Synthesis of a Parallel Structure Manipulator Control Algorithm

The work is devoted to the construction of a mathematical model of the dynamics of a manipulator of a parallel structure with three degrees of freedom, based on reducing the kinetic and potential energies of the manipulator to a quadratic form relative to three independent generalized coordinates and velocities. The mechanical system of the manipulator includes seven masses: the masses of three housings, the masses of three actuator rods and the concentrated mass of a spherical hinge with a gripper and a load. The lengths of the actuating links, the angles of rotation of the actuating cylinders relative to the absolute coordinate system, and the Cartesian coordinates of the spherical hinge, on which stationary holonomic connections are superimposed, are used as generalized coordinates of the manipulator. Using the Lagrange formalism with indefinite multipliers, taking into account holonomic connections, a system of twelve nonlinear differential equations with respect to twelve generalized coordinates and nine Lagrange multipliers is formed. The parameters of dynamic models are determined using the method of quadratic approximation of functions over a given time interval for each type of basic displacement. The resulting system of linear algebraic equations is used to synthesize optimal control forces. Using the methods of calculus of variations, optimal control forces are determined that ensure the program law of motion of the manipulator grip from the condition of minimal heat losses of the drive electric motors. The results of mathematical modeling when moving the manipulator grip along a spatial straight line are presented.