Kinematics as a science of geometry of motion describes motion by means of position, orientation, and their time derivatives. The focus of this article aims screw theory approach for the solution of inverse kinematics problem. The kinematic elements are mathematically assembled through screw theory by using only the base, tool, and workpiece coordinate systems-opposite to conventional Denavit-Hartenberg approach, where at least n þ 1 coordinate frames are needed for a robot manipulator with n joints. The inverse kinematics solution in Denavit-Hartenberg convention is implicit. Instead, explicit solutions to inverse kinematics using the Paden-Kahan subproblems could be expressed. This article gives step-by-step application of geometric algorithm for the solution of all the cases of Paden-Kahan subproblem 2 and some extension of that subproblem based on subproblem 2. The algorithm described here covers all of the cases that can appear in the generalized subproblem 2 definition, which makes it applicable for multiple movement configurations. The extended subproblem is used to solve inverse kinematics of a manipulator that cannot be solved using only three basic Paden-Kahan subproblems, as they are originally formulated. Instead, here is provided solution for the case of three subsequent rotations, where last two axes are parallel and the first one does not lie in the same plane with neither of the other axes. Since the inverse kinematics problem may have no solution, unique solution, or many solutions, this article gives a thorough discussion about the necessary conditions for the existence and number of solutions.
One of the most important factors which influence on the dynamical behavior of the linear motor servo drives for CNC machine tools is position loop gain or Kv factor. From the magnitude of the Kv-factor depends tracking or following error. In multi-axis contouring the following errors along the different axes may cause form deviations of the machined contours. Generally position loop gain Kv should be high for faster system response and higher accuracy, but the maximum gains allowable are limited due to undesirable oscillatory responses at high gains and low damping factor. Usually Kv factor is experimentally tuned on the already assembled machine tool. This paper presents a simple method for analytically calculation of the position loop gain Kv. A combined digital-analog model of the 4-th order of the position loop is presented. In order to ease the calculation, the 4-th order system is simplified with a second order model. With this approach it is very easy to calculate the Kv factor for necessary position loop damping. The difference of the replacement of the 4-th order system with second order system is presented with the simulation program MATLAB. Analytically calculated Kv factor is function of the nominal angular frequency and damping D of the linear motor servo drive electrical parts (motor and regulator), as well as sampling period T. :The influence of nonlinearities was taken with the correction factor. Our investigations have proven that experimentally tuned Kv factor differs from analytically calculated Kv factor less than 10%, which is completely acceptable
In this paper a model for the mechatronic position servo system with disturbance forces is presented. Static and dynamic stiffness for the proposed model is analyzed. An equation for analytical calculation of the static stiffness is developed. Correctness of the proposed equation is experimentally verified. Simulation of the influence of some parameters on the static and dynamic mechatronic position servo system stiffness is performed with simulation program MATLAB & SIMULINK.
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