A parametric divergence-free vector field method for the optimization of composite structures with curvilinear fibers

A modified path-following control system using the vector field method for an underactuated autonomous surface ship model is proposed in the presence of static obstacles. With this integrated system, autonomous ships are capable of following the predefined path, while avoiding the obstacles automatically. It is different from the methods in most published papers, which usually study path-following and obstacle collision avoidance, separately. This paper considers the coupled path following and collision avoidance task as a whole. Meanwhile, the paper also shows the heading control design method in the presence of static obstacles. To obtain a strong stability property, a nonlinear autopilot is designed based on the manoeuvring tests of the free-running ship model. The equilibrium point of the controller is globally exponentially stable. For the guidance system, a novel vector field method was proposed, and the proof shows the coupled guidance and control system is uniform semi-global exponentially stable (USGES). To prevent the obstacles near the predefined path, the proposed guidance law is augmented by integrating the repelling field of obstacles so that it can control the ship travel toward the predefined path through the obstacles safely. The repelling field function is given considering the obstacle shape and collision risk using the velocity obstacle (VO) algorithm. The simulations and ship model test were performed to validate the integrated system of autonomous ships.

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A vector field method for relativistic transport equations with applications

Analysis & PDE ◽

10.2140/apde.2017.10.1539 ◽

2017 ◽

Vol 10 (7) ◽

pp. 1539-1612 ◽

Cited By ~ 10

Author(s):

David Fajman ◽

Jérémie Joudioux ◽

Jacques Smulevici

Keyword(s):

Vector Field ◽

Field Method ◽

Transport Equations ◽

Vector Field Method

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Quasi periodic flow near a codimension one singularity of a divergence free vector field in dimension three

Lecture Notes in Mathematics - Dynamical Systems and Turbulence, Warwick 1980 ◽

10.1007/bfb0091908 ◽

1981 ◽

pp. 75-89 ◽

Cited By ~ 10

Author(s):

Henk Broer

Keyword(s):

Vector Field ◽

Periodic Flow ◽

Codimension One ◽

Divergence Free Vector ◽

Divergence Free

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Singular perturbed vector field method applied to combustion in diesel engine: Continuous case with thermal runaway

Applied Mathematical Modelling ◽

10.1016/j.apm.2018.05.016 ◽

2018 ◽

Vol 61 ◽

pp. 604-617

Author(s):

OPhir Nave ◽

Shlomo Hareli

Keyword(s):

Vector Field ◽

Diesel Engine ◽

Field Method ◽

Thermal Runaway ◽

Continuous Case ◽

Vector Field Method

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Analyticity and metric transitivity on the torus

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385798108192 ◽

1998 ◽

Vol 18 (3) ◽

pp. 717-723

Author(s):

SOL SCHWARTZMAN

Keyword(s):

Vector Field ◽

Periodic Orbit ◽

Differential Equations ◽

Entire Plane ◽

Transitive Flow ◽

Divergence Free Vector ◽

Divergence Free ◽

Open Question ◽

Set Of Points

Suppose we are given an analytic divergence free vector field $(X,Y)$ on the standard torus. We can find constants $a$ and $b$ and a function $F(x,y)$ of period one in both $x$ and $y$ such that $(X,Y)=(a-F_y,b+F_x)$. For a given $F$, let $P$ be the map sending $(x,y)$ into $(F_y(x,y),-F_x(x,y))$. Let $A$ be the image of the torus under this map and let $B$ be the image under this map of the set of points $(x,y)$ at which $F_{xx}F_{yy}-(F_{xy})^2$ vanishes. For any point $(a,b)$ in the complement of the interior of $A$, the flow on the torus arising from the differential equations $dx/dt=a-F_y(x,y)$, $dy/dt=b+F_x(x,y)$ is metrically transitive if and only if $a/b$ is irrational. For any point in $A$ but not in $B$ the flow is not metrically transitive. Moreover, if $a/b$ is irrational but the flow on the torus is not metrically transitive and we use our differential equations to define a flow in the entire plane (rather than on the torus), this flow has a nonstationary periodic orbit. It is an open question whether a point $(a,b)$ in the interior of $A$ can give rise to a metrically transitive flow.

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