Golf swing analyzing apparatus and method of analyzing golf swing

Description:
This application is based upon and claims the benefit of priority from the prior Japanese Patent Application No. 2012-223327 filed on Oct. 5, 2012, and the prior Japanese Patent Application No. 2012-223326 filed on Oct. 5, 2012, the entire contents of which are incorporated herein by reference. 
     BACKGROUND 
     The present invention relates to a golf swing analyzing apparatus and a method of analyzing golf swings. 
     A golf swing analyzing apparatus is generally known as disclosed in Japanese Patent Application Publication No. 2010-11926, for example. The golf swing analyzing apparatus utilizes an optical motion capture system for capturing an image of a swing of a golfer. Markers are fixed to specific positions of the golfer and/or a golf club for the capture of the image of the swing. The movement of the markers is recorded as an image for determining the moving paths of the specific positions. In addition, a golf swing analyzing apparatus utilizing an acceleration sensor is also generally known as disclosed in Japanese Patent Application Publication No. 11-169499, for example. An acceleration sensor is attached to the golf club. The form of the golf swing is analyzed based on the acceleration measured by the acceleration sensor. 
     The golf swing analyzing apparatus utilizing an optical motion capture system as disclosed in Japanese Patent Application Publication No. 2010-11926 requires tremendously large equipment so that it is hard to realize the measurement in the field. A golf swing analysis utilizing an inertial sensor such as an acceleration sensor is recently proposed as disclosed in Japanese Patent Application Publication No. 11-169499. However, the golf swing analysis utilizing an acceleration sensor cannot usefully present a relative angle between the arm and the golf club to users. 
     SUMMARY 
     An aspect of the invention relates to a golf swing analyzing apparatus, comprising: an arithmetic section operating to process the output of a first inertial sensor and the output of a second inertial sensor to calculate a relative angle between a forearm of a golfer and a golf club, the first inertial sensor being attached to a portion of the upper body of the golfer, the second inertial sensor being attached to the golf club. 
     Another aspect of the invention relates to a method of analyzing golf swings, comprising: processing the output of a first inertial sensor and the output of a second inertial sensor to calculate a relative angle between a forearm of a golfer and a golf club, the first inertial sensor being attached to a portion of the upper body of the golfer, the second inertial sensor being attached to the golf club. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic view illustrating the structure of a golf swing analyzing apparatus according to one exemplary embodiment of the invention. 
         FIG. 2  is a schematic view illustrating the relationship between a three-dimensional double pendulum model and a golfer grasping a golf club. 
         FIG. 3  is a block diagram schematically illustrating the structure of an arithmetic unit. 
         FIG. 4  is a block diagram schematically illustrating a part of the arithmetic unit. 
         FIG. 5  is a graph illustrating the result of the analysis on a golf swing of a golf teaching professional, specifically the change of the relative angle between the forearm and the golf club along the elapse of time. 
         FIG. 6  is a graph illustrating the result of the analysis on a golf swing of an amateur golfer, specifically the change of the relative angle between the forearm and the golf club along the elapse of time. 
         FIG. 7  is a graph illustrating the result of the analysis on a golf swing of a golf teaching professional, specifically the change of the total energy change rate along the elapse of time. 
         FIG. 8  is a schematic view illustrating the attitude of the golf teaching professional and the golf club at the zero crossing. 
         FIG. 9  is a graph illustrating the result of the analysis on a golf swing of an amateur golfer, specifically the change of the total energy change rate along the elapse of time. 
         FIG. 10  is a schematic view illustrating the attitude of the amateur golfer and the golf club at the zero crossing. 
     
    
    
     DESCRIPTION OF EXEMPLARY EMBODIMENTS 
     According to at least one aspect of the invention, a golf swing analyzing apparatus and a method of analyzing golf swings are provided to usefully present a relative angle between an arm and a golf club. 
     (a) An aspect of the invention relates to a golf swing analyzing apparatus comprising an arithmetic section operating to process the output of a first inertial sensor and the output of a second inertial sensor to calculate the relative angle between a forearm of a golfer and the golf club, the first inertial sensor being attached to a portion of the upper body of the golfer, the second inertial sensor being attached to the golf club. 
     It is preferable to fix the relative angle between the forearm and the golf club at an initial stage from the top in a golf swing. If a golfer is capable of loosening the wrist to allow a natural rotation of the golf club relative to the forearm, the golfer is supposed to enjoy an increased head speed. The golf swing analyzing apparatus is configured to present the relative angle between the forearm and the golf club to a user. The observation of the relative angle between the forearm and the golf club enables discovery of the form of golf swing which results in an efficient transfer of the energy to the golf club. Indices are in this manner provided for the form of golf swing. For example, repetition of changing the form in combination with the subsequent observation realizes a superior improvement effected on the form of golf swing through try and error. 
     (b) The golf swing analyzing apparatus may operate to utilize a three-dimensional double pendulum model to calculate the relative angle, the portion of the upper body of the golfer forming a first link of the three-dimensional double pendulum model, the golf club forming a second link of the three-dimensional double pendulum model. A golf swing is in this manner fitted into a model. The three-dimensional double pendulum model kinetically represents the movement of a golf swing with a relatively high accuracy. The golf swing is in this manner effectively analyzed. 
     (c) A fulcrum of the first link may be located at the center of a line connecting the shoulders of the golfer, the joint between the first link and the second link being located on the grip of the golf club. A golf swing is thus analyzed with a higher accuracy. 
     (d) Each of the first inertial sensor and the second inertial sensor may include an acceleration sensor and a gyro sensor. The acceleration sensor and the gyro sensor enable a precise detection of the acceleration and the angular velocity for the calculation of the relative angle. 
     (e) The golf swing analyzing apparatus may operate to process the output of the first inertial sensor and the output of the second inertial sensor to calculate a total energy change rate for the portion of the upper body of the golfer. The derivation of the total energy change rate contributes to discovery of the form of golf swing which results in an efficient transfer of the energy to the golf club. Indices are in this manner provided for the form of golf swing. 
     (f) The golf swing analyzing apparatus may include an energy change rate inversion detecting section configured to detect the inversion of the positive/negative signs of the total energy change rate for the portion of the upper body of the golfer. The detection of the inversion contributes to discovery of the form of golf swing which results in an efficient transfer of the energy to the golf club. Indices are in this manner provided for the form of golf swing. For example, repetition of changing the form in combination with the subsequent observation realizes a superior improvement effected on the form of golf swing through try and error. 
     In particular, relating the timing of the zero crossing to the change of the relative angle contributes to a further improvement of the form of golf swing. 
     (g) The golf swing analyzing apparatus may include an image data generating section generating an image data for displaying the change of the relative angle. The change of the relative angle serves to provide an index for improvement of the form of golf swing. 
     (h) The image data generating section may generate an image data for displaying an image of information on the timing of the inversion superimposed on an image of the relative angle. The displayed image serves to provide an index for improvement of the form of golf swing. 
     (i) Another aspect of the invention relates to a method of analyzing golf swings, comprising: processing the output of a first inertial sensor and the output of a second inertial sensor to calculate the relative angle between a forearm of a golfer and the golf club, the first inertial sensor being attached to a portion of the upper body of the golfer, the second inertial sensor being attached to the golf club. 
     It is preferable to fix the relative angle between the forearm and the golf club at an initial stage from the top in a golf swing. If a golfer is capable of loosening the wrist to allow a natural rotation of the golf club relative to the forearm, the golfer is supposed to enjoy an increased head speed. The golf swing analyzing apparatus is configured to present the relative angle between the forearm and the golf club to a user. The observation of the relative angle between the forearm and the golf club enables discovery of the form of golf swing which results in an efficient transfer of the energy to the golf club. Indices are in this manner provided for the form of golf swing. For example, repetition of changing the form in combination with the subsequent observation realizes a superior improvement effected on the form of golf swing through try and error. 
     (j) Still another aspect of the invention relates to a method of displaying an analysis on a golf swing, comprising: displaying an image including the change of the relative angle between a forearm of a golfer and a golf club, and information on a timing of the inversion of positive/negative signs of the total energy change rate for a portion of the upper body of the golfer. 
     A detailed description will be made below on an exemplary embodiment of the invention referring to the attached drawings. It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are not restrictive of the invention as claimed, and all elements of the exemplary embodiment may not be indispensable to a solution of the invention. 
     (1) Structure of Golf Swing Analyzing Apparatus 
       FIG. 1  schematically illustrates the structure of a golf swing analyzing apparatus  11  according to one embodiment of the invention. The golf swing analyzing apparatus  11  includes a first inertial sensor  12  and a second inertial sensor  13 , for example. The first and second inertial sensors  12 ,  13  individually include an acceleration sensor and a gyro sensor assembled therein. The acceleration sensor is configured to detect the acceleration in the directions of three axes of an orthogonal coordinate system. The gyro sensor is configured to detect the angular velocity around each of three axes of an orthogonal coordinate system. The first and second inertial sensors  12 ,  13  output detection signals. The detection signals specify the magnitude of the acceleration and the angular velocity for the individual axes of an orthogonal coordinate system. The acceleration sensors and the gyro sensors are expected to detect the acceleration and the angular velocity with a relatively high accuracy. The first inertial sensor  12  is attached to an arm  15  or a hand, for example, to the left arm for the right-handed golfer. Here, although the first inertial sensor  12  is attached to a forearm of a golfer, the first inertial sensor  12  may be attached to a brachium of a golfer. The second inertial sensor  13  is attached to a golf club  14 . Preferably, the second inertial sensor  13  is attached to the grip or shaft of the golf club  14 . The first and second sensors  12 ,  13  may respectively be fixed to the arm  15  and the golf club  14  in an immobilized manner. Here, a detection axis of the second inertial sensor  13  is set in parallel with the longitudinal axis of the golf club  14 . It should be noted that the first inertial sensor  12  may be mounted to the upper body of a golfer, especially to the shoulders, although the first inertial sensor  12  is attached to the arm  15  in this embodiment. 
     The golf swing analyzing apparatus  11  includes an arithmetic unit  16 . The first and second inertial sensors  12 ,  13  are connected to the arithmetic unit  16 . An interface circuit  17  is connected to the arithmetic unit  16  for the connection of the first and second inertial sensors  12 ,  13 . The interface circuit  17  may be connected to the first and second inertial sensors  12 ,  13  with or without wires. The arithmetic unit  16  receives the detection signals from the first and second inertial sensors  12 ,  13 . 
     A storage unit  18  is connected to the arithmetic unit  16 . For example, a golf swing analyzing software program  19  and related data are stored in the storage unit  18 . The arithmetic unit  16  executes the golf swing analyzing software program  19  to realize a method of analyzing golf swings. The storage unit  18  may include a dynamic random access memory (DRAM), a large capacity storage unit, a non-volatile memory, and the like. For example, the DRAM temporarily holds the golf swing analyzing software program  19  for the realization of the method of analyzing golf swings. The golf swing analyzing software program  19  and data are stored in the large capacity storage unit such as a hard disk drive unit (HDD). A relatively small program such as a basic input/output system (BIOS) and relatively small data may be stored in the non-volatile memory. 
     An image processing circuit  21  is connected to the arithmetic unit  16 . The arithmetic unit  16  supplies image data to the image processing circuit  21 . A display unit  22  is connected to the image processing circuit  21 . An interface circuit, not depicted, is connected to the image processing circuit  21  for the connection of the display unit  22 . The image processing circuit  21  supplies imaging signals to the display unit  22  in accordance with the supplied image data. The imaging signals determine images displayed on the screen of the display unit  22 . A liquid crystal display or any other type of a flat panel display may be utilized as the display unit  22 . Here, the arithmetic unit  16 , the storage unit  18  and the image processing circuit  21  are provided in the form of a computer apparatus, for example. 
     An input device  23  is connected to the arithmetic unit  16 . The input device  23  at least includes alphabetical keypads and numeric keypads. The input device  23  is utilized to input alphabetical information and numeric information to the arithmetic unit  16 . The input device  23  may be a keyboard, for example. 
     (2) Three-Dimensional Double Pendulum Model 
     The arithmetic unit  16  defines an imaginary space. The imaginary space is formed as a three-dimensional space. As depicted in  FIG. 2 , the three-dimensional space has an absolute reference coordinate system Σxyz. A three-dimensional double pendulum model  31  is constructed in the three-dimensional space in accordance with the absolute reference coordinate system Σxyz. The three-dimensional double pendulum model  31  includes a first link  32  and a second link  33 . The end of the first link  32  is coupled to a fulcrum  34  (coordinate x 0 ). The first link  32  thus acts as a spherical pendulum around the fulcrum  34 . The fulcrum  34  may move. The end of the second link  33  is coupled to the other end of the first link  32  at a joint  35  (coordinate x 1 ) functioning as a ball joint. The second link  33  thus acts as a spherical pendulum around the joint  34  relative to the first link  32 . It is required to identify the mass m 1 , m 2  of the first and second links  32 ,  33 , the inertia tensor J 1  of the first link  32  around the fulcrum  34 , the inertia tensor J 2  of the second link  33  around the joint  35  in the three-dimensional double pendulum model. Here, the absolute reference coordinate system Σxyz serves to locate the centroid  36  of the first link  32  at the coordinate x g1 , the centroid  37  of the second link  33  at the coordinate x g2 , and the club head  38  at the coordinate x h2 . 
     The three-dimensional double pendulum model  31  corresponds to a representation of a golfer and the golf club  14 . The fulcrum  34  of the first link  32  corresponds to the central position between the shoulders in the upper body of the golfer. The joint  35  represents the grip. The second link  33  represents the golf club  14 . The first inertial sensor  12  is fixed to the arm  15  of the golfer. The central position between the shoulders can be fixed relative to the first inertial sensor  12 . The absolute reference coordinate system Σxyz serves to locate the first inertial sensor  12  at the coordinate x s1 . The second inertial sensor  13  is fixed to the second link  33 . The absolute reference coordinate system Σxyz serves to locate the second inertial sensor  13  at the coordinate x s2 . The first inertial sensor  12  and the second inertial sensor  13  individually output acceleration signals and angular velocity signals. The acceleration signals from the first inertial sensor  12  and the second inertial sensor  13  respectively specify the acceleration including the effect of the gravity g as follows:
 
( {umlaut over (x)}   s1   −g ),( {umlaut over (x)}   s2   −g )  [Mathematical Expression 1]
 
     The angular velocity signals from the first inertial sensor  12  and the second inertial sensor  13  respectively specify the angular velocity ω 1 , ω 2 . 
     The arithmetic unit  16  fixes a local coordinate system Σ s1  to the first inertial sensor  12 . The local coordinate system Σ s1  has the origin coincident with the origin of the detection axes of the first inertial sensor  12 . The local coordinate system Σ s1  locates the joint  35  on the y-axis. Accordingly, the position l sj1  of the joint  35  is identified as the coordinate (0, l sj1y , 0) in the local coordinate system Σ s1 . Likewise, the position l s0  of the fulcrum  34  and the position l sg1  of the centroid  36  are identified as the coordinate (l s0x , l s0y , l s0z ) and the coordinate (l sg1x , l sg1y , l sg1z ) in the local coordinate system Σ s1 . 
     The arithmetic unit  16  likewise fixes a local coordinate system Σ s2  to the second inertial sensor  13 . The local coordinate system Σ s2  has the origin coincident with the origin of the detection axes of the second inertial sensor  13 . The longitudinal axis of the golf club  14  coincides with the y-axis of the local coordinate system Σ s2 . Accordingly, the position l sj2  of the joint  35  is identified as the coordinate (0, l sj2y , 0) in the local coordinate system Σ s2 . Likewise, the position l sg2  of the centroid  37  and the position l sh2  of the club head  38  are identified as the coordinate (0, l sg2y , 0) and the coordinate (0, l sh2y , 0) in the local coordinate system Σ s2 . 
     (3) Structure of Arithmetic Unit 
       FIG. 3  schematically illustrates the structure of the arithmetic unit  16 . The arithmetic unit  16  includes a component calculating section  44 . The acceleration signals and the angular velocity signals are input to the component calculating section  44  from the first inertial sensor  12  and the second inertial sensor  13 . The component calculating section  44  calculates, based on the supplied acceleration signals and the supplied angular velocity signals, componential values required in the calculation of the energy change rate. The component calculating section  44  obtains various values from the storage unit  18  for the calculation of the energy change value. 
     The component calculating section  44  includes a first force calculating section  45 . The first force calculating section  45  calculates the first inter-joint force F 2  acting on the second link  33 . The first force calculating section  45  obtains the acceleration signals from the second inertial sensor  13  and a first mass data of the golf club  14  for the calculation of the first inter-joint force F 2 . The first mass data specifies the mass m 2  of the golf club  14 . The first mass data may previously be stored in the storage unit  18 . The first inter-joint force F 2  is calculated in accordance with the following mathematical expression:
 
 F   2   =m   2 ( {umlaut over (x)}   g2   −g )  [Mathematical Expression 2]
 
     In this case, the following component represents the acceleration of the centroid  37  of the second link  33 :
 
( {umlaut over (x)}   g2   −g )  [Mathematical Expression 3]
 
     The constant g represents the gravity. The acceleration of the centroid  37  is determined based on the measurement of the second inertial sensor  13 . The first force calculating section  45  outputs a first inter-joint force signal specifying the value of the first inter-joint force F 2 . 
     The component calculating section  44  includes a second force calculating section  46 . The second force calculating section  46  calculates the second inter-joint force F 1  acting on the first link  32 . The second force calculating section  46  obtains the acceleration signals from the first inertial sensor  12 , a second mass data and the first inter-joint force signals for the calculation of the second inter-joint force F 1 . The second mass data specifies the mass m 1  of the arm  15 . The second mass data may previously be stored in the storage unit  18 . The second inter-joint force F 1  is calculated in accordance with the following mathematical expression:
 
 F   1   =m   1 ( {umlaut over (x)}   g1   −g )+ F   2   [Mathematical Expression 4]
 
     In this case, the following component represents the acceleration of the centroid  36  of the first link  32 :
 
( {umlaut over (x)}   g1   −g )  [Mathematical Expression 5]
 
     The acceleration of the centroid  36  is determined based on the measurement of the first inertial sensor  12 . The second force calculating section  46  outputs a second inter-joint force signal specifying the value of the second inter-joint force F 1 . 
     The component calculating section  44  includes a first torque calculating section  47 . The first torque calculating section  47  calculates torque τ 2  acting on the second link  33  around the joint  35 . The first torque calculating section  47  obtains the angular velocity signals from the second inertial sensor  13 , a first inertia tensor data, a first position data, a second position data and the first inter-joint force signals for the calculation of the torque τ 2 . The first inertia tensor data specifies the inertia tensor J 2  of the golf club  14 . The first position data specifies the position l sj2  of the joint  35  in the local coordinate system Σ s2 . The second position data specifies the position l sg2  of the centroid  37  in the local coordinate system Σ s2 . The first inertia tensor data, the first position data and the second position data may previously be stored in the storage unit  18 . The first inter-joint force signals may be supplied from the first force calculating section  45 . The torque τ 2  is calculated in accordance with the following mathematical expression:
 
τ 2   =J   2 {dot over (ω)} 2 +ω 2   ×J   2 ω 2   +∥l   sg2   −l   sj2   ∥e   l2   ×F   2   [Mathematical Expression 6]
 
     Here, the unit vector e l2 , determines the longitudinal direction from the grip end to the club head of the golf club  14 . The first torque calculating section  47  outputs a first torque signal specifying the value of the torque τ 2 . 
     The component calculating section  44  includes a second torque calculating section  48 . The second torque calculating section  48  calculates torque τ 1  acting on the first link  32  around the fulcrum  34 . The second torque calculating section  48  obtains the angular velocity signals from the first inertial sensor  12 , a second inertia tensor data, a third position data, a fourth position data, a fifth position data, the first inter-joint force signals, the second inter-joint force signals and the first torque signal for the calculation of the torque τ 1 . The second inertia tensor data specifies the inertia tensor J 1  of the arm  15 . The third position data specifies the position l s0  of the fulcrum  34  in the local coordinate system Σ s1 . The fourth position data specifies the position l sj1  of the joint  35  in the local coordinate system Σ s1 . The fifth position data specifies the position l sg1  of the centroid  36  in the local coordinate system Σ s1 . The second inertia tensor data and the third to fifth position data may previously be stored in the storage unit  18 . The first inter-joint force signals may be supplied from the first force calculating section  45 . The second inter-joint force signals may be supplied from the second force calculating section  46 . The torque τ 1  is calculated in accordance with the following mathematical expression:
 
τ 1   =J   1 {dot over (ω)} 1 +ω 1   ×J   1 ω 1   +∥l   sg1   −l   s0   ∥e   l1   ×F   1   +∥l   sj1   −l   sg1   ∥e   l1 ×(− F   2 )+τ 2   [Mathematical Expression 7]
 
     Here, the unit vector e l1  determines the longitudinal direction of the first link  32 . The second torque calculating section  48  outputs a second torque signal specifying the value of the torque τ 1 . 
     The component calculating section  44  includes a first velocity calculating section  49 . The first velocity calculating section  49  calculates the velocity of the movement of the fulcrum  34 . The first velocity calculating section  49  obtains the acceleration signals and the angular velocity signals from the first inertial sensor  12  and the third position data for the calculation of the velocity. The first velocity calculating section  49  operates to calculate the acceleration of the fulcrum  34  in accordance with the following mathematical expression:
 
 {umlaut over (x)}   0   ={umlaut over (x)}   s1   +{dot over (ω)}   1   ×l   s0 +ω 1 ×(ω 1   ×l   s0 )  [Mathematical Expression 8]
 
     The calculated acceleration is subjected to integration in accordance with the following mathematical expression:
 
 {dot over (x)}   0   =∫{umlaut over (x)}   0   dt   [Mathematical Expression 9]
 
     This calculation results in the velocity of the movement of the fulcrum  34  (coordinate x 0 ). It should be understood that the initial velocity equals zero in this case. The first velocity calculating section  49  outputs a first velocity signal specifying the velocity of the movement of the fulcrum  34 . 
     The component calculating section  44  includes a second velocity calculating section  51 . The second velocity calculating section  51  calculates the velocity of the movement of the joint  35 . The second velocity calculating section  51  obtains the acceleration signals and the angular velocity signals from the first inertial sensor  12  and the fourth position data for the calculation of the velocity. The second velocity calculating section  51  operates to calculate the acceleration of the joint  35  in accordance with the following mathematical expression:
 
 {umlaut over (x)}   1   ={umlaut over (x)}   s1 +{dot over (ω)} 1   ×l   sj0 +ω 1 ×(ω 1   ×l   sj0 )  [Mathematical Expression 10]
 
     The calculated acceleration is subjected to integration in accordance with the following mathematical expression:
 
 {dot over (x)}   1   =∫{umlaut over (x)}   1   dt   [Mathematical Expression 11]
 
     This calculation results in the velocity of the movement of the joint  35  (coordinate x 1 ). It should be understood that the initial velocity equals zero in this case. The second velocity calculating section  51  outputs a second velocity signal specifying the velocity of the movement of the joint  35 . 
     The arithmetic unit  16  includes an energy change rate calculating section  52 . The angular velocity signals are input to the energy change rate calculating section  52  from the first inertial sensor  12  and the second inertial sensor  13 . The first and second inter-joint force signals, the first and second torque signals and the first and second velocity signals are likewise input to the energy change rate calculating section  52  from the component calculating section  44 . The energy change rate calculating section  52  calculates some energy change rates based on the input signals. 
     The energy change rate calculating section  52  includes a first calculating section  53 . The first calculating section  53  calculates the energy change rate of the first energy amount generated at the arm  15  of the golfer. The first calculating section  53  obtains the second torque signals from the component calculating section  44  and the angular velocity signals from the first inertial sensor  12 . The energy change rate of the first energy amount is calculated based on the torque τ 1  and the angular velocity ω 1  in accordance with the following mathematical expression:
 
τ 1   T ω 1   [Mathematical Expression 12]
 
     The first energy corresponds to the inflow energy flowing into the arm  15  resulting from the swing of the golfer. The first calculating section  53  outputs a first energy change rate signal specifying the energy change rate of the first energy amount. 
     The energy change rate calculating section  52  includes a second calculating section  55 . The second calculating section  55  calculates the energy change rate of the second energy amount transferred to the golf club  14  from the arm  15  of the golfer. The second calculating section  55  obtains the first inter-joint force signals and the second velocity signals from the component calculating section  44 . The energy change rate of the second energy amount is calculated based on the first inter-joint force F 2  and the velocity of the joint  35  in accordance with the following mathematical expression:
 
 F   2   T   {dot over (x)}   1   [Mathematical Expression 13]
 
     The second calculating section  55  outputs a second energy change rate signal specifying the energy change rate of the second energy amount. 
     The energy change rate calculating section  52  includes a third calculating section  57 . The third calculating section  57  calculates the energy change rate of the third energy amount resulting from the second inter-joint force F 1  of the first link  32 , namely of the arm  15  of the golfer. The third calculating section  57  obtains the second inter-joint force signals and the first velocity signal from the component calculating section  44 . The energy change rate of the third energy amount is calculated in accordance with the following mathematical expression:
 
 F   1   T   {dot over (x)}   0   [Mathematical Expression 14]
 
     The third calculating section  57  outputs a third energy change rate signal specifying the energy change rate of the third energy amount. 
     The energy change rate calculating section  52  includes a fourth calculating section  58 . The fourth calculating section  58  calculates the energy change rate of the fourth energy amount resulting from the torque τ 2  acting on the golf club  14 . The fourth calculating section  58  obtains the first torque signals from the component calculating section  44  and the angular velocity signals from the first inertial sensor  12 . The energy change rate of the fourth energy amount is calculated in accordance with the following mathematical expression:
 
τ 2   T ω 1   [Mathematical Expression 15]
 
     The fourth calculating section  58  outputs a fourth energy change rate signal specifying the energy change rate of the fourth energy amount. 
     The arithmetic unit  16  includes an energy change rate inversion detecting section  61 . The energy change rate inversion detecting section  61  determines the timing of the zero crossing of the total energy change rate signal. Here, “zero crossing” means the time point of the total energy change rate signal crossing the “zero” value, or the time point of the inversion from the positive sign to the negative sign of the total energy change rate, or the time point of the balance between the positive value and the negative value of the total energy change rate. The total energy change rate is calculated based on the energy change rate of the first energy amount, the energy change rate of the second energy amount, the energy change rate of the third energy amount and the energy change rate of the fourth energy amount in accordance with the following mathematical expression:
 
 Ė   1   =F   1   T   {dot over (x)}   0   −F   2   T   {dot over (x)}   1 +τ 1   T ω 1 −τ 2   T ω 1   [Mathematical Expression 16]
 
     The energy change rate inversion detecting section  61  outputs a zero-crossing signal specifying the change of the total energy change rate along the elapse of time. The time point of the zero crossing is identified based on the change along the elapse of time. 
     The arithmetic unit  16  includes an image data generating section  62 . The image data generating section  62  is connected to the energy change rate inversion detecting section  62 . The zero crossing signal is input to the image data generating section  62  from the energy change rate inversion detecting section  61 . The image data generating section  62  generates, based on the supplied zero crossing signal, a first image data for visualizing the total energy change rate signal along the elapsed time. The first image data is output toward the image processing circuit  21 . 
     As depicted in  FIG. 4 , the arithmetic unit  16  includes a first attitude calculating section  65  and a second attitude calculating section  66 . The first attitude calculating section  65  calculates the attitude of the first inertial sensor  12 . The angular velocity signal is supplied to the first attitude calculating section  65  from the first inertial sensor  12  for the calculation of the attitude. The detection axes are established in the first inertial sensor  12  in accordance with the orthogonal sensor coordinate system for the generation of the angular velocity signal. The angular velocity signal specifies, in accordance with the orthogonal sensor coordinate system, the angular velocity ω x  around the x-axis, the angular velocity ω y  around the y-axis, and the angular velocity ω z  around the z-axis. The first inertial sensor  12  is configured to define the change of the attitude of the first inertial sensor  12  as a rotation matrix per a unit time. For example, if the attitude at the time t is expressed as the rotation matrix R t , the attitude at the time (t+1) is defined as the rotation matrix R t+1  in accordance with the following mathematical expression: 
     
       
         
           
             
               
                 
                   
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                       t 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Mathematical 
                     ⁢ 
                     
                         
                     
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                     Expression 
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                     ⁢ 
                     17 
                   
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     Here, the following mathematical expression represents the angular velocity at the time (t+1):
 
ω=(ω x ,ω y ,ω z )  [Mathematical Expression 18]
 
     The component dt corresponds to the interval of sampling in the first inertial sensor  12 . The first attitude calculating section  65  outputs a first attitude data specifying the rotation matrix R S1  defining the attitude of the first inertial sensor  12  in the absolute reference coordinate system Σxyz. 
     An initial attitude data is supplied to the first attitude calculating section  65  for the calculation of the first inertial sensor  12 . The initial attitude data may be stored in the storage unit  18 . The initial attitude data specifies the rotation matrix R 0  for the initial attitude of the first inertial sensor  12 . The rotation matrix R 0  describes the relationship between the absolute reference coordinate system Σxyz and the orthogonal sensor coordinate system. The rotation matrix R 0  functions to convert the coordinate values of the orthogonal sensor coordinate system to the coordinate values of the absolute reference coordinate system Σxyz. The product of the rotation matrix R 0  for the initial attitude and the rotation matrix R t+1  at the time (t+1) describes the change of the attitude of the first inertial sensor  12  along the elapsed time in the absolute reference coordinate system Σxyz. The rotation matrix R 0  for the initial attitude is determined based on the attitude of the first inertial sensor  12  at the beginning of a golf swing. Here, a predetermined value is set for the rotation matrix R 0  for the initial attitude. Alternatively, the initial attitude of the first inertial sensor  12  may be determined based on the angle of elevation and the direction angle. The angle of elevation may be measured based on the output from the acceleration sensor, for example, and the direction angle may be determined based on the output from a magnetic sensor, for example. 
     The second attitude calculation section  66  likewise calculates the attitude of the second inertial sensor  13 . The second attitude calculating section  66  outputs a second attitude data specifying the rotation matrix R S2  defining the attitude of the second inertial sensor  13  in the absolute reference coordinate system Σxyz. 
     The arithmetic unit  16  includes a first vector calculating section  67  and a second vector calculating section  68 . The output from the first attitude calculating section  65  is supplied to the first vector calculating section  67 . The output from the second attitude calculating section  66  is supplied to the second vector calculating section  68 . The first and second vector calculating sections  67 ,  68  calculate the vector r 1 , r 2  in the y-axis of the first and second inertial sensors  12 ,  13 , respectively, based on the rotation matrices R S1 , R S2  in accordance with the following mathematical expression:
 
 r   1 =(0,1,0) R   S1  
 
 r   2 =(0,1,0) R   S2   [Mathematical Expression 19]
 
     The first and second vector calculating sections  67 ,  68  output vector data, respectively. The vector data specify the vector r 1 , r 2  in the y-axis of the first and second inertial sensors  12 ,  13 , respectively. 
     The arithmetic unit  16  includes a relative angle calculating section  69 . The vector data are supplied to the relative angle calculating section  69  from the first and second vector calculating sections  67 ,  68 . The relative angle calculating section  69  determines the relative angle θ between the vector r 1  and the vector r 2  based on the vector r 1 , r 2  in accordance with the following mathematical expression: 
     
       
         
           
             
               
                 
                   θ 
                   = 
                   
                     acos 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             r 
                             1 
                           
                           · 
                           
                             r 
                             2 
                           
                         
                         
                           
                              
                             
                               r 
                               1 
                             
                              
                           
                           · 
                           
                              
                             
                               r 
                               2 
                             
                              
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Mathematical 
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                     20 
                   
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     The relative angle calculating section  69  outputs a relative angle data. The relative angle data specifies the relative angle θ in the absolute reference coordinate system Σxyz. The relative angle data is supplied to the image data generating section  62 . 
     The image data generating section  62  generates a second image data for visualizing the change of the relative angle θ along the elapsed time. 
     (4) Performance of Golf Swing Analyzing Apparatus 
     A brief description will be made on the performance of the golf swing analyzing apparatus  11 . First of all, the golf swing of a golfer is measured. Required information is input to the arithmetic unit  16  through the input device  23  prior to the measurement of a golf swing. Here, one is instructed to input the information including, according to the three-dimensional double pendulum model  31 , the mass m 1 , m 2  of the first and second links  32 ,  33 , the inertia tensor J 1  of the first link  32  around the fulcrum x 0 , the inertia tensor J 2  of the second link  33  around the joint x 1 , the length l 1  of the first link  32  between the fulcrum x 0  and the joint x 1 , the length l g1  between the fulcrum x 0  and the centroid x g1  of the first link  32 , the length l g2  between the joint x 1  and the centroid x g2  of the second link  33 , a unit vector e l1  in the axial direction of l 1 , a unit vector e l2  in the axial direction of l 2 , the position l s0  of the fulcrum  34  in the local coordinate system Σ s1 , the position l sj1  of the joint  35  in the local coordinate system Σ s1 , the rotation matrix R 0  for the initial attitude of the first inertial sensor  12 , and the rotation matrix R 0  for the initial attitude of the second inertial sensor  13 . The input information is controlled under a predetermined identifier, for example. The identifier may be utilized to discriminate a predetermined golfer. 
     The first and second inertial sensors  12 ,  13  are attached to the arm  15  of the golfer and the golf club  14 , respectively, prior to the measurement of a golf swing. The left arm may be selected if the golfer is right-handed. The left arm of the right-handed golfer usually keeps straight to the utmost without bending at the elbow from the beginning of a golf swing to the impact. The first and second inertial sensors  12 ,  13  are fixed to the arm  15  and the golf club  14 , respectively, in an immobilized manner. 
     The first and second inertial sensors  12 ,  13  start operating to measure prior to the execution of a golf swing. The first and second inertial sensors  12 ,  13  are forced to take predetermined positions in predetermined attitudes, respectively, at the beginning of the measurement. These positions and attitudes correspond to those defined by the rotation matrices R 0  for the initial attitude. Synchronization is established between the first inertial sensor  12  and the second inertial sensor  13  during the measurement. The first and second inertial sensors  12 ,  13  keep operating to continuously measure the acceleration and the angular velocity at predetermined intervals. The size of the intervals determines the resolution of the measurement. The detection signals of the first and second inertial sensors  12 ,  13  may be transmitted to the arithmetic unit  16  in a realtime fashion, or temporarily be stored in storage devices respectively incorporated in the first and second inertial sensors  12 ,  13 . In the latter case, the detection signals may be transmitted to the arithmetic unit  16  with or without wires after the completion of the golf swing. 
     The arithmetic unit  16  executes the analysis of the golf swing in response to the receipt of the detection signals. The analysis may be effected between the beginning of the golf swing and the finish of the golf swing, or between the beginning of the golf swing and the impact. The arithmetic unit  16  thus operates to calculate the relative angle θ and the total energy change rate. The image data generating section  62  operates to generate the first and second image data in response to the calculation of the mentioned relative angle θ and the total energy change rate. The first and second image data are input to the image processing circuit  21 . As a result, expected images are displayed on the screen of the display unit  22 . 
     The inventors have observed the performance of the golf swing analyzing apparatus  11 . The golf swing of an amateur golfer and the golf swing of a golf teaching professional are compared with each other in the observation. The inventors observed the relative angle θ for the golf teaching professional. As depicted in  FIG. 5 , the inventors have confirmed that the relative angle θ moderately decreased from the top of the golf swing to the impact of the golf teaching professional. In particular, the observation revealed that the inclination of the decrease enlarged after the relative angle θ exceeded 100°. On the other hand, as depicted in  FIG. 6 , the inventors have confirmed that the relative angle θ was kept at 80° for the duration before a predetermined time point in the golf swing of the amateur golfer. The relative angle θ suddenly decreased after the predetermined time point in the golf swing of the amateur golfer. The observation of the relative angle θ in this manner between the arm  15  and the golf club  14  enables discovery of the form of golf swing which results in an efficient transfer of the energy to the golf club. An index is in this manner provided for improvement of the form of golf swing. For example, repeated changes of the form in combination with a subsequent observation enable an accelerated improvement of the form of golf swing through try and error. 
     The inventors have also observed the total energy change rate signal for the golf teaching professional. As depicted in  FIG. 7 , the inventors found the zero point, the zero crossing in  FIG. 7 , of the total energy change rate of the arm  15  at an early stage of the golf swing of the golf teaching professional. As depicted in  FIG. 8 , the inventors have confirmed the total energy change rate of the arm  15  exhibiting the transition from the positive value to the negative value at a relatively high position in the downswing of the golf club  14 . It has been confirmed that the pendulum movement of the golf club  14  around the joint  35  started at an early stage of the golf swing of the golf teaching professional. On the other hand, as depicted in  FIG. 9 , the inventors have found the zero point, the zero crossing in  FIG. 9 , of the total energy change rate of the arm  15  immediately before the impact in the golf swing of the amateur golfer. As depicted in  FIG. 10 , the inventors have confirmed the total energy change rate of the arm  15  exhibiting the transition from the positive value to the negative value at a relatively low position in the downswing of the golf club  14 . The pendulum movement of the golf club  14  around the joint  35  is expected to contribute to improvement of the transferring ratio η of the energy. The observation of the zero crossing of the total energy change rate enables discovery of the form of golf swing which results in an efficient transfer of the energy to the golf club  14 . An index is in this manner provided for improvement of the form of golf swing. For example, repeated changes of the form in combination with a subsequent observation enable an accelerated improvement of the form of golf swing through try and error. In addition, as depicted in  FIGS. 5 and 6 , correlation of the timing of the zero crossing with the relative angle θ between the arm  15  and the golf club  14  enables a contribution to a further accelerated improvement of the form of golf swing. 
     The golf swing analyzing apparatus  11  allows establishment of the three-dimensional double pendulum model  31  including a predetermined portion of the upper body of a golfer, namely the arm  15  as the first link  32  and the golf club  14  as the second link  33 . A golf swing is in this manner fitted into a model. The three-dimensional double pendulum model  31  kinetically represents the movement of a golf swing with a relatively high accuracy. The golf swing is in this manner effectively analyzed. And further, the fulcrum  34  of the first link  32  is located at the center of a line connecting the shoulders of the golfer. The joint  35  between the first link  32  and the second link  33  is located on the grip of the golf club  14 . A golf swing is thus analyzed with a higher accuracy. 
     It should be noted that it is easily conceivable to a person having ordinary skills in the art to make various modification on the embodiment substantially within the scope of the novel features and effects of the invention although the exemplary embodiment has been described above in detail. The scope of the invention covers all the modifications. For example, the terminology at least once used to mean a broader or similar meaning in the subject specification and attached drawings may have the identical coverage even in the other part of the specification and drawings. In addition, the components and operation of the golf swing analyzing apparatus  11 , the first and second inertial sensors  12 ,  13 , the arithmetic unit  16 , and the like may not be limited to ones described in the embodiment, and various modification may be made.