Abstract. Nonlinear hybrid dynamical systems are the main focus of this paper. A modeling framework is proposed, feedback control strategies and numerical solution methods for optimal control problems in this setting are introduced, and their implementation with various illustrative applications are presented. Hybrid dynamical systems are characterized by discrete event and continuous dynamics which have an interconnected structure and can thus represent an extremely wide range of systems of practical interest. Consequently, many modeling and control methods have surfaced for these problems. This work is particularly focused on systems for which the degree of discrete/continuous interconnection is comparatively strong and the continuous portion of the dynamics may be highly nonlinear and of high dimension. The hybrid optimal control problem is defined and two solution techniques for obtaining suboptimal solutions are presented (both based on numerical direct collocation for continuous dynamic optimization): one fixes interior point constraints on a grid, another uses branch-and-bound. These are applied to a robotic multi-arm transport task, an underactuated robot arm, and a benchmark motorized traveling salesman problem.

After tooth extraction, the alveolar bone undergoes a remodeling process, which leads to horizontal and vertical bone loss. These resorption processes complicate dental rehabilitation, particularly in connection with implants. Various methods of guided bone regeneration (GBR) have been described to retain the original dimension of the bone after extraction. Most procedures use filler materials and membranes to support the buccal plate and soft tissue, to stabilize the coagulum and to prevent epithelial ingrowth. It has also been suggested that resorption of the buccal bundle bone can be avoided by leaving a buccal root segment (socket shield technique) in place, because the biological integrity of the buccal periodontium (bundle bone) remains untouched. This method has also been described in connection with immediate implant placement. The present case report describes three consecutive cases in which a modified method was applied as part of a delayed implantation. The latter was carried out after six months, and during re-entry the new bone formation in the alveolar bone and the residual ridge was clinically evaluated as proof of principle. It was demonstrated that the bone was clinically preserved with this method. Possibilities and limitations are discussed and directions for future research are disclosed.

Numerical methods for optimal control of hybrid dynamical systems are considered where the discrete dynamics and the nonlinear continuous dynamics are tightly coupled. A decomposition approach for numerically solving general mixed-integer continuous optimal control problems (MIOCPs) is discussed. In the outer optimization loop a branch-and-bound binary tree search is used for the discrete variables. The multiple-phase optimal control problems for the continuous state and control variables in the inner optimization loop are solved by a sparse direct collocation transcription method. A genetic algorithm is applied to improve the performance of the branch-and-bound approach by providing a good initial upper bound on the MIOCP performance index. Results are presented for motorized traveling salesmen problems, new benchmark problems in hybrid optimal control.

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Based on a nonlinear hybrid dynamical systems model a new planning method for optimal coordination and control of multiple unmanned vehicles is investigated. The time dependent hybrid state of the overall system consists of discrete (roles, actions) and continuous (e.g. position, orientation, velocity) state variables of the vehicles involved. The evolution in time of the system's hybrid state is described by a hybrid state automaton. The presented approach enables a tight and formal coupling of discrete and continuous state dynamics, i.e. of dynamic role and action assignment and sequencing as well as of the physical motion dynamics of a single vehicle modeled by nonlinear differential equations. The planning problem of determining optimal hybrid state trajectories that minimize a cost function as time or energy for optimal multi-vehicle cooperation subject to constraints including the vehicle's motion dynamics is transformed to a mixed-binary dynamic optimization problem being solved numerically. The numerical method consists of an inner iteration where multiphase optimal control problems are solved using a direct collocation method and an outer iteration based on a branch-and-bound search of the discrete solution space. The approach presented in this paper is applied to the scenarios of optimal simultaneous waypoint or target sequencing and dynamic trajectory planning for a team of unmanned aerial vehicles in a plane and to optimal role assignment and physics-based trajectories in robot soccer.Keywords: multi-vehicle task allocation and trajectory planning, nonlinear hybrid dynamical systems, mixed-integer optimal control, multiple motorized salesmen problem