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Page 110                             Wu. Intell Robot 2021;1(2):99-115  I http://dx.doi.org/10.20517/ir.2021.11



                                                                                                −1
                                                                                     ∂f /∂K   [Hz/Nm⋅rad ]
                                                                                      1  act,2
                                                      −1
                                           ∂f /∂K   [Hz/Nm⋅rad ]                                     7
                                            1  act,1      0.96
                                                          0.94                                       6
                                                               600
                      600
                                                          0.92
                      500                                                                            5
                                                          0.9  400
                      400
                                                          0.88  z [mm]                               4
                    z [mm]  200                           0.86  200
                      300
                                                          0.84                                       3
                      100
                                                                0
                       0                                  0.82
                                                                                                     2
                     −100                                 0.8
                                                          0.78    800                          800
                        800                          800             600                    600      1
                           600                    600     0.76          400              400
                         y [mm]  400          400  x [mm]          y [mm]                    x [mm]
                                  200      200                             200       200
                                      0  0                                      0  0
                                      (a)                                       (b)
                          Figure 6. Sensitivities of the first-order natural frequency to the joint stiffness: (a) Joint 1; and (b) Joint 2.

                                                                                                    9
                                                    f  [Hz]                               f  [Hz]
                                                                                          2
                                                    1
                                                           8
                                                                                                    8.5
                      600                                       600
                      500                                  7.5  500                                 8
                                                                400
                      400                                       300
                     z [mm]  300                           7  z [mm]  200                           7.5
                      200
                      100                                  6.5  100                                 7
                                                                 0
                       0
                     −100                                  6   −100                                 6.5
                        800                          800          800                          800  6
                           600                    600      5.5       600                    600
                               400             400                      400             400
                          y [mm]                                   y [mm]   200      200     x [mm]
                                  200      200   x [mm]                                             5.5
                                      0  0                                      0  0
                                      (a)                                       (b)
                        Figure 7. The frequencies with payload at constant-orientation [0,   /2, 0]: (a) first order; and (b) second order.
                                      Table 4. The mean frequency (Hz) within the dexterous workspace
                                                  ¯    1  ¯    2  ¯    3  ¯    4  ¯    5  ¯    6
                               Natural frequency  7.9404  8.9579  19.0263  84.9292  136.1418  300.6579
                               Frequency with payload  5.9538  6.5358  15.2772  55.8998  86.6291  276.5956
               Assuming that the motion of the robotic arm follows the trajectory (unit: mm) defined by


                                                     = 750 + 750(cos      − 1)
                                                     = 750(1 − cos     )                              (56)
                                                     = 600(1 − cos     )
               where the end-effector keeps constant-orientation [0,   , 0] and the motion period    = 0.5 s is divided into
               1024 intervals, Figure 8 shows the displacement responses of the end-effector, from which it is seen that the
               linear elastic displacement responses are close, whenever the robotic arm is under loaded and unloaded work-
               ing modes. The angular displacements of the end-effector generate relatively large differences. The largest
               deformations appear around 0.3 s where the end-effector is located in the middle layer of the workspace, ap-
               proximately    = 250 mm.

               Figure 9 shows the comparison of the joint angular displacements between the numerical simulation and ex-
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