Dynamics and control of a chain pendulum on a cart

A geometric form of Euler-Lagrange equations is developed for a chain pendulum, a serial connection of n rigid links connected by spherical joints, that is attached to a rigid cart. The cart can translate in a horizontal plane acted on by a horizontal control force while the chain pendulum can undergo complex motion in 3D due to gravity. The configuration of the system is in (S
²
)
ⁿ
×ℝ
²
. We examine the rich structure of the uncontrolled system dynamics: the equilibria of the system correspond to any one of 2
ⁿ
different chain pendulum configurations and any cart location. A linearization about each equilibrium, and the corresponding controllability criterion is provided. We also show that any equilibrium can be asymptotically stabilized by using a proportional-derivative type controller, and we provide a few numerical examples.