Grasp and Orientation Control of an Object by Two Euler–Bernoulli Arms With Rolling Constraints

2019 ◽  
Vol 141 (12) ◽  
Author(s):  
Takahiro Endo ◽  
Nobuhiro Shiratani ◽  
Kaiyo Yamaguchi ◽  
Fumitoshi Matsuno
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Lyapunov Functional ◽  
Bending Moment ◽  
Ode Model ◽  
Rotational Angle ◽  
Orientation Control ◽  
Grasping And Manipulation ◽  
Flexible Arms

Abstract This paper focuses on grasping and manipulation of an object by two one-link flexible arms. By taking rolling constraints between the arm tip and the grasped object, the arms have the potential to grasp and manipulate an object at the same time. To realize grasping and manipulation by two flexible arms, a boundary controller is derived from a Lyapunov functional related to the total energy of a dynamic model described by a hybrid partial differential equation-ordinary differential equation (PDE-ODE) model. The derived controller consists of the bending moment at the root of the arm, the rotational angle, and the angular velocity of the motor. In particular, the controller does not need the feedback of the information of the grasped object, and thus, it is easy to implement the controller. Further, it is shown that the derived controller realizes stable grasping and orientation control of the object as well as vibration control of the arms. Finally, experiments and numerical simulations are conducted to investigate the validity of the derived boundary controller.

scholarly journals Continuity equations and ODE flows with non-smooth velocity

2014 ◽  
Vol 144 (6) ◽  
pp. 1191-1244 ◽  
Author(s):  
Luigi Ambrosio ◽  
Gianluca Crippa
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Bounded Variation ◽  
Velocity Fields ◽  
Well Posedness ◽  
Continuity Equations ◽  
Quantitative Estimates ◽  
The Cauchy Problem

In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable ‘weak differentiability’ assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.

scholarly journals Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
R. Naz ◽  
F. M. Mahomed
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Power Law ◽  
Fourth Order ◽  
Applied Load ◽  
Mass Density ◽  
Load Case ◽  
Order Ordinary Differential Equation ◽  
General Power

We study a dynamic fourth-order Euler-Bernoulli partial differential equation having a constant elastic modulus and area moment of inertia, a variable lineal mass densityg(x), and the applied load denoted byf(u), a function of transverse displacementu(t,x). The complete Lie group classification is obtained for different forms of the variable lineal mass densityg(x)and applied loadf(u). The equivalence transformations are constructed to simplify the determining equations for the symmetries. The principal algebra is one-dimensional and it extends to two- and three-dimensional algebras for an arbitrary applied load, general power-law, exponential, and log type of applied loads for different forms ofg(x). For the linear applied load case, we obtain an infinite-dimensional Lie algebra. We recover the Lie symmetry classification results discussed in the literature wheng(x)is constant with variable applied loadf(u). For the general power-law and exponential case the group invariant solutions are derived. The similarity transformations reduce the fourth-order partial differential equation to a fourth-order ordinary differential equation. For the power-law applied load case a compatible initial-boundary value problem for the clamped and free end beam cases is formulated. We deduce the fourth-order ordinary differential equation with appropriate initial and boundary conditions.

Multidimensional recursive filters via a helix

Geophysics ◽  
1998 ◽  
Vol 63 (5) ◽  
pp. 1532-1541 ◽  
Author(s):  
Jon Claerbout
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Wide Class ◽  
Prediction Error ◽  
General Purpose ◽  
Recursive Filtering ◽  
Recursive Filters ◽  
Computational Advantage ◽  
Estimation Problems

Wind a wire onto a cylinder to create a helix. I show that a filter on the 1-D space of the wire mimics a 2-D filter on the cylindrical surface. Thus 2-D convolution can be done with a 1-D convolution program. I show some examples of 2-D recursive filtering (also called 2-D deconvolution or 2-D polynomial division). In 2-D as in 1-D, the computational advantage of recursive filters is the speed with which they propagate information over long distances. We can estimate 2-D prediction‐error filters (PEFs) that are assured of being stable for 2-D recursion. Such 2-D and 3-D recursions are general‐purpose preconditioners that vastly speed the solution of a wide class of geophysical estimation problems. The helix transformation also enables use of the partial‐differential equation of wave extrapolation as though it were an ordinary‐differential equation.

On the Approximate Solutions of Heat-Conduction Problems

1963 ◽  
Vol 85 (3) ◽  
pp. 203-207 ◽  
Author(s):  
Fazil Erdogan
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Approximate Solutions ◽  
Integral Transforms ◽  
Partial Differential ◽  
The Given

Integral transforms are used in the application of the weighted residual methods to the solution of problems in heat conduction. The procedure followed consists in reducing the given partial differential equation to an ordinary differential equation by successive applications of appropriate integral transforms, and finding its solution by using the weighted-residual methods. The undetermined coefficients contained in this solution are functions of transform variables. By inverting these functions the coefficients are obtained as functions of the actual variables.

scholarly journals Effects of spatial heterogeneity on bacterial genetic circuits

2019 ◽  
Author(s):  
Carlos Barajas ◽  
Domitilla Del Vecchio
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Spatial Heterogeneity ◽  
Mixed Models ◽  
Mixed Model ◽  
Spatial Effects ◽  
Genetic Circuits ◽  
Association Rate Constant ◽  
Ode Model ◽  
Association Rate

AbstractIntracellular spatial heterogeneity is frequently observed in bacteria, where the chromosome occupies part of the cell’s volume and a circuit’s DNA often localizes within the cell. How this heterogeneity affects core processes and genetic circuits is still poorly understood. In fact, commonly used ordinary differential equation (ODE) models of genetic circuits assume a well-mixed ensemble of molecules and, as such, do not capture spatial aspects. Reaction-diffusion partial differential equation (PDE) models have been only occasionally used since they are difficult to integrate and do not provide mechanistic understanding of the effects of spatial heterogeneity. In this paper, we derive a reduced ODE model that captures spatial effects, yet has the same dimension as commonly used well-mixed models. In particular, the only difference with respect to a well-mixed ODE model is that the association rate constant of binding reactions is multiplied by a coefficient, which we refer to as the binding correction factor (BCF). The BCF depends on the size of interacting molecules and on their location when fixed in space and it is equal to unity in a well-mixed ODE model. The BCF can be used to investigate how spatial heterogeneity affects the behavior of core processes and genetic circuits. Specifically, our reduced model indicates that transcription and its regulation are more effective for genes located at the cell poles than for genes located on the chromosome. The extent of these effects depends on the value of the BCF, which we found to be close to unity. For translation, the value of the BCF is always greater than unity, it increases with mRNA size, and, with biologically relevant parameters, is substantially larger than unity. Our model has broad validity, has the same dimension as a well-mixed model, yet it incorporates spatial heterogeneity. This simple-to-use model can be used to both analyze and design genetic circuits while accounting for spatial intracellular effects.Abstract FigureHighlightsIntracellular spatial heterogeneity modulates the effective association rate constant of binding reactions through a binding correction factor (BCF) that fully captures spatial effectsThe BCF depends on molecules size and location (if fixed) and can be determined experimentallySpatial heterogeneity may be detrimental or exploited for genetic circuit designTraditional well-mixed models can be appropriate despite spatial heterogeneityStatement of significanceA general and simple modeling framework to determine how spatial heterogeneity modulates the dynamics of gene networks is currently lacking. To this end, this work provides a simple-to-use ordinary differential equation (ODE) model that can be used to both analyze and design genetic circuits while accounting for spatial intracellular effects. We apply our model to several core biological processes and determine that transcription and its regulation are more effective for genes located at the cell poles than for genes located on the chromosome and this difference increases with regulator size. For translation, we predict the effective binding between ribosomes and mRNA is higher than that predicted by a well-mixed model, and it increases with mRNA size. We provide examples where spatial effects are significant and should be considered but also where a traditional well-mixed model suffices despite severe spatial heterogeneity. Finally, we illustrate how the operation of well-known genetic circuits is impacted by spatial effects.

scholarly journals On the link between finite difference and derivative of polynomials

2017 ◽  
Author(s):  
Kolosov Petro
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Differential Calculus ◽  
Divided Difference ◽  
Applied Mathematics ◽  
Numerical Differentiation ◽  
Discrete Analog ◽  
Derivatives Of

The main aim of this paper to establish the relations between forward, backward and central finite (divided) differences (that is discrete analog of the derivative) and partial & ordinary high-order derivatives of the polynomials.MSC 2010: 46G05, 30G25, 39-XXarXiv:1608.00801Keywords: Finite difference, Derivative, Divided difference, Ordinary differential equation, Partial differential equation, Partial derivative, Differential calculus, Difference Equations, Numerical Differentiation, Finite difference coefficient, Polynomial, Power function, Monomial, Exponential function, Exponentiation, arXiv, Preprint, Calculus, Mathematics, Mathematical analysis, Numerical methods, Applied Mathematics

scholarly journals About Some Possible Implementations of the Fractional Calculus

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 893 ◽  
Author(s):  
María Pilar Velasco ◽  
David Usero ◽  
Salvador Jiménez ◽  
Luis Vázquez ◽  
José Luis Vázquez-Poletti ◽  
...  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Fractional Calculus ◽  
Panoramic View ◽  
Partial Differential ◽  
Dirac Equations ◽  
Basic Equations

We present a partial panoramic view of possible contexts and applications of the fractional calculus. In this context, we show some different applications of fractional calculus to different models in ordinary differential equation (ODE) and partial differential equation (PDE) formulations ranging from the basic equations of mechanics to diffusion and Dirac equations.

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