A new type of the manipulator dynamics equations for the synthesis of adaptive PID controller for program gripping

A new type of equations for the plane revolute-joints manipulator dynamics is derived according to the Appell's formalism in which the partial derivative of the acceleration energy is taken with respect to the coordinate acceleration, and on the left side of the equations, the parameters of the manipulator gripper position and their second time derivatives are received by identical transformations. This type of dynamics equations allows synthesizing an adaptive PID controller with a programmed gripping without solving the inverse kinematics problem. We consider an example of writing such equation and synthesizing, on its basis, an adaptive PID controller with a program gripping. In this example, the kinematic structure of the manipulator is a special case of the structure of a six-link manipulator with an angular coordinate system where certain joints are fixed, which corresponds to manipulations in the vertical plane. In conclusion, we describe how the obtained results can be extended to arbitrary manipulators.