Polynomial Curve Research Articles

Modeling and Simulation of High Voltage Power Lines under Transient and Persistent Faults

The work proposes an original method for modeling and simulating the triggering of 110 kV interconnection power lines in case of common faults, such as transient or persistent faults. Urban and industrial areas, surrounding urban areas, require a high energy consumption that is being supplied through 110 kV overhead power lines, responsible for distributing power to the industrial and domestic consumers. High voltage distribution power lines are most prone to failure, due to their exposure, affecting a large number of consumers if a fault occurs. Faults of power lines in service certify that currently there is no perfectly controllable operation mode in terms of load rating, environmental factors, insulation resistance, or mechanical resistance, which would allow total avoidance of faults, it is only possible to reduce the impact they have on the network as a whole. Mathematical models have been developed to determine the experimental voltage and current responses describing the fault propagation, expressed as a 7th-degree polynomial curve, as a second-order transfer function or as the Gaussian model type. By comparing these mathematical models, the most probable answers that can lead to the development of a control structure for rapid identification of a fault were obtained, with the possibility of triggering the line protection relay. In the final part of the manuscript, the viability of applying artificial intelligence techniques, for the approached fault management application, is proven. The developed control structure evaluates the nature of the fault and determines a faster reaction of the line protection causing an increase in the performance of the distribution service.

Adaptive Bézier Degree Reduction and Splitting for Computationally Efficient Motion Planning

As a parametric polynomial curve family, Bézier curves are widely used in safe and smooth motion design of intelligent robotic systems from flying drones to autonomous vehicles to robotic manipulators. In such motion planning settings, the critical features of high-order Bézier curves such as curve length, distance-to-collision, maximum curvature/velocity/acceleration are either numerically computed at a high computational cost or inexactly approximated by discrete samples. To address these issues, in this article we present a novel computationally efficient approach for adaptive approximation of high-order Bézier curves by multiple low-order Bézier segments at any desired level of accuracy that is specified in terms of a Bézier metric. Accordingly, we introduce a new Bézier degree reduction method, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">parameterwise matching reduction</i> , which approximates Bézier curves more accurately compared to the standard least squares and Taylor reduction methods. We also propose a new Bézier metric, called the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">maximum control-point distance</i> , that can be computed analytically, has a strong equivalence relation with other existing Bézier metrics, and defines a geometric relative bound between Bézier curves. We provide extensive numerical evidence to demonstrate the effectiveness of our proposed Bézier approximation approach. As a rule of thumb, based on the degree-one matching reduction error, we conclude that an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ n^{\rm{th}} $</tex-math></inline-formula> -order Bézier curve can be accurately approximated by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$3(n-1)$</tex-math></inline-formula> quadratic and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$6(n-1)$</tex-math></inline-formula> linear Bézier segments, which is fundamental for Bézier discretization.

A dynamic lane-changing decision and trajectory planning model of autonomous vehicles under mixed autonomous vehicle and human-driven vehicle environment

How to complete a lane changing process considering various variables has always been a critical issue in the field of autonomous driving. Developing a lane-changing decision model with full consideration of the surrounding vehicles and related decision-based trajectory planning model that comprehensively weighs safety and efficiency are conducive to the driving of autonomous vehicles (AVs) under mixed autonomous vehicle and human-driven vehicle (AV–HV) environment. Under the mixed AV–HV environment, we optimize a multi-player dynamic game model considering the status of surrounding vehicles to ensure the accurate execution of lane-changing decision of AVs. Lane changing trajectory of AV is planned based on polynomial curves, which can be dynamically updated according to the real-time status of vehicles and game results. Then, a computational experiment basing on the lane changing vehicles data from NGSIM (Next Generation Simulation) is performed with proposed models. The simulation results show that the lane-changing decision and trajectory planning model developed in our research have good adaptability to lane changing process in different scenarios, which can effectively measure the driving intention of surrounding vehicles and dynamically plan a smooth trajectory line considering safety and efficiency.

Industrial Robot Contouring Control Based on Non-Uniform Rational B-Spline Curve

This paper presents a novel algorithm about the industrial robot contouring control based on the NURBS (non-uniform rational B-spline) curve. First, aiming at the error between the industrial robot’s actual trajectory and the desired trajectory, the contour error is proposed as the trajectory evaluation index, and the estimation algorithm of contour error based on the tangent approximation is proposed. Based on the tangent approximation algorithm, the estimation algorithm of contour error in the local task coordinate frame is proposed to realize the transformation from the Cartesian coordinate frame to the local task coordinate frame. Second, according to the configuration of the industrial robot, a modified cross-coupling control scheme based on the local task coordinate frame is designed. Finally, the Bernoulli’s lemniscate curves are constructed by NURBS curve and five-order polynomial curve, respectively, and they are symmetrical. The contrast experiment is designed using the two types of constructed Bernoulli’s lemniscate curves as the incentive trajectory. Through the analysis and comparison between the obtained uniaxial tracking error and the contour error curve of the two incentive trajectories, it is concluded that the incentive trajectory constructed by the NURBS curve has better contour control performance than that constructed by the five-order polynomial curve. The results drawn from this paper lay a certain foundation for the future high-precision contouring control of industrial robots.

Construction and Evaluation of Pythagorean Hodograph Curves in Exponential-Polynomial Spaces

In the past few decades polynomial curves with Pythagorean hodograph (PH curves) have received considerable attention due to their usefulness in various CAD/CAM areas, manufacturing, numerical control machining, and robotics. This work deals with classes of PH curves built upon exponential-polynomial spaces (EPH curves). In particular, for the two most frequently encountered exponential-polynomial spaces, we first provide necessary and sufficient conditions to be satisfied by the control polygon of the Bézier-like curve in order to fulfill the PH property. Then, for such EPH curves, fundamental characteristics like parametric speed or arc length are discussed to show the interesting analogies with their well-known polynomial counterparts. Differences and advantages with respect to ordinary PH curves become commendable when discussing the solutions to application problems like the interpolation of first-order Hermite data. Finally, a new evaluation algorithm for EPH curves is proposed and shown to compare favorably with the celebrated de Casteljau--like algorithm and two recently proposed methods: Woźny and Chudy's algorithm and the dynamic evaluation procedure by Yang and Hong.

Positivization time of a COVID-19 rapid antigen self-test predicts SARS-CoV-2 viral load: a proof of concept.

This proof of concept study was aimed to validate the hypothesis that the time of positivization of SARS-CoV-2 self-performed rapid diagnostic tests (RDTs) may reflect the actual viral load in the specimen. A SARS-CoV-2 positive sample with high viralload was diluted and concomitantly assayed with molecular assay (Xpert Xpress SARS-CoV-2) and RDT (COVID-VIRO ALL IN RDT). The (mean cycle threshold; Ct) values and RDT positivization times of these dilutions were plotted and interpolated by calculating the best fit. The parameters of this equation were then used for converting the positivization times into RDT-estimated SARS-CoV-2 Ct values in routine patient samples. The best fit between measured and RDT-estimated Ct values could be achieved with a 2-degree polynomial curve. The RDT-estimated Ct values exhibited high correlation (r=0.996) and excellent Deming fit (y=1.01×x- 0.18) with measured Ct values. In 30 consecutive patients with positive RDT test, the correlation between RDT positivization time and measured Ct value was r=0.522 (p=0.003). The correlation of RDT-estimated and measured Ct values slightly improved to 0.577 (Deming fit: y=0.44×x+11.08), displaying a negligible bias (1.0; 95% CI,-0.2 to 2.2; p=0.105). Concordance of RDT-estimated and measured Ct values at the <20 cut-off was 80%, with 0.84 sensitivity and 0.73 specificity. This proof of concept study demonstrates the potential feasibility of using RDTs for garnering information on viral load in patients with acute SARS-CoV-2 infection.

A Multivariate Computational Approach With Hybrid Graphene Oxide Sensor Array for Partial Fulfillment of Breath Acetone Sensing

Breath analysis is becoming more prominent in the field of medical science because of its noninvasive nature. In the human breath, there exist thousands of volatile organic components; however, only a limited number are correlated with a specific disease. Acetone is the disease marker for diabetes, and an acetone concentration of more than 1800 ppb in a person’s breath indicates diabetes. In this study, different concentrations of acetone were detected in a dummy exhaled human breath consisting of a large number of interfering volatile organic compounds (VOCs), relative humidity (70%), and synthetic air. An array of six sensors comprised of hybridized graphene oxide (GO) field-effect transistors (FETs) was fabricated and tested for the detection of acetone within the concentration range of approximately 400 ppb to 80 ppm. A GO channel was hybridized with noble metal nanoparticles (NPs) like Au, Pd, and Pt along with WO3 flower-like nanostructures to enhance the acetone selective behavior under suitable back gate voltage in the FET structured sensors. A wide variety of feature vectors were extracted from the sensor data and used via linear discriminant analysis (LDA) and principal component analysis (PCA) to discriminate acetone with different stages of interfaces. Then, the exact acetone concentrations were quantified with a polynomial curve fitting in the sensitivity versus concentration characteristics of all the individual sensors. Acetone concentration in a complex VOC mixture was detected with an average 6.65% relative error by implementing the above technique.

No-Reference Video Quality Assessment using novel hybrid features and two-stage hybrid regression for score level fusion

This paper presents a novel No-Reference Video Quality Assessment (NR-VQA) model that utilizes proposed 3D steerable wavelet transform-based Natural Video Statistics (NVS) features as well as human perceptual features. Additionally, we proposed a novel two-stage regression scheme that significantly improves the overall performance of quality estimation. In the first stage, transform-based NVS and human perceptual features are separately passed through the proposed hybrid regression scheme: Support Vector Regression (SVR) followed by Polynomial curve fitting. The two visual quality scores predicted from the first stage are then used as features for the similar second stage. This predicts the final quality scores of distorted videos by achieving score level fusion. Extensive experiments were conducted using five authentic and four synthetic distortion databases. Experimental results demonstrate that the proposed method outperforms other published state-of-the-art benchmark methods on synthetic distortion databases and is among the top performers on authentic distortion databases. The source code is available at https://github.com/anishVNIT/two-stage-vqa.

CFD analysis of a wire-wrapped infinite sub-channel using temperature-dependent LBE properties

Once a significant number of LMFRs (liquid-metal cooled fast nuclear reactors) employ wire-wrapped fuel assemblies, considerable attention should be directed to this engineering feature. Given the many contact points and tight gaps, wire-wrapped fuel assemblies are difficult to analyze without numerical tools, out of which CFD (computational fluid dynamic) is the leading tool in the field. This work presents a CFD analysis of a wire-wrapped infinite sub-channel, based on MYRRHA’s fuel assembly geometry and with LBE (lead bismuth eutectic) coolant properties implemented as temperature-dependent polynomial curves. Periodic boundary conditions were applied, and the turbulence model chosen was k-w SST. All calculations were run on ANSYS Fluent R19.3. Following good engineering practices for CFD simulations, a GCI (grid convergence index) analysis was carried out, in order to ensure grid/mesh independence on the results. Ultimately, the main goal of this work was to evaluate if the implementation of temperature-dependent thermal-physical properties for LBE coolant would lead to any different results, when compared to the static properties. However, no significant result change was observed in this case, only an increase in computational time, which leads to the conclusion that for small domains with periodic boundaries, the implementation of temperature-dependent coolant properties is not justifiable.

S-Velocity Profile of Industrial Robot Based on NURBS Curve and Slerp Interpolation

This paper presents a novel algorithm for industrial robot trajectory planning based on the NURBS(Non-Uniform Rational B-Spline) curve and Slerp interpolation aiming at the problems that the trajectory of a six-axis industrial robot is not smooth enough in the operation process, the posture planning process is non-uniform, and the six-axis industrial robot starts and stops frequently. Firstly, aiming at the first problem, the trajectory planning algorithm based on the NURBS curve is presented to improve the smoothness of the trajectory curve. Combined with Slerp posture planning based on quaternion description, which realizes the uniform change of posture on the robot’s end-effector. Secondly, aiming at the second problem, the S-velocity planning algorithm is presented in the interpolation interval of the robot, which realizes the operation process of complex curves continuously, and improves the operation quality. Finally, this paper uses Bernoulli’s lemniscate as the incentive trajectory, and the contrast experiment of trajectory planning between two incentive profiles is designed, which are the NURBS curve and the five-order polynomial curve. The result of the experiment indicates that the planning algorithm proposed in this paper could effectively improve the smoothness of trajectory in a Cartesian workspace, decrease the impact and tremulous in a Cartesian workspace, and effectively improve the performance of the robot working process. The results drawn from this paper lay a certain foundation for the future high-precision control of industrial robots.

Polynomial curves with projections to PH curves

Despite the fact that the orthogonal projection of a spatial Pythagorean hodograph (PH) curve into the plane is not a planar PH curve in general, we can find special cases such that the PH property is preserved when the curve is projected. In Farouki et al. (2021) the authors studied how to generate spatial PH curves with planar PH projections. Their approach and presented results motivated us to continue and extend this investigation. We study geometric conditions under which a spatial curve is projected to a PH curve. For this purpose, we introduced a suitable geometric characterization of the curves with PH property via intersection multiplicity of the associated curves described by the hodograph mapping with the absolute conic. As a consequence we will show that a generic polynomial curve of degree higher than five possesses no parallel projection to a PH curve. On contrary, for a spatial cubic there are finitely many ways how to orthogonally project it to a planar PH cubic. And the same holds for oblique parallel projections of spatial quintics. Hence these cases are examined in more detail.

An improved algorithm for three-dimensional reconstruction of nonwoven microstructures by using multi-focus images

The filtration performance of nonwovens, such as virus protection and air purification, depends largely on their microstructure, that is, their internal structure. With the development of manufacturing techniques, nonwovens can be produced with complex microstructures to deliver low air resistance and high filtration efficiency. Therefore, it seems important to reconstruct an accurate three-dimensional (3D) model to describe the various microstructures for performance evaluation and product development. Multi-focus image fusion is one of the most effective ways used for the 3D nonwoven reconstruction. However, the overlapping and crossing of fibers in multi-focus images taken on an optional microscope make 3D fiber reconstruction incomplete and discontinuous. In this study, we developed a novel fiber segmentation and 3D nonwoven reconstruction algorithm to restore fiber continuity. The fiber web image was firstly thinned into the fiber mid-axis or skeleton, and fiber fragments were detected by searching the intersections. The features, such as the angle and depth, of each fiber fragment were obtained using Harris corner detection. A cluster technique was used to group the head-points of the fiber fragments that have similar features into one cluster. All fiber fragments in the same cluster are fitted with a polynomial curve to form a complete fiber in the 3D space. The experimental results show that this new method is suitable for the reconstruction of nonwovens with complex structures, such as fiber bonding or crossing, and can automatically detect and connect fiber fragments in 3D space. It provides an effective way for the subsequent simulation/evaluation of filtration characteristics.

Robust fault-tolerant control design for polynomial fuzzy systems

This paper focuses on the robust H∞ fault-tolerant control issue for a family of polynomial fuzzy systems subject to actuator faults and external disturbances. Especially, asynchronous fuzzy membership function technique is considered to minimize the expenses of controller implementation. To derive the robust stability conditions, actuator fault factors are taken into the account in the closed-loop system formulation. Moreover, to utilize the exact shape information of fuzzy membership functions in the stability criterion, they are approximated as sum-of-square polynomials with the aid of polynomial curve fitting method. In addition, to reduce the conservatism, a generalized polynomial Lyapunov function is considered. However, there are some non-convex terms in the stability conditions due to the construction of the generalized Lyapunov function that consists of all state-dependent polynomial matrix. Thus, some existing convex methods for general polynomial fuzzy systems are rendered inapplicable. To deal with this issue, the actual non-convex stability conditions converts into a convex optimization problem by introducing some additional convex constraints. Moreover, the membership-function-dependent robust stability conditions of the closed-loop system are provided as sum-of-square convex polynomial constraints. Lastly, the usefulness and superiority of the designed controller are illustrated by the simulation examples.

MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves

This paper studies the polynomial basis that generates the smallest n -simplex enclosing a given n th -degree polynomial curve in R n . Although the Bernstein and B-Spline polynomial bases provide feasible solutions to this problem, the simplexes obtained by these bases are not the smallest possible, which leads to overly conservative results in many CAD (computer-aided design) applications. We first prove that the polynomial basis that solves this problem (MINVO basis) also solves for the n th -degree polynomial curve with largest convex hull enclosed in a given n -simplex. Then, we present a formulation that is independent of the n -simplex or n th -degree polynomial curve given. By using Sum-Of-Squares (SOS) programming, branch and bound, and moment relaxations, we obtain high-quality feasible solutions for any n ∈ N , and prove (numerical) global optimality for n = 1 , 2 , 3 and (numerical) local optimality for n = 4 . The results obtained for n = 3 show that, for any given 3 rd -degree polynomial curve in R 3 , the MINVO basis is able to obtain an enclosing simplex whose volume is 2.36 and 254.9 times smaller than the ones obtained by the Bernstein and B-Spline bases, respectively. When n = 7 , these ratios increase to 902.7 and 2 . 997 ⋅ 1 0 21 , respectively. • The MINVO basis finds the smallest n -simplex enclosing a polynomial curve. • The MINVO basis finds the polynomial curve with largest convex hull in an n -simplex. • Improvements up to several orders of magnitude with respect to Bernstein/B-Spline. • Numerical global optimality is proven for n = 1,2,3. • High-quality feasible solutions are obtained for any degree n .

Process Modelling and Simulation of Key Volatile Compounds of Maillard Reaction Products Derived from Beef Tallow Residue Hydrolysate Based on Proxy Models.

The hydrolysis time is directly related to the flavor of the Maillard reaction, but existing proxy models cannot simulate and model the variation curves of vital volatile components. This study developed a predictive model for modelling and simulating key volatile compounds of Maillard reaction products (MRPs) derived from beef tallow residue hydrolysate. Results showed the degree of hydrolysis increased with hydrolysis time, and the most significant improvement in the roast flavor and overall acceptance was when hydrolyzing 4 h. Based on flavor dilution value and the relative odor activity value, nine key volatile components were identified, and 2-ethyl-3,5-dimethylpyrazine with roast flavor was the highest. Compared with Polynomial Curve Fitting (PCF) and Cubic Spline Interpolation (CSI), key volatile compounds of MRPs could be better modeled and simulated by the Curve Prediction Model (CPM). All results suggested that CPM could predict the changes in key volatile components produced by MRPs.

Toolpath generation for ultraprecision machining: Freeform surface blending

This paper is the final part in a two-part series investigating toolpath generation for freeform optics manufacturing. Building on the surface generation techniques already presented, this paper introduces a 3D blending surface that has the form of a continuous polynomial curve and is used to smoothly fill in the space between two non-intersecting, polar boundaries and two surfaces of the form Z(X,Y). The proposed blending surface renders it possible to smoothly join two surfaces defined over different domains together, as well as to smooth over the discontinuity created where two surfaces intersect. It is shown that increasing the order of the blending polynomial increases the continuity of the overall piecewise, blended surface. As a corollary, it is possible to increase the overall continuity of a toolpath along its cutting direction. To illustrate the robustness of the proposed surface blending technique, experimental data is presented for the toolpath generation, diamond machining, and measurement of a freeform microlens array. Designed with no exploitable symmetries or simplifications and very difficult to model with traditional CAD software, this surface is comprised of an array of freeform lenslets superimposed and blended onto a freeform base surface. All lenslets are individually tilted about the X and Y-axes such that they are normal to the base surface at their respective locations, and all blends have the same form as the intersection curve between each respective lenslet and base surface. Machined in brass to optical quality, this C3 continuous surface represents a general case of the proposed surface generation and blending techniques.

A Framework for Soil Salinity Monitoring in Coastal Wetland Reclamation Areas Based on Combined Unmanned Aerial Vehicle (UAV) Data and Satellite Data

Soil salinization is one of the most important causes of land degradation and desertification, often threatening land management and sustainable agricultural development. Due to the low resolution of satellites, fine mapping of soil salinity cannot be completed, while high-resolution images from UAVs can only achieve accurate mapping of soil salinity in a small area. Therefore, how to realize fine mapping of salinity on a large scale based on UAV and satellite data is an urgent problem to be solved. Therefore, in this paper, the most relevant spectral variables for soil salinity were firstly determined using Pearson correlation analysis, and then the optimal inversion model was established based on the screened variables. Secondly, the feasibility of correcting satellite data based on UAV data was determined using Pearson correlation analysis and spectral variation trends, and the correction of satellite data was completed using least squares-based polynomial curve fitting for both UAV data and satellite data. Finally, the reflectance received from the vegetated area did not directly reflect the surface reflectance condition, so we used the support vector machine classification method to divide the study area into two categories: bare land and vegetated area, and built a model based on the classification results to realize the advantages of complementing the accurate spectral information of UAV and large-scale satellite spectral data in the study areas. By comparing the modeling inversion results using only satellite data with the inversion results based on optimized satellite data, our method framework could effectively improve the accuracy of soil salinity inversion in large satellite areas by 6–19%. Our method can meet the needs of large-scale accurate mapping, and can provide the necessary means and reference for soil condition monitoring.

Preliminary Study on the Dissolved Oxygen Recovery Process in Freshwater Ecosystems under the Coupling Effect of Oxygen-Consuming Pollutants and Temperature

Widespread deoxygenation of the oceans has caused great concern, but few large studies of changes in dissolved oxygen (DO) concentrations in freshwater ecosystems and the factors driving such changes have been performed. Here, DO and oxygen-consuming pollutant (OCP) data for freshwater ecosystems in China from 2008 to 2018 were used to assess changes in DO concentrations and the mechanisms involved. DO concentrations in freshwater ecosystems in China recovered markedly in the study period. The DO concentrations have linearly increased (R2 = 0.95, p < 0.01) in that period. The chemical oxygen demand (CODMn) and NH3–N concentrations have significantly decreased [polynomial fitted curves (n = 2): R2 = 0.77, p < 0.01 for CODMn and R2 = 0.95, p < 0.01 for NH3–N]. The polynomial curves fitted to the CODMn, NH3–N, and DO data indicated that there are thresholds for oxygen-rich and deoxygenation processes dominated by OCPs in freshwater ecosystems. Decreasing OCP concentrations will not promote DO recovery if the CODMn and NH3–N concentrations are <4 and <0.4 mg L–1, respectively. The effect of temperature on the DO concentration in freshwater was obscured by inputs of large amounts of OCPs. The discovery of thresholds for the effect of OCPs led to the division of the DO recovery process into stages by increases in the temperature.

On planar polynomial geometric interpolation

In the paper, the planar polynomial geometric interpolation of data points is revisited. Simple sufficient geometric conditions that imply the existence of the interpolant are derived in general. They require data points to be convex in a certain discrete sense. Since the geometric interpolation is based precisely on the known data only, one may consider it as the parametric counterpart to the polynomial function interpolation. The established result confirms the Höllig–Koch conjecture on the existence and the approximation order in the planar case for parametric polynomial curves of any degree stated quite a while ago.

Hydrodynamic performance analysis of flapping hydrofoil for single-body architecture wave glider

Wave energy is directly converted into the propulsion of wave gliders by using the flapping hydrofoils. The propulsive performance of the wave glider has relationships with some parameters, especially spring constant of the submerged flapping hydrofoil mechanism and wave frequency. A numerical model of the submerged passive flapping hydrofoil based on the float response of the single-body architecture wave glider was established to analyze its propulsion in this paper. The float response was analyzed by using Advanced Quantitative Wave Analysis (AQWA) software under different wave frequencies in time domain and frequency domain. The numerical model of the hydrofoil was established, and the forward speed under different wave frequencies and spring constants were analyzed by using ANSYS-FLUENT software. By means of the polynomial curve fit, the corresponding equation describing the response of average forward speed was established. Tank experiment and sea trial were carried out to verify the accuracy of the model and the propulsive performance of the single-body architecture wave glider. The results showed that the speed fluctuation of the flapping hydrofoil was smaller as wave frequency increased under four wave frequencies. At the same time, wave frequency had a close relationship with spring constant, and there was an optimal spring constant under different wave frequencies to maximize the navigation speed of the whole system.