For the linear tracking simulation, from the initial position of the end-effector, for tracking the trajectory of a point moving to a target (
X = 0.2 m,
Y = 0 m,
Z = 0 m) positioned on a horizontal line, as depicted in
Fig. 8, the trajectory of a point moving to a target (
X = 0.1 m,
Y = 0 m,
Z = −0.2 m) positioned diagonally, as presented in
Fig. 9, was determined.
The plots of the end-effector position and the position error of the horizontal tracking presented in
Fig. 8 and of the diagonal tracking in
Fig. 9 are indicated in
Figs. 10 and
11, respectively. The two figures in the upper part of
Fig. 10 display the position and position errors in the
X and
Z directions, respectively. In the
X and
Z directions, the maximum position errors are approximately 3.3 mm and 1 mm or less, respectively. After the movement, the position error in the steady state is 0.1 mm in the
X direction and 0.8 mm in the
Z direction. The figures in the upper part of
Fig. 11 display the position and position errors in the
X direction and
Z directions. The maximum position error in the
X direction is approximately 2.3 mm, and it is 2 mm in the
Z direction. After the movement, the position error in the steady state is 0.1 mm in the
X direction and 0.8 mm in the
Z direction. After completing the movement in both the horizontal and vertical directions, the error in the
Z direction is greater than that in the
X direction. As the manipulator moves further away, the length of the moment arm increases at a particular gravity, and the torque loaded on the specific joint increases. With the increase in the torque, the position error of the joint increases. However, as expressed in
Eq. (7), when there is an error in
qd, the target position of the joint, and
q, the joint position. When the velocity input increases to compensate the error, the error becomes less than 1 mm in the steady state in the
Z-direction. During the movement, the end-effector velocity is determined in proportion to the position and error of the point to be tracked by the manipulator, as written in
Eq. (9). Accordingly, it can be estimated that a position error will occur in proportion to the acceleration of the object to be tracked, which can be seen by comparing the position error variation in the
X direction with acceleration and that in the
Z direction without acceleration. Compared to the
X direction position error graph, in the
Z direction position error graph, since there is no acceleration of the object, we can see that the position error is small.