Performance Analysis of Numerical Integration Methods in the Trajectory Tracking Application of Redundant Robot Manipulators

Abstract

Differential kinematic is one of the most important solution methods in robot kinematics. The main advantage of the differential kinematic method is that it can be easily implemented any kind of mechanisms. Also, an accurate and efficient kinematic based trajectory tracking application can be easily implemented by using this method. In differential kinematic method, we use Jacobian as a mapping operator in the velocity space. Inversion of Jacobian matrix transforms the desired trajectory velocities, which are the linear and angular velocities of the end effector, into the joint velocities. The joint velocities are required to be integrated to obtain the pose of the robot manipulator. This integration can be evaluated by using numerical integration methods, since the inverse kinematic equations are highly complex and nonlinear. Therefore, the performance of the trajectory tracking application of the robot manipulator is directly affected by the chosen numerical integration method. This paper compares the performances of numerical integration methods in the trajectory tracking application of redundant robot manipulators. Several widely used numerical integration methods are implemented into the trajectory tracking application of the 7-DOF redundant robot manipulator named PA-10 and simulation results are given.

Closed-Loop Inverse Kinematics Algorithm with Implicit Numerical Integration

“…In the first case, the CLIK algorithm was solved using the explicit Euler [8], implicit Euler [10,11], explicit trapezoid [9] and implicit trapezoid [10,11] methods for different α feedback parameters. The initial value of α was zero, then it was increased by 0.1 in each step until it reached the value α = 21.…”

“…The authors in [9] considered the application of higher-order explicit methods. However, implicit methods may yield better tracking performance [10,11], thus we consider mostly implicit methods here, i.e.…”

“…becomes faster as α is increased, until the stability margin is reached, at which point the algorithm becomes unstable, thus the increase of the performance by changing the feedback parameter is limited. However, further increase in the tracking performance can be achieved by replacing the numerical integration step (5) by a different technique [9][10][11] that will be discussed in the upcoming sections.…”

“…in [9][10][11]. Second-order, explicit trapezoid method was proposed in [9], while implicit methods with an algorithm to calculate the implicit solutions were proposed in [10,11]. In [11], explicit Euler, implicit Euler, explicit trapezoid and implicit trapezoid methods were implemented and compared to each other, and it turned out that the trapezoid methods yield better performance, and the implicit trapezoid method has the best performance properties.…”

“…in [9][10][11]. Second-order, explicit trapezoid method was proposed in [9], while implicit methods with an algorithm to calculate the implicit solutions were proposed in [10,11].…”