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Wu. Intell Robot 2021;1(2):99-115  I http://dx.doi.org/10.20517/ir.2021.11           Page 101
























                                    Figure 1. The 5-dof lightweight robotic arm and its coordinate systems  [38] .

                                            Table 1. D–H parameters of the 5-dof robotic arm
                                               Joint                  [mm]        [mm]       
                                               1       /2  0      250       1
                                               2     0    600     0         2
                                               3       /2  0      0         3
                                               4     −  /2  0     600       4
                                               5       /2  0      150       5

               2.   KINEMATICS OF THE LIGHTWEIGHT ROBOTIC ARM
               The lightweight robotic arm under study has five degrees of freedom (dof) [38] , which adopts a modular design
               approach, as shown in Figure 1. The revolute joints are composed of CPU series gearboxes of Harmonic Drive
               and Maxon motor with gearhead to enhance the torque capabilities, except Joint 4 with geared motor. The
               actuators of joints are controlled by Maxon EPOS controllers. The Controller Area Network (CANopen) bus
               is adopted to build the communications between motors and controllers, and A CAN–USB interface is used
               to establish the communications between CANopen bus and the PC [38] . In accordance with the Denavit–
               Hartenberg (D–H) convention [39] , the Cartesian coordinate systems are established accordingly.


               2.1.  Kinematics of robotic arm
               Throughout this work, i, j, and k stand for the unit vectors of the   -axis,   -axis, and   -axis, respectively. The
               transformation matrix in forward kinematics of the end-effector in reference frame is expressed as

                                            [     ]   5                [   −1    −1  ]
                                             R q     ∏
                                       0                  −1      −1      R     q   
                                       A 5 =       =      A    ;  A    =                               (1)
                                             0   1                       0     1
                                                       =1
               with

                                                −1                                                     (2a)
                                                 R    = R(     −1 ,       )R(      ,       )
                                                −1   [                   ]                            (2b)
                                                 q    =       cos              sin             
               where D–H parameters are given in Table 1, and the inverse geometry problem for this robotics is well docu-
                                   [8]
               mented in the literature .
               2.2.  Kinematic jacobian matrix
               The velocities between the joints and end-effector are mapped with the Kinematic Jacobian matrix

                                                        ¤   −1                                         (3)
                                                           = J v      
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