Geometry and Direct Kinematics to MP3R with 4×4 Operators

Abstract

Anthropomorphic mechatronic systems are the most widely used robotics systems worldwide today in industry and in all automated environments. These systems are best suited to the modern automation and mechatronisation needs of the modern world, being mobile, dynamic, light, robust, complex, technologically simple, easy to design and manufactured, implemented, maintained and used in almost any industrial site, both in machine building and in special environments, such as chemical, toxic, dyeing, underwater, nuclear, in space.... Anthropomorphic robots are flexible, dynamic, stable, lightweight, fast, fast, inexpensive, easy-to-install, mechanical, mechanical, mechanical and mechanical systems with a pleasant appearance, modern industrial design and easy to design and implement in any workplace, imposed. In this study we will present the 4×4 operators and the way they can be implemented and used in the complex matrix calculations in order to simplify the complex matrix algebraic numerical methods. 4×4 operators are designed to simplify algebraic matrix calculations by making difficult matrix operations simpler and easier to approach. The method of using 4×4 operators is meant to introduce a degree in addition to the matrices to facilitate algebraic operations. The kinematics of serial manipulators and robots will be exemplified for the 3R kinematic model. The fixed coordinate system was denoted by x₀O₀y₀z₀. The mobile systems (rigid) of the three mobile elements (1, 2, 3) have indices 1, 2 and 3. Their orientation has been chosen conveniently. Known kinematic parameters in the direct kinematics are the absolute rotation angles of the three moving elements: φ₁₀, φ₂₀, φ₃₀, angles related to the rotation of the three actuators (electric motors) mounted in the kinematic rotation couplers. The output parameters are the three absolute coordinates x_M, y_M, z_M of point M, i.e., the kinematic parameters (coordinates) of the endeffector (the actuator element (the final), which can be a grasping hand, a solder tip, painted , cut, etc ...). The 3×3 matrix is transformed into 4×4 (it is a mathematical operator) by adding two zero vectors (formed from three elements 0), one line and the other column, and adding one element 1 to the main diagonal (the last element). The matched T₀₁ matrix becomes T₀₁⁴. The column vector matrix (consisting of three elements) undergoes a minimal transformation receiving a fourth fixed value 1 if it is only used for matrix products. The convenient form of matrix A₁₂ is A₁₂^c.