The closed form solution of the inverse kinematic problem of 3r positioning manipulators

In this thesis we contribute to the inverse kinematics solution of serial positional manipulators with three rotational joints (3R). This type of linkage is frequently used for 6R manipulators, where three – mostly the last – joint axes intersect at one point, decoupling with this structure the positional from the orientational part of the forward kinetics. In order to solve the inverse kinematics problem, i.e. to find the joint values for a given point in the work (task) space, we introduce a geometrical approach. The "core equation" obtained this way helps us not just to find the inverse kinematics solutions but also to select the Denavit-Hartenberg parameters in order to satisfy our requirements concerning different properties (accessibility characteristics, ability of the manipulator to change posture without meeting a singularity, stability of kinematic properties and solvability). During our work, the occurring singularities became a crucial issue, therefore we devote the second part to their analysis. We deal with joint (configuration) space singularities, because they contain more information (e.g. on singularity crossing) than their images in the work space and their analysis is more straightforward since direct kinematics is much simpler than inverse kinematics. Besides these calculations, which form the background for further analysis, we also prove a theorem about the accessibility characteristics around singular points of higher multiplicity, which was stated in the literature and the numerical calculations supported it too, but the proof was not found. Furthermore we present several algorithms, like calculating the singularities in the joint space without artifacts (which occur at discontinuities), determining special type singularities and calculating efficiently the pseudo-singular points. Finally we turn to the classification of 3R-type manipulators, which is still not a well discovered field, in spite of its importance. Applying the extended resultant theory we are able to get an accurate description of the boundaries representing non-generic manipulators, which separate different generic manipulator classes, where genericity means stable kinematics properties.