Path Space Research Articles

Path planning for ships assisted by the icebreaker in ice-covered waters in the Northern Sea Route based on optimal control

This paper aims to develop an innovative model based on optimal control to optimize the shipping path of the fleet, including icebreakers and following ships in continuous time and space in the Northern Sea Route (NSR) waters, considering the influence of breakable and unbreakable ice. Path planning assumptions of ship navigation in ice-covered waters are summarized by considering navigation efficiency, navigation cost, icebreaking cost, and safety. The objective function of the model based on optimal control is formulated to convert the path planning problem into a multi-objective optimization problem. The cost functions based on path planning assumptions are specified in the objective function. Finally, dynamic programming and numerical simulation are used to solve the multi-objective optimization functions. Then, the optimal path in continuous time and space in ice-covered waters in the NSR is obtained. A case study of four scenarios, considering breakable and unbreakable sea ice conditions in the NSR, is conducted to verify the strength of the proposed model. The case study results show that the proposed model can optimize the path of ships in ice-covered waters. The method proposed in this research could guide ship fleet path planning for the bridge team and improve icebreaker management in the NSR.

Restraint of the mid-spatial frequency error on optical surfaces by multi-jet polishing.

Nowadays, the mid-spatial frequency (MSF) error existing in the optical surface after polishing is still a great challenge for the ultra-precision manufacturing of optical components. MSF error severely deteriorates the performances of optical components such as causing small-angle scattering and reducing imaging contrast. In this paper, multi-jet polishing (MJP) was proposed to restrain the MSF error, whose tool influence function (TIF) was relatively more complicated and adjustable than the TIFs of other tools. The results demonstrated that MJP had a superior ability to reduce the ripple error, and the path spacing and nozzle orientation angle both had a significant effect on the MSF error of the polished surface. The optimization of nozzle orientation angle under different path spacings was conducted to achieve a high surface quality. This study contributes to the ultra-precision manufacturing of optical components, achieving a low MSF error together with high finishing efficiency.

Dark path holonomic qudit computation

Non-adiabatic holonomic quantum computation is a method used to implement high-speed quantum gates with non-Abelian geometric phases associated with paths in state space. Due to their noise tolerance, these phases can be used to construct error resilient quantum gates. We extend the holonomic dark path qubit scheme in [M.-Z. Ai {\t et al.}, Fundam. Res. {\bf 2}, 661 (2022)] to qudits. Specifically, we demonstrate one- and two-qudit universality by using the dark path technique. Explicit qutrit ($d=3$) gates are demonstrated and the scaling of the number of loops with the dimension $d$ is addressed. This scaling is linear and we show how any diagonal qudit gate can be implemented efficiently in any dimension.

Study of a High-Index Dielectric Non-Hermitian Metasurface and Its Application in Holograms.

We demonstrate that a high-index dielectric Si metasurface with a designed chiral unit structure possesses an exceptional point (EP) when it is described by a non-Hermitian Hamiltonian associated with the transmission matrix. By encircling any path in the parameter space around the EP, topologically protected 2π-phase accumulation occurs. These typical non-Hermitian properties are ascribed to complex scattering phenomena related to the coupling between electric and magnetic dipolar modes from the high-index dielectric Si metasurface. The topologically guaranteed entire 2π-phase accumulation and chiral distinction around the EP open up many promising possibilities in nanophotonic device designing; for instance, phase-only and polarization multiplexing holograms are realized in this work.

An analytical method for corner smoothing of five-axis linear paths using the conformal geometric algebra

The chain of linear path segments is widely used in five-axis machining, but the tangential discontinuity of the path reduces machining efficiency and accuracy. This paper proposes an analytical G 3 continuous corner smoothing method based on the conformal geometric algebra. Based on the locations of the tooltip point and a tool axis point, the toolpath is represented in the 6-dimensional Euclidean space. With the help of the representation of conformal transformations and circles in conformal geometric algebra, the path in the 6-dimensional space is smoothed by G 3 continuous circle-based splines under the constraints of the maximum deviation error tolerance. The proposed approach can generate a smooth toolpath that passes through the discrete cutter locations given in the original linear segments analytically. The cycle time of machining is improved thanks to the small curvature and G 3 continuity of the toolpath. The effectiveness and efficiency of the proposed method are validated by simulations and experiments. • A new analytical method for smoothing of 5-axis linear paths based on the conformal geometric algebra is proposed. • The proposed method generates G 3 continuous paths that satisfy the smooth error tolerance and pass through the initial discrete cutter locations. • The proposed method performs high efficiency and significantly improves the feedrate of the tool.

Inverse relationship between diffusion coefficient and mass for a free particle system: Approach by using maximum caliber principle and Monte Carlo simulations.

A derivation of the diffusion equation is presented using the maximum caliber principle and the continuity equation for a system composed of paths traveled by a free particle in a time interval. By identifying the diffusion coefficient in the obtained diffusion equation, it is shown that there is an inverse proportionality relationship concerning the particle's mass so that a higher mass is related to lower diffusion, and the lower mass is connected to the higher diffusion. This relationship is also shown using Monte Carlo simulations to sample the path space for a free particle system and then using the time slicing equation to obtain the probability of the particle position for each time showing the diffusion behavior for different masses.

Large deviations for Markov jump processes in periodic and locally periodic environments

The paper deals with a family of jump Markov process defined in a medium with a periodic or locally periodic microstructure. We assume that the generator of the process is a zero order convolution type operator with rapidly oscillating locally periodic coefficient and, under natural ellipticity and localization conditions, show that the family satisfies the large deviation principle in the path space equipped with Skorokhod topology. The corresponding rate function is defined in terms of a family of auxiliary periodic spectral problems. It is shown that the corresponding Lagrangian is a convex function of velocity that has a superlinear growth at infinity. However, neither the Lagrangian nor the corresponding Hamiltonian need not be strictly convex, we only claim their strict convexity in some neighbourhood of infinity. It then depends on the profile of the generator kernel whether the Lagrangian is strictly convex everywhere or not.

Distance-as-time in physical aging.

Although it has been known for half a century that the physical aging of glasses in experiments is described well by a linear thermal-history convolution integral over the so-called material time, the microscopic definition and interpretation of the material time remains a mystery. We propose that the material-time increase over a given time interval reflects the distance traveled by the system's particles. Different possible distance measures are discussed, starting from the standard mean-square displacement and its inherent-state version that excludes the vibrational contribution. The viewpoint adopted, which is inspired by and closely related to pioneering works of Cugliandolo and Kurchan from the 1990s, implies a "geometric reversibility" and a "unique-triangle property" characterizing the system's path in configuration space during aging. Both of these properties are inherited from equilibrium, and they are here confirmed by computer simulations of an aging binary Lennard-Jones system. Our simulations moreover show that the slow particles control the material time. This motivates a "dynamic-rigidity-percolation" picture of physical aging. The numerical data show that the material time is dominated by the slowest particles' inherent mean-square displacement, which is conveniently quantified by the inherent harmonic mean-square displacement. This distance measure collapses data for potential-energy aging well in the sense that the normalized relaxation functions following different temperature jumps are almost the same function of the material time. Finally, the standard Tool-Narayanaswamy linear material-time convolution-integral description of physical aging is derived from the assumption that when time is replaced by distance in the above sense, an aging system is described by the same expression as that of linear-response theory.

Stochastic paths controlling speed and dissipation.

Natural processes occur in a finite amount of time and dissipate energy, entropy, and matter. Near equilibrium, thermodynamic intuition suggests that fast irreversible processes will dissipate more energy and entropy than slow quasistatic processes connecting the same initial and final states. For small systems, recently discovered thermodynamic speed limits suggest that faster processes will dissipate more than slower processes. Here, we test the hypothesis that this relationship between speed and dissipation holds for stochastic paths far from equilibrium. To analyze stochastic paths on finite timescales, we derive an exact expression for the path probabilities of continuous-time Markov chains from the path summation solution to the master equation. We present a minimal model for a driven system in which relative energies of the initial and target states control the speed, and the nonequilibrium currents of a cycle control the dissipation. Although the hypothesis holds near equilibrium, we find that faster processes can dissipate less under far-from-equilibrium conditions because of strong currents. This model serves as a minimal prototype for designing kinetics to sculpt the nonequilibrium path space so that faster paths produce less dissipation.

Time-Optimal Trajectory Planning of 6-DOF Manipulator Based on Fuzzy Control

Currently, the teaching programming or offline programming used by an industrial manipulator can manually set the running speed of the manipulator. In this paper, to consider the running speed and stability of the manipulator, the time-optimal trajectory planning (TOTP) of the manipulator is transformed into a nonlinear optimal value search problem under multiple constraints, and a time-search algorithm based on fuzzy control is proposed, so that the end of the manipulator can run along the given path in Cartesian space for the shortest time, and the angular velocity and angular acceleration of each joint is within a limited range. In addition, a simulation model of a 6-DOF manipulator is established in MATLAB, taking a straight-line trajectory of the end of the manipulator in Cartesian space as an example, and the effectiveness and efficiency of the algorithm proposed in this paper are proved by comparing the execution time with the bisection algorithm and the traditional gradient descent method.

A Trajectory Planning Approach for Delta Robots Considering Both Motion Smoothness and Dynamic Stress

Abstract By considering both motion smoothness and dynamic stress, a trajectory planning method for delta robots, with the goal of determining the two optimal normalized time factors that dominate the motion path in operation space, is presented in this article. First, based on the semianalytical elastodynamics model of parallel robots, which considers the compliance of the limbs and joints, a dynamic stress model of the kinematic chain was built. Two indices were proposed to reflect the motion smoothness and dynamic stress. A sensitivity analysis and an optimization of the normalized time factors for a modified fifth-order B-spline approach were conducted in isight and matlab. The results show that the two normalized time factors have important impacts on the motion smoothness and dynamic stress. A comparison showed that the trajectory planning approach based on the modified fifth-order B-spline reduced the dynamic stress while improving the motion smoothness. The approach proposed in this work can also be applied in trajectory planning for other parallel or hybrid robots.

Spatial and Temporal Evolution of Vegetation Water Consumption in Arid and Semi-Arid Areas against the Background of Returning Farmland to Forestland

Sustainable development in arid and semi-arid areas is largely constrained by water resources. Expanding ecological space is considered an effective way to conserve water resources. The innovation of this study is the analysis of water consumption in different land-use types from a complete watershed scale, which can evaluate space management against the background of returning farmland to forestland during the past 20 years, and provide suggestions for future space management in semi-arid areas. Based on meteorological data and GIS technology, the current study quantitatively analyzes the spatial and temporal variation characteristics of the water consumption of different vegetation growth stages in the Yanhe River Basin by using the improved Penman formula. The results show that the water consumption of vegetation in the Yanhe River Basin increased from 0.44 km³ in 2000 to 0.68 km³ in 2020. The water consumption of vegetation showed obvious spatial heterogeneity, with the highest value in the central Baota area (1.094 km³) followed by the western Ansai region (0.727 km³), whereas the consumption in the eastern Yanchang area is relatively low (0.483 km³). In addition, the annual average water consumption is (0.381 km³). The cultivated land consumes the most water (0.21 km3), while the woodland consumes the least (0.072 km³). The water consumption per unit area of forested land is the highest, reaching 190 m, and the water consumption of low-coverage grassland is the lowest, only reaching 50 m. Vegetation distribution change could be the main influencing factor of vegetation water consumption change in the Yanhe River Basin. Through the establishment of the sustainable development path of ecological space with water as the core, the high-quality development of ecological environments in arid and semi-arid areas will be achieved.

$\mathrm{SL}_{2}(\mathbb{R})$-developments and signature asymptotics for planar paths with bounded variation

The signature transform, defined by the formal tensor series of global iterated path integrals, is a homomorphism between the path space and the tensor algebra that has been studied in geometry, control theory, number theory as well as stochastic analysis. An elegant isometry conjecture states that the length of a bounded variation path \gamma can be recovered from the asymptotics of its normalised signature: \text{Length}(\gamma)= \lim_{n\rightarrow\infty} \Big\Vert n! \int_{0<t_{1}<\cdots<t_{n}<T} d\gamma_{t_{1}}\otimes\cdots\otimes d\gamma_{t_{n}} \Big\Vert^{1/n}. This property depends on a key topological non-degeneracy notion known as tree-reducedness (namely, with no tree-like pieces). Existing arguments have relied crucially on \gamma having a continuous derivative under the unit speed parametrisation. In this article, we prove the above isometry conjecture for planar paths by assuming only local bounds on the angle of \gamma' (which ensures the absence of tree-like pieces). Our technique is based on lifting the path onto the special linear group \mathrm{SL}_{2}(\mathbb{R}) and analysing the behaviour of the associated angle dynamics at a microscopic level.

Contact angle ageing and anomalous capillary flow of a molten metal

Recent capillary flow experiments, conducted on a combined wetting/non-wetting assemble, consistently feature an anomalous flow over the non-wetting substrate: (i) apparent abrupt or gradual recession stages in the motion of the contact line, (ii) non-monotonic abrupt changes in the receding contact angle, and (iii) contact angle overshoot above the nominal equilibrium contact angle. We find that such behavior of a liquid metal alloy cannot be explained by the standard capillary flow models. However, a model that includes the ageing of the equilibrium contact angle predicts all the observed features qualitatively. We use the phase field formulation for capillary flows with a diffusive motion of the triple line to accommodate the novel diffusive boundary condition with the time-evolving quasi-equilibrium contact angle. We discover that the observed anomalies in capillary flow are qualitatively explained by two factors: (1) time evolution (ageing) of the quasi-equilibrium contact angle and (2) high viscosity of the partially molten braze. We also discover that for the given flow geometry, the transition from the initial to the final configuration may follow two distinct topological paths: one is characterized by a coalescence of liquid–solid contact domains, the other by a contact separation. The selection of the two paths in the configurational space is dependent on both contact ageing parameters and viscosity.

On the characterization of Brownian bridge measure on the pinned path space over a compact Riemannian manifold

In this paper, we focus on the characterization of a Brownian bridge measure on the pinned path space over a compact Riemannian manifold. In the case when the Riemannian manifold is simply connected, we prove that the integration by parts formula can characterize the Brownian bridge measure. Otherwise, we show that it is not always true by constructing an illustrating example.

Imagination as a fundamental function of the hippocampus.

Imagination is a biological function that is vital to human experience and advanced cognition. Despite this importance, it remains unknown how imagination is realized in the brain. Substantial research focusing on the hippocampus, a brain structure traditionally linked to memory, indicates that firing patterns in spatially tuned neurons can represent previous and upcoming paths in space. This work has generally been interpreted under standard views that the hippocampus implements cognitive abilities primarily related to actual experience, whether in the past (e.g. recollection, consolidation), present (e.g. spatial mapping) or future (e.g. planning). However, relatively recent findings in rodents identify robust patterns of hippocampal firing corresponding to a variety of alternatives to actual experience, in many cases without overt reference to the past, present or future. Given these findings, and others on hippocampal contributions to human imagination, we suggest that a fundamental function of the hippocampus is to generate a wealth of hypothetical experiences and thoughts. Under this view, traditional accounts of hippocampal function in episodic memory and spatial navigation can be understood as particular applications of a more general system for imagination. This view also suggests that the hippocampus contributes to a wider range of cognitive abilities than previously thought. This article is part of the theme issue 'Thinking about possibilities: mechanisms, ontogeny, functions and phylogeny'.

A new definition of rough paths on manifolds

Smooth manifolds are not the suitable context for trying to generalize the concept of rough paths on a manifold. Indeed, when one is working with smooth maps instead of Lipschitz maps and trying to solve a rough differential equation, one loses the quantitative estimates controlling the convergence of the Picard sequence. Moreover, even with a definition of rough paths in smooth manifolds, ordinary and rough differential equations can only be solved locally in such case. In this paper, we first recall the foundations of the Lipschitz geometry, introduced in Rough Paths on Manifolds (Cass, T., Litterer, C. & Lyons, T.), along with the main findings that encompass the classical theory of rough paths in Banach spaces. Then we give what we believe to be a minimal framework for defining rough paths on a manifold that is both less rigid than the classical one and emphasized on the local behaviour of rough paths. We end by explaining how this same idea can be used to define any notion of coloured paths on a manifold.

Identifying some regularities of radio frequency propagation of a radar system by analyzing different environmental effects

Free space path loss is a function of frequency and propagation distance and the RF signal propagates at light speed in all directions in free space. The performance evaluation of wireless and radar communication technologies is related to understanding the propagation environments. This work presents the modeling of several RF propagation properties that include atmospheric attenuation due to rain, free space path loss, gas, and fog, as well as multipath propagations caused by ground bounces. The methodology discusses the developed model according to the series of (ITU) International Telecommunication Union references to radio wave propagation. This work discusses the Free Space Path Losses (FSPL), and Propagation Losses (PL) due to the atmosphere, precipitation, snow, rain, clouds, fog, atmospheric lensing and absorption, and polarization mismatch. The work also discusses the vertical coverage diagram and radar propagation factor. The obtained results demonstrate that the PL increases with frequency and range, at a 90-degree roll angle, the attenuation approaches infinity, and as the altitude rises, the amount of attenuation caused by lensing decreases. The analysis of attenuation at 1 km vs. frequency variations, at roughly 60 GHz, indicated a high absorption owing to air gas. Lensing attenuations are also offered as secondary outputs for convenience, the wideband channels present higher performance crossways and a wide range of target height as expected. When the target height increases, the influence of multi-path fading approximately vanishes entirely due to the variation increasing in the spreading delay between the bounce and direct pathway signals. This will reduce the coherence sum between these two samples on receiving by the target

Central measures of continuous graded graphs: the case of distinct frequencies

We define a class of continuous graded graphs similar to the graph of Gelfand–Tsetlin patterns, and describe the set of all ergodic central measures of discrete type on the path spaces of such graphs. The main observation is that an ergodic central measure on a subgraph of a Pascal-type graph can often be obtained as the restriction of the standard Bernoulli measure to the path space of the subgraph. This observation dramatically changes the approach to finding central measures also on discrete graphs, such as the famous Young graph. The simplest example of this type is given by the theorem on the weak limits of normalized Lebesgue measures on simplices; these are the so-called Cesàro measures, which are concentrated on the sequences with prescribed Cesàro limits (this limit parametrizes the corresponding measure). More complicated examples are the graphs of continuous Young diagrams with fixed number of rows and the graphs of spectra of infinite Hermitian matrices of finite rank. We prove existence and uniqueness theorems for ergodic central measures and describe their structure. In particular, our results (1) give a new spectral description of the so-called infinite-dimensional Wishart measures (in Biometrika 20A(1–2):32–52, 1928)—ergodic unitarily invariant measures of discrete type on the set of infinite Hermitian matrices; (2) describe the structure of continuous analogs of measures on discrete graded graphs. New problems and connections which appear are to be considered in new publications.

Study on the Spatial Characteristics and Protection of Streets and Alleys in Ancient Towns in Southern Anhui

This study takes the typical ancient towns in southern Anhui Province as an example, and applies the quantitative research method of space syntax to analyze the spatial patterns of streets and alleys in Hongcun Town, Xidi Town and Sanhe Ancient Town. First, it analyzes the symbiotic relationship between a series of ecological laws, social culture and material space formed in the formation and development of ancient towns. This paper analyzes the general characteristics of street space in ancient towns from the aspects of street structure, communication mode with people and space scale, and discusses the optimization path and strategy of street space in historical towns. It provides new ideas for the renewal and protection of street space in ancient towns in southern Anhui, and reflects the concept of cultural self-confidence in the new era and the concept of comprehensive, coordinated and sustainable development.