This paper presents a new method of solving the inverse kinematics problem of manipulators with the help of the biquaternion theory of kinematics control. Application of the method reduces solving of 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 generalized coordinates. 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 the inverse kinematics problem will be solved. In this paper an algorithm for solving the inverse kinematics of Stanford robot arm is introduced. Control laws used in the algorithm are valid for any manipulator. A numerical solution of the inverse kinematics problem of Stanford robot arm has been found. It proves efficiency of application of the biquaternion theory of kinematics control for solving of the inverse kinematics problem of manipulators. Given examples of the numerical solution demonstrate dependency between the solution results (obtained values of the phase coordinates, solution time) and the input parameters, such as initial pose (position and orientation) of the end effector of a manipulator, accuracy of the solution and dual feedback gain. Graphs of the changes of the generalized coordinates, the main and moment parts of the biquaternion of the end effector error pose, the main and moment parts of the control and tensor were built.