Persistent rigid-body motions and Study's "Ribaucour" problem

In this work we show that the concept of a one-parameter persistent rigid-body motion is a slight generalisation of a class of motions called Ribaucour motions by Study. This allows a simple description of these motions in terms of their axode surfaces. We then investigate other special rigid-body motions, and ask if these can be persistent. The special motions studied are line-symmetric motions and motions generated by the moving frame adapted to a smooth curve. We are able to find geometric conditions for the special motions to be persistent and, in most cases, we can describe the axode surfaces in some detail. In particular, this work reveals some subtle connections between persistent rigid-body motions and the classical differential geometry of curves and ruled surfaces

Morphoelastic rods Part 1: A single growing elastic rod

A theory for the dynamics and statics of growing elastic rods is presented. First, a single growing rod is considered and the formalism of three-dimensional multiplicative decomposition of morphoelasticity is used to describe the bulk growth of Kirchhoff elastic rods. Possible constitutive laws for growth are discussed and analysed. Second, a rod constrained or glued to a rigid substrate is considered, with the mismatch between the attachment site and the growing rod inducing stress. This stress can eventually lead to instability, bifurcation, and buckling

Path planning of multiple autonomous vehicles

Safe and simultaneous arrival of constant speed, constant altitude UAVs on target is solved by design of paths of equal lengths. The starting point of the solution is the well-known Dubins path which is composed of circular arcs and line segments, thus requiring only one simple manoeuvre - constant rate turn. An explicit bound can be imposed on the rate during the design and the resulting paths are the minimum time solution of the problem. However, transition between arc and line segment entails discontinuous changes in lateral accelerations (latax), making this approach impractical for real fixed wing UAVs. Therefore, the Dubins solution is replaced with clothoid and also a novel one, based on quintic Pythagorean Hodograph (PH) curves, whose latax demand is continuous. The clothoid solution is direct as in the case of the Dubins path. The PH path is chosen for its rational functional form. The clothoid and the PH paths are designed to have lengths close to the lengths of the Dubins paths to stay close to the minimum time solution. To derive the clothoid and the PH paths that way, the Dubins solution is first interpreted in terms of Differential Geometry of curves using the path length and curvature as the key parameters. The curvature of a Dubins path is a piecewise constant and discontinuous function of its path length, which is a differential geometric expression of the discontinuous latax demand involved in transitions between the arc and the line segment. By contrast, the curvature of the PH path is a fifth order polynomial of its path length. This is not only continuous, also has enough design parameters (polynomial coefficients) to meet the latax (curvature) constraints (bounds) and to make the PH solution close to the minimum time one. The offset curves of the PH path are used to design a safety region along each path. The solution is simplified by dividing path planning into two phases. The first phase produces flyable paths while the second phase produces safe paths. Three types of paths are used: Dubins, clothoid and Pythagorean Hodograph (PH). The paths are produced both in 2D and 3D. In two dimensions, the Dubins path is generated using Euclidean and Differential geometric principles. It is shown that the principles of Differential geometry are convenient to generalize the path with the curvature. Due to the lack of curvature continuity of the Dubins path, paths with curvature continuity are considered. In this respect, initially the solution with the Dubins path is extended to produce clothoid path. Latter the PH path is produced using interpolation technique. Flyable paths in three dimensions are produced with the spatial Dubins and PH paths. In the second phase, the flyable paths are tuned for simultaneous arrival on target. The simultaneous arrival is achieved by producing the paths of equal lengths. Two safety conditions: (i) minimum separation distance and (ii) non-intersection of paths at equal distance are defined to maneuver in free space. In a cluttered space, an additional condition, threat detection and avoidance is defined to produce safe paths. The tuning is achieved by increasing the curvature of the paths and by creating an intermediate way-point. Instead of imposing safety constraints, the flyable paths are tested for meeting the constraints. The path is replanned either by creating a new way-point or by increasing the curvature between the way-points under consideration. The path lengths are made equal to that of a reference path.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Abstract Shape Synthesis From Linear Combinations of Clelia Curves

This article outlines several families of shapes that can be produced from a linear combination of Clelia curves. We present parameters required to generate a single curve that traces out a large variety of shapes with controllable axial symmetries. Several families of shapes emerge from the equation that provide a productive means by which to explore the parameter space. The mathematics involves only arithmetic and trigonometry making it accessible to those with only the most basic mathematical background. We outline formulas for producing basic shapes, such as cones, cylinders, and tori, as well as more complex families of shapes having non-trivial symmetries. This work is of interest to computational artists and designers as the curves can be constrained to exhibit specific types of shape motifs while still permitting a liberal amount of room for exploring variations on those shapes

Collective dynamics and control of a fleet of heterogeneous marine vehicles

Cooperative control enables combinations of sensor data from multiple autonomous underwater vehicles (AUVs) so that multiple AUVs can perform smarter behaviors than a single AUV. In addition, in some situations, a human-driven underwater vehicle (HUV) and a group of AUVs need to collaborate and preform formation behaviors. However, the collective dynamics of a fleet of heterogeneous underwater vehicles are more complex than the non-trivial single vehicle dynamics, resulting in challenges in analyzing the formation behaviors of a fleet of heterogeneous underwater vehicles. The research addressed in this dissertation investigates the collective dynamics and control of a fleet of heterogeneous underwater vehicles, including multi-AUV systems and systems comprised of an HUV and a group of AUVs (human-AUV systems). This investigation requires a mathematical motion model of an underwater vehicle. This dissertation presents a review of a six-degree-of-freedom (6DOF) motion model of a single AUV and proposes a method of identifying all parameters in the model based on computational fluid dynamics (CFD) calculations. Using the method, we build a 6DOF model of the EcoMapper and validate the model by field experiments. Based upon a generic 6DOF AUV model, we study the collective dynamics of a multi-AUV system and develop a method of decomposing the collective dynamics. After the collective dynamics decomposition, we propose a method of achieving orientation control for each AUV and formation control for the multi-AUV system. We extend the results and propose a cooperative control for a human-AUV system so that an HUV and a group of AUVs will form a desired formation while moving along a desired trajectory as a team. For the post-mission stage, we present a method of analyzing AUV survey data and apply this method to AUV measurement data collected from our field experiments carried out in Grand Isle, Louisiana in 2011, where AUVs were used to survey a lagoon, acquire bathymetric data, and measure the concentration of reminiscent crude oil in the water of the lagoon after the BP Deepwater Horizon oil spill in the Gulf of Mexico in 2010.Ph.D

Key developments in geometry in the 19th Century

This paper describes several key discoveries in the 19th century that led to the modern theory of manifolds in the twentieth century: intrinsic differential geometry, projective geometry and higher dimensional manifolds and Riemannian geometry