Hierarchical trajectory refinement for a class of nonlinear systems

Trajectory generation for nonlinear control systems is an important and difficult problem. In this paper, we provide a constructive method for hierarchical trajectory refinement. The approach is based on the recent notion of φ-related control systems. Given a control affine system satisfying certain assumptions, we construct a φ-related control system of smaller dimension. Trajectories designed for the smaller, abstracted system are guaranteed, by construction, to be feasible for the original system. Constructive procedures are provided for refining trajectories from the coarser to the more detailed system

Foundations of multiple black hole evolutions

We present techniques for long-term, stable, and accurate evolutions of multiple-black-hole spacetimes using the `moving puncture' approach with fourth- and eighth-order finite difference stencils. We use these techniques to explore configurations of three black holes in a hierarchical system consisting of a third black hole approaching a quasi-circular black-hole binary, and find that, depending on the size of the binary, the resulting encounter may lead to a prompt merger of all three black holes, production of a highly elliptical binary (with the third black hole remaining unbound), or disruption of the binary (leading to three free black holes). We also analyze the classical Burrau three-body problem using full numerical evolutions. In both cases, we find behaviors distinctly different from Newtonian predictions, which has important implications for N-body black-hole simulations. For our simulations we use analytic approximate data. We find that the eighth-order stencils significantly reduce the numerical errors for our choice of grid sizes, and that the approximate initial data produces the expected waveforms (after a rescaling of the puncture masses) for black-hole binaries with modest initial separations.Comment: Revtex 4, 13 pages, 15 figure

Hierarchical Trajectory Generation for a Class of Nonlinear Systems

Trajectory generation and motion planning for nonlinear control systems is an important and difficult problem. In this paper, we provide a constructive method for hierarchical trajectory generation and hierarchical motion planning. The approach is based on the recent notion of φ-related control systems. Given a control affine system satisfying certain assumptions, we project the trajectory planning problem onto a φ-related control system of smaller dimension. Trajectories designed for the smaller, abstracted system are guaranteed, by construction, to be feasible for the original system. Constructive procedures are provided for refining trajectories of the coarser system to the more detailed system