{
  "id": "1211.4604",
  "version": "v1",
  "published": "2012-11-19T22:00:59.000Z",
  "updated": "2012-11-19T22:00:59.000Z",
  "title": "Dynamics and Control of a Chain Pendulum on a Cart",
  "authors": [
    "Taeyoung Lee",
    "Melvin Leok",
    "N. Harris McClamroch"
  ],
  "comment": "7 pages, 4 figures",
  "categories": [
    "math.OC"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-19T22:00:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "equilibrium",
      "chain pendulum configurations",
      "undergo complex motion",
      "horizontal control force",
      "rich structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.4604L"
    }
  }
}