Continuum of motion equations and control laws for underactuated mechanical systems

Abstract

As sizes, lengths, or shapes of a system grow large or shrink to zero, a system will approach limiting forms. As the parameter is allowed to grow or shrink, the system could resemble a simpler system. The sufficient conditions for when the equations of motion will morph from the original system to a target system will be presented. The ball and arc equations of motion morph to those of the ball and beam as the arc’s radius is allowed to grow. The equations of motion for the rotary pendulum, pendubot, and two-link robot manipulator will morph to the equations of motion of the inverted pendulum cart. The effect of a parameter growing large or shrinking to zero has on the controller for the original system will not be fully investigate in this work. A case for when controller morphing might be possible will be examined. A controller for the rotary pendulum will morph to a controller that stabilizes the inverted pendulum cart. Next, a controller for the pendubot will be morphed that does not stabilize the dimensionless inverted pendulum cart. Lastly, a controller for a fully actuated two-link robot manipulator will be morphed to a stabilizing controller for a fully actuated inverted pendulum cart.