@phdthesis{Mauder2012,
  author    = {Mauder, Markus},
  title     = {Time-Optimal Control of the Bi-Steerable Robot: A Case Study in Optimal Control of Nonholonomic Systems},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-75036},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2012},
  abstract  = {In this thesis, time-optimal control of the bi-steerable robot is addressed. The bi-steerable robot, a vehicle with two independently steerable axles, is a complex nonholonomic system with applications in many areas of land-based robotics. Motion planning and optimal control are challenging tasks for this system, since standard control schemes do not apply. The model of the bi-steerable robot considered here is a reduced kinematic model with the driving velocity and the steering angles of the front and rear axle as inputs. The steering angles of the two axles can be set independently from each other. The reduced kinematic model is a control system with affine and non-affine inputs, as the driving velocity enters the system linearly, whereas the steering angles enter nonlinearly. In this work, a new approach to solve the time-optimal control problem for the bi-steerable robot is presented. In contrast to most standard methods for time-optimal control, our approach does not exclusively rely on discretization and purely numerical methods. Instead, the Pontryagin Maximum Principle is used to characterize candidates for time-optimal solutions. The resultant boundary value problem is solved by optimization to obtain solutions to the path planning problem over a given time horizon. The time horizon is decreased and the path planning is iterated to approximate a time-optimal solution. An optimality condition is introduced which depends on the number of cusps, i.e., reversals of the driving direction of the robot. This optimality condition allows to single out non-optimal solutions with too many cusps. In general, our approach only gives approximations of time-optimal solutions, since only normal regular extremals are considered as solutions to the path planning problem, and the path planning is terminated when an extremal with minimal number of cusps is found. However, for most desired configurations, normal regular extremals with the minimal number of cusps provide time-optimal solutions for the bi-steerable robot. The convergence of the approach is analyzed and its probabilistic completeness is shown. Moreover, simulation results on time-optimal solutions for the bi-steerable robot are presented.},
  subject      = {Mobiler Roboter},
  language  = {en}
}
@phdthesis{Jordan2008,
  author    = {Jordan, Jens},
  title     = {Reachable sets of numerical iteration schemes : a system semigroup approach},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-28416},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2008},
  abstract  = {We investigate iterative numerical algorithms with shifts as nonlinear discrete-time control systems. Our approach is based on the interpretation of reachable sets as orbits of the system semigroup. In the first part we develop tools for the systematic analysis of the structure of reachable sets of general invertible discrete-time control systems. Therefore we merge classical concepts, such as geometric control theory, semigroup actions and semialgebraic geometry. Moreover, we introduce new concepts such as right divisible systems and the repelling phenomenon. In the second part we apply the semigroup approach to the investigation of concrete numerical iteration schemes. We extend the known results about the reachable sets of classical inverse iteration. Moreover, we investigate the structure of reachable sets and systemgroup orbits of inverse iteration on flag manifolds and Hessenberg varieties, rational iteration schemes, Richardson's method and linear control schemes. In particular we obtain necessary and sufficient conditions for controllability and the appearance of repelling phenomena. Furthermore, a new algorithm for solving linear equations (LQRES) is derived.},
  subject      = {Nichtlineare Kontrolltheorie},
  language  = {en}
}