In this paper we propose a new technique for verification by simulation of continuous and hybrid dynamical systems with uncertain initial conditions. We provide an algorithmic methodology that can, in most cases, verify that the system avoids a set of bad states by conducting a finite number of simulation runs starting from a finite subset of the set of possible initial conditions. The novelty of our approach consists in the use of sensitivity analysis, developed and implemented in the context of numerical integration, to efficiently characterize the coverage of sampling trajectories.

We introduce the notion of low-discrepancy curves and use it to solve the problem of optimally covering space. In doing so, we extend the notion of low-discrepancy sequences in such a way that sufficiently smooth curves with low discrepancy properties can be defined and generated. Based on a class of curves that cover the unit square in an efficient way, we define induced low discrepancy curves in Riemannian spaces. This allows us to efficiently cover an arbitrarily chosen abstract surface that admits a diffeomorphism to the unit square. We demonstrate the application of these ideas by presenting concrete examples of low-discrepancy curves on some surfaces that are of interest in robotics.

ABSTRACT The Fast Fourier Transform (FFT) correlation approach to protein-protein dock-ing can evaluate the energies of billions of docked conformations on a grid if the energy is de-scribed in the form of a correlation function. Here, this restriction is removed, and the approach is efficiently used with pairwise interactions potentials that substantially improve the docking results. The basic idea is approximating the interaction matrix by its eigenvectors correspond-ing to the few dominant eigenvalues, resulting in an energy expression written as the sum of a few correlation functions, and solving the problem by repeated FFT calculations. In addi-tion to describing how the method is implemented, we present a novel class of structure based pairwise intermolecular potentials. The DARS (Decoys As the Reference State) potentials are extracted from structures of protein-protein complexes and use large sets of docked conforma-tions as decoys to derive atom pair distributions in the reference state. The current version of the DARS potential works well for enzyme-inhibitor complexes. With the new FFT-based program, DARS provides much better docking results than the earlier approaches, in many cases generating 50 % more near-native docked conformations. Although the potential is far from optimal for antibody-antigen pairs, the results are still slightly better than those given by an earlier FFT method. The docking program PIPER is freely available for non-commercial applications. Key words: Fast Fourier Transform; rigid body docking; intermolecular potentials; structure based potentials; scoring funtion.

Abstract. The problem of generating uniform deterministic samples over the rotation group, SO(3), is fundamental to many fields, such as computational structural biology, robotics, computer graphics, astrophysics. We present the best-known method to date for constructing incremental, deterministic grids on SO(3); it provides the: 1) lowest metric distortion for grid neighbor edges, 2) optimal dispersionreduction with each additional sample, 3) explicit neighborhood structure, and 4) equivolumentic partition of SO(3) by the grid cells. We also demonstrate the use of the sequence on motion planning problems. 1

In its original formulation, the motion planning problem considers the search of a robot path from an initial to a goal configuration. The study of motion planning has advanced significantly in recent years, in large part due to the development of highly successful sampling and searching techniques. Recent advances have influenced sampling-based motion planning algorithms to be used in disparate areas such as humanoid robotics, automotive manufacturing, architecture, computational geography, computer graphics, and computational biology. Many of these methods work well on a large set of problems, however, they have weaknesses and limitations. This thesis expands the basic motion planning techniques to include critical concerns that are not covered by the motion planning algorithms that are in widespread use now. The technical approach is organized around three main thrusts: 1) the development of efficient nearest neighbor searching techniques for spaces arising in motion planning; 2) the development of uniform sampling techniques on these spaces to allow resolution completeness in sampling-based planning algorithms; and 3) the development of guided sampling techniques for efficient exploration on such spaces. We show that most of the modern motion planners incorporate one or more of these components; therefore, addressing these core issues in motion planning does not only lead to a more fundamental understanding of the problem, but also to

correlation approach to protein–protein docking can evaluate the energies of billions of docked conformations on a grid if the energy is described in the form of a correlation function. Here, this restriction is removed, and the approach is efficiently used with pairwise interaction potentials that substantially improve the docking results. The basic idea is approximating the interaction matrix by its eigenvectors corresponding to the few dominant eigenvalues, resulting in an energy expression written as the sum of a few correlation functions, and solving the problem by repeated FFT calculations. In addition to describing how the method is implemented, we present a novel class of structure-based pairwise intermolecular potentials. The DARS (Decoys As the Reference State) potentials are extracted from structures of protein–protein complexes and use large sets of docked conformations as decoys to derive atom pair distributions in the reference state. The current version of the DARS potential works well for enzyme–inhibitor complexes. With the new FFT-based program, DARS provides much better docking results than the earlier approaches, in many cases generating 50 % more near-native docked conformations. Although the potential is far from optimal for antibody–antigen pairs, the results are still slightly better than those given by an earlier FFT method. The docking program PIPER is freely available for noncommercial applications. Proteins 2006;65:392–406. VC 2006 Wiley-Liss, Inc. Key words: Fast Fourier Transform; rigid body docking; intermolecular potentials; structure based potentials; scoring function