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.
We solve the problem of generating symmetric, periodic minimum energy gaits for a 5-link biped robot moving in the sagittal plane of forward motion. We seek to approximate natural walking motion through the minimization of actuation energy. The model we use has considerably more structure than those previously studied. This forces us to a fully nonlinear minimum energy path planning problem on a 14dimensional state space. Also a large number of constraints must be considered, including contact and collision effects. Our solution required development of various symbolic, dynamical algorithms relating to multibody systems and use of powerful numerical optimal control software. Solving the minimum energy walking problem including saturation and algebraic constraints amounts to solving a Hamilton-Jacobi-Bellman type equation along the optimal path. We use the path planning software DIRCOL which can handle the many constraints, as well as the high degree of nonlinearity and high dimensionality. A newly available version of this software provided a substantial decrease in computing time required for generating solutions. We discuss numerical optimization and other modeling issues in this paper.
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