## Abstract

Abstract. Motivated by the path planning problem for robotic systems this paper

considers nonholonomic path planning on the Euclidean group of motions SE(n)

which describes a rigid bodies path in n-dimensional Euclidean space. The problem is formulated as a constrained optimal kinematic control problem where the cost function to be minimised is a quadratic function of translational and angular velocity inputs. An application of the Maximum Principle of optimal control leads to a set of Hamiltonian vector field that define the necessary conditions for optimality and consequently the optimal velocity history of the trajectory. It is illustrated that the systems are always integrable when n = 2 and in some cases when n = 3. However, if they are not integrable in the most general form of the cost function they can be rendered integrable by considering special cases. This implies that it is possible to reduce the kinematic system to a class of curves defined analytically. If the optimal motions can be expressed analytically in closed form then the path planning problem is reduced to one of parameter optimisation where the parameters are optimised to match prescribed boundary conditions.This reduction procedure is illustrated for a simple wheeled robot with a sliding constraint and a conventional slender underwater vehicle whose velocity in the lateral directions are constrained due to viscous damping.

considers nonholonomic path planning on the Euclidean group of motions SE(n)

which describes a rigid bodies path in n-dimensional Euclidean space. The problem is formulated as a constrained optimal kinematic control problem where the cost function to be minimised is a quadratic function of translational and angular velocity inputs. An application of the Maximum Principle of optimal control leads to a set of Hamiltonian vector field that define the necessary conditions for optimality and consequently the optimal velocity history of the trajectory. It is illustrated that the systems are always integrable when n = 2 and in some cases when n = 3. However, if they are not integrable in the most general form of the cost function they can be rendered integrable by considering special cases. This implies that it is possible to reduce the kinematic system to a class of curves defined analytically. If the optimal motions can be expressed analytically in closed form then the path planning problem is reduced to one of parameter optimisation where the parameters are optimised to match prescribed boundary conditions.This reduction procedure is illustrated for a simple wheeled robot with a sliding constraint and a conventional slender underwater vehicle whose velocity in the lateral directions are constrained due to viscous damping.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science |

Subtitle of host publication | Towards Autonomous Robotic Systems |

Publisher | Springer |

Number of pages | 12 |

Volume | 6856 |

Edition | 1st |

ISBN (Print) | 978-3-642-23231-2 |

Publication status | Published - 17 Aug 2011 |

Event | 12th Conference Towards Autonomous Robotic Systems 2011 - Sheffield , United Kingdom Duration: 31 Aug 2011 → 2 Sep 2011 |

### Conference

Conference | 12th Conference Towards Autonomous Robotic Systems 2011 |
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Country/Territory | United Kingdom |

City | Sheffield |

Period | 31/08/11 → 2/09/11 |

## Keywords

- algorithmic learning
- autonomous robots
- mobile robot navigation
- personal robots
- robot agents
- robot emotions
- robot routing
- artificial intelligence
- HCI
- image processing