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Wu. Intell Robot 2021;1(2):99-115  I http://dx.doi.org/10.20517/ir.2021.11           Page 105


               The kinetic energy of the upper/lower links and the wrist link can be expressed as

                                                 1  (                       )
                                                  =  v M    v    + v M    v    + v M    v             (27)
                                                                        
                                                       
                                                               
                                                 2
               with
                                                        [            ]
                                                        R 1 I    R  0
                                                  M    =      1                                       (28a)
                                                           0           1 3
                                                       [            ]
                                                        R 3 I    R  0
                                                  M    =     3                                       (28b)
                                                           0          1 3
                                                        [             ]
                                                         R 4 I    R  0
                                                  M    =      4                                       (28c)
                                                           0           1 3
               where the subscripted I,   , and v stand for the moment of inertia, mass, and velocities in the Cartesian space,
               respectively, and
                                              v    = E      ;  v    = E      ;  v    = E       ¤      (29)
                                                     ¤
                                                               ¤
               with
                                    [                       ]
                                      z 0        z 1      0 3
                               E    =                                                                 (30a)
                                    z 0 × q     z 1 × (q    − q 1 )  0 3
                                   [                                      ]
                                      z 0        z 1          z 2      0 3×2
                               E    =                                                                (30b)
                                    z 0 × q     z 1 × (q    − q 1 )  z 2 × (q    − q 2 )  0 3×2
                                    [                                                    ]
                                       z 0        z 1          z 2           z 3      0 3×1
                               E    =                                                                 (30c)
                                     z 0 × q 4  z 1 × (q 4 − q 1 )  z 2 × (q 4 − q 2 )  z 3 × (q 4 − q 2 )  0 3×1
               where q    and q    are the position vector of the centers of the mass of the upper and lower links, respectively.
               Equation (27) can be cast in a matrix form as follows:

                                                           1
                                                             ¤   
                                                            =     M       ¤                           (31)
                                                           2
               with
                                                                                                      (32)
                                             M    = E M    E    + E M    E    + E M    E   
                                                                        
                                                     
                                                               
               Similarly, the kinetic energy of the end-effector can be obtained as
                                                                  [           ]
                                               1                   RI    R     0 3
                                                =  v M    v       ;  M    =                           (33)
                                                      
                                               2                    0 3         1 3
               where I    is the moment of inertia of the end-effector and       is the mass.

               From the total kinetic energy of the robotic arm    =       +       +      , the mass matrix M for the robotic arm can
               be expressed as
                                                           −           −1
                                                M = M    + J  (M    + M    )J                         (34)

               3.3.  Dynamic equation and analysis
               The dynamic equation of the robotic arm can be formulated as

                                               M¥ u + C¤ u + Ku = f − M¤ v       = F                  (35)
               where C is the damping matrix, F is the resultant force, and u and ¥ u are the elastic displacement and accelera-
               tion, respectively. Since damping can only slightly influence the natural frequency and mode of free vibrations,
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