Non-natural metrics on the tangent bundle

Natural metrics provide a way to induce a metric on the tangent bundle from the metric on its base manifold. The most studied type is the Sasaki metric, which applies the base metric separately to the vertical and horizontal components. We study a more general class of metrics which introduces interactions between the vertical and horizontal components, with scalar weights. Additionally, we explicitly clarify how to apply our and other induced metrics on the tangent bundle to vector fields where the vertical component is not constant along the fibers. We give application to the Special Orthogonal Group SO(3) as an example.Published versio

Attitude control on manifolds via optimization and contractions with automatic gain tuning

The attitude (or orientation) of an object is often crucial in its ability to perform a task, whether the task is driving a car, flying an aircraft, or focusing a satellite. In traditional control approaches, the attitude is often parameterized by Euler angles or unit quaternions which exhibit problems such as gimbal lock or ambiguity in representation, respectively. These complications prevent the controllers from achieving global stability and worse they may cause real physical harm due to unexpected large motions. More recent works have achieved global stability and avoided these system failures by working directly on the configuration manifold, but these approaches are generally complex or lack automatic, user-friendly ways to tune them. The goal of this dissertation is to develop simple geometric attitude controllers that are globally, exponentially stable and can be automatically tuned. By simple, we mean that the controllers are computationally efficient for real time implementation on embedded computers and the tuning parameters have geometric interpretations. These properties make the controllers user friendly and practical for real hardware implementation even on fast dynamical systems. Furthermore, we aim to obtain an automatic tuning procedure that ensures convergence, and can also quantify and optimize performance guarantees. We achieve our goal through four major contributions. The first is a substantial generalization on the theory of classical Riemannian metrics for tangent bundles which provides the ability to compare and combine attitude and velocity terms in the stability analysis, allowing us to consider a larger set of feasible controller gains. The second contribution is a framework to study the stability of attitude systems on manifolds and to automatically tune the controller gains by combining Riemannian geometry, contraction theory, and offline optimization. The third contribution is the development of a globally, exponentially stable attitude controller. This controller overcomes the topological limitation that prevents continuous, time-invariant controllers from achieving global stability by using a time-varying intermediate reference trajectory. The fourth contribution is the improvement of the proposed controllers by way of point-wise-in-time quadratic programming

Development of an autonomous robotic air hockey player

Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (page 45).Air hockey is a widely played sport in the United States and is expanding to other countries. It is usually played on a frictionless table between two or more people. The dependency of another individual makes it impossible for a person to practice or play alone. An autonomous two-link manipulator was designed and developed to allow a person to individually play air hockey. The two-link manipulator consisted of a mechanical, electrical, and software system. The mechanical system was designed to actuate the two-link manipulator through geared motors and a belt transmission while being self-contained. The electrical system was designed to provide feedback of the physical system and motor control capabilities. The software system was designed to integrate the information from the mechanical and electrical systems to create the controls and behavior of the mechanical arm. The autonomous two-link manipulator proved to be a robust design. It had a 68.5 percent hit rate, suggesting that the design will be capable of playing air hockey.by Bee Vang.S.B