Closed-Loop Inverse Kinematics Algorithm with Implicit Numerical Integration

The closed-loop inverse kinematics algorithm is a numerical approximation of the solution of the inverse kinematics problem, which is a central problem of robotics. The accuracy of this approximation, i.e. the convergence of the numerical solution to the real solution can be increased by increasing the value of a feedback gain parameter. However, this can lead to unstable operation if the stability margin is reached. The accuracy of the closedloop inverse kinematics algorithm is increased here by replacing the numerical integration with second-order and implicit numerical integration techniques. The application of implicit Euler, explicit trapezoid, implicit trapezoid and the weighted average method is considered, and an iteration is presented to calculate the implicit solutions. Simulation results show that implicit second-order methods give the best results. However, they decrease the stability margin due to the iteration required to calculate the implicit solution. The stability margin of the algorithms with different numerical integration techniques is analyzed, and it turns out that the implicit trapezoid method has the most desirable properties.