Interplay between dynamics and geometry in integrable systems and engineering problems

The geometry of the phase space imposes restrictions on the dynamics of the system, and the system’s dynamics reflect the geometric properties of the space. In this thesis, we study several geometric objects/structures which either come from the phase space or arise from certain dynamical systems and look into their influences on the dynamics.In Chapter 2 we study interactions between Hamiltonian monodromy and Maslov indices, and generalize these interactions to nonHamiltonian integrable systems. In Chapter 3 we explore the bundle structures (e.g. the bundle of Lagrangian planes, the Maslov S1 bundles) over symplectic manifolds behind/related to the concept of Maslov indices and study the implications of the structural properties of the bundles on the symplectic dynamics on the manifolds. In Chapters 4 and 5, we discuss a particular type of singular fibers in integrable systems and the foliation structures in a vicinity of the fibers. We show how such a fiber fails to be homeomorphic to S2 (and hence can only be a pinched torus), and provide detailed discussions on the construction of a small neighbourhood of a pinched torus, and, the structure of the restricted Maslov S1 bundle over such a fiber. In Chapter 6 we discuss the topology of domains of attraction of stable manifolds with uniform asymptoticity and In Chapter 7 we discuss a particular case in which the communicationnetwork of a robotic swarm changes its topology in the movement

A Bubble-breaking Phenomenon in the Variation of a Swarm Communication Network

We discuss a specific circumstance in which the topology of the communication network of a robotic swarm has to change during the movement. The variation is caused by a topological obstruction which emerges from certain geometric restrictions on both the environment and the swarm.Comment: 12 page

Maslov S^{1} Bundles and Maslov Data

In this paper we consider the Maslov S^{1}bundles of a symplectic manifold (M,\omega), which refer to both the determinant bundle (denoted by \Gamma_{J}) of the unitary frame bundle and the bundle \Gamma_{J}^{2}=\Gamma_{J}\big/\{\pm1\}. The sympletic action of a compact Lie group G on M can be lifted to group actions on the principal S^{1} bundles \Gamma_{J} and \Gamma_{J}^{2}. In this work we study the interplay between the geometry of the Maslov S^{1} bundles and the dynamics of the group action on M. We show that when M is a homogeneous G-space and the first real Chern class c_{\Gamma} is nonvanishing, \Gamma_{J} and \Gamma_{J}^{2} are also homogeneous G-spaces. We also show that when [\omega]=r\cdot c_{\Gamma} for some real number r, then the G action is Hamiltonian and the Hamiltonians assume particular forms. In the end, we study a function called the \beta-Maslov data of a symplectic S^{1} action with respect to a connection 1-form \beta on \Gamma_{J}^{2}, which serves as the nonintegrable version of the notion of Maslov indices when \Gamma_{J}^{2} is not a trivial bundle

Integrated Path Following and Collision Avoidance Using a Composite Vector Field

Path following and collision avoidance are two important functionalities for mobile robots, but there are only a few approaches dealing with both. In this paper, we propose an integrated path following and collision avoidance method using a composite vector field. The vector field for path following is integrated with that for collision avoidance via bump functions, which reduce significantly the overlapping effect. Our method is general and flexible since the desired path and the contours of the obstacles, which are described by the zero-level sets of sufficiently smooth functions, are only required to be homeomorphic to a circle or the real line, and the derivation of the vector field does not involve specific geometric constraints. In addition, the collision avoidance behaviour is reactive; thus, real-time performance is possible. We show analytically the collision avoidance and path following capabilities, and use numerical simulations to illustrate the effectiveness of the theory

On Wilson’s theorem about domains of attraction and tubular neighborhoods

In this paper, we show that the domain of attraction of a compact asymptotically stable submanifold in a finite-dimensional smooth manifold of an autonomous system is homeomorphic to the submanifold’s tubular neighborhood. The compactness of the submanifold is crucial, without which this result is false; two counterexamples are provided to demonstrate this

Refining dichotomy convergence in vector-field guided path-following control

In the vector-field guided path-following problem, the desired path is described by the zero-level set of a sufficiently smooth real-valued function and to follow this path, a (guiding) vector field is designed, which is not the gradient of any potential function. The value of the aforementioned real-valued function at any point in the ambient space is called the level value at this point. Under some broad conditions, a dichotomy convergence property has been proved in the literature: the integral curves of the vector field converge either to the desired path or the singular set, where the vector field attains a zero vector. In this paper, the property is further developed in two respects. We first show that the vanishing of the level value does not necessarily imply the convergence of a trajectory to the zero-level set, while additional conditions or assumptions identified in the paper are needed to make this implication hold. The second contribution is to show that under the condition of real-analyticity of the function whose zero-level set defines the desired path, the convergence to the singular set (assuming it is compact) implies the convergence to a single point of the set, dependent on the initial condition, i.e. limit cycles are precluded. These results, although obtained in the context of the vector- field guided path-following problem, are widely applicable in many control problems, where the desired sets to converge to (particularly, a singleton constituting a desired equilibrium point) form a zero-level set of a Lyapunov(-like) function, and the system is not necessarily a gradient system

Singularity-free Guiding Vector Field for Robot Navigation

Most of the existing path-following navigation algorithms cannot guarantee global convergence to desired paths or enable following self-intersected desired paths due to the existence of singular points where navigation algorithms return unreliable or even no solutions. One typical example arises in vector-field guided path-following (VF-PF) navigation algorithms. These algorithms are based on a vector field, and the singular points are exactly where the vector field diminishes. In this paper, we show that it is mathematically impossible for conventional VF-PF algorithms to achieve global convergence to desired paths that are self-intersected or even just simple closed (precisely, homeomorphic to the unit circle). Motivated by this new impossibility result, we propose a novel method to transform self-intersected or simple closed desired paths to non-self-intersected and unbounded (precisely, homeomorphic to the real line) counterparts in a higher-dimensional space. Corresponding to this new desired path, we construct a singularity-free guiding vector field on a higher-dimensional space. The integral curves of this new guiding vector field is thus exploited to enable global convergence to the higher-dimensional desired path, and therefore the projection of the integral curves on a lower-dimensional subspace converge to the physical (lower-dimensional) desired path. Rigorous theoretical analysis is carried out for the theoretical results using dynamical systems theory. In addition, we show both by theoretical analysis and numerical simulations that our proposed method is an extension combining conventional VF-PF algorithms and trajectory tracking algorithms. Finally, to show the practical value of our proposed approach for complex engineering systems, we conduct outdoor experiments with a fixed-wing airplane in windy environment to follow both 2D and 3D desired paths.Comment: Accepted for publication in IEEE Trransactions on Robotics (T-RO

Loops of Infinite Order and Toric Foliations

In 2005 Dullin et al. proved that thenonzero vector of Maslov indices is an eigenvector with eigenvalue1 of the monodromy matrices of an integrable Hamiltonian system.We take a close look at the geometry behind this result and extendit to the more general context of possibly non-Hamiltonian systems.We construct a bundle morphism definedon the lattice bundle of an (general) integrable system, which canbe seen as a generalization of the vector of Maslov indices. The nontriviality of this bundle morphism implies the existence of common eigenvectors with eigenvalue 1of the monodromy matrices, and gives rise to a corank 1 toric foliationrefining the original one induced by the integrable system. Furthermore,we show that, in the case where the system has 2 degrees of freedom,this implies the existence of a compatible free 1 action on the regular part of the system

The Domain of Attraction of the Desired Path in Vector-Field Guided Path Following

In the vector-field-guided path-following problem, a sufficiently smooth vector field is designed such that its integral curves converge to and move along a 1-D geometric desired path. The existence of singular points where the vector field vanishes creates a topological obstruction to global convergence to the desired path and some associated topological analysis has been conducted in our previous work. In this article, we further show that the domain of attraction of the desired path, which is a compact asymptotically stable 1-D embedded submanifold of an n -dimensional ambient manifold M , is homeomorphic to Rn−1×S1 , and not just homotopy equivalent to S

Topological Analysis of Vector-Field Guided Path Following on Manifolds

A path-following control algorithm enables a system's trajectories to converge to and evolve along a given geometric desired path. There exist various such algorithms, but many of them can only guarantee local convergence to the desired path in its neighborhood. In contrast, the control algorithms using a well-designed guiding vector field can ensure almost global convergence of trajectories to the desired path. In this paper, we first generalize the guiding vector field from the Euclidean space to a general smooth Riemannian manifold. Then we are motivated by the observation that singular points of the guiding vector field exist in many examples where the desired path is homeomorphic to the unit circle, but it is unknown whether the existence of singular points always holds in general. In the n-dimensional Euclidean space, we provide an affirmative answer, and conclude that it is impossible to guarantee global convergence to desired paths that are homeomorphic to the unit circle. Furthermore, we show that there always exist diverging trajectories starting from the boundary of a ball containing the desired path in an $n$-dimensional Euclidean space wher