On the control of a two-dimensional multi-link inverted pendulum: the form of the dynamic equations from choice of co-ordinate system

Abstract

The inverted pendulum is a mechanical idealization which poses an interesting and, with an arbitrary level of degree of freedom, difficult problem in control. This paper, noting that the model may be taken as a simple two-dimensional approximation to a robot manipulator, proceeds to investigate how the relative form and complexity of the dynamic equations of motion, fundamental to the application of control theory, may be dictated by choice of co-ordinate system. Two such systems, both appropriate from the nature of the problem, are compared in a theoretical formulation using the method of Lagrange, and it is shown that one—the commonly accepted robotic kinematic notation of Denavit and Hartenberg—leads here to an unnecessarily complex set of closed form equations. It is also demonstrated how additional physical features may be incorporated into the equations to yield an improved approximation, and so a modified control problem.