A geometrical approach to the motion planning problem for a submerged rigid body

The main focus of this paper is the motion planning problem for a deeply submerged rigid body. The equations of motion are formulated and presented by use of the framework of differential geometry and these equations incorporate external dissipative and restoring forces. We consider a kinematic reduction of the affine connection control system for the rigid body submerged in an ideal fluid, and present an extension of this reduction to the forced affine connection control system for the rigid body submerged in a viscous fluid. The motion planning strategy is based on kinematic motions; the integral curves of rank one kinematic reductions. This method is of particular interest to autonomous underwater vehicles which can not directly control all six degrees of freedom (such as torpedo shaped AUVs) or in case of actuator failure (i.e., under-actuated scenario). A practical example is included to illustrate our technique

Symplectic integrators in sub-Riemannian geometry: the Martinet case

We compare the performances of symplectic and non-symplectic integrators for the computation of normal geodesics and conjugate points in sub-Riemannian geometry at the example of the Martinet case. For this case study we consider first the flat metric, and then a one parameter perturbation leading to non integrable geodesics. From our computations we deduce that near the abnormal directions a symplectic method is much more efficient for this optimal control problem. The explanation relies on the theory of backward error analysis in geometric numerical integration

Minimum fuel round trip from a L2L_2 Earth-Moon Halo orbit to Asteroid 2006 RH120_{120}

International audienceThe goal of this paper is to design a spacecraft round trip transfer from a parking orbit to Asteroid 2006\;RH$_{120}$, during its capture time by Earth's gravity, while maximizing the final mass or equivalently minimizing the delta-v. The parking orbit is chosen as a Halo orbit around the Earth-Moon $L_2$ libration point. The round-trip transfer is composed of three portions: a rendezvous transfer departing from the parking orbit to reach 2006\;RH$_{120}$, a lock-in portion with the spacecraft following the asteroid orbit, and finally a return transfer to $L_2$. An indirect method based on the maximum principle is used for our numerical calculations. To partially address the issue of local minima, we restrict the control strategy to reflect an actuation corresponding to up to three constant thrust arcs during each portion of the transfer. The model considered here is the circular restricted four-body problem (CR4BP) with the Sun considered as a perturbation of the Earth-Moon circular restricted three body problem. A shooting method is applied to solve numerically this problem, and the rendezvous point to and departure point from \RH\ are optimized using a time discretization of the trajectory of \RH

Creación de mapas batimétricos usando vehículos submarinos autónomos en el río Magdalena

The goal is to develop a guidance and navigation algorithm for an AUV to perform high resolution scanning of the constantly changing river bed of the Magdalena River, the main river of Colombia, from the river mouth to a distance of 10 Km upriver, which is considered to be the riskiest section to navigate. Using geometric control we design the required thrust for an under-actuated autonomous underwater vehicle to realize the desired mission.El objetivo es desarrollar un algoritmo de orientación y navegación para un AUV (Autonomous Underwater Vehicle) para realizar el escaneado de alta resolución del cambiante lecho del río Magdalena, principal río de Colombia, desde su desembocadura hasta una distancia de 10 Km río arriba, que se considera la sección de mayor riesgo para navegar. Usando control geométrico se diseñó el empuje necesario para unvehículo submarino autónomo subactuado para realizar la misión deseada

COVID-19 Heterogeneity in Islands Chain Environment

As 2021 dawns, the COVID-19 pandemic is still raging strongly as vaccines finally appear and hopes for a return to normalcy start to materialize. There is much to be learned from the pandemic's first year data that will likely remain applicable to future epidemics and possible pandemics. With only minor variants in virus strain, countries across the globe have suffered roughly the same pandemic by first glance, yet few locations exhibit the same patterns of viral spread, growth, and control as the state of Hawai'i. In this paper, we examine the data and compare the COVID-19 spread statistics between the counties of Hawai'i as well as examine several locations with similar properties to Hawai'i

A Grobman-Hartman Theorem for Control Systems

http://www.springerlink.comInternational audienceWe consider the problem of locally linearizing a control system via topological transformations. According to [2, 3], there is no naive generalization of the classical Grobman-Hartman theorem for ODEs to control systems: a generic control system, when viewed as a set of underdetermined differential equations parametrized by the control, cannot be linearized using pointwise transformations on the state and the control values. However, if we allow the transformations to depend on the control at a functional level (open loop transformations), we are able to prove a version of the Grobman-Hartman theorem for control systems

Topological versus Smooth Linearization of Control Systems

This note deals with «Grobman-Hartman like» theorems for control systems (or in other words under-determined systems of ordinary differential equations- ). The main results (proved elsewhere) is that when a control system is topologically conjugate to a linear controllable one, then it is also «almost» differentiably conjugate. We focus on the meaning of this result, and on an open question resulting from it

On the Grobman-Hartman theorem for control systems

We consider the problem of topological linearization of control systems, i.e. local equivalence to a linear controllable system via transformations that are topological but nto necessarily differentiable. On the one hand we prove that, when point-wise transformations are considered (static feedback transformations), topological linearization implies smooth linearization, at least away from singularities. On the other hand, if we allow the transformation to depend on the control at a functional level so as to define a flow (open loop transformations), we prove a version of the Grobman-Hartman theorem for control systems