ANALYTICAL SOLUTION OF THE INVERSE AND DIRECT KINEMATICS PROBLEM FOR THE SPATIAL ANGULAR MANIPULATOR ABB WITH SUBSEQUENT 3D MODELING

This article proposes an algorithm for solving the inverse kinematics problem of a six-degree angular manipulator based on the ABB robot model. The algorithm is based on analytical derivations obtained by the authors, which consist of a sequence of expressions that allow determining all six rotation angles of the kinematic pair parts that connect the chains by a given law of movement of the output chain – the gripper. Initially, the law of its movement – functions of time of Cartesian coordinates of the pole and orientation parameters – is determined from the conditions of performing a production task by the gripper. Any point of the gripper can be chosen as the pole. Quaternion or rotation matrix can be used as orientation parameters. It is shown how a rotation matrix and quaternion can be constructed according to the laws of change of Cartesian coordinates of three gripper points. A distinctive feature of the manipulator discussed here is the presence of a common rotation axis point for the last three chains. This allows determining the law of movement of this point from the law of movement of the gripper. After which the laws of change of rotation angles of the first three chains are uniquely determined. At the same time, the orientation task of these chains is solved. To determine the angles of the last three links, a matrix or quaternion equation is compiled, based on the equivalence of two sequences of chain rotations. The first sequence is a combination of already known rotations from the third chain through the second and first chain to the absolute coordinate system and, finally, from the absolute system to the gripper. The parameters of these rotations are known. The second sequence consists of 3 rotations at unknown angles of the last three chain from the third again to the gripper. These unknown angles are determined from such an equation. The presentation of the proposed algorithm takes into account the ABB manipulator but can be applied to manipulators of the PUMA class. The algorithm is implemented using a special computer algebra system KidyM and a developed C++ program, which uses OpenGL and SolidWorks engineering graphic packet that allow 3D visualization of the calculation results.