A Lagrange--d'Alembert integrator is a geometric numerical method for finding numerical solutions to the Lagrange--d'Alembert equations for mechanical systems with nonholonomic constraints that are linear in the velocities. The integrator is developed from geometry and principles that are analogues of the continuous theory.<p>Using discrete analogues of the symplectic form and momentum map, the resulting methods are symplectic and momentum preserving whenever the continuous system is symplectic and momentum preserving. In addition, it is possible to, in principle, generate Lagrange--d'Alembert integrators of any method order.

A Lagrange--d'Alembert integrator is a geometric numerical method for finding numerical solutions to the Lagrange--d'Alembert equations for mechanical systems with nonholonomic constraints that are linear in the velocities. The integrator is developed from geometry and principles that are analogues of the continuous theory.<p>Using discrete analogues of the symplectic form and momentum map, the resulting methods are symplectic and momentum preserving whenever the continuous system is symplectic and momentum preserving. In addition, it is possible to, in principle, generate Lagrange--d'Alembert integrators of any method order.

Atmospheric sensing tasks, including measuring the thermodynamic state (pressure, temperature, and humidity) and kinematic state (wind velocity) of the atmospheric boundary layer (ABL) can aid in numerical weather prediction, help scientists assess climatological and topological features over a region, and can be incorporated into flight path planning and control of small aircraft. Small uncrewed aerial vehicles (UAVs) are becoming an attractive platform for atmospheric sensing tasks as they offer increased maneuverability and are low-cost instruments when compared to traditional atmospheric sensing methods such as ground-based weather stations and weather balloons. In situ measurements using a UAV can be obtained for the thermodynamic state of the ABL using dedicated sensors that directly measure pressure, temperature, and humidity whereas the kinematic state (wind velocity) can be measured directly, using, for example, a five-hole Pitot probe or a sonic anemometer mounted on an aircraft, or indirectly. Indirect measurement methods consider the dynamics of the aircraft and use measurements from its operational sensor suite to infer wind velocity. This work is concerned with the design of the invariant extended Kalman filter (invariant EKF) for indirect wind estimation using a small, fixed-wing uncrewed aerial vehicle. Indirect wind estimation methods are classified as model-based or model-free, where the model refers to the aerodynamic force and moment model of the considered aircraft. The invariant EKF is designed for aerodynamic model-free wind estimation using a fixed-wing UAV in horizontal-plane flight and the full six degree of freedom UAV. The design of the invariant EKF relies on leveraging the symmetries of the dynamic system in the estimation scheme to obtain more accurate estimates where convergence of the filter is guaranteed on a larger set of trajectories when compared to conventional estimation techniques, such as the conventional extended Kalman filter (EKF). The invariant EKF is applied on both simulated and experimental flight data to obtain wind velocity estimates where it is successful in providing accurate wind velocity estimates and outperforms the conventional EKF. Overall, this work demonstrates the feasibility and effectiveness of implementing an invariant EKF for aerodynamic model-free indirect wind estimation using only the available measurements from the operational sensor suite of a UAV. / Doctor of Philosophy / Atmospheric sensing tasks, such as obtaining measurements of the pressure, temperature, humidity, and wind velocity of the atmospheric boundary layer (ABL), the lowest part of the atmosphere, have historically been dominated by the use of ground-based weather stations and deployment of weather balloons. Uncrewed aerial vehicles (UAVs) are emerging as an attractive, cost-effective platform for measuring desired quantities in the ABL. A UAV provides increased maneuverability when compared to fixed ground-based sensors and weather balloons as it can fly in different patterns and over any specified region within physical limits. Measurements of the ABL can help atmospheric scientists improve numerical weather prediction by providing more temporally and spatially dense data, in addition to helping assess climatological or topological features such as how the flow of wind varies over different types of terrain. A UAV can measure wind velocity directly or indirectly. Direct wind velocity measurements require mounting a dedicated wind sensor on a UAV and indirect measurement methods require only knowledge of the UAV's motion model with measurements from sensors already onboard to support automated flight. This work is concerned with designing an estimator for indirect wind velocity estimation using a small, fixed-wing UAV and only measurements from its operational sensor suite. The estimator, the invariant extended Kalman filter, leverages the symmetries of the system to provide estimates of the state or extended state of the system which can include position, velocity, and wind velocity. A system with symmetry is one that is unchanged by actions or transformations such as translation and rotation. The knowledge that the system remains unchanged under some transformations is used in the design of the invariant EKF. This estimator is then implemented for indirect wind estimation on both simulated and experimental flight data where it, in general, outperforms a conventional estimation method–the extended Kalman filter. The work presented in this dissertation demonstrates the effectiveness of implementing an invariant EKF for indirect wind estimation using a small, fixed-wing UAV and measurements from its operational sensor suite.

In this thesis is initiated a more systematic geometric exploration of energy shaping. Most of the previous results have been dealt wih particular cases and neither the existence nor the space of solutions has been discussed with any degree of generality. The geometric theory of partial differential equations originated by Goldschmidt and Spencer in late 1960s is utilized to analyze the partial differential equations in energy shaping. The energy shaping partial differential equations are described as a fibered submanifold of a $ k $-jet bundle of a fibered manifold. By revealing the nature of kinetic energy shaping, similarities are noticed between the problem of kinetic energy shaping and some well-known problems in Riemannian geometry. In particular, there is a strong similarity between kinetic energy shaping and the problem of finding a metric connection initiated by Eisenhart and Veblen. We notice that the necessary conditions for the set of so-called $ \lambda $-equation restricted to the control distribution are related to the Ricci identity, similarly to the Eisenhart and Veblen metric connection problem. Finally, the set of $ \lambda $-equations for kinetic energy shaping are coupled with the integrability results of potential energy shaping. The procedure shows how a poor design of closed-loop metric can make it impossible to achieve any flexibility in the character of the possible closed-loop potential function. The integrability results of this thesis have been used to answer some interesting questions about the energy shaping. In particular, a geometric proof is provided which shows that linear controllability is sufficient for energy shaping of linear simple mechanical systems. Furthermore, it is shown that all linearly controllable mechanical control systems with one degree of underactuation can be stabilized using energy shaping feedback. The result is geometric and completely characterizes the energy shaping problem for these systems. Using the geometric approach of this thesis, some new open problems in energy shaping are formulated. In particular, we give ideas for relating the kinetic energy shaping problem to a problem on holonomy groups. Moreover, we suggest that the so-called Fakras lemma might be used for investigating the stabilization condition of energy shaping. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-09-02 12:12:55.051

Light transport simulation aims at the numerical computation of the propagation of visible electromagnetic energy in macroscopic environments. In this thesis, we develop the foundations for a modern theory of light transport simulation, unveiling the geometric structure of the continuous theory and providing a formulation of computational techniques that furnishes remarkably efficacy with only local information. Utilizing recent results from various communities, we develop the physical and mathematical structure of light transport from Maxwell's equations by studying a lifted representation of electromagnetic theory on the cotangent bundle. At the short wavelength limit, this yields a Hamiltonian description on six-dimensional phase space, with the classical formulation over the space of "positions and directions" resulting from a reduction to the five-dimensional cosphere bundle. We establish the connection between light transport and geometrical optics by a non-canonical Legendre transform, and we derive classical concepts from radiometry, such as radiance and irradiance, by considering measurements of the light energy density. We also show that in idealized environments light transport is a Lie-Poisson system for the group of symplectic diffeomorphisms, unveiling a tantalizing similarity between light transport and fluid dynamics. Using Stone's theorem, we also derive a functional analytic description of light transport. This bridges the gap to existing formulations in the literature and naturally leads to computational questions. We then address one of the central challenges for light transport simulation in everyday environments with scattering surfaces: how are efficient computations possible when the light energy density can only be evaluated pointwise? Using biorthogonal and possibly overcomplete bases formed by reproducing kernel functions, we develop a comprehensive theory for computational techniques that are restricted to pointwise information, subsuming for example sampling theorems, interpolation formulas, quadrature rules, density estimation schemes, and Monte Carlo integration. The use of overcomplete representations makes us thereby robust to imperfect information, as is often unavoidable in practical applications, and numerical optimization of the sampling locations leads to close to optimal techniques, providing performance which considerably improves over the state of the art in the literature.

Light transport simulation aims at the numerical computation of the propagation of visible electromagnetic energy in macroscopic environments. In this thesis, we develop the foundations for a modern theory of light transport simulation, unveiling the geometric structure of the continuous theory and providing a formulation of computational techniques that furnishes remarkably efficacy with only local information. Utilizing recent results from various communities, we develop the physical and mathematical structure of light transport from Maxwell's equations by studying a lifted representation of electromagnetic theory on the cotangent bundle. At the short wavelength limit, this yields a Hamiltonian description on six-dimensional phase space, with the classical formulation over the space of "positions and directions" resulting from a reduction to the five-dimensional cosphere bundle. We establish the connection between light transport and geometrical optics by a non-canonical Legendre transform, and we derive classical concepts from radiometry, such as radiance and irradiance, by considering measurements of the light energy density. We also show that in idealized environments light transport is a Lie-Poisson system for the group of symplectic diffeomorphisms, unveiling a tantalizing similarity between light transport and fluid dynamics. Using Stone's theorem, we also derive a functional analytic description of light transport. This bridges the gap to existing formulations in the literature and naturally leads to computational questions. We then address one of the central challenges for light transport simulation in everyday environments with scattering surfaces: how are efficient computations possible when the light energy density can only be evaluated pointwise? Using biorthogonal and possibly overcomplete bases formed by reproducing kernel functions, we develop a comprehensive theory for computational techniques that are restricted to pointwise information, subsuming for example sampling theorems, interpolation formulas, quadrature rules, density estimation schemes, and Monte Carlo integration. The use of overcomplete representations makes us thereby robust to imperfect information, as is often unavoidable in practical applications, and numerical optimization of the sampling locations leads to close to optimal techniques, providing performance which considerably improves over the state of the art in the literature.

This report provides an introduction to geometric mechanics, which seeks to model the behavior of physical mechanical systems using differential geometric objects. In addition to its elegance as a method of representation, this formulation also admits the application of powerful analytical techniques from geometry as an aid to understanding these systems. In particular, it reveals the fundamental role that symplectic geometry plays in mechanics (something which is not at all obvious from the traditional Newtonian formulation), and in the case of systems exhibiting symmetry, leads to an elucidation of conservation and reduction laws which can be used to simplify the analysis of these systems. The contribution here is primarily one of exposition. Geometric mechanics was developed as an aid to understanding physics, and we have endeavored throughout to highlight the physical principles at work behind the mathematical formalism. In particular, we show quite explicitly the entire development of mechanics from first principles, beginning with Newton's laws of motion and culminating in the geometric reformulation of Lagrangian and Hamiltonian mechanics. Self-contained presentations of this entire range of material do not appear to be common in either the physics or the mathematics literature, but we feel very strongly that this is essential in order to understand how the more abstract mathematical developments that follow actually relate to the real world. We have also attempted to make many of the proofs contained herein more explicit than they appear in the standard references, both as an aid in understanding and simply to make them easier to follow, and several of them are original where we feel that their presentation in the literature was unacceptably opaque (this occurs primarily in the presentation of the geometric formulation of Lagrangian mechanics and the appendix on symplectic geometry). Finally, we point out that the fields of geometric mechanics and symplectic geometry are vast, and one could not hope to get more than a fragmentary glimpse of them in a single work, which necessiates some parsimony in the presentation of material. The subject matter covered herein was chosen because it is of particular interest from an applied or engineering perspective in addition to its mathematical appeal. / text

We extend the concept of central configurations to the N-body problem in spaces of nonzero constant curvature. Based on the work of Florin Diacu on relative equilib- ria of the curved N-body problem and the work of Smale on general relative equilibria, we find a natural way to define the concept of central configurations with the effective potentials. We characterize the ordinary central configurations as constrained critical points of the cotangent potential, which helps us to establish the existence of ordi- nary central configurations for any given masses. After these fundamental results, we study central configurations on H2, ordinary central configurations in S3, and special central configurations in S3 in three separate chapters. For central configurations on H2, we generalize the theorem of Moulton on geodesic central configurations, the theorem of Shub on the compactness of central configurations, the theorem of Conley on the index of geodesic central configurations, and the theorem of Palmore on the lower bound for the number of central configurations. We show that all three-body central configurations that form equilateral triangles must have three equal masses. For ordinary central configurations in S3, we construct a class of S3 ordinary central configurations. We study the geodesic central configurations of two and three bodies. Three-body non-geodesic ordinary central configurations that form equilateral trian- gles must have three equal masses. We also put into the evidence some other classes of central configurations. For special central configurations, we show that for any N ≥ 3, there are masses that admit at least one special central configuration. We then consider the Dziobek special central configurations and obtain the central con- figuration equation in terms of mutual distances and volumes formed by the position vectors. We end the thesis with results concerning the stability of relative equilibria associated with 3-body special central configurations. We find that these relative equilibria are Lyapunov stable when confined to S1, and that they are linearly stable on S2 if and only if the angular momentum is bigger than a certain value determined by the configuration. / Graduate

This thesis proposes a general and unified methodological framework suitable for studying the locomotion of a wide range of robots, especially bio-inspired. The objective of this thesis is twofold. First, it contributes to the classification of locomotion robots by adopting the mathematical tools developed by the American school of geometric mechanics.Secondly, by taking advantage of the recursive nature of the Newton-Euler formulation, it proposes numerous efficient tools in the form of computational algorithms capable of solving the external direct dynamics and the internal inverse dynamics of any locomotion robot considered as a mobile multi-body system. These generic tools can help the engineers or researchers in the design, control and motion planning of manipulators as well as locomotion robots with a large number of internal degrees of freedom. The efficient algorithms are proposed for discrete and continuous robots. These methodological tools are applied to numerous illustrative examples taken from the bio-inspired robotics such as snake-like robots, caterpillars, and others like snake-board, etc.

We have fully classified the momentum polytopes of the SU(3) action on CP(2)xCP(2) and CP(2)xCP(2) xCP(2), both actions with weighted symplectic forms, and their corresponding transition momentum polytopes. For CP(2)xCP(2) the momentum polytopes are distinct line segments. The action on CP(2)xCP(2) xCP(2), has 9 different momentum polytopes. The vertices of the momentum polytopes of the SU(3) action on CP(2)xCP(2) xCP(2), fall into two categories: definite and indefinite vertices. The reduced space corresponding to momentum map image values at definite vertices is isomorphic to the 2-sphere. We show that these results can be applied to assess the dynamics by introducing and computing the space of allowed velocity vectors for the different configurations of two-vortex systems.