Relative Velocity Research Articles

Crash risk associated with powered two-wheeler filtering maneuvers in urban mixed and weak-lane-disciplined traffic streams

The paper studies the crash risk associated with the filtering Powered Two-Wheelers (PTWs) using the trajectory data. A methodology is proposed to automatically identify the filtering PTWs from the trajectory data. A state-of-the-art surrogate safety measure, considering both proximity and evasive actions, was used for identifying the PTW conflicts. The results revealed that the side-swipe conflicts, particularly involving larger vehicles, are predominantly observed during filtering maneuvers. However, the results revealed no significant correlation between the accepted lateral gap and relative speed during PTW filtering. These aspects highlight the hazardous driving behavior of PTW riders while engaging in the filtering maneuvers. The side-swipe crash risk during PTW filtering is about three times higher than the rear-end and side-swipe crash risk during non-filtering. The study suggests that improving infrastructure by implementing designated PTW lanes, where feasible, can reduce the interaction of PTWs with larger vehicles, thereby mitigating the risk of severe crashes.

Force-velocity profiling in elite luge athletes during the simulated luge start

Introduction & Purpose The idea of some athletes having high strength levels but being too slow for a particular movement while others are fast but not strong enough for the demands of their sport exists among many coaches. Therefore, planning decisions are often based on the measurement results of speed and force capabilities and the interest in the relationships between additional load, force, velocity and power during the required movement is high. The luge start is a ballistic movement that requires both high forces and velocities to reach maximal performance (i.e. fast starting times). Examining the load, force, velocity and power capabilities of elite luge athletes during the luge start could deliver insights for training prescription to improve individual weak points and improve performance. This study investigated the relationship between the calculated maxima of force-velocity (FV), load-velocity (LV) and power-velocity (PV) profiles during a simulated luge start and on-ice start times in elite luge athletes. Methods Eleven Austrian national luge team athletes (1 female, 10 male, 25 [4.4] y, 182.4 [7.3] cm, 86.65 [10.4] kg) participated in this study. Measurements were conducted during a single testing session on a specially designed start simulator. Each athlete performed starts against 5 increasing resistances relative to their bodyweight (0%, 25%, 50%, 75%, 100%). For each pull, relative maximum force (Fmax), maximum power (Pmax) and maximum velocity (Vmax) were calculated using the measurements of a linear position transducer (GymAware PowerTool). An experimental approach was used to create the FV-, LV- and PV-profiles showing the highest R2. On-ice start times [s] were measured by photoelectric sensors (ALGE). A correlation matrix (p < 0.05) was used to analyze the relationships between the theoretical Pmax, Fmax (i.e. theoretical isometric maximum force), theoretically unloaded Vmax (Vmax(F0)), theoretical relative maximum loads (Lmax(V0)) and starting times obtained from the profiles. Results Ten linear FV profiles and 11 exponential LV were created (Figure 1; FV: R2mean = 0.9532 ± 0.06; LV: R2mean = 0.9918 ± 0.01). There were significant negative relationships between Vmax(F0) and on-ice starting times (r(9) = -.74, p < .05; r(9) = -.71, p < .05; r(9) = -.74, p < .05) and significant positive correlations between Lmax(V0) and ice start times (r(9) = .71, p < .05; r(9) = .71, p < .05; r(9) = .72, p < .05). Discussion The FV relationship during a simulated luge start can be represented mostly through a linear regression line, supporting the findings of former studies which showed a quasilinear relationship between force and velocity in multi-joint movements (Bobbert, 2012; Jaric, 2015). In line with earlier approaches, PV relationships could be represented quite accurately by use of second-degree polynomial regressions (Allison et al., 2013; Padulo et al., 2017). Contradictory to the linear LV relationships found in other movements (García-Ramos et al., 2018), the luge start LV profile can be better represented by exponential regression curves which could be due to the minimized friction in the luge start setup. Conclusion The abilities of accelerating high external loads and producing high velocities both affect luge start time.

Quiet Please: Detrending Radial Velocity Variations from Stellar Activity with a Physically Motivated Spot Model

For solar-type stars, spots and their associated magnetic regions induce radial velocity perturbations through the Doppler rotation signal and the suppression of convective blueshift, collectively known as rotation modulation. We developed the Rotation–Convection (RC) model: a method of detrending and characterizing rotation modulation using only cross–correlation functions or one-dimensional spectra without the need for continuous high-cadence measurements. The RC method uses a simple model for the anomalous radial velocity induced by an active region and has two inputs: stellar flux (or a flux proxy) and the relative radial velocity between strongly and weakly absorbed wavelengths (analogous to the bisector–inverse slope). On NEID solar data (3 month baseline), the RC model lowers the amplitude of rotationally modulated stellar activity to below the meter–per–second level. For the standard star HD 26965, the RC model detrends the activity signal to the meter–per–second level for HARPS, EXPRES, and NEID observations, even though the temporal density and time span of the observations differ by an order of magnitude between the three data sets. In addition to detrending, the RC model also characterizes the rotation–modulation signal. From comparison with the Solar Dynamics Observatory, we confirmed that the model accurately recovers and separates the rotation and convection radial velocity components. We also mapped the amplitude of the rotation and convection perturbations as a function of height within the stellar atmosphere. Probing stellar atmospheres with our revised spot model will fuel future innovations in stellar activity mitigation, enabling robust exoplanet detection.

Average optimal minimum velocity threshold: A practical variable to increase the accuracy of one-repetition maximum estimation during the free-weight back squat

ABSTRACT The prediction of one-repetition maximum (1RM) using the submaximal load–velocity relationship (LVR) is highly relevant for the field of strength and conditioning. The optimal minimum velocity threshold (MVT) was recently proposed to increase the accuracy of 1RM predictions. However, using the average optimal MVT would allow for more practical estimations. LVRs of the free-weight back squat were obtained in 53 participants, throughout 2 sessions. LVRs were obtained using the multi- and two-point methods. Estimations of 1RM were made based on the average actual MVT (1RM velocity) and the average optimal MVT. The accuracy of 1RM predictions was examined using absolute-percent error and Bland-Altman plots. Cross-validation was performed using a leave-one-out approach. The number of selected loads did not affect the slope, y-intercept, optimal MVT or the accuracy of 1RM predictions. Predictions based on the average optimal MVT displayed greater accuracy than those obtained with the average actual MVT (~6% vs. ~8% absolute-percent error, respectively). However, wide 95% limits of agreement (LoA) were found between actual and estimated 1RM using both approaches (~13%1RM). The average optimal MVT offers more accurate 1RM estimations than the average actual MVT. However, errors prove substantial, making it challenging to precisely track minor changes in 1RM.

Candidate Runaway Stars in the Cluster NGC 2180

The relative proper motions and tangential velocities of four stars from the cluster NGC 2180 are investigated using Gaia DR3 data. Our approach is to subtract the mean proper motion of the fields (the stars within a 1” radius around each target star) and the cluster from the motion of the target stars to calculate the relative proper motions with respect to their field and the cluster, respectively. From the relative proper motions of the target stars with respect to the field stars and the distance to the center of the cluster, we estimated the relative tangential velocities of our target stars. We use the astrometric excess noise information to filter the field stars from star-like objects, such as unresolved binaries. To eliminate foreground contaminations of the field stars used to figure out the relative proper motions of the target stars, we made a parallax cut, Color Magnitude Diagram (CMD) cut, and proper motion (PM) selection. The flight times of the target stars to the cluster NGC 2180 (kinematic ages) are calculated using angular separations from the cluster and their relative proper motions with respect to the cluster. All of the target stars have kinematic ages less than the age of the cluster. The stars, ID1 and ID3 have and with respect to their fields, which corresponds to the relative velocities respectively. These two stars could be runaways from the cluster because they have flight times from the cluster agreed with late ejection from the cluster and have larger proper. The remaining two stars, ID 2 and ID 4 have and with respect to the fields, which corresponds to the relative velocities and respectively, and could not be runaways as they have smaller proper motions, although their flight times to the cluster agreed with late ejection from the cluster.

Heating patterns and temperature distribution of projectile surface in lunar regolith penetration

During penetration, a large quantity of friction-induced heat is generated, significantly increasing the projectile surface temperature. Considering that the temperature variation depends on the physical properties of the target being penetrated, understanding this relationship can aid in extraterrestrial material behavior for detection and analysis efforts. The study investigated the patterns of heat generation and the distribution of temperature on the surface of a projectile as it penetrates lunar regolith. For discrete medium penetration, large deviations appear in temperature prediction due to particle extrusion flow. Thus, a heat flux density model on the projectile surface by introducing a relative velocity factor (RVF) for correction was established. The particle flow characteristics simulation and fitting model of the introduced factor were also obtained. We constructed a theoretical relationship between the resistance and stress model parameters using dynamic modeling. Experimental projectiles recording penetration acceleration and temperature at the points of interest on the projectile surface were designed and tested to obtain recorded data. The temperature field in this process was simulated in COMSOL software to calculate the projectile surface's heat flux density and temperature distribution. The results indicate that the developed model is effective. This research infers the physical characteristics of the penetrating target under specified penetration conditions and provides more dimensional information for lunar regolith exploration.

Spin orbit resonance cascade via core shell model: application to Mercury and Ganymede

We discuss a model describing the spin orbit resonance cascade. We assume that the body has a two-layer (core–shell) structure; it is composed of a thin external shell and an inner and heavier solid core that are interacting due to the presence of a viscous friction. We assume two sources of dissipation: a viscous one, depending on the relative angular velocity between core and shell and a tidal one, smaller than the first, due to the viscoelastic structure of the core. We show how these two sources of dissipation are needed for the capture in spin–orbit resonance. The shell and the core fall in resonance with different time scales if the viscous coupling between them is big enough. Finally, the tidal dissipation of the viscoelastic core, decreasing the eccentricity, brings the system out of the resonance in a third very long time scale. This mechanism of entry and exit from resonance ends in the 1 : 1 stable state.

PM1.0 removal enhancement in cooperated electric-ionic field by negative air ions carbon fiber electrode

The study focuses on improving the removal efficiency of submicron particles using negative air ions (NAIs), which are challenging to remove. To enhance this efficiency, cascaded carbon fiber tubular (CFT) chargers were designed to boost particle charging. Through experiments and numerical simulations, the chargers’ enhancement mechanisms, cooperation effects, and overall performance were evaluated. The optimal configuration for the CFT charger was identified as a gap distance (d = 2.0 r0) and grounded electrode length (L0 = 4πr03) that ensures high collection efficiency (96.7 % for 0.5 μm particles at 4 m/s) and ultra-low ozone generation (7.25 ppb). The study found that while the average electric field strength (E¯) and ion density significantly influence charging, an unusual improvement in collection efficiency was observed the charger with d = 2.0 r0 at u = 3 m/s, likely due to the potential predominance of the maximum electric field strength (Emax) at relative higher velocity. Besides, the CFT charger demonstrated balanced performance compared to other reported works, although higher relative humidity (80 ± 5 % RH) inhibits removal efficiency, which can be mitigated by reducing gas velocity.

A Two-month Mini-moon: 2024 PT5 Captured by Earth from September to November

Near-Earth objects (NEOs) that follow horseshoe paths, and approach our planet at close range and low relative velocity, may undergo mini-moon events in which their geocentric energy becomes negative for hours, days or months, but without completing one revolution around Earth while bound. An example of NEO experiencing such a temporarily captured flyby is 2022 NX1, which was a short-lived mini-moon in 1981 and 2022. Here, we show that the recently discovered small body 2024 PT5 follows a horseshoe path and it will become a mini-moon in 2024, from September 29 until November 25.

Spatiotemporal Variation Characteristics of Ozone and Identification of Key Influencing Factors Based on Random Forest Model： A Case Study of Chuzhou City

The cause of ozone pollution is a complex scientific problem. Studying the spatiotemporal variation characteristics of O3 at different time scales and analyzing the key influencing factors of O3 concentration is of great significance for the precise formulation of urban air pollution control measures and the improvement of urban air quality. Based on the analysis of the spatiotemporal variation characteristics of O3 concentration in Chuzhou City, we studied the 12 ozone-influencing factors of meteorology and pollutants at multiple time scales using Spearman correlation analysis and a random forest model. The results showed that： ① The O3 pollution level of Chuzhou City showed an aggravating trend, and the O3 concentration distribution showed a spatial pattern of "high in the southeast and low in the northwest." ② From February to May, SO2 concentration had a strong impact on the increase in O3 concentration. From June to September, PM2.5 and PM10 were significantly positively correlated with ozone and had a greater impact. ③ Relative humidity, temperature, and wind speed had a significant impact on O3, whereas barometric pressure and hourly rainfall had a weak impact. ④ The O3 pollution mechanism in Chuzhou City changed from "pollutant-controlled" to "meteorology-controlled." ⑤ Among meteorological and pollutant factors, the three influencing factors that had the greatest influence on O3 concentration were temperature, wind speed, and relative humidity, with PM10 concentration, PM2.5 concentration, and SO2 concentration also contributing. All of the above six influencing factors had a significant nonlinear relationship with the O3 concentration.

Molecular simulation of CO2/N2 injection on CH4 adsorption and diffusion

As an efficient and clean energy, coalbed methane development and utilization have deep significance in promoting energy conservation and emission reduction, reducing greenhouse gas emissions. Therefore, molecular simulation was utilized to study the influence of N2/CO2 on the adsorption and diffusion of methane in coal under different gas injection methods and to elucidate the influence of varying gas injection methods on the efficiency of coalbed methane extraction, which provides a basis for the efficient development of coalbed methane. The results show that the adsorption effect of gases in coal decreases with the increase of temperature and increases with the rise of pressure, and the adsorption performance of the three gases in coal shows the law of CO2 > CH4 > N2. In addition, the injection of CO2/N2 had an obvious inhibition effect on CH4 adsorption, and the inhibition effect of CO2 was more significant, and the inhibition effect on CH4 adsorption reached the maximum when the two gases were mixture injected. In terms of diffusion, compared with separate injection, mixed injection of N2 + CO2 promotes CH4 diffusion more effectively, which can be reflected in the relative concentration distribution and velocity distribution. The injection of N2 helps to increase the porosity of coal, and the injection of CO2 and N2 + CO2 will lead to the decrease of porosity, but the mixed gas injection has less effect than the injection of CO2 alone.

Collision dynamics between breather and lump-type localized waves in the (3+1)-dimensional shallow water wave equation

In this paper, the collision dynamics of breather and lump-type localized waves in the (3+1)-dimensional shallow water wave equation are investigated in detail. Firstly, the auto-Bäcklund transformation and the linear representation of the (3+1)-dimensional shallow water wave equation are derived in virtue of the truncated Painlevé expansion method, which provide convenience in solving the (3+1)-dimensional shallow water wave equation. Secondly, based on the linear representation and the principle of linear superposition, the rational solutions in the exponential and polynomial forms are constructed. Tuning the free parameters of the rational solutions, localized waves of various patterns are obtained such as breather, lump-type localized waves and their hybrid structure. The anomalous inelastic interaction phenomenons of breather and lump-type localized waves are exhibited. Thirdly, combining the large-time behaviors of solution with the velocity relationship of localized waves, the dynamical properties and the classification of localized wave solutions are discussed in detail. Finally, we discuss the bound state of breather and lump-type localized waves under the velocity resonance condition, three different types of lump-breather molecules are displayed. The obtained results further enrich the structures and dynamical behaviors of localized waves. It is expected that the interaction phenomena taking place in the (3+1)-dimensional shallow water wave equation will be helpful in predicting or controlling some related shallow water wave phenomena.

Barred active galactic nucleus galaxies in paired systems: Exploring the impact on nuclear activity

To unveil the influence of galaxy-galaxy interactions on the material transport driven by galactic bars toward the central regions of active galactic nucleus (AGN) galaxies, and to assess the efficiency of the combined mechanisms of interactions and bars in fueling massive black holes, we meticulously examine barred active galaxies in paired systems. Our study focuses on barred AGN galaxies in pairs with projected separations of $r_ p \,kpc $ and relative radial velocities of $ V< 500 \,km $ within $z<0.1$, identified by the Sloan Digital Sky Survey (SDSS). To quantify the impact of interactions on material transport by galactic bars, we also constructed a suitable control sample of barred active galaxies without paired companions, matched in redshift, absolute r-band magnitude, stellar mass, color, and stellar age distributions. Additionally, we calculated the structural characteristics of galactic bars through two-dimensional image modeling, considering that bars exhibit a wide range of shapes and sizes, which may influence their ability to channel material. From this study, we clearly found that nuclear activity (derived from the $Lum OIII $) increases as the projected separations between galaxy pair members decrease. Notably, barred AGN galaxies in close pairs ($r_p 25 kpc $) exhibit significantly higher nuclear activity compared to galaxies in the control sample. Additionally, barred galaxies with a close pair companion show enhanced nuclear activity across all ranges of luminosity, stellar mass, and color. We also found that barred AGN galaxies with longer bar structures exhibit more efficient nuclear activity compared to those with shorter bars. This trend is especially pronounced in barred AGN galaxies within close pair systems, which show a significant excess of high $Lum OIII $ values. Furthermore, we examined the central nuclear activity in barred AGNs undergoing major and minor interactions. Our findings show a clear escalation in nuclear activity as the pair projected separations decrease, particularly pronounced in major systems. Additionally, nuclear activity distributions in barred AGN samples within major and minor pairs exhibit similar trends. However, a significant deviation occurs among barred AGN galaxies in close pair systems within major interactions, showing a substantial excess of high $Lum OIII $ values. This result is also reflected in the analysis of the accretion strength onto central black holes. These findings indicate that external perturbations from a nearby galaxy companion can influence gas flows induced by galactic bars, leading to increased nuclear activity in barred AGN galaxies within pair systems. Thus, the coexistence of both — bars and interactions — significantly amplifies central nuclear activity, thereby influencing the accretion processes onto massive black holes.

Numerical study of downwash flow on rice plant protection drone with computational fluid dynamics method

Unnamed Unmanned Aerial Vehicles (UAVs) are increasingly being utilized in various industries, including agriculture, to support the growing demand for food. UAVs streamline work processes and are particularly useful in the spraying method for plant protection. This study aims to analyze the characteristics of the downwash flow, which are influenced by factors such as flight altitude, airfoil profile, and the flying speed of the drone. Unlike previous studies that used 6-blade UAVs, this research focused on a 4-blade configuration. The study employed Computational Fluid Dynamics (CFD) to analyze drone geometry and input boundary conditions based on environmental factors. The drone's flying altitude significantly impacted downwash flow, particularly concerning In Ground Effect (IGE) and Out of Ground Effect (OGE) conditions. Unlike previous research, this study considered the airfoil profile of the propeller, which, along with the drag and lift coefficients from the airfoil geometry, affected the downwash flow. The drone's flying speed, related to the relative wind speed around its working area, also influenced pressure distribution and downwash flow speed. These factors significantly impacted downwash flow and determined the distribution of plant protection droplets on the rice field. The results indicated that increasing flight altitude reduced the ground effect, affecting the quadcopter's downwash. Similarly, flight speed had a similar effect on downwash as altitude. Based on these findings, the study recommended a flight altitude of 2 m and a speed of 2 m/s for optimal downwash and proper distribution of plant protection.

Association of meteorological variables with leaf spot and fruit rot disease incidence in eggplant and YOLOv8-based disease classification

Eggplant is one of the major vegetables consumed worldwide. Several fungal, bacterial, and viral diseases challenge the yield and quality of eggplant. The incidence of plant diseases is strongly influenced by weather factors such as temperature, humidity, rainfall, and wind speed. Mattu Gulla (MG) is a GI-tagged traditional variety of eggplant grown in Mattu village of the Udupi district in Karnataka state, India, with a cultural legacy of more than four centuries. In this study, we investigated the relationships between weather parameters and disease incidence in Mattu Gulla. Leaf spot (LS) and fruit rot (FR) are the major diseases affecting this plant variety. The influence of plant age and weather parameters on the modulation of the disease incidence (%) [DI (%)] of leaf spot and fruit rot was recorded and analyzed via correlation and regression. Prediction equations for disease incidence was derived via regression. A significant negative correlation was observed between the leaf spot DI (%) and minimum temperature (Min. temp), and a positive correlation was observed between the DI (%) and fruit rot. In the case of FR, the DI (%) is also significantly positively correlated with wind speed (WS), temperature, maximum relative humidity (RH I), rainfall (RF), and wind speed (WS). An RH I of 86–87 % was favorable for the incidence of fruit rot in the field. Regression analysis revealed a significant association between Min. temp and leaf spot DI (%), and in the case of fruit rot DI (%), the association was with Min. temp and WS. An android application, “Leaf Guard,” has been developed for AI-based disease detection in eggplant. During testing, the accuracy of the trained model reached 98.2 %.

Three-Dimensional Lorentz-Invariant Velocities

Lorentz invariance underlies special relativity, and the energy formula and relative velocity formula are well known to be invariant under a Lorentz transformation. Here, we determine the functional forms in terms of four arbitrary functions for those three dimensional velocity fields that are automatically invariant under the most general fully three-dimensional Lorentz transformation. For general three-dimensional motion, using rectangular Cartesian coordinates (x,y,z), we determine the first-order partial differential equations for the three velocity components u(x,y,z,t), v(x,y,z,t) and w(x,y,z,t) in the x−, y− and z−directions respectively. These partial differential equations and the associated partial differential relations connecting energy and momentum are fully compatible with the Lorentz-invariant energy–momentum relations and appear not to have been given previously in the literature. We determine the spatial and temporal dependence of the functional forms for those three-dimensional velocity fields that are automatically invariant under three-dimensional Lorentz transformations. An interesting special case gives rise to families of particle paths for which the magnitude of the velocity is the speed of light. This is indicative of the abundant possibilities existing in the “fast lane”.

Constraining bosonic dark matter-baryon interactions from neutron star collapse

Dark matter (DM) may be captured around a neutron star (NS) through DM-nucleon interactions. We observe that the enhancement of such capturing is particularly significant when DM-nucleon scattering cross-section depends on the relative velocity and/or momentum transfer. This increment could potentially lead to the formation of a black hole within the typical lifetime of the NS. As the black hole grows through the accretion of matter from the NS, it ultimately results in the collapse of the host. Utilizing the existing pulsar data J0437-4715 and J2124-3858, we derive the stringent constraints on the DM-nucleon scattering cross-section across a broad range of DM masses.

Energetic explosions from collisions of stars at relativistic speeds in galactic nuclei

Aims. We investigated collisions that could occur between stars moving near the speed of light around supermassive black holes (SMBHs) with mass M• ≳ 108 M⊙, without being tidally disrupted. Within this approximate SMBH mass range, for sun-like stars, the tidal-disruption radius is smaller than the SMBH’s event horizon; therefore we did not anticipate tidal disruption events (TDEs). Methods. Differential collision rates were calculated by defining probability distribution functions for various parameters of interest, such as the impact parameter, distance from the SMBH at the time of the collision, the relative velocity between the two colliding stars, and the masses of the two colliding stars. The relative velocity parameter was drawn from an appropriate distribution function for SMBHs. We integrated over all these parameters to arrive at a total collision rate for a galaxy with a specific SMBH mass. We then considered how the stellar population in the vicinity of the SMBH was depleted and replenished over time, and calculated the effect this can have on the collision rate over time. We further calculated the differential collision rate as a function of the total energy released, the energy released per unit mass lost, and the galactocentric radius. Results. The overall rate for collisions taking place within the inner ∼1 pc of galaxies with M• = 108, 109, and 1010 M⊙ are Γ ∼ 2.2 × 10−3, 2.2 × 10−4, and 4.7 × 10−5 yr−1, respectively. The most common collisions would release energies on the order of ∼1049 − 1051 ergs, with the energy distribution peaking at higher energies in galaxies with more massive SMBHs. In addition, we examined sample light curves for collisions with varying parameters, and find that the peak luminosity could reach or even exceed that of superluminous supernovae (SLSNe), albeit in the case of light curves with much shorter durations. Conclusions. Weaker events may initially be mistaken for low-luminosity supernovae. In addition, we note that these events would likely create streams of debris that would accrete onto the SMBH, potentially creating accretion flares that may resemble TDEs.

Characterization of friction between metal-polymer material pairs with different speeds and surface pressures

Abstract Engineering design requires accurate knowledge of the friction conditions of material pairs. The friction coefficient (static or dynamic) depends on many parameters in addition to the pairs of materials used (e.g. surface roughness, temperature, humidity, surface pressure, etc.). In this study, based on our own measurements, we present the friction conditions of an aluminum (Al 6082-T6) – polypropylene homopolymer surface pair under different surface loads (5-15 MPa) and different relative speed differences (4-8–48-96 mm/min). The results of the experiments show that the static and dynamic friction coefficients decrease with the increase of surface load. As relative speed increases, the dynamic coefficient of friction increases, while the static coefficient of friction can be considered constant.

Prescribed-Time Network-Based Deployment of Nonlinear Multiagent Systems: A Discrete-Space PDE Method.

This article attempts to design the prescribed-time time-varying deployment schemes for first-order and second-order nonlinear multiagent systems (MASs). We assume that all agents can obtain the information of their current and final relative positions with their neighbors, and the final absolute velocities (as well as their current and final relative velocities, the final absolute accelerations for the second-order MASs) through a communication network, whereas two boundary agents are able to obtain their current and final absolute positions (as well as their current and final absolute velocities for the second-order MASs). The neighbor relationship of all agents is described by a spatial variable and two static-feedback controllers are introduced, which can be expressed as a second-order space difference of the spatial variable. Then, the deployment of MASs can be transformed into the stabilization of discrete-space partial differential equation (PDE) systems. Three virtual agents are introduced to constitute the Dirchlet and Neumann boundary conditions. Several algebraic inequality criteria are derived to guarantee that the prescribed-time time-varying deployment can be achieved within a prescribed time under the Dirchlet and mixed boundary conditions. Unlike the published results, our results are derived based on the discrete-space PDE systems instead of continuous-space PDE systems, which is consistent with the discrete spatial distribution of agents. Finally, two numerical examples are given to illustrate the effectiveness of our results.