Solution to the Problems of Direct and Inverse Kinematics of the Robots-Manipulators Using Dual Matrices and Biquaternions on the Example of Stanford Robot Arm. Part 1

Solution to the problems of direct kinematics with the help of dual cosine matrices and Klifford biquaternions is demonstrated on the example of Stanford robot arm. Derivation of the kinematics equations is performed. The obtained kinematics equations and the solution to the direct kinematics problem are used for solving the inverse kinematics problem. The new method of solving the inverse kinematics problem is based on biquaternion theory of kinematics control of free rigid body motion by using the feedback principal. Application of the method reduces solving. Cauchy problem for differential kinematic equations of a manipulator motion. Vectors of the angular and linear velocities contained in these equations are considered as controls. They are formed according to the feedback principal as certain functions of the generalized coordinates so that every chosen end effector position is asymptotically stable in the whole. In this case any particular solution to the differential kinematics equations will aspire in asymptotically stable way to the desired point in the space of the generalized coordinates corresponding to the target position of the end effector of a manipulator. As the result of solving of Cauchy problem for any given initial values of the generalized coordinates from their operational range the generalized coordinates, will finally take the values corresponding to the desired position of the end effector, so that the inverse kinematics problem will be solved. The advantages of the new method are the following: the method gives a unique solution (if there is such) for the chosen control law and given initial position; it ensures high accuracy solutions and high performance; but, above all, it is non-iterative. The paper extends and supplements the results presented in [1, 2].

робот-манипулятор, прямая задача кинематики, матрица дуальных направляющих косинусов, бикватернион, кинематические уравнения, обратная задача кинематики, robot-manipulator, Stanford robot arm, direct kinematics problem, dual direction cosine matrix, biquaternion, kinematics equations, inverse kinematics problem

1. Ломовцева Е. И., Челноков Ю. Н. Дуальные матричные и бикватернионные методы решения прямой и обратной задач кинематики роботов-манипуляторов на примере стэнфордского манипулятора. I // Изв. Сарат. ун-та. Сер. Математика. Механика. Информатика. 2013. Т. 13., вып. 4. С. 82-89.

2. Ломовцева Е. И., Челноков Ю. Н. Дуальные матричные и бикватернионные методы решения прямой и обратной задач кинематики роботов-манипуляторов на примере стэнфордского манипулятора. II // Изв. Сарат. ун-та. Сер. Математика. Механика. Информатика. 2014. Т. 14., вып. 1. С. 88-95.

3. Челноков Ю. Н. Бикватернионное решение кинематической задачи управления движением твердого тела и его приложение к решению обратных задач кинематики роботов-манипуляторов // Изв. РАН. Механика твердого тела. 2013. № 1. С. 38-58.

4. Фу К., Гонсалес Р., Ли К. Робототехника. М.: Мир, 1989. 621 с.

5. Челноков Ю. Н. Кватернионные и бикватернионные модели и методы механики твердого тела и их приложения. Геометрия и кинематика движения. М.: ФИЗМАТЛИТ. 2006. 511 с.