The Denavit-Hartenberg transformation matrices below describe the successive joints from 1 to 6 for a particular...

 

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The Denavit-Hartenberg transformation matrices below describe the successive joints from 1 to 6 for a particular robot manipulator with end-effector. Ca -Sa 0 0.5Cα 0 1 0 0 -Sẞ 0 CB 0 Sa Ca 0 0.5Sa 1 0 0 0.45 CB 0 sp 0 A₁ A₂ = A3 0 0 1 0.3 00-1 a 0 1 0 0.25 0 0 0 1 00 0 1 0 0 0 1 A4 1 0 0 0 -Cy 0 -Sy 0 C8 -S8 0 0 0 0 -1 0 -S7 0 Cy 0 SE Сб 0 0 = A₂ A6 0 1 0 b 0 1 0 0 0 0 1 0.1 0 0 0 1 0 00 1 0 00 1 Here a and b are prismatic joint variables and α, ß, y and are revolute joint variables. For a particular position control operation of this manipulator the following joint variable values are employed: α=77°, a = 0.75m, ß = 22°, b = 0.45m, y = 43°, &= 167° (a) Using these joint values, derive the T6 Jacobian for the robot manipulator's end- effector defined by the transformation matrices given above. (b) [13] Hence or otherwise, calculate the resulting differential motion of the end- effector from the following differential motions of the joints: De [-2, 0.05, -5, 0.05, 2, -3]T [2] The Denavit-Hartenberg transformation matrices below describe the successive joints from 1 to 6 for a particular robot manipulator with end-effector. Ca -Sa 0 0.5Cα 0 1 0 0 -Sẞ 0 CB 0 Sa Ca 0 0.5Sa 1 0 0 0.45 CB 0 sp 0 A₁ A₂ = A3 0 0 1 0.3 00-1 a 0 1 0 0.25 0 0 0 1 00 0 1 0 0 0 1 A4 1 0 0 0 -Cy 0 -Sy 0 C8 -S8 0 0 0 0 -1 0 -S7 0 Cy 0 SE Сб 0 0 = A₂ A6 0 1 0 b 0 1 0 0 0 0 1 0.1 0 0 0 1 0 00 1 0 00 1 Here a and b are prismatic joint variables and α, ß, y and are revolute joint variables. For a particular position control operation of this manipulator the following joint variable values are employed: α=77°, a = 0.75m, ß = 22°, b = 0.45m, y = 43°, &= 167° (a) Using these joint values, derive the T6 Jacobian for the robot manipulator's end- effector defined by the transformation matrices given above. (b) [13] Hence or otherwise, calculate the resulting differential motion of the end- effector from the following differential motions of the joints: De [-2, 0.05, -5, 0.05, 2, -3]T [2]