Tail-Enabled Spiraling Maneuver for Gliding Robotic Fish

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Feitian Zhang ◽  
Fumin Zhang ◽  
Xiaobo Tan
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Three Dimensional ◽  
Relative Equilibria ◽  
Basins Of Attraction ◽  
Robotic Fish ◽  
Experimental Results ◽  
Underwater Gliders ◽  
Local Asymptotic Stability ◽  
Turning Radius

Gliding robotic fish, a new type of underwater robot, combines both strengths of underwater gliders and robotic fish, featuring long operation duration and high maneuverability. In this paper, we present both analytical and experimental results on a novel gliding motion, tail-enabled three-dimensional (3D) spiraling, which is well suited for sampling a water column. A dynamic model of a gliding robotic fish with a deflected tail is first established. The equations for the relative equilibria corresponding to steady-state spiraling are derived and then solved recursively using Newton's method. The region of convergence for Newton's method is examined numerically. We then establish the local asymptotic stability of the computed equilibria through Jacobian analysis and further numerically explore the basins of attraction. Experiments have been conducted on a fish-shaped miniature underwater glider with a deflected tail, where a gliding-induced 3D spiraling maneuver is confirmed. Furthermore, consistent with model predictions, experimental results have shown that the achievable turning radius of the spiraling can be as small as less than 0.4 m, demonstrating the high maneuverability.

Gliding Robotic Fish: An Underwater Sensing Platform and Its Spiral-Based Tracking in 3D Space

2017 ◽  
Vol 51 (5) ◽  
pp. 71-78 ◽  
Author(s):  
Feitian Zhang ◽  
Osama Ennasr ◽  
Xiaobo Tan
Keyword(s):  
Robotic Fish ◽  
Spiral Trajectory ◽  
Underwater Gliders ◽  
3D Space ◽  
Sensing Platform ◽  
Locomotion Pattern ◽  
New Type ◽  
Turning Radius ◽  
Spiral Trajectories ◽  
Curve Tracking

AbstractGliding robotic fish are a new type of underwater robot that combines the advantages of energy efficiency of underwater gliders and high maneuverability of robotic fish. Tail-enabled spiraling, as a novel locomotion pattern of gliding robotic fish, uses a buoyancy-driven mechanism and features a small turning radius. This paper investigates the spiral trajectory characteristics from the viewpoint of differential geometry and exploits them for curve tracking in the 3D space. The influences of control inputs on spiral trajectories are investigated through both simulation and experiments. A simulation example using a combined feedforward and feedback controller illustrates the proposed curve-tracking approach.

scholarly journals On the robust Newton’s method with the Mann iteration and the artistic patterns from its dynamics

2021 ◽  
Author(s):  
Krzysztof Gdawiec ◽  
Wiesław Kotarski ◽  
Agnieszka Lisowska
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Basins Of Attraction ◽  
Picard Iteration ◽  
Mann Iteration ◽  
Iteration Parameter ◽  
Root Finding ◽  
Base Method ◽  
Fabric Design ◽  
Root Finding Method

AbstractThere are two main aims of this paper. The first one is to show some improvement of the robust Newton’s method (RNM) introduced recently by Kalantari. The RNM is a generalisation of the well-known Newton’s root finding method. Since the base method is undefined at critical points, the RNM allows working also at such points. In this paper, we improve the RNM method by applying the Mann iteration instead of the standard Picard iteration. This leads to an essential decrease in the number of root finding steps without visible destroying the sharp boundaries among the basins of attractions presented in polynomiographs. Furthermore, we investigate visually the dynamics of the RNM with the Mann iteration together with the basins of attraction for varying Mann’s iteration parameter with the help of polynomiographs for several polynomials. The second aim of this paper is to present the intriguing polynomiographs obtained from the dynamics of the RNM with the Mann iteration under various sequences used in this iteration. The obtained polynomiographs differ considerably from the ones obtained with the RNM and are interesting from the artistic perspective. Moreover, they can easily find applications in wallpaper or fabric design.

JULIA SETS OF GENERALIZED NEWTON'S METHOD

Fractals ◽  
2007 ◽  
Vol 15 (04) ◽  
pp. 323-336 ◽  
Author(s):  
XINGYUAN WANG ◽  
TINGTING WANG
Keyword(s):  
Newton’S Method ◽  
Iteration Method ◽  
Newton's Method ◽  
Julia Sets ◽  
Basins Of Attraction ◽  
Principal Value ◽  
Extraneous Fixed Points ◽  
Sets Theory ◽  
Steffensen Method ◽  
Generalized Newton’S Method

The Julia sets theory of generalized Newton's method is analyzed and the Julia sets of generalized Newton's method are constructed using the iteration method. From the research we find that: (1) the basins of attraction of the Julia sets of generalized Newton's method depend on the roots of the equation and their orders and also the existence of the extraneous fixed points; (2) the Steffensen method is an exception to the law given in (1); and (3) if the order of the root is decimal, then the different choice of the range of the principal value of the phase angle will cause a different evolvement of the Julia sets.

Numerical Simulation of a Kinetic Cell for in-situ Combustion. A Parametric Study and History Matching Aspects

2017 ◽  
Vol 15 (6) ◽  
Author(s):  
Carlos G. Aguilar-Madera ◽  
L. Molina-Espinosa ◽  
F. Velasco-Tapia
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
History Matching ◽  
Experimental Results ◽  
Petroleum Reservoir ◽  
Oil Viscosity ◽  
In Situ Combustion ◽  
Oil Fraction ◽  
Matching Techniques

Abstract The in-situ combustion method is an enhanced oil recovery technique based on the injection of air in petroleum reservoirs with the aim to burn a portion of hydrocarbons. This reduces the oil viscosity improving substantially the oil mobility. Simultaneously other phenomena take place as: distillation, segregation, oil upgrading, among others. In this work, a mathematical model to simulate oil combustion for kinetic cell experiments is presented. The model includes four-phases, nine components and four chemical reactions: coke formation, heavy oil fraction combustion, light oil fraction combustion and coke combustion. This formulation is commonly used to simulate in-situ combustion projects at combustion tubes- and petroleum reservoir-scales. The mass and energy balances were formulated leading to one set of highly coupled ordinary differential equations, which was numerically solved. The predictive model capabilities were tested by comparison with lab data, and it was found that CO and CO2 productions, oxygen uptake and cell temperature evolution agree well with experimental results. At one preliminary stage, the parameters fitting experimental results were inferred by individual manipulation until the best results were found. These parameters were perturbed in order to identify those parameters dominating the global dynamic of process. We found that energy activations and the mass density of oil components are the dominant parameters. We suggest that history matching processes must be focused over these parameters, and for this end, the implementation of advanced computational routines to solve multivariable inverse problems is recommended. In this work, we developed two automatic history matching techniques: one process based on Newton’s method and the second one based on evolutionary algorithms. The Newton’s method showed problems to find the minimum error, meanwhile the evolutionary algorithm was able to optimize the dominant parameters, but at the expense of slow convergence.

Influence of the multiplicity of the roots on the basins of attraction of Newton’s method

2013 ◽  
Vol 66 (3) ◽  
pp. 431-455 ◽  
Author(s):  
José M. Gutiérrez ◽  
Luis J. Hernández-Paricio ◽  
Miguel Marañón-Grandes ◽  
M. Teresa Rivas-Rodríguez
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Basins Of Attraction

FINITENESS OF THE AREA OF BASINS OF ATTRACTION OF RELAXED NEWTON METHOD FOR CERTAIN HOLOMORPHIC FUNCTIONS

2004 ◽  
Vol 14 (12) ◽  
pp. 4177-4190 ◽  
Author(s):  
FİGEN ÇİLİNGİR
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Basin Of Attraction ◽  
Complex Function ◽  
Holomorphic Functions ◽  
Basins Of Attraction ◽  
Rational Map ◽  
Finite Area ◽  
The Basin Of Attraction ◽  
Parabolic Fixed Point

For a nonconstant function F and a real number h∈]0, 1] the relaxed Newton's method NF,h of F is an iterative algorithm for finding the zeroes of F. We show that when relaxed Newton's method is applied to complex function F(z)=P(z)eQ(z), where P and Q are polynomials, the basin of attraction of a root of F has finite area if the degree of Q exceeds or equals 3. The key point is that NF,h is a rational map with a parabolic fixed point at infinity.

Three-dimensional swimming robotic fish with slide-block structure: design and realization

Robotica ◽  
2013 ◽  
Vol 32 (5) ◽  
pp. 823-834
Author(s):  
Yongnan Jia ◽  
Long Wang
Keyword(s):  
Mechanical Design ◽  
Block Structure ◽  
Three Dimensional ◽  
Vertical Plane ◽  
Structure Design ◽  
Robotic Fish ◽  
Experimental Results ◽  
Design Parameters ◽  
Slide Block ◽  
The Impact

SUMMARYThis paper focuses on the mechanism design of a slide-block structure and its application on a biomimetic modular robotic fish for three-dimensional swimming. First, as a barycenter-adjustor, the slide-block structure is integrated into a mechanical design of a robotic fish, which is constructed by a control module, a driving module, and a fan-shaped caudal fin. The three-dimensional locomotion of robotic fish is decomposed into two-dimensional locomotion in horizontal plane and ascent–descent locomotion in vertical plane. Both the kinematics of the horizontal swim and the dynamics of the ascent–descent swim are analyzed by the curve fitting method. Finally, experimental results validate the three-dimensional swimming capability of the robotic fish. Furthermore, the impact of two design parameters on the swimming capability of the robotic fish is discussed by the experimental method. The experimental results confirm that the robotic fish with one driving module and a fan-shaped low-aspect-ratio caudal foil can produce higher propulsive speed than other parameter combinations.

On Global Aspects of Real Newton’s Method in Synthesis of Linkages

2006 ◽  
Author(s):  
Jin Xie ◽  
Kaiyin Yan ◽  
Yong Chen
Keyword(s):  
Dynamical System ◽  
Newton’S Method ◽  
Newton's Method ◽  
Global Analysis ◽  
Basins Of Attraction ◽  
Random Motion ◽  
Iterative Root ◽  
Degree Polynomial ◽  
Root Finding ◽  
Deterministic Dynamical System

Nonlinear equations arise from the synthesis of linkages. Newton’s method is one of the most accessible and easiest to implement of the iterative root-finding algorithms for these equations. As a discrete deterministic dynamical system, Newton’s method contains subsystems which have highly random motion. In a so-called chaotic zone, there is a rapid interchange between the basins of attraction for each root of the equation. Choosing initial points from such chaotic zone, one can obtain a certain number of roots or possible all of them under the Newton’s method. In this paper, how to locate the chaotic zones is addressed following the global analysis of real Newton’s method. It is show that there exist four chaotic zones for a general 4th degree polynomial. As an example, the equation derived from exact synthesis for five positions is solved.

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