The Dynamics of a Four Degree of Freedom Gimballed Platform

摘要

The platform here considered is cradeled within three gimbals whose purpose is to provide the various degrees of freedom desired. Two approaches are used to obtain the equation of motion: The torque-momentum approach, and the application of Lagranges equations. In the torque-momentum approach, Euler's equations are obtained for each rigid body of the system, with action torques on each body manifesting themselves as reaction torques on the adjacent body. A combination of all the equations then essentially puts the system together to form the system equations. Such a method is involved, but more readily reveals the dynamics of the inner workings of the system than does the application of Lagranges equations. The system readily adapts itself to the use of Lagranges equations, which are simplified by the constant potential characteristic. The system possesses four degrees of freedom, therefore four equations of motion emerge. Since the system is described in terms of displacements about the bearing axes, such displacements become naturally the generalized coordinates in Lagranges equations. The generalized torques then represent the friction type torques along the various bearing axes. Equations obtained by the torque-momentum approach do not appear initially as friction type torque equations, rather they manifest inertial reactions of the system to other components of torque. However an appropriate grouping of these equations to the friction type torque form readily admits comparison with the results of the Lagrange approach. Equations presented herein have been so adjusted and checked.