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 $(\Sph^2)^n \times \Re^2$. 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. 
Taeyoung Lee, Melvin Leok, N. Harris McClamroch
math.OC