arXiv subjects
Publications and source records attributed to Bohuan Lin.
In this paper we study a path-following problem on $R^3$ with a non-holonomic constraint. The geometric structure associated to the velocity constraint is explored, and general principles for constructing guiding vector fields are obtained, fulfilling the path-following requirements on a neighborhood of the desired path while allowing the design of vector fields to be conducted in global coordinates.
We investigate a class of algebras on $\mathbb{R}^3$ arising and generalized from the algebraic structure of magnetic gradient fields induced by systems of synchronous magnets with identical dipole moments (i.e., $\mathbf{M}_i=\mathbf{M},\,\forall i$). We show that when there is a $2$ dimensional sub-algebra, the linear structure associated to such an algebra admits a certain type of decompositions, which allows the locating of the dipole moment $\bar{\mathbf{M}}$ that yields the strongest translational force(s) on a test magnet $\mathfrak{m}$. Upper bounds to the strength of this magnetic force are then established.
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 one-dimensional 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 paper, we strengthen the result in our previous work by showing that the domain of attraction of the desired path, which is a compact asymptotically stable one-dimensional embedded submanifold of an $n$-dimensional ambient manifold $\mathcal{M}$, is homeomorphic to $\mathbb{R}^{n-1} \times \mathbb{S}^1$, and not just homotopy equivalent to $\mathbb{S}^1$. This result is extended for a $k$-dimensional compact manifold for $k \ge 2$.
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.
We define Maslov $S^1$ bundles over a symplectic manifold $(M,\omega)$. These are the determinant bundle $\Gamma_J$ of the unitary frame bundle defined by an almost complex structure compatible with $\omega$, and the bundle $\Gamma_J^2 = \Gamma_J \big/ \{\pm1\}$. We analyze the properties of the Maslov $S^1$ bundles $\Gamma_J$ and $\Gamma_J^2$, focusing on the interplay between their geometry and the dynamics of a symplectic action of a compact Lie group $G$ on $M$ which induces lifted $G$ actions on $\Gamma_J$ and on $\Gamma_J^2$. 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. Moreover, we give an alternative proof of the fact that when $[\omega]=r\,c_{\Gamma}$ for some real number $r$, then the symplectic $G$ action on $(M,\omega)$ is Hamiltonian. When the Maslov $S^1$ bundle $\Gamma_J^2$ is trivial, then an index generalizing the Maslov index can be defined. This is no longer true if $\Gamma_J^2$ is not trivial. However, if $G=S^1$ acts symplectically on $(M,\omega)$ we define a quantity that we call Maslov data which serves as a non-integrable version of the notion of Maslov index in the case where $\Gamma_J^2$ is not trivial, and we associate the Maslov data at fixed points of the $G=S^1$ action to their resonance type. Finally, we consider three applications motivated by the study of integrable Hamiltonian systems. First, we discuss conditions under which an $S^1$ symmetry of a two degrees of freedom integrable Hamiltonian system can be extended to a $\mathbb T^2$ symmetry. Second, we show that the Maslov $S^1$ bundles over Lagrangian pinched tori are trivial. Third, we consider $S^2 \times S^2$ as a symplectic manifold with an $S^1$ action corresponding to simultaneous rotations of the two spheres, and we compute the corresponding Maslov data.
Accurately following a geometric desired path in a two-dimensional space is a fundamental task for many engineering systems, in particular mobile robots. When the desired path is occluded by obstacles, it is necessary and crucial to temporarily deviate from the path for obstacle/collision avoidance. In this paper, we develop a composite guiding vector field via the use of smooth bump functions, and provide theoretical guarantees that the integral curves of the vector field can follow an arbitrary sufficiently smooth desired path and avoid collision with obstacles of arbitrary shapes. These two behaviors are reactive since path (re)-planning and global map construction are not involved. To deal with the common deadlock problem, we introduce a switching vector field, and the Zeno behavior is excluded. Simulations are conducted to support the theoretical results.
A path-following control algorithm enables a system's trajectories under its guidance 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; here, "almost" means that in some cases, a measure-zero set of trajectories converge to the singular set where the vector field becomes zero (with all other trajectories converging to the desired path). In this paper, we first generalize the guiding vector field from the Euclidean space to a general smooth Riemannian manifold. This generalization can deal with path-following in some abstract configuration space (such as robot arm joint space). Then we show several theoretical results from a topological viewpoint. Specifically, 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 (i.e., is inherent in the topology of the desired path). In the $n$-dimensional Euclidean space, we provide an affirmative answer, and conclude that it is not possible to guarantee global convergence to desired paths that are homeomorphic to the unit circle. Furthermore, we show that there always exist \emph{non-path-converging trajectories} (i.e., trajectories that do not converge to the desired path) starting from the boundary of a ball containing the desired path in an $n$-dimensional Euclidean space where $n \ge 3$. Examples are provided to illustrate the theoretical results.
In this paper, we show that the domain of attraction of a compact asymptotically stable submanifold of a finite-dimensional smooth manifold of an autonomous system is homeomorphic to its tubular neighborhood. The compactness of the attractor is crucial, without which this result is false; two counterexamples are provided to demonstrate this.
In 2005 Dullin et al. proved that the non-zero vector of Maslov indices is an eigenvector with eigenvalue 1 of the monodromy matrices of an integrable Hamiltonian system. We take a close look at the geometry behind this result and extend it to a more general context. We construct a bundle morphism defined on the lattice bundle of an (general) integrable system, which can be seen as a generalization of the vector of Maslov indices. The non-triviality of this bundle morphism implies the existence of common eigenvectors with eigenvalue 1 of the monodromy matrices, and gives rise to a corank 1 toric foliation refining 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 global existence of a free S^{1} action.
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.