We consider the problem of finding optimal gaits for a quadruped robot. Paths are sought which minimize the actuation energy required for walking in an attempt to approximate natural motion. The number of possible gaits for a quadruped is quite large when one considers varied orders of leg motion, different liftoff times, and various ground contact combinations for the legs. The problem is treated as a fully nonlinear optimal hybrid path planning problem on a 22dimensional state space. Modeling aspects, our numerical approach, and experimental results are discussed in this paper.

. We investigate nonlinear programs that have a nonempty but possibly unbounded Lagrange multiplier set and that satisfy the quadratic growth condition. We show that such programs can be transformed, by relaxing the constraints and adding a linear penalty term to the objective function, into equivalent nonlinear programs that have differentiable data and a bounded Lagrange multiplier set and that satisfy the quadratic growth condition. As a result we can define, for this type of problem, algorithms that are linearly convergent, using only first-order information, and superlinearly convergent. 1. Introduction. Recently, there has been renewed interest in analyzing and modifying sequential quadratic programming (SQP) algorithms for constrained nonlinear optimization for cases where the traditional regularity conditions do not hold [5, 14, 13, 25, 30]. This research is partly motivated by the fact that large-scale nonlinear programming problems tend to be almost degenerate (have large co...

We examine the influence of optimality measures on the benchmarking process, and show that scaling requirements lead to a convergence test for nonlinearly constrained solvers that uses a mixture of absolute and relative error measures. We show that this convergence test is well behaved at any point where the constraints satisfy the Mangasarian-Fromovitz constraint qualification and also avoids the explicit use of a complementarity measure. Our computational experiments explore the impact of this convergence test on the benchmarking process with performance profiles. 1

The paper presents a data and task paxallel environment for parallelizing low-level image processing applications on distributed memory systems. Image processing operators axe paxallelized by data decomposition using algorithmic skeletons. At the application level we use task decomposition, based on the Image Application Task Graph.

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. 1

In this thesis, we study optimal anytime stochastic search algorithms (SSAs) for solving general constrained nonlinear programming problems (NLPs) in discrete, continuous and mixed-integer space. The algorithms are general in the sense that they do not assume di#erentiability or convexity of functions. Based on the search algorithms, we develop the theory of SSAs and propose optimal SSAs with iterative deepening in order to minimize their expected search time. Based on the optimal SSAs, we then develop optimal anytime SSAs that generate improved solutions as more search time is allowed. Our SSAs

Numerical simulation and optimization of gaits for quadruped robots based on nonlinear multibody dynamics models of legged locomotion have made progress recently. A fully threedimensional dynamical model of Sony’s four-legged robot is used to state an optimal control problem for a symmetric, dynamically stable gait. The optimal control problem is solved by a sparse direct collocation method. Numerical problems related to the high-index differential algebraic equations of motion are avoided by substituting the differential algebraic equations by an equivalent set of reduced dynamics ordinary differential equations. Numerical and experimental results validate the model and the methods used for gait generation. 1

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In the present work we explore an adaptive discretization scheme for dynamic optimization problems formulated on moving horizons. The proposed method is embedded into a solution methodology where the dynamic optimization problem is approximated by a hierarchy of successively rened nite dimensional problems. Information on the solution of the coarser approximations is used to initialize the employed NLP solver and to construct a fully adaptive, problem dependent discretization where the nite dimensional spaces are spanned by biorthogonal wavelets arising from B-splines. We demonstrate exemplarily that the proposed strategy is capable to identify accurate discretization meshes which are more economical than uniform meshes with respect to the ratio of approximation quality vs. number of trial functions used. Keywords: dynamic optimization, realtime optimization, model predictive control, wavelets, adaptive discretization 1

In recent years, much work has been done on implementing a variety of algorithms in nonlinear programming software. In this paper, we analyze the performance of several stateof -the-art optimization codes on large-scale nonlinear optimization problems. Extensive numerical results are presented on di#erent classes of problems, and features of each code that make it e#cient or ine#cient for each class are examined. 1.