# American Institute of Mathematical Sciences

2011, 2011(Special): 312-321. doi: 10.3934/proc.2011.2011.312

## Invariant feedback design for control systems with lie symmetries - A kinematic car example

 1 Saarland University, Chair of Systems Theory and Control Engineering, Germany, Germany 2 TU Dresden, Institut für Regelungs- und Steuerungstheorie, Germany

Received  July 2010 Revised  February 2011 Published  October 2011

Invariant tracking control design for control systems in state representation with classical Lie point symmetry is considered. The relevance of the invariance aspect is motivated by an exemplary control design for the kinematic car. Introducing the perpendicular distance between the position of the rear axle midpoint and the reference trajectory in combination with the contouring error as tracking error, an invariant feedback w.r.t. the action of $SE$(2) is derived. Generally, the use of compatible tracking errors allows the application of well-known feedback design approaches yielding an error dynamics that is invariant w.r.t. the group action. A normalization procedure allowing the construction of invariant tracking errors is recalled for which a geometric interpretation is presented.
Citation: Carsten Collon, Joachim Rudolph, Frank Woittennek. Invariant feedback design for control systems with lie symmetries - A kinematic car example. Conference Publications, 2011, 2011 (Special) : 312-321. doi: 10.3934/proc.2011.2011.312
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