Point-like Research Articles

On the “contact” hyperfine interaction

When Enrico Fermi analyzed the magnetic interaction between the electron and Na, Cs, and Rb nuclei in 1930, he used the Dirac equation to compute the energy of an electron interacting with a point charge and magnetic dipole (the nucleus) fixed at the origin. After the mathematical dust had settled, he found an interaction that appeared to act only at a single point: the center of the electron wavefunction; it has been called a contact interaction. In 1936, Casimir analyzed the magnetic interaction of proton and neutron spins in the ground state of the deuteron using the Schrödinger equation and classical electrodynamics. Using symmetries appropriate only to s states, and performing an integration by parts, he found that they, too, seemed to interact only at a single common point of the proton and neutron: their center-of-mass. When applied to hydrogen, his result agreed with Fermi's. We present an expanded version of Casimir's important but little-known calculation.

Learning Electronic Polarizations in Aqueous Systems.

The polarization of periodically repeating systems is a discontinuous function of the atomic positions, a fact which seems at first to stymie attempts at their statistical learning. Two approaches to build models for bulk polarizations are compared: one in which a simple point charge model is used to preprocess the raw polarization to give a learning target that is a smooth function of atomic positions and the total polarization is learned as a sum of atom-centered dipoles and one in which instead the average position of Wannier centers around atoms is predicted. For a range of bulk aqueous systems, both of these methods perform perform comparatively well, with the former being slightly better but often requiring an extra effort to find a suitable point charge model. As a challenging test, we also analyze the performance of the models at the air-water interface. In this case, while the Wannier center approach delivers accurate predictions without further modifications, the preprocessing method requires augmentation with information from isolated water molecules to reach similar accuracy. Finally, we present a simple protocol to preprocess the polarizations in a data-driven way using a small number of derivatives calculated at a much lower level of theory, thus overcoming the need to find point charge models without appreciably increasing the computation cost. We believe that the training strategies presented here help the construction of accurate polarization models required for the study of the dielectric properties of realistic complex bulk systems and interfaces with ab initio accuracy.

A modified Lorentz force law for point-like charged particles in classical electrodynamics

In classical electrodynamics, the well-known Lorentz force law falls short of providing a satisfactory result for the trajectory of point-like charged particles when considering that particle’s own self-force. While there have been many historical attempts, Gralla, Harte and Wald developed a new model for classical charged particles that is free from pathologies while being consistent with Maxwell’s equations and conserves stress-energy. Expanding upon this approach, we derive a relativistically correct, modified Lorentz force law in vector form, which includes radiation reaction, and spin- and magnetic moment-dependent correction terms, suitable to be included in classical electrodynamics lectures and beam dynamics simulation tools. As by-products we obtain evolution equations for mass, spin angular momentum and the radiated power. We compare the new equations to the classical ones and use the new equations to conduct numerical simulations, showing that the results are free of any nonphysical artifacts, and which might be possible to test in future experiments at particle accelerators. The new equations foster improved insight into beam dynamics.

Development of a MATLAB‐based graphical user interface to illustrate Coulomb's law and Gauss's law

AbstractCoulomb's law and Gauss's law are two fundamental laws in electrostatics that involve electric charges and electric fields. These concepts that are not visible physically in real life can be challenging for students to understand, especially when it involves complex charge distributions. This paper presents a MATLAB‐based graphical user interface (GUI) to visualise the concept of Coulomb's law and Gauss's law as well as a serious snake game to reinforce users' knowledge. This MATLAB GUI features a diverse range of 3‐dimensional (3D) visualisations illustrating electric field attributes such as electric force, charge distribution and electric flux. These visualisations dynamically adapt to user‐defined parameters, offering a rich and varied exploration of the electric field's characteristics without confining the scope to a specific count of models. In Coulomb's law section, the GUI plots the electric forces in 3‐D for dipole, three‐point charges, and unlimited charge. The Gauss's law section consists of windows for illustration of the fundamentals of Gauss's law for the electric field, and the illustration for the application of Gauss's law in finding the electric field for a point charge, infinite line charge and infinite plane charge. The serious snake game developed in this work allows students to engage in quizzes related to Coulomb's law and Gauss's law to serve as a tool for reinforcing their learning outcomes. The MATLAB‐based GUI was chosen as the platform due to its excellent visualisation capabilities, ease of programming and capability to act as a powerful tool to enhance the learning of Coulomb's law and Gauss's law for students.

Lorentz contraction of electric field lines for a point charge in uniform motion

Abstract We examine a logical foundation of depicting a Lorentz contraction of a Coulomb field (an electric field of a point charge in uniform motion) by means of the ‘Lorentz contracted’ field lines. Two existing arguments for a contraction of field lines sound appealing and lead to very simple calculations yielding the correct results. However, one of them is a victim to subtle logical weaknesses, as it relies on ascribing a degree of physical reality to the electric field lines. The other one correctly proves what it sets out to prove. But it does not provide a proof, or even a suggestion, of an additional result that can be obtained by a new poof that we present here. Though our idea is very simple, the calculations used to prove it—based on a little known, half a century old result by Tsien—are somewhat more involved than those from past arguments.

Vertical vibration model for the roll system of a six-high rolling mill based on the Timoshenko beam theory

During the production of cold-rolled strips, the high-speed operation of rolling mills generates vibrations that significantly impact the production efficiency, the strip quality, and the rolling mill service life. Therefore, investigating the vibration characteristics of rolling mills is of paramount importance. Previous studies predominantly treated the rolls as point masses, overlooking variations in the vibrations along the lengths of the rolls. In this study, the working, intermediate, and backup rolls were regarded as continuous elastic bodies, and they were simplified as Timoshenko beams. The modal superposition method was used to solve the vertical vibration equations for the rolls of a six-high rolling mill; thus, the natural frequencies and modal shapes for the rolls were obtained for both free and forced vibrations. A finite-element model validation confirmed that the accuracy of the novel Timoshenko beam six-high rolling mill vibration model introduced in this paper is significantly higher than that of the traditional Euler beam model when the roller length-to-diameter ratio is relatively small. Further analysis based on this model indicates that the forced vibration of the rolls is primarily influenced by fluctuations in rolling force. The amplitude of forced vibration is affected by factors such as the friction coefficient in the deformation zone, rolling speed, and cumulative rolling distance of the work rolls. These findings provide a precise representation of the characteristics of forced vibrations in rolling mills and offer valuable theoretical insight to ensure stable rolling mill operation.

Deceleration of kicked objects due to the Galactic potential

Various stellar objects experience a velocity kick at some point in their evolution. These include neutron stars and black holes at their birth, or binary systems when one of the two components goes supernova. For most of these objects, the magnitude of the kick and its impact on the object dynamics remains a topic of debate. We investigate how kicks alter the velocity distribution of objects born in the Milky Way disc, both immediately after the kick and at later times, and whether these kicks are encoded in the observed population of Galactic neutron stars. We simulated the Galactic trajectories of point masses on circular orbits in the disc after being perturbed by an isotropic kick, with a Maxwellian distribution where $ km/s. Then, we simulated the motion of these point masses for $200$ Myr. These trajectories were then evaluated, either for the Milky Way population as a whole or for those passing within two kiloparsecs of the Sun, to get the time evolution of the velocities. During the first $20$ Myr, the bulk velocity of kicked objects becomes temporarily aligned with the cylindrical radius, implying an anisotropy in the velocity orientations. Beyond this age, the velocity distribution shifts towards lower values and settles to a median of $ km/s. Around the Sun, the distribution also loses its upper tail, primarily due to unbound objects escaping the Galaxy. We compared this to the velocities of Galactic pulsars and find that pulsars show a similar evolution with characteristic age. The shift in the velocity distribution is due to bound objects spending most of their orbits at larger radii after the kick. They are, therefore, decelerated by the Galactic potential. We find the same deceleration for nearby objects and the total population, and conclude that it is also observed in Galactic pulsars. Because of this effect, the (scalar) speeds of old neutron stars provide little information about their kicks at birth.

Box integrals with fermion bubbles for low-energy measurements of the weak mixing angle

The Moller experiment and the P2 experiment aim at measuring the weak mixing angle at low scales. The Moller experiment uses e-e-→e-e-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^- e^- \\rightarrow e^- e^-$$\\end{document}-scattering, the P2 experiment uses e-N→e-N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^- N \\rightarrow e^- N$$\\end{document}-scattering. In both cases, two-loop electroweak corrections have to be taken into account, and here in particular diagrams which give rise to large logarithms. In this paper we compute a set of two-loop electroweak Feynman integrals for point-like particles, which are obtained from a box integral by the insertion of a light fermion loop. By rationalising all occurring square roots we show that these Feynman integrals can be expressed in terms of multiple polylogarithms. We present the results in a form, which makes the large logarithms manifest. We provide highly efficient numerical evaluation routines for these integrals.

Vibration Analysis of Uniform Flexible Beams under Large Deformation with Uniformly Continuous Mass Elements Using the Equivalent Pseudolinear System

This paper analyses vibration of uniform flexible simply supported rectangular isotropic beam under large deformation with uniformly distributed mass elements. The method of equivalent pseudolinear systems was employed. The deformation of the beam was assumed to be large and the beam was also assumed to be inextensible. The expressions for elastic and nonlinear bending moments were determined. The numerical values of horizontal displacements of the movable support, and the equivalent lengths at various depth-to-breadth (aspect) ratios were determined. The corresponding equivalent pseudolinear systems for various nonlinear bending moment diagrams at various aspect ratios were determined. Consequently the concentrated loads yielding the equivalent pseudolinear systems were converted to point masses using the gravitational acceleration; and subsequently a unit load system was applied successively and independently at each mass point. The Vereshchagin’s method was applied to determine the displacements for canonical equations of motion. Non-trivial solution of the canonical equations at each aspect ratio was sought for the desired eigenvalues. The first mode frequencies at aspect ratios, β = 1.00 and β = 1.25 are complex eigenvalues. Also the natural frequencies exhibit hard-spring type with aspect ratios; and the fundamental frequencies for all the aspect ratios are at seventh mode. Conclusively, the dynamic stability of the flexible rectangular beam of 0.25m-breadth and 15m-undeformed length is not guaranteed when the aspect ratio is less than or equal to 1.25. It is also concluded that deepness of the flexible rectangular beam loaded with uniformly distributed mass elements influences the vibratory characteristics.

Systematic improvement of empirical energy functions in the era of machine learning.

The impact of targeted replacement of individual terms in empirical force fields is quantitatively assessed for pure water, dichloromethane (CH Cl ), and solvated K and Cl ions. For the electrostatic interactions, point charges (PCs) and machine learning (ML)-based minimally distributed charges (MDCM) fitted to the molecular electrostatic potential are evaluated together with electrostatics based on the Coulomb integral. The impact of explicitly including second-order terms is investigated by adding a fragment molecular orbital (FMO)-derived polarization energy to an existing force field, in this case CHARMM. It is demonstrated that anisotropic electrostatics reduce the RMSE for water (by 1.4 kcal/mol), CH Cl (by 0.8 kcal/mol) and for solvated Cl clusters (by 0.4 kcal/mol). An additional polarization term can be neglected for CH Cl but further improves the models for pure water (by 1.0 kcal/mol) and hydrated Cl (by 0.4 kcal/mol), and is key for solvated K , reducing the RMSE by 2.3 kcal/mol. A 12-6 Lennard-Jones functional form performs satisfactorily with PC and MDCM electrostatics, but is not appropriate for descriptions that account for the electrostatic penetration energy. The importance of many-body contributions is assessed by comparing a strictly 2-body approach with self-consistent reference data. Two-body interactions suffice for CH Cl whereas water and solvated K and Cl ions require explicit many-body corrections. Finally, a many-body-corrected dimer potential energy surface exceeds the accuracy attained using a conventional empirical force field, potentially reaching that of an FMO calculation. The present work systematically quantifies which terms improve the performance of an existing force field and what reference data to use for parametrizing these terms in a tractable fashion for ML fitting of pure and heterogeneous systems.

Analysis of infinite plane assumption and point charge assumption in the particle–plane electrostatic interaction model by re-expansion method and image multipole method

The solutions for the electrostatic interacting force between a particle and a plane typically assume that the plane is infinite and the particle can be approximated as a point charge. Nevertheless, the prerequisites of underlying assumptions of these calculations remain unspecified, resulting in the inability to assess the reliability of the results and constraining the practical applicability of analytical models. In this paper, the authors introduce the particle–coated plane electrostatic interaction model by employing the re-expansion method. Additionally, the multipole–coated plane electrostatic interaction model is presented using the image multipole method. An analysis of the assumptions’ prerequisites is conducted, considering a relative error of less than 2 % as the approximate standard. The findings indicate that, irrespective of the material properties of the particle and plane, the radius of the planar plate must exceed three times the distance between the particle’s center and the plane to satisfy the infinite plane assumption. When the distance between the particle’s center and the plane surpasses three times the radius, it is permissible to approximate the charged particle as a point charge. The study presented in this paper establishes a standard for prior estimation of the departure range in existing analytical models related to particle–plane electrostatic interaction. This work also serves as a theoretical basis for the practical application of these models in engineering.

Implications of vibration characteristics towards mass detection using graphene nanoribbon-based bridge configured mass sensor

Owing to its astounding mechanical properties, graphene nanoribbon (GNR) based nano-resonator is aimed as mass sensor with doubly clamped nano beam subjected to pair of point mass adsorbed at different positions over its surface. Though pristine GNR is revealed with non-piezoelectric behaviour inherently, it can be turned into piezoelectric by suitable adatoms. The present analysis investigates the performance of monolayer based (F-, Li- and K-) doped GNRs in the form of vibration attributes such as eigen-frequencies, Q-factor and Force sensitivities. Euler-Bernoulli thin beam theory is applied to model the doped GNR. The obtained results highlight the shift in the eigen-frequency due to adsorbed masses and their positions over the surface of GNRs. Moreover, the Q-factor and Force sensitivities are also determined for the proposed mass sensors. The results indicate that the eigen-frequencies at higher modes are insignificant to the Q-factor. For the fundamental mode, the magnitude of Q-factor varies in the range of 4.39-4.52 for the proposed mass sensor with respect to the positions and magnitudes of pair of mass. The force sensitivity for the doped GNR is attained at a level of aN/√Hz. Though, force sensitivity rises with the eigen-modes, the sensitivity with the minimum magnitude to be preferred for a sensor as it reflects to early detection of mass. This suggests the fundamental mode to operate the mass sensor. For the fundamental mode, the sensitivity of Li doped GNR is ranging from 103.60–117.58 aN/√Hz, while it is ranging from 109.67-124.45 aN/√Hz and 125.31–142.20 aN/√Hz for K- and F- doped GNRs, respectively. The results of the present work are compared with the available literatures. From the results, significant effect of adatom is observed on the vibration characteristics of the GNRs. The findings of the present work can significantly be used to design the GNR based bridge configured mass sensor.

Fabrication of high-performance recrystallized silicon carbide ceramic membrane based on particle packing optimization

Overcoming the challenges associated with achieving high uniformity and connectivity of pore channels in ceramic membranes, we designed silicon carbide ceramic membrane derived from the recrystallization process based on the Dinger-Funk equation of the closest-packing model with various grain grading. Furthermore, the effects of particle size distribution on the resulting microstructure and pore architecture of the ceramic membrane was also explored. The findings corroborated the critical importance of raw material particle size distribution in controlling pore size distribution and morphology. After sintering at 1900oC, the silicon carbide ceramic membrane, benefiting from ideal particle packing, exhibited a remarkably uniform pore structure. Notably, the most probable pore size constituted over 70%, while achieving an open porosity of 51.3% even without the addition of pore-forming agents. The silicon carbide ceramic membrane also demonstrated exceptional hydrophilicity (water contact angle:∼0o), impressive water permeation (1210 L·m-2·h-1·bar-1), coupled with efficient turbidity removal (∼100%) in carbon black wastewater treatment applications. Additionally, membrane regeneration proved effective using a dilute NaOH solution backwash, achieving a flux recovery efficiency of 98%. This strategy had directive significance for designing high-performing silicon carbide ceramic membranes.

Impact of battery swapping in the passenger sector: EV adoption, emissions, and energy mix

Transport sector electrification will play a critical role in achieving India's net zero emission target by 2070; however, the adoption of EVs is hindered by challenges such as long charging times, high ownership costs, and limited range. Battery swapping (BS) offers quick recharging, unlike point charging (PC). This study examines BS subscription fees in India's passenger sector (buses, four-wheelers, three-wheelers, and two-wheelers). Furthermore, the assessment of its effects on EV adoption, the energy mix, and CO2 emissions in India's passenger sector is analysed using the global change analysis model (GCAM). The results show that BS is cost-effective for four-wheelers and two-wheelers without subsidies. Providing similar subsidies to BS as to PC achieves cost parity or lower costs for three-wheelers and buses compared to PC. BS could reduce CO2 emissions by 9 % by 2070, with a 6 % increase in EV adoption and an 8 % rise in electricity demand. With BS subsidies, EV penetration may increase by 10 %, resulting in a 14 % reduction in CO2 emissions compared to PC. The study suggests BS is a viable option but may not entirely replace PC.

The Influence of Nonlinear High-Intensity Dynamic Processes on the Standing Wave Precession of a Non-Ideal Hemispherical Resonator.

The properties of small size, low noise, high performance and no wear-out have made the hemispherical resonator gyroscope a good choice for high-value space missions. To enhance the precision of the hemispherical resonator gyroscope for use in tasks with large angular velocities and angular accelerations, this paper investigates the standing wave precession of a non-ideal hemispherical resonator under nonlinear high-intensity dynamic conditions. Based on the thin shell theory of elasticity, a dynamic model of a hemispherical resonator is established by using Lagrange's second kind equation. Then, the dynamic model is equivalently transformed into a simple harmonic vibration model of a point mass in two-dimensional space, which is analyzed using a method of averaging that separates the slow variables from the fast variables. The results reveal that taking the nonlinear terms about the square of the angular velocity and the angular acceleration in the dynamic equation into account can weaken the influence of the 4th harmonic component of a mass defect on standing wave drift, and the extent of this weakening effect varies with the dimensions of the mass defects, which is very important for steering the development of the high-precision hemispherical resonator gyroscope.

The Jacobi-Sobolev, Laguerre-Sobolev, and Gegenbauer-Sobolev differential equations and their interrelations

Recently, the author determined the higher-order differential operator having the Jacobi-Sobolev polynomials as its eigenfunctions for certain eigenvalues. These polynomials form an orthogonal system with respect to an inner product equipped with the Jacobi measure on the interval [ − 1 , 1 ] with parameters α ∈ N 0 , β > − 1 and two point masses N, S>0 at the right end point of the interval involving functions and their first derivatives. The first purpose of the present paper is to reveal how the Jacobi-Sobolev equation reduces to the differential equation satisfied by the Laguerre-Sobolev polynomials on the positive half line via a confluent limiting process as β → ∞ . Secondly, we explicitly establish the differential equation for the symmetric Gegenbauer-Sobolev polynomials by employing a quadratic transformation of the argument. Each of the three differential operators involved is of order 4 α + 10 and symmetric with respect to the corresponding Sobolev inner product.

Structural properties of different sphere packings with arbitrary porosities for planetary-science applications

Granular solids in planetary science are found in the regolith that covers planetary surfaces as well as in the bulk of rubble-pile asteroids, comets and planetesimals. To help understand the physics of these planetary bodies, we aim at deriving the structural properties of granular packings over a large range of porosities. Relevant to fluid flow and gas diffusion are the void spaces inside the granular packings so that we analyze the mean free path of point-like particles, their diffusion constant and their total traveled path lengths. For mechanical and heat-transport properties, the coordination number and the absolute chain length of the inter-connected particles are important. Generally, we also derive the homogeneity and isotropy of the granular solids. We compare granular packings generated by four algorithms for porosities in the range between 85 and 42%, which are the upper and lower limits for natural packings of equal-sized spheres. All produced sphere packings arrive at very similar quantities for the mean free path, the free path probability distribution function, the diffusion constant and the total traveled path length for the entire range of porosities. Hence, transport processes governed by the void-space properties are independent of the specific generation algorithm for the granular packing. In contrast, heat conduction or mechanical stresses almost exclusively depend on the existence and properties of particle contacts and particle chains in the network of spheres. In this case, the four algorithms deliver very different results.Graphic abstract

Grain boundary segregation in iron doped strontium titanate: From dilute to concentrated solid solutions

Strontium titanate, a perovskite oxide, is a frequently studied material for a large variety of applications. When acceptor-doped (with Fe, for example), the material is useful for its mixed oxygen and electronic conductivity, with potential use in oxygen transport membranes or as a cathode for solid oxide fuel cells. A barrier to conductivity in perovskites is the presence of space charge regions at the grain boundaries, which form due to the segregation of charged point defects. Typically, space charge theory assumes bulk dopant concentrations beneath the dilute limit, however concentrated solid-solutions are often utilized in applications. The current work aims to address this disparity: grain boundary segregation in strontium ferrite-strontium titanate solid-solutions is analyzed at three compositions, with Fe contents ranging from near the dilute limit to well above the dilute limit (Fe contents of 2 %, 5 %, and 25 % on the B-site of the perovskite). Electrochemical impedance spectroscopy shows an increase in material conductivity as Fe is added. High-resolution STEM imaging and spectral mapping is utilized, showing that Fe segregates to the grain boundary core, contrary to what is expected from space charge theory. As the Fe content is increased, the amount of Fe segregated to the boundary increases significantly, but the segregation width of Fe at the boundary remains consistent.

On n-Dimensional Maxwell and Dirac Equations in Curved Space-Time and Its Applications in SO(P,Q) Group Theoretic Image Processing

Maxwell equations in p time and q spatial dimensions are formulated. Properties of the Green’s function for the associated (p,q)-wave operator are derived based on contour integration. The notions of electric and magnetic fields in (p,q)-dimensions is explored by analogy with four dimensional physics and the problem of far field computation of the radiation field generated by charges and currents in \((1,n-1)\) space-time is analyzed. SO(p,q) invariance properties of the Maxwell equations are deduced and used to formulate SO(p,q)-group theoretic image processing problems in (p,q)-dimensions. Dirac’s equation in (p,q)-dimensional space-time is derived using the Clifford algebra of the Dirac gamma matrices. SO(p,q)-invariance of the Dirac equation based on the spinor representation of \(SO(p,q)\) is mentioned. The Maxwell’s equations in (p,q)-dimensional curved space-time is analyzed using perturbation theory in the context of maximally symmetric spaces which play a fundamental role in \((p,q)\)-dimensional cosmological models for homogeneous and isotropic spaces. The Einstein-Maxwell equations for (p,q)-dimensional gravity and electromagnetism is studied and used to derive the equations of motion of point charges carrying mass moving under mutual gravitational and electromagnetic interactions in general relativity. Finally, Dirac’s equation in (p,q)-dimensional curved space-time interacting with the (p,q)-dimensional electromagnetic field is looked at. \(U(1)\)-Gauge, local SO(p,q) Lorentz and diffeomorphism invariance of this equation is analyzed. Local \(SO(p,q)\) invariance of the curved space-time Dirac equation is deduced based on transformation properties of the Dirac matrices under the spinor representation. The appendix presents some applications of group representation theoretic statistical image processing on a manifold to the situation when the image field is described by the six component electromagnetic field tensor on which the Lorentz group acts.

Research on two stage obstacle-avoidance trajectory planning and trajectory tracking control in curves

The existing obstacle-avoidance trajectory planning and trajectory tracking control algorithms have limitations such as long-time consumption, high failure rate in dynamic traffic environments, and insufficient trajectory tracking accuracy in curved roads. Based on the above problems, this paper designs a two stage obstacle-avoidance trajectory planner based on nonlinear optimization theory. In first stage Part-NLP, only considering the safety obstacle avoidance, a point mass model and linearization constraints are established to quickly solve the initial trajectory. In the second stage Full-NLP, considering smooth soft constraints comprehensively, the initial trajectory is optimized by establishing driving corridors and a lightweight iterative framework. In control module, this paper selects a linear quadratic form lateral trajectory tracking controller, and the parameters were optimized through the carnivorous plant algorithm. The joint simulation results show that in dynamic traffic environment of curved roads, the two stage planner proposed can accurately plan safe and smooth obstacle avoidance trajectories, and there is a significant reduction in time consumption compared to traditional NLP algorithms. The control strategy can accurately track the planned trajectories, with lateral error controlled within plus or minus 0.1 m, heading error controlled within plus or minus 0.15 rad, speed tracking error controlled within plus or minus 0.15 m/s, and vehicle yaw angle error controlled within plus or minus 0.04 rad; the hardware-in-loop test results indicate that the controller can achieve real-time and accurate trajectory tracking.