Solution of the closed-loop inverse kinematics algorithm using the Crank-Nicolson method

The closed-loop inverse kinematics algorithm is a numerical method used to approximate the solution of the inverse kinematics problem of robot manipulators based on the explicit Euler integration, that is a simple numerical integration technique. It is known from the theory of numerical techniques that the Crank-Nicolson method gives results with better convergence properties than the explicit Euler method, however it needs the application of the implicit Euler method, that needs the solution of the differential inverse kinematics problem, which is not known in advance in the robotics problem. An iterative method is proposed to calculate the implicit Euler solution, and the results are used to carry out the numerical integration using the Crank-Nicolson method. Experimental results show that the application of the Crank-Nicolson method yields better convergence for the differential inverse positioning problem of planar nonredundant and redundant manipulators than the application of the explicit Euler method. Both the case of reaching a single end effector point and path tracking are analyzed and the results are compared to the results acquired using the explicit and implicit Euler methods separately; the convergence is faster if the goal is to reach a desired end effector configuration, while the tracking error is smaller in the case of path tracking if the numerical integration is done in the proposed way.