Patent Publication Number: US-8996170-B2

Title: Actuating apparatus

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is based upon and claims the benefit of priority from Japanese Patent Application No. 2012-191970 filed on Aug. 31, 2012, the contents of which are incorporated herein by reference. 
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to an actuating apparatus for actuating joints of a robot. 
     2. Description of the Related Art 
     Japanese Laid-Open Patent Publication No. 2005-349555 discloses an apparatus for controlling a robot arm having a flexible passive joint with improved controllability against model parameter errors and disturbances. 
     SUMMARY OF THE INVENTION 
     The apparatus disclosed in Japanese Laid-Open Patent Publication No. 2005-349555 is capable of imparting rigidity (e.g., elasticity) in directions of rotation of respective joints. However, the apparatus is unable to determine a level of rigidity in the directions of rotation of the respective joints, such that a required level of rigidity can be maintained at a certain task position (e.g., a hand tip position) on a robot arm. 
     It is an object of the present invention to provide an actuating apparatus for determining a level of rigidity in directions of rotation of respective joints, such that a required level of rigidity can be maintained at a certain task position (e.g., a hand tip position) on a robot arm. 
     According to the present invention, there is provided an actuating apparatus comprising an actuator including a link having a plurality of joints, a plurality of rotary prime movers for actuating the joints, and a plurality of flexible transmitting assemblies disposed respectively between the joints and the rotary prime movers, each of the flexible transmitting assemblies including a variable rigidity element a rigidity of which is variable in directions of rotation of the joints, and a controller for controlling the actuator, wherein the controller includes a rigidity threshold value calculator for calculating rigidity threshold values for the joints, based on a required rigidity for a predetermined task position on the link and a coefficient matrix determined based on rotational angles of the rotary prime movers. 
     In the above actuating apparatus, the rigidity threshold value calculator may use, as the rigidity threshold values for the joints, a diagonal element of a matrix, which is obtained by calculating the product of a transposed matrix of the coefficient matrix, a matrix representing the rigidity at the task position, and the coefficient matrix. 
     In the above actuating apparatus, the controller may further include a rigidity determiner for determining a rigidity of the joints, which is equal to or greater than the rigidity threshold values for the joints, and a rigidity change controller for changing the rigidity of the variable rigidity elements of each of the joints, such that the rigidity of the joints will be equivalent to the rigidity of the joints determined by the rigidity determiner. 
     In the above actuating apparatus, the flexible transmitting assembly may include a variable viscosity coefficient element for varying a viscosity coefficient in directions of rotation of the joints, and the controller may further include an inherent frequency calculator for using, as a self inherent vibrational component of the joints, the square root of a diagonal element of a matrix produced by calculating the product of a matrix representing the rigidity of the variable rigidity element, which is changed by the rigidity change controller, and an inverse matrix of an inertia matrix of the actuator, which is determined based on the rotational angles of the rotary prime movers, a viscosity coefficient calculator for calculating a viscosity coefficient of the joints, using the self inherent vibrational component of the joints and a diagonal element of the inertia matrix, and a viscosity change controller for changing a viscosity coefficient of the variable viscosity coefficient element of the joints, such that the viscosity coefficient of the joints will be equivalent to the viscosity coefficient of the joints calculated by the viscosity coefficient calculator. 
     According to the present invention, there also is provided an actuating apparatus comprising an actuator including a link having a plurality of joints, a plurality of rotary prime movers for actuating the joints, and a plurality of flexible transmitting assemblies disposed respectively between the joints and the rotary prime movers, the flexible transmitting assemblies each including a variable viscosity coefficient element a viscosity coefficient of which is variable in directions of rotation of the joints, and a controller for controlling the actuator, wherein the controller includes an inherent frequency calculator for using, as a self inherent vibrational component of the joints, the square root of a diagonal element of a matrix produced by calculating the product of a matrix representing the rigidity of the joints and an inverse matrix of an inertia matrix of the actuator, which is determined based on rotational angles of the rotary prime movers, a viscosity coefficient calculator for calculating a viscosity coefficient of the joints, using a self inherent vibrational component of the joints and a diagonal element of the inertia matrix, and a viscosity change controller for changing a viscosity coefficient of the variable viscosity coefficient element of the joints, such that the viscosity coefficient of the joints will be equivalent to the viscosity coefficient of the joints calculated by the viscosity coefficient calculator. 
     According to the present invention, since rigidity threshold values for the joints are calculated based on a required rigidity for the predetermined task position on the link, and the coefficient matrix is determined based on rotational angles of the rotary prime movers, the rigidity threshold values of the joints can be determined so as to satisfy the required rigidity for the predetermined task position. 
     According to the present invention, furthermore, the square root of the diagonal element of the matrix obtained by calculating the product of the matrix representing the rigidity of the joints and the inverse matrix of the inertia matrix of the actuator, which is determined based on the rotational angles of the rotary prime movers, is used as a self inherent vibrational component of the joints. Further, the viscosity coefficient of the joints is calculated using the self inherent vibrational component of the joints and the diagonal element of the inertia matrix. Consequently, vibrations of the joints can be suppressed. 
     The above and other objects, features, and advantages of the present invention will become more apparent from the following description when taken in conjunction with the accompanying drawings in which a preferred embodiment of the present invention is shown by way of illustrative example. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram showing an actuating apparatus according to an embodiment of the present invention; 
         FIG. 2  is a schematic view of the structure of a joint actuating system of an actuator of the actuating apparatus according to the embodiment; and 
         FIG. 3  is a functional block diagram of a controller of the actuating apparatus according to the embodiment. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     An actuating apparatus according to a preferred embodiment of the present invention will be described in detail below with reference to the accompanying drawings. 
       FIG. 1  shows an actuating apparatus  10  according to an embodiment of the present invention. As shown in  FIG. 1 , the actuating apparatus  10  includes an actuator  12 , and a controller  14  for controlling the actuator  12 . The actuator  12  includes a link  20  having a plurality of joints  16  ( 16   1 ,  16   2 ) and a plurality of connectors  18  ( 18   1 ,  18   2 ). The connectors  18  are interconnected through the joints  16 . The joint  16   1  at the foundation of the link  20  is mounted on a base  19 . 
     As shown in  FIG. 1 , the distance that a predetermined task position, e.g., a hand tip position A in the present embodiment, on the link  20  has traveled is denoted by Δx, the rotational angle through which the joint  16   1  is turned is denoted by Δθ 1 , and the rotational angle through which the joint  16   2  is turned is denoted by Δθ 2 . The rigidity at the hand tip position A is denoted by Kx. 
     In  FIG. 1 , the actuator  12  is shown as having two joints  16  ( 16   1 ,  16   2 ) and two connectors  18  ( 18   1 ,  18   2 ). However, the actuator  12  according to the present embodiment will be described below as having n joints  16  and n connectors  18 . Each of the n joints  16  will be referred to as a joint  16   1r  and each of the n connectors  18  will be referred to as a connector  18   1 . The rotational angle of a joint  16   1  will be referred to as Δθ 1 , where i is represented by i=1, 2, . . . , n, which indicates the number of the joint  16 , the connector  18 , or the rotational angle. 
       FIG. 2  schematically shows the configuration of a joint actuating system  30  for actuating each of the joints  16  of the actuator  12  according to the present embodiment. As shown in  FIG. 2 , the joint actuating system  30  includes a motor (rotary prime mover)  32 , a drive gear  34 , a driven gear  36 , and a flexible transmitting assembly  38 . The joint  16  is operatively connected to the motor  32  through the flexible transmitting assembly  38 . 
     The motor  32  is an electric motor, which is supplied with electric energy from a non-illustrated electric power supply in order to generate a torque that serves to rotate an output shaft  32   a , which is connected to the motor  32 , about its axis. The drive gear  34  is mounted on the output shaft  32   a . The motor  32  is combined with a rotary encoder  40  for detecting an angle of rotation of the motor  32 . 
     The driven gear  36  is held in mesh with the drive gear  34 , so that the driven gear  36  is rotated upon rotation of the drive gear  34 . The driven gear  36  rotates at a rotational speed, which is lower than the rotational speed of the drive gear  34 . Therefore, the drive gear  34  and the driven gear  36  jointly make up a speed reducer. 
     The flexible transmitting assembly  38  includes a variable rigidity element  50 , a variable viscosity coefficient element  52 , a rigidity changer  54 , a viscosity coefficient changer  56 , screw shafts  58 ,  60 , and nuts  62 ,  64 . The variable rigidity element  50  comprises a nonlinear spring, the rigidity of which in the directions of rotation of the joint  16  varies when the spring is displaced. The variable viscosity coefficient element  52  comprises a nonlinear damper, the viscosity coefficient of which in the directions of rotation of the joint  16  varies when the damper is displaced. 
     The rigidity changer  54  comprises a drive source for rotating the screw shaft  58  based on a voltage applied thereto. Similarly, the viscosity coefficient changer  56  comprises a drive source for rotating the screw shaft  60  based on a voltage applied thereto. The rigidity changer  54  and the viscosity coefficient changer  56  are supplied with electric energy from a non-illustrated electric power supply. The nut  62 , which is threaded over the screw shaft  58 , is movable axially along the screw shaft  58  to the left or the right in  FIG. 1  when the screw shaft  58  is rotated about its axis by the rigidity changer  54 . The nut  64 , which is threaded over the screw shaft  60 , is movable axially along the screw shaft  60  to the left or the right in  FIG. 1  when the screw shaft  60  is rotated about its axis by the viscosity coefficient changer  56 . 
     If the voltage applied to the rigidity changer  54  and the viscosity coefficient changer  56  is increased, the rigidity changer  54  and the viscosity coefficient changer  56  rotate the screw shafts  58 ,  60  respectively in directions to move the nuts  62 ,  64  to the right in  FIG. 1 . If the voltage applied to the rigidity changer  54  and the viscosity coefficient changer  56  is decreased, the rigidity changer  54  and the viscosity coefficient changer  56  rotate the screw shafts  58 ,  60  respectively in directions to move the nuts  62 ,  64  to the left in  FIG. 1 . If the voltage applied to the rigidity changer  54  and the viscosity coefficient changer  56  remains constant, the respective screw shafts  58 ,  60  are not rotated, thereby keeping the nuts  62 ,  64  at rest. If voltages are not applied to the rigidity changer  54  and the viscosity coefficient changer  56 , the rigidity changer  54  and the viscosity coefficient changer  56  rotate the screw shafts  58 ,  60  respectively in directions to move the nuts  62 ,  64  to the leftmost position in  FIG. 1 . 
     When the nut  62  moves to the right, the nut  62  presses the variable rigidity element  50 , thereby displacing the variable rigidity element  50  to the right over a distance commensurate with the distance that the nut  62  moves. On the other hand, when the nut  62  moves to the left, the nut  62  releases the variable rigidity element  50 , thereby displacing the variable rigidity element  50  to the left over a distance commensurate with the distance that the nut  62  moves. When the nut  64  moves to the right, the nut  64  presses the variable viscosity coefficient element  52 , thereby displacing the variable viscosity coefficient element  52  to the right over a distance commensurate with the distance that the nut  62  moves. On the other hand, when the nut  64  moves to the left, the nut  64  releases the variable viscosity coefficient element  52 , thereby displacing the variable viscosity coefficient element  52  to the left over a distance commensurate with the distance that the nut  62  moves. Therefore, the rigidity of the variable rigidity element  50  and the viscosity coefficient of the variable viscosity coefficient element  52  can be varied depending on the voltages applied respectively to the rigidity changer  54  and the viscosity coefficient changer  56 . 
     In  FIG. 2 , the variable rigidity element  50  is shown as applying a force depending on the rigidity and the displacement thereof to the left or right. Further, the variable viscosity coefficient element  52  is shown as applying a force depending on the viscosity coefficient and the speed of displacement thereof to the left or right. Actually, however, the variable rigidity element  50  and the variable viscosity coefficient element  52  apply respective forces in a direction of rotation of the joint  16 . Thus, the rigidity of the variable rigidity element  50  refers to the rigidity in the direction of rotation of the joint  16 , and the viscosity coefficient of the variable viscosity coefficient element  52  refers to the viscosity coefficient in the direction of rotation of the joint  16 . If the voltage applied to the rigidity changer  54  is increased, the rigidity of the variable rigidity element  50  in the direction of rotation of the joint  16  increases. If the voltage applied to the viscosity coefficient changer  56  is increased, the viscosity coefficient of the variable viscosity coefficient element  52  in the direction of rotation of the joint  16  increases. The variable rigidity element  50  and the variable viscosity coefficient element  52  transmit a torque, which is output from the driven gear  36 , to the joint  16 , thereby rotating the joint  16 . 
     As shown in  FIG. 1 , if the link  20  is supported at a certain posture by a matrix Kr (N·m/rad) representing the rigidity in the direction of rotation of the plural joints  16 , then a matrix representing the rigidity of the hand tip position A on the link  20  is denoted by Kx, and the distance by which the hand tip position A is displaced is denoted by Δx. In this case, the force F applied to the hand tip position A is expressed by the following equation (1): 
                   F   =         K   ⁢           ⁢     x   ·   Δ     ⁢           ⁢   x     ⁢     
     ⇒     (           f   1               f   2             ⋮             f   n           )       =       (           k   ⁢           ⁢     x   11           …         k   ⁢           ⁢     x     1   ⁢           ⁢   n                 ⋮       ⋱       ⋮             k   ⁢           ⁢     x     n   ⁢           ⁢   1             …         k   ⁢           ⁢     x     n   ⁢           ⁢   n               )     ⁢     (           Δ   ⁢           ⁢     x   1                 Δ   ⁢           ⁢     x   2               ⋮             Δ   ⁢           ⁢     x   n             )                 (   1   )               
where f 1 , f 2 , . . . , f n  represent components in respective directions of the force F applied to the hand tip position A, e.g., an x-axis direction, a y-axis direction, a direction of rotation about the x-axis, and a direction of rotation about the y-axis, and Δx 1 , Δx 2 , . . . , Δx n  represent components in respective directions of a change Δx of the hand tip position A, e.g., components in the x-axis direction and the y-axis direction. The number of components of force F, and the number of components of the change Δx of the hand tip position A can be set to any desired value. The elements of the matrix Kx are represented by kx gh  (g, h=1, 2, . . . , n).
 
     If the angle of rotation of a joint  16  is denoted by Δθ, then a torque τ applied to the joint  16  is calculated according to the following equation (2) from the angle of rotation detected by the rotary encoder  40 . 
                   τ   =         K   ⁢           ⁢     r   ·   Δ     ⁢           ⁢   θ     ⁢     
     ⇒     (           τ   1               τ   2             ⋮             τ   n           )       =       (           k   ⁢           ⁢     r   11           …         k   ⁢           ⁢     r     1   ⁢           ⁢   n                 ⋮       ⋱       ⋮             k   ⁢           ⁢     r     n   ⁢           ⁢   1             …         k   ⁢           ⁢     r     n   ⁢           ⁢   n               )     ⁢     (           Δ   ⁢           ⁢     θ   1                 Δ   ⁢           ⁢     θ   2               ⋮             Δ   ⁢           ⁢     θ   n             )                 (   2   )               
where τ is a vector representation of respective torques τ i  applied to each of the joints  16   i , and Δθ is a vector representation of respective angles of rotation Δθ i  of the joints  16   i . The elements of the matrix Kr are represented by kr gh  (g, h=1, 2, . . . , n).
 
     Δx, Δθ can be expressed according to the following equation (3), using a Jacobian matrix (coefficient matrix) J. The Jacobian matrix J is determined depending on the hand tip position A and rotational angles of the motors  32  that actuate the respective joints  16 . 
     
       
         
           
             
               
                 
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       x 
                     
                     = 
                     
                       
                         J 
                         · 
                         Δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   where 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     J 
                     = 
                     
                       ( 
                       
                         
                           
                             
                               j 
                               11 
                             
                           
                           
                             … 
                           
                           
                             
                               j 
                               
                                 1 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋱ 
                           
                           
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                               j 
                               
                                 n 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               j 
                               
                                 n 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     The torque τ applied to each joint  16  and the force F applied to the hand tip position A can be expressed according to the following equation (4), using a transposed matrix of the Jacobian matrix J.
 
τ= J   T   ·F   (4)
 
     Using equations (1) through (4), therefore, the following equation (5) can be derived.
 
 Kr=J   T   ·Kx·J   (5)
 
     As described above, the matrix Kr representing minimum rigidity, which is required for the plural joints  16 , can be determined from the matrix Kx, which represents the required rigidity for the hand tip position A on the link  20 . In order to make the rigidity of the hand tip position A on the link  20  equal to or greater than the required rigidity Kx, a matrix Km representing rigidity of the plural joints  16  must at least satisfy the inequality Km≧Kr. The inequality Km≧Kr indicates that the respective elements (rigidity) km gh  of the matrix Km are equal to or greater than the respective elements (rigidity threshold) kr gh  of the corresponding matrix Kr. In other words, each of the elements should satisfy the inequality km gh ≧kr gh . 
     If the matrix representing rigidity of the plural joints  16  is represented by Km, then when the plural joints  16  are rotated, the torques τ that are applied to the plural joints  16  can be expressed according to the following equation (6): 
     
       
         
           
             
               
                 
                   τ 
                   = 
                   
                     
                       
                         K 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           m 
                           · 
                           Δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         θ 
                       
                       ⁢ 
                       
                         
 
                       
                       ⇒ 
                       
                         ( 
                         
                           
                             
                               
                                 τ 
                                 1 
                               
                             
                           
                           
                             
                               
                                 τ 
                                 2 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 τ 
                                 n 
                               
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   m 
                                   11 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   m 
                                   
                                     1 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
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                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   m 
                                   
                                     n 
                                     ⁢ 
                                     
                                         
                                     
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                               … 
                             
                             
                               
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                                 ⁢ 
                                 
                                     
                                 
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                         ) 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   n 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     From equation (6), the torque τ 1  applied to the joint  16   1  is expressed by τ 1 =km 11 ×Δθ i +km 12 ×Δθ 2 + . . . +km in ×Δθ n . Therefore, the torque τ 1  applied to the joint  16   1  is affected by components (components in second and subsequent terms of the above equation), which are represented by the angles of rotation Δθ i  of the joints  16   i  other than the joint  16   1 . 
     Similarly, the torque τ 2  applied to the joint  16   2  is expressed by τ 2 =km 21 ×Δθ 1 +km 22 ×Δθ 2 + . . . +km 2n ×Δθ n . Therefore, the torque τ 2  applied to the joint  16   2  is affected by components (components in first, third, and subsequent terms of the above equation), which are represented by the angles of rotation Δθ i  of the joints  16   i  other than the joint  16   2 . 
     Concerning the torques τ i , components thereof other than the rotational angles Δθ i  of the joints  16   i  are regarded as disturbances, and the diagonal elements km ii  of the matrix Km are extracted, i.e., elements other than the diagonal elements of the matrix Km are set to 0, thereby removing the disturbances. 
     Therefore, km 11 , which is a diagonal element of the matrix Km, represents the rigidity of the joint  16   1 , and km 22 , which is another diagonal element of the matrix Km, represents the rigidity of the joint  16   2 . In this manner, the diagonal elements km ii  of the matrix Km represent rigidities of the joints  16   i . Torques τ i  applied to the joints  16   i  are expressed by the product km ii ×Δθ i . 
     With the rigidities of the joints  16   i  being expressed as respective rigidities km ii , the required rigidity for the hand tip position A can be satisfied. 
     Inasmuch as the rigidities km ii  of the joints  16   i  cannot be set to values higher than the mechanical rigidities k i  of the joints  16   i , the rigidities km ii  of the joints  16   i  must satisfy the relationship k i ≧km ii ≧kr ii . The mechanical rigidities k i  represent maximum rigidities in the directions of rotation of the variable rigidity elements  50  of the joints  16   i . 
     An equation of motion for the joints  16  is expressed as follows: 
                         M   ·   dd     ⁢           ⁢   Δθ     =         -   Km     ·   Δ     ⁢           ⁢   θ       ⁢     
     ⁢   where   ⁢     
     ⁢     M   =     (           m   11         …         m     1   ⁢           ⁢   n               ⋮       ⋱       ⋮             m     n   ⁢           ⁢   1           …         m     n   ⁢           ⁢   n             )               (   7   )               
and where M represents an inertia matrix indicating a moment of inertia (inertia) of the actuator  12 , and ddΔθ represents a second-order differential of Δθ, which is indicative of a rotational angular acceleration. Equation (7) can be modified and rewritten as the following equation (8). In equation (8), elements of the matrix M −1 ·Km are represented by s gh  (g, h=1, 2, . . . , n). The inertia matrix M is determined by the angles of rotation of the motors  32 , which actuate the respective joints  16 .
 
     
       
         
           
             
               
                 
                   
                     
                       dd 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     = 
                     
                       
                         
                           
                             
                               - 
                               
                                 M 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             · 
                             Km 
                             · 
                             Δ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         ⁢ 
                         
                           
 
                         
                         ⇒ 
                         
                           ( 
                           
                             
                               
                                 
                                   dd 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
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                                     θ 
                                     1 
                                   
                                 
                               
                             
                             
                               
                                 
                                   dd 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     θ 
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 ⋮ 
                               
                             
                             
                               
                                 
                                   dd 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
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                                     n 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   s 
                                   11 
                                 
                               
                               
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                                     1 
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                                     1 
                                   
                                 
                               
                             
                             
                               
                                 
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                                   ⁢ 
                                   
                                       
                                   
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                                     2 
                                   
                                 
                               
                             
                             
                               
                                 ⋮ 
                               
                             
                             
                               
                                 
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                                     θ 
                                     n 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   where 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         M 
                         
                           - 
                           1 
                         
                       
                       · 
                       Km 
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               s 
                               11 
                             
                           
                           
                             … 
                           
                           
                             
                               s 
                               
                                 1 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
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                               s 
                               
                                 n 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                           
                           
                             … 
                           
                           
                             
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                                 n 
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                                 n 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     From equation (8), the rotational angular acceleration ddΔθ 1  of the joint  16   1  is expressed by ddΔθ 1 =−(s 11 ×Δθ 1 +s 12 ×Δθ 2 + . . . +s 1n ×Δθ n ). The rotational angular acceleration ddΔθ 1  of the joint  16   1  is affected by components (components in second and subsequent terms of the above equation), which are represented by rotational angular accelerations ddΔθ n  of the joints  16   i  other than the joint  16   1 . 
     Similarly, the rotational angular acceleration ddΔθ 2  of the joint  16   2  is expressed by ddΔθ 2 =−(s 21 ×Δθ 1 +s 22 ×Δθ 2 + . . . +s 2n ×Δθ n ). The rotational angular acceleration ddΔθ 2  of the joint  16   2  is affected by components (components in first, third, and subsequent terms of the above equation), which are represented by rotational angular accelerations ddΔθ n  of the joints  16   i  other than the joint  16   2 . 
     Concerning the rotational angular accelerations ddΔθ i , components thereof other than the rotational angles Δθ i  of the joint  16   i  are regarded as disturbances, and the diagonal elements s ii  of the matrix M −1 ·Km are extracted, thereby removing such disturbances. Accordingly, self inherent vibrational components (self inherent frequencies) ω i  of the joints  16   i  can be expressed according to the following equation (9):
 
ω i =√{square root over ( s   ii )}  (9)
 
     Therefore, the viscosity coefficients d i  of the joints  16   i  can be expressed according to the following equation (10). In equation (10), m ii  represents diagonal elements of the inertia matrix M, and ζ represents a damping ratio.
 
 d   i =2 ·m   ii ·ω i ζ(0&lt;ζ&lt;1)  (10)
 
     The viscosity coefficient d 1  serves as a viscosity coefficient for the joint  16   1 , whereas the viscosity coefficient d 2  serves as a viscosity coefficient for the joint  16   2 . Thus, the viscosity coefficients d i  serve as respective viscosity coefficients for the plural joints  16   i . 
     With the viscosity coefficients of the joints  16   i  being represented as respective viscosity coefficients d i , vibrations of the joints  16   i  having determined rigidities km ii  can be suppressed. 
       FIG. 3  is a functional block diagram of the controller  14 . As shown in  FIG. 3 , the controller  14  includes a Jacobian matrix determiner  80 , a rigidity threshold value calculator  82 , a rigidity determiner  84 , a rigidity change controller  86 , an inertia matrix determiner  88 , an inherent frequency calculator  90 , a viscosity coefficient calculator  92 , and a viscosity change controller  94 . 
     The Jacobian matrix determiner  80  determines a Jacobian matrix J that depends on the rotational angles Δθ i  of the motors  32 , which are calculated by the rotary encoders  40  of the joints  16   i  and the hand tip position A. The hand tip position A may be detected by a gyro sensor or the like, which is disposed in the hand tip position A, or alternatively, may be calculated from the rotational angles Δθ 1  of the joints  16   i . Calculations for the Jacobian matrix J are well known in the art and will not be described below. The Jacobian matrix determiner  80  outputs the determined Jacobian matrix to the threshold value calculator  82 . 
     The threshold value calculator  82  calculates rigidity threshold values Kr of the plural joints  16  based on a required rigidity Kx for the hand tip position A, and the Jacobian matrix J determined by the Jacobian matrix determiner  80 . More specifically, the threshold value calculator  82  calculates rigidity threshold values Kr of the plural joints  16  according to equation (5). The elements kr gh  of the matrix Kr serve as rigidity threshold values. 
     The rigidity determiner  84  determines a matrix Km, which is equal to or greater than the matrix Kr. More specifically, the rigidity determiner  84  determines a matrix Km such that the elements km gh  of the matrix Km are equal to or greater than corresponding elements kr gh  of the matrix Kr. The rigidity determiner  84  outputs the determined matrix Km to the rigidity change controller  86  and the inherent frequency calculator  90 . 
     The rigidity change controller  86  changes rigidities of the variable rigidity elements  50  of the plural joints  16  by applying voltages corresponding to the determined matrix Km to the rigidity changers  54  of the plural joints  16 . In the present embodiment, the elements km gh , other than the diagonal components, of the matrix Km are regarded as disturbances. The rigidity change controller  86  extracts the diagonal elements km ii  of the matrix Km, and applies voltages to the rigidity changers  54  of the plural joints  16 , so that the rigidities of the joints  16   i  are represented by the diagonal elements km ii  of the determined matrix Km. 
     The diagonal elements km ii  of the determined matrix Km are set to satisfy the relationship k i ≧km ii ≧kr ii , where k i  represents mechanical rigidities of the joints  16   i , i.e., maximum rigidities of the variable rigidity elements  50  of the joints  16   i . 
     The inertia matrix determiner  88  calculates an inertia matrix M representing moments of inertia of the actuator  12 , based on rotational angles Δθ i  of the motors  32 , which are calculated by the rotary encoders  40  of the joints  16   i . Calculations for obtaining the inertia matrix M are well known in the art and will not be described below. The inertia matrix determiner  88  outputs the calculated inertia matrix M to the inherent frequency calculator  90  and the viscosity coefficient calculator  92 . 
     The inherent frequency calculator  90  uses, as self inherent vibrational components ω i  of the joints  16   i , a square root of the diagonal elements s ii  of a matrix, which is obtained by calculating the product of an inverse matrix M −1  of the inertia matrix M and the matrix Km. More specifically, the inherent frequency calculator  90  calculates self inherent vibrational components ω i  of the joints  16   i  according to equation (9), and based on the inertia matrix M and the determined matrix Km representing rigidities of the plural joints  16 . The inherent frequency calculator  90  outputs the calculated self inherent vibrational components ω i  to the viscosity coefficient calculator  92 . 
     The inherent frequency calculator  90  may determine a matrix by calculating the product of a matrix in which elements other than the diagonal elements km ii  of the matrix Km are set to 0, and the inverse matrix M −1  of the inertia matrix M. This is because the diagonal elements of the determined matrix are of the same value as the diagonal elements s ii  of the matrix produced by calculating the product of the matrix Km and the inverse matrix M. 
     The viscosity coefficient calculator  92  calculates viscosity coefficients d i  for each of the joints  16   i  using the self inherent vibrational components ω i  of the joints  16   i  and the diagonal elements m ii  of the inertia matrix M. More specifically, the viscosity coefficient calculator  92  calculates viscosity coefficients d i  for the respective joints  16   i  according to equation (10). The viscosity coefficient calculator  92  outputs the calculated viscosity coefficients d i  to the viscosity change controller  94 . 
     The viscosity change controller  94  changes viscosity coefficients of the variable viscosity coefficient elements  52  of the plural joints  16  by applying voltages corresponding to the calculated viscosity coefficients d i  to the viscosity coefficient changers  56  of the respective joints  16   i . More specifically, the viscosity change controller  94  applies voltages to the viscosity coefficient changers  56  of each of the joints  16   i , so that the viscosity coefficients of the joints  16   i  will reach the calculated viscosity coefficients d i . 
     As described above, since rigidity threshold values kr ii  of the joints  16   i  are calculated based on a required rigidity Kx for the predetermined hand tip position A on the link  20 , and based on the Jacobian matrix J, which is determined based on rotational angles Δθ of the motors  32  that actuate the plural joints  16 , the rigidity threshold values kr ii  of the joints  16   i  can be determined in order to satisfy the required rigidity for the hand tip position A. 
     Furthermore, the square root of the diagonal elements s ii  of a matrix obtained by calculating the product of the matrix M representing the rigidity of the plural joints  16  and an inverse matrix M −1  of the inertia matrix M of the actuator  12 , which is determined based on the rotational angles Δθ of the plural motors  32 , is used as self inherent vibrational components ω i  of the joints  16   i . Further, the viscosity coefficients d i  of the joints  16   i  are calculated using the self inherent vibrational components ω i  of the joints  16   i  and the diagonal elements m ii  of the inertia matrix M. Consequently, vibrations of the joints  16  can be suppressed. 
     While a preferred embodiment of the present invention has been described, the technical scope of the present invention is not limited to the illustrated embodiment. It will be obvious to those skilled in the art that various changes and modifications may be made to the above embodiment. Such changes and modifications should be interpreted as being covered by the technical scope of the present invention as set forth in the appended claims.