Methods Of Differential Geometry Research Articles

基于微分几何的蛇形机器人动力学与控制统一模型

Whereas the inputs for a snakelike robot are torques, the dynamics system is a nonlinear control system. With increasing modules in a snakelike robot, its nonlinear control system becomes complex and inconvenient for regulation and control. In this paper, the differential geometry method is used, and the Euler-Lagrange equations are extended to equations under any base. Thus, the dynamics equations are reduced to the standard affine control system, and the dynamics-control unified model is derived; this simplifies the regulation and control of the snakelike robot. Based on the unified model, a partial feedback linearization method is developed, and the head trajectory controller is designed. The configuration space of a snakelike robot corresponds to the manifold space, the velocity corresponds to the tangent space, the torque space corresponds to the cotangent space, and the kinematic energy provides a Riemann measure on the manifold. Thus, the dynamics of a snakelike robot can be described by Riemann geometry. Additionally, the passive wheels installed under the snakelike robot introduce the velocity constraint, which constrains the velocity space to a subspace of the tangent space. That is, the velocity space forms a distribution, and the dynamics system becomes a nonholonomic dynamics system. For a snakelike robot with passive wheels, the configuration is a Riemann manifold with a distribution. In the distribution, the appropriate base can be chosen to simplify the dynamics. In this paper, a base model is built based on the fiber bundle theory. Any set base is only a section in the fiber bundle. The orthogonal normalization technique is adopted to derive a set base that can simplify the dynamics calculation, and the dynamics-control unified model is derived. Finally, a nine-module snakelike robot is used as an example of the partial feedback linearization method.

Material description of fluxes in terms of differential forms

The flux of a certain extensive physical quantity across a surface is often represented by the integral over the surface of the component of a pseudo-vector normal to the surface. A pseudo-vector is in fact a possible representation of a second-order differential form, i.e. a skew-symmetric second-order covariant tensor, which follows the regular transformation laws of tensors. However, because of the skew-symmetry of differential forms, the associated pseudo-vector follows a transformation law that is different from that of proper vectors, and is named after the Italian mathematical physicist Gabrio Piola (1794–1850). In this work, we employ the methods of Differential Geometry and the representation in terms of differential forms to demonstrate how the flux of an extensive quantity transforms from the spatial to the material point of view. After an introduction to the theory of differential forms, their transformation laws, and their role in integration theory, we apply them to the case of first-order transport laws such as Darcy’s law and Ohm’s law.

The Equation of the Contact Line of the Involute Curvilinear-Tooth Cylindrical Gear Pump for the Agricultural Tractor

In order to derive the equation of the contact line of the involute curvilinear-tooth cylindrical gear pump for the agricultural tractor, the tooth surface of the involute curvilinear-tooth cylindrical gear is firstly generated as that of the spur or helical gear. Then the equation of the tooth surface is derived from changing the settings and orientations of the coordinate systems after the equation of the tooth profile in an arbitrary radial section is calculated by the methods of differential geometry. Based on the equation of the tooth surface, the meshing equation of the two gears is further acquired and then the equation of the contact line. Finally, the tooth surface and the contact line are simulated with mathematical software. The results suggests that the contact line between two curvilinear-tooth cylindrical gears is an arc line in the surface of action; and this line, shaped as an arc line in the generating plane of the tooth surface, is longer than that of the spur or helical gear with the same face width.

Static characteristics of electromechanical systems with second order sliding mode

The article is devoted to analytical investigation of the steady state points of generalized closed electromechanical systems that are experiencing the oscillatory processes. The main purpose of this paper is to develop the method for steady-state values determining of the state vector components of a closed system, depending on its control algorithms and equations of motion. Methods of differential geometry and modern control theory allow converting these equations, representing them to the different phase spaces. Transition to the space, the coordinates of which are interconnected by differential dependencies, is done in the work by the feedback conversions and a system of equations of a generalized electromechanical object motion in controlled Brunovsky form is obtained. Based on the analysis of the system it is found the equation of static equilibrium of the object, and it is shown that this equation equating to controller algorithm simplifies the determination of steady-state values of the state variables of the object. Using the proposed method is illustrated by the definition of control error and finding the static characteristics of control loop of DC electric drive position. This material can be useful for specialists in the field of electromechanical systems of automation and control systems of dynamic objects.

The coordinate system for force control.

The primary objective of this study was to establish the coordinate frame for force control by observing how parameters of force that are not explicitly specified by a motor task vary across the workspace. We asked subjects to apply a force of a specific magnitude with their hand. Subjects could complete the task by applying forces in any direction of their choice in the transverse plane. They were tested with the arm in seven different configurations. To estimate whether contact forces are represented in extrinsic or intrinsic coordinates, we applied the parallel transport method of differential geometry to the net joint torques applied during the task. This approach allowed us to compare the force variability observed at different arm configurations with the force variability that would be expected if the control system were applying an invariant pattern of joint torques at the tested configurations. The results indicate that for the majority of the subjects, the predominant pattern was consistent with an invariant representation in joint coordinates. However, two out of eleven subjects also demonstrated a preference for extrinsic representation. These findings suggest that the central nervous system can represent contact forces in both coordinate frames, with a prevalence toward intrinsic representations.

Structural design and experimental investigation on filament wound toroidal pressure vessels

Composite toroidal pressure vessels have widely been used in many areas. In this paper a novel design approach for determining the winding parameters of filament wound toroidal pressure vessels was presented based on differential geometry and finite element method (FEM). The winding angle developments and the thickness distribution along the meridional direction were derived using the geodesic equation and fiber stability criteria. Then, a FEM-based design procedure was developed taking into account the variations of the laminate thickness and winding angle, in order to determine the optimal initial winding angle and lay-up. In addition, the rationality and accuracy of the present method was evaluated through the hydraulic test. The results show that the obtained fiber trajectories have good wind ability without fiber stacking or bridging. The results predicted using finite element simulation are consistent with the experimental data. Therefore, the present model and method provide an important reference for design and production of filament wound toroidal pressure vessels.

Model01: Quantifying the Risk of Incremental Model Changes

DV01, the dollar value impact of a 1 basis point change in a yield curve is one of the most basic and widely used measures of market risk. Similar quantities and sensitivities have been defined for other parameters or “risk drivers” for a given, fixed model. In this present work we provide a meaningful and intuitive notion of Model01, which attempts to capture the analogue of a “1 basis point” bump in the space of models, beyond simple parametric changes. Importantly, our technique successfully calibrates each of these bumped models to the same set of liquid reference contracts. This turns out to be fundamental for a proper assessment of Model01 among exotic portfolios, and allows a meaningful comparison of the Model01 dollar value against the total reference price of a portfolio. Using the same procedure, it is possible to compute Model01 for single trades or portfolios of multiple trades, across different asset types and underliers, due to the flexibility of Weighted Monte Carlo techniques, on which it relies. The literature on quantification of model risk is limited. We will highlight the main features of R. Cont on model uncertainty and P. Glasserman and X. Xu on model risk, and compare Model01 against these approaches. The contributions from the present effort are manifold. We put forward the Hellinger distance as a more intuitive and genuine metric in the space of probability distributions, in contrast with relative entropy, which is more commonly used in the financial literature. We highlight connections between Model01 and an active area of research called information-geometry. This field applies the methods of differential geometry to the problem of statistical inference, and in particular, the problem of defining intuitive notions of distances across probability distributions. The problem at hand is strongly related, with a meaningful quantification of model uncertainty calling for a normalized measure of distances in the space of models. A fruitful connection and further explorations of these ideas should come then as no surprise. We motivate an interpretation of relative entropy as a distance squared. We employ this to analyze P. Glasserman and X. Xu techniques and cast them in rescaled units, revealing a linear dependence of their risk profiles which can be explained and proved. To the authors' best knowledge, these contributions represent novel undertakings in the financial literature, with the Hellinger distance, the interpretation of relative entropy and the introduction of information-geometry techniques being put forward in the context of model sensitivity for the first time in this present work. We apply the techniques of information geometry to shed light into relative entropy and the Hellinger distance, and reveal a natural Riemannian geometric structure in the parameter space of the alternative, bumped models used to compute Model01. The application of these concepts further enriches the interpretation of our results. Armed with these techniques, we demonstrate that Model01 intrinsically depends on the effective marginal covariance between a set of liquid instruments used for model calibration, and a portfolio of more exotic products. As the level of “exoticity'' or illiquidity of this portfolio increases, so does Model01. Model01 represents a flexible tool, potentially applicable to a wide variety of financial instruments, providing a glimpse into portfolios' model sensitivity, in a unified way.

Robust synchronization of coupled neural oscillators using the derivative-free nonlinear Kalman Filter

A synchronizing control scheme for coupled neural oscillators of the FitzHugh-Nagumo type is proposed. Using differential flatness theory the dynamical model of two coupled neural oscillators is transformed into an equivalent model in the linear canonical (Brunovsky) form. A similar linearized description is succeeded using differential geometry methods and the computation of Lie derivatives. For such a model it becomes possible to design a state feedback controller that assures the synchronization of the membrane's voltage variations for the two neurons. To compensate for disturbances that affect the neurons' model as well as for parametric uncertainties and variations a disturbance observer is designed based on Kalman Filtering. This consists of implementation of the standard Kalman Filter recursion on the linearized equivalent model of the coupled neurons and computation of state and disturbance estimates using the diffeomorphism (relations about state variables transformation) provided by differential flatness theory. After estimating the disturbance terms in the neurons' model their compensation becomes possible. The performance of the synchronization control loop is tested through simulation experiments.

Large-displacement strain theory and its application to graphene

Under the application of a force a material will deform and, hence, the crystal lattice will experience strain. This induced strain will alter the electronic properties of the material. In particular, strain in graphene generates an artificial vector potential that, if spatially varying, admits a pseudomagnetic field. Current theories for spatially varying strain use linear or finite strain theory, whose derivation is based on small displacements of infinitesimal length vectors. Here we apply a differential geometry method to derive a strain theory for large displacements of finite length vectors. This method gives a finite displacement term whose contribution is comparable to that of the linear strain term. Furthermore, we show that a ``domain-wall''-like pseudomagnetic-field profile can be generated when a wide graphene ribbon is subjected to a pair of opposing point forces (point stretch). The resulting field is a function of the new finite displacement term only and displays a maximum strength of over three times that which is predicted by the linear strain theory. These results extend the current theories of strain, which are based on the transformation of infinitesimal length vectors, to finite length vectors, thus providing an accurate description of pseudomagnetic-field structures in strained materials.

Derivatives of Siegel modular forms and modular connections

We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi–Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomorphic derivative operator of the Siegel modular forms. In order to get a holomorphic derivative operator, we introduce a weaker notion, called modular connection, on the Siegel upper half plane. Then we show that on a Siegel upper half plane there exists at most one holomorphic $${{\rm Sp}(2g, {\mathbb {Z}})}$$ -modular connection in some sense, and get a possible holomorphic derivative operator of Siegel modular forms.

Topology of drift trajectories of charged particles in a tokamak

The topology of drift orbits in a tokamak is analyzed in the entire cross section of the device both near the magnetic axis and at the periphery of the plasma column. The use of invariants of the drift equations (the generalized momentum, magnetic moment, and total energy) as variables for the entire cross section of the plasma column and self-similar variables near the magnetic axis makes it possible to comprehensively classify closed drift orbits in a tokamak. When describing orbits of different types and domains of their existence, discriminant and locus curves obtained by the methods of differential geometry are used to determine the ranges in which the invariants vary. The influence of the nonuniformity of the longitudinal current on the drift trajectories of fast particles is studied. The works in which, together with known types of orbits, trajectories along which particles leave the plasma column and can fall on the chamber wall are analyzed.

Feedback Linearization Control of Electric Arc Furnace System Based on dSPACE Simulation

In this paper, the method of feedback linearization control based on dSPACE simulation for electric arc furnace system is discussed. With the linear feedback method of differential geometry dealing with non-linear part of electric arc furnace system, the controller was designed. Then the hardware-in-the-loop simulation system was built, and several simulations was done. Simulation results show that the feedback linearization control has better performance.

Application of differential geometry in agricultural vehicle dynamics

This paper deals with the application of differential geometry methods to a precise calculation of the length of trajectory of an agricultural mechanism that moves on a sloping terrain. We obtained technical exciting function from experimental measurements, out of which we obtained the function of Euler’s parameters by using computer processing. The processing of these parameters provided translational and angular velocities of the gravity centre of the systemic vehicle MT8-222, which performed the determined mounted manoeuvres. We obtained differential equations that describe the function of a spatial curve by the application of differential geometry methods. The length of the curve is obtained by a numerical solution of the differential equations formed. We used Dormand-Prince numerical method for the numerical solution. Next, we evaluated the error of the numerical integration for every calculation by reason of the stability of computation. We also addressed the geometric characteristics of the curves such as the radius of curvature. The mounted manoeuvres as well as the corresponding velocities, trajectories, and radiuses of curvature were processed in a graphic way.

Quantum Matter in Quantum Space-Time

Quantum matter in quantum space-time is discussed using general properties of energy-conservation laws. As a rather radical conclusion, it is found that standard methods of differential geometry and quantum field theory on curved space-time are inapplicable in canonical quantum gravity, even at the level of effective equations.

Invariants of non-developable ruled surfaces in Euclidean 3-space

In this paper, using the classical methods of differential geometry, we define invariants of non-developable ruled surfaces in Euclidean 3-space, called structure functions, and show kinematics meaning of these invariants. We also generalize the notion of the angle of pitch of a closed ruled surface to any non-developable ruled surface. Then we discuss the properties of these invariants and give a kind of classification of the non-developable ruled surfaces in Euclidean 3-space with the theories of these invariants.

Algorithm of optimal surface deployment in wireless sensor networks

Node deployment is a basic problem in sensor networks,which directly relates to the performance of the entire network.Most existing researches on sensor network node deployment are for the case of two-dimensional planar and three dimensions space,but very few researches for three-dimensional surface deployment scenario.This paper proposed an algorithm of optimal surface deployment in wireless sensor networks.First by mathematical or differential geometry method for three-dimensional surface it constructed mathematical model,and then through the centroid of the three-dimensional surface Voronoi subdivision partitions,an error function was proposed to evaluate the superiority of deployment method.Finally compared with other surface deployment methods,the performance of the proposed algorithm in this paper is superior.

Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method

The present paper investigates the static behavior of doubly-curved laminated composite shells and panels. A two dimensional General Higher-order Equivalent Single Layer (GHESL) approach, based on the Carrera Unified Formulation (CUF), is proposed. The geometry description of the middle surface of shells and panels is computed by means of differential geometry tools. All structures have been solved through the generalized differential quadrature numerical methodology. A three dimensional stress recovery procedure based on the shell equilibrium equations is used to calculate through-the-thickness quantities, such as displacements components and the strain and stress tensors. Several lamination schemes, loadings and boundary conditions are considered in the worked out applications. The numerical results are compared with the ones obtained with commercial finite element codes. New profiles, concerning displacements, strains and stresses, for doubly-curved multi-layered shell structures are presented for the first time by the authors.

信息几何理论与应用研究进展

Information geometry is the fundamental and cutting-edge discipline that explores statistical problems on Riemannian manifolds of probability distributions using the methods of differential geometry. It has been identified as the second generation of modern information theory pioneered by Shannon, and exhibits great potential in developing the field of information science and systems theory. This paper begins defining the scientific content of information geometry from the intrinsic geometrical structures of parameterized families of probability distributions as well as geometric properties of information; it points out theoretical advantages and methodological innovations of information geometry compared with classical statistics and information theory. Next, the paper briefly introduces connections between information geometry and differential geometry, and elaborates the history of information geometry as well as its applications to various areas such as neural networks, statistical inference, communications and coding, systems and control theory, physics and medical imaging, among others. In particular, the applications of information geometry to signal processing, including the latest results of geometric methods of signal detection, nonlinear parameter estimation, and filtering, are comprehensively introduced. The basic ideas and methods of information geometry are also summarized. Finally, by viewing prospects for the development of information geometry, several open problems of information geometry in applications to signal processing are proposed.

LQR Control of Permanent Magnet Synchronous Motor Based on Exact State Feedback Linearization

Permanent magnet synchronous motor (PMSM) is a typical nonlinear system with significant coupling. This paper proposes a linear quadratic regulator (LQR) to control the PMSM with exact state feedback linearization. Based on the mathematical model of PMSM, the corresponding affine nonlinear system is constructed and the exact linearization conditions are discussed using differential geometry theory. Applying appropriate coordinate transformation and state feedback transformation, the affine nonlinear model is linearized into a linear system. Then, an optimal controller is designed to control the decoupled linear plant with LQR. Simulations and experiments are performed to verify the effectiveness of the proposed control strategy. The results demonstrate the feasibility of the differential geometry method and reveal the high performance and advantages of the proposed nonlinear controller in dynamic response and steady-stage error regulating characteristics

Simultaneous positioning and non-minimum phase vibration suppression of slewing flexible-link manipulator using only joint actuator

This paper presents a new global terminal sliding mode control (GTSMC) methodology for the simultaneous positioning and vibration suppression of slewing a flexible-link manipulator. The proposed control scheme is realized using only a joint actuator without an extra actuator (for example, the widely used piezoelectric zirconate titanate), and the non-minimum phase vibration control problem is solved. Firstly, based on the differential geometry method, the initial dynamic system is decomposed into two subsystems, namely an input-output subsystem and a zero-dynamics subsystem. Secondly, a continuous GTSMC strategy without chattering phenomenon is designed for the input-output subsystem, and the limited convergence time is deduced. Moreover, the eigenvalues of the zero-dynamics subsystem can be pointed by setting proper controller parameters for Lyapunov asymptotical stability. Finally, simulation and experimental results demonstrated the efficacy and feasibility of the proposed control methodology.