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We used an optimal control method involving covariant control equations as optimality conditions, to command the actuators of robot manipulators. These form a coupled system of second order nonlinear ordinary differential equations when associated with the robot motion equations. By solving this system, the control action required to take the robot from an initial to a final state is optimized in a prescribed time. However, the target set of equations exhibited stiffness. Therefore, an adequate solution could only be found for short trajectory durations with readily available numerical methods. We examined a time discretization procedure based on cubic and quintic Hermite finite elements which exhibited superconvergence properties for interpolation. This motivated us to develop a time integration algorithm based on Hermite’s technique, where motion and control equations were perturbed to solve the optimal control problem. The optimal motion of a robotic manipulator was simulated using this algorithm. Our method was compared with a commercial differential equations solver on the basis of specific indicators. It outperformed the commercial solver by effectively solving the stiff set of equations for longer trajectory durations, with the cubic elements performing better than the quintic ones in this sense. The convergence analysis of our method confirmed that the quintic elements are more precise at the cost of increased computational burden, but converge at a lower rate than expected. Controlled motion experiments on a robotic manipulator validated our methodology. Trajectories were smoothly tracked and results exposed further methodology improvements.

In this paper, a new method of online planning high smooth and time-optimal trajectory for robotic manipulators that applies an adaptive elite genetic algorithm with singularity avoidance (AEGA-SA) is presented. The strategy is designed as a combination of the time-optimal trajectory planning with quintic polynomial in Cartesian space. For improving optimization performance, elitist group and adaptive adjustment mechanisms are used based on genetic algorithm (GA) framework. In the meantime, GA is combined with singularity avoidance mechanism to avoid the singularities appearing in the trajectory, improves the recognition capability of optimum solution. Experimental results show that, the proposed approach is more effective and better performance than the original GA and its variants, with ensuring a both smooth and efficiency performance for the robotic manipulators.

Different from urban environment with high-precision map, off-road environment lacks rich priori road structure information and precise location information, which is a huge challenge for autonomous driving. To solve these problems, this paper proposes a real-time decision-making and local trajectory planning method based on a predefined global reference line. According to the reference line, a cluster of parallel offset curves with corresponding obstacle and deviation costs are generated. Then, an appropriate parallel line with minimum cost is selected as the preliminary decision, and the target position with a matched speed on this line is choosed for local planning. Finally, the lateral and longitudinal trajectories planning is carried out with the initial constraint and target constraint of the quintic polynomials. The proposed method is verified in a 3.6 km bumpy off-road environment, including straight road, curves, continuous U-turn and other driving conditions. The maximum autonomous driving speed of an small-medium-sized platform can reach 30 km/h under the condition of stable and smooth driving.

Applying the Slime Mould Algorithm (SMA) in optimization and coding it with Python programming language, we found that runtime warnings occurred because of the inherent limits of arctanh functions. Considering the circumstances that the maximum residual errors would be constrained and the maximum allowed iteration number would be bothering, we therefore proposed an improvement to the SMA with exponential functions. Three kinds of benchmark functions were introduced to verify the suitability, accessibility, and capability. Experiment results demonstrated that the improved SMA with exponential functions would be suitable and accessible. Comparing with the standard SMA, the improved algorithm would result in better performance with exception that they perform similar when optimizing the Quintic function, the representative of those kinds of benchmark functions with basins in their three-dimensional profiles.

Many soft robotic systems for manipulation have been developed for their compliance and their smooth obstacles avoidance capabilities contrary to rigid systems. In order to increase human task aid, a new class of robots known as mobile soft manipulators has thus emerged, where the path planning of the whole system, both soft manipulator and its mobile platform, remains a real scientific challenge. For that purpose, in this paper, we suggest a curvature control algorithm for that class of hyper-redundant mobile soft manipulator approaching and handling objects with simultaneous obstacle avoidance based on potential field of velocity. The algorithm is based on principle of sliding mode, which converges in a finite time. The soft manipulator kinematics is modeled in 2D by the mean of Pythagorean Hodograph (PH) quintic curve theory with length constraints. The result of this modeling and control is applied on Festo Robotino-XT, mobile and soft manipulator robot.

Many fields of science rely on the collection of samples and estimation of true population distributions from those samples. There are several effective nonparametric methods for approximating a true distribution from empirical data, however it is unclear which methods produce the best approximations in practice. This work presents a case study on the effectiveness of various distribution approximations. Results show that piecewise linear approximations produce the smallest maximum absolute error, while the classic empirical distribution function (EDF) produces the smallest median absolute error as well as the smallest first quartile and minimum absolute error when approximating a distribution from a sample. When building distribution prediction models, the piecewise quintic and cubic approximations produce the lowest absolute error at most error percentiles. This case study encourages more research on the best methods of fitting empirical data with smooth functions to generate accurate distribution approximations.

The propagation of elliptic optical beam in (1 + 2)-dimensional nonlocal competing cubic–quintic (CQ) nonlinear media was investigated analytically and numerically. The evolution equations for the parameters of the optical beam and the critical powers of stable propagation in x - and y -directions are obtained. Both the analytical and numerical solutions show that the stable elliptic optical solitons can be formed in nonlocal cubic, quantic, and competing CQ nonlinear media. The stability of soliton depends on the initial power, beamwidth, and characteristic length of the response function. In particular, there are bistable elliptic optical solitons in nonlocal media with self-focusing cubic and self-defocusing quintic nonlinearities when these nonlocal and nonlinear parameters are in the appropriate value domain.

For selecting appropriate permutation polynomials (PPs) in practical applications, it is necessary to know the coefficients of the polynomial since a brute-force exhaustive search is impractical when the number of PPs is large. Previous results give the conditions on the coefficients of a polynomial of degree up to four so that it is a PP modulo a given positive integer. For polynomials of degree higher than four, we only know the conditions so that they are PPs modulo a power of two. In [13] all PPs of degree no more than six are generated using an algorithm based on normalized PPs, two previous important theorems about PPs and the Chinese remainder theorem. In this paper, we propose a coefficient test for quintic permutation polynomials (5-PPs) over integer rings which, unlike the algorithm from [13] , allows to decide directly whether a polynomial of degree five or less is PP. Using the proposed coefficient test, the coefficients of PPs modulo a given positive integer can be obtained in a desired order, which is tractable in computer processing.

We numerically simulate dynamic propagation of finite-energy Airy pulses in anomalous and normal regions of optical fiber and analyze the effects of quintic nonlinearity and initial chirp on evolution properties. Numerical results show that the effects of quintic nonlinearity on finite-energy Airy pulse imposed by initial chirp in two regions are entirely different. In anomalous dispersion region, soliton pulses shed from all finite-energy Airy pulses whether they are chirped or not, and quintic nonlinearity has profound effects on the propagation dynamics of the soliton pulse. However, in normal dispersion region, none soliton pulses are generated from Airy pulses. Depending on the peak intensity varying with the propagation distance, differing from the cases in the abnormal dispersion region, the effect of quintic nonlinearity has a slight impact on the evolution of Airy pulse.

We numerically simulated dynamic propagation of in-phase and out-of-phase finite-energy Airy pulses in anomalous regions of optical fibers and analyzed the effects of cubic–quintic (CQ) competing nonlinearity and separation parameter on evolution properties. Numerical results showed entirely different effects of quintic nonlinearity on in-phase and out-of-phase Airy pulses that vary with different separation parameters. For in-phase Airy pulses, a central quasi-soliton pulse is shed from all finite-energy Airy pulses when separation parameter is certain. CQ competing nonlinearity exhibits profound effects on propagation dynamics of soliton pulses. However, for out-of-phase Airy pulses, two quasi-soliton pulses and more dispersion wave pulses are symmetrically shed from the main lobe of out-of-phase Airy pulses. CQ competing nonlinearity slightly impacts evolution of out-of-phase Airy pulses depending on peak intensity, which varies with propagation distance and unlike in cases of in-phase Airy pulses.

The live-working robot is placed on the aerial working platform to connect the drainage wire. Due to the sloshing of the manipulator base on the platform, the effective operation is seriously affected by the end position error of the manipulator. Aiming at this problem, a trajectory planning method based on feedforward compensation combined with quintic polynomial interpolation is proposed in this paper. The angle and angular velocity constraints of every joint of the manipulator are determined by establishing the kinematic model of the manipulator. Analyze the sloshing characteristics of the base of the manipulator, and combine the movement time of the manipulator from the initial point to the target point to predict the position error of the end of manipulator. The feedforward compensation is performed to correct the target point, and quintic polynomial interpolation is used to plan the compensation end trajectory. Based on the analysis of the sloshing law of the base of the manipulator, the effectiveness of the method is verified by simulation experiments, and the target point can be reached accurately.

By considering lane change maneuver as primarily a Partial Observed Markov Decision Process (POMDP) and motion planning problem, this paper presents an interactive model with a Recurrent Neural Network (RNN) approach to determine the adversarial or cooperative intention probability of following vehicle in target lane. To make proper and efficient lane change decision, Deep Q-value network (DQN) is applied to solve POMDP with expected global maximum reward. Then quintic polynomials-based motion planning algorithm is used to obtain both optimal lateral and longitudinal trajectory for autonomous vehicle to pursuit. Experimental results demonstrate the capability of the proposed model to execute lane change maneuver with comfortable and safety reference trajectory at an appropriate time instance and traffic gap in various highway traffic scenarios.

This paper presents our newest findings in planning a dynamically and kinematically feasible center of mass motion for bipedal walking robots. We use a simplified robot model to incorporate multi-body dynamics and kinematic limits, while still being able to meet hard real-time requirements. The vertical center of mass motion is obtained through interpolation of a quintic spline whose control points are projected onto the kinematically feasible region. Subsequently, the horizontal motion is computed from multi-body dynamics which we approximate by solving an overdetermined boundary value problem via spline collocation based on quintic polynomials. The proposed algorithm is an improvement of our previous method, which used a parametric torso height optimization for vertical and cubic spline collocation for horizontal components. The novel center of mass motion improves stability, especially for stepping up and down platforms. Moreover, the new method leads to a less complex overall algorithm since it removes the necessity of manually tuned parameters and strongly simplifies the incorporation of boundary values. Lastly, the new approach is more efficient, which leads to a significantly reduced total runtime. The proposed method is validated through successfully conducted simulations and experiments on our humanoid robot platform, LoLA.

Trajectory planning ensures that the robot arm moves smoothly and quickly to the target position. The difficulty is to use inverse kinematics to convert the path point into the joint angle and fit a smooth function to each joint. This paper studies the polynomial trajectory planning of the seven-DOF cooperative robot KUKA LBR iiwa. Kinematics of robot arm is analyzed in the coordinate system established by denavit-hartenberg method. In the joint space, the trajectory planning of the robot arm is performed using the cubic polynomial and the quintic polynomial function. The simulation results show that the quintic polynomial method effectively solves the problem of acceleration discontinuity, and obtains a continuous smooth trajectory curve of each joint, which visually verifies the effect of trajectory planning and provides an efficient and feasible trajectory planning method.

Path planning is a crucial task in automated driving. Due to the complexity of dynamically changing driving environment, the planned path generally requires the capability to adjust itself in real time to avoid obstacles detected in its way. The use of an optimization method is able to generate a collision-free and smooth path. However, its high computation burden limits its direct application to online path planning. This paper proposed a time-efficient online table-lookup approach to deal with this dilemma. Given discrete target points, this approach is capable to form a quintic-spline path with second-order geometric $(\boldsymbol{G}^{2}-)$ continuity using a look-up table. The look-up table was generated beforehand in the reference space with minimization on curvature variation. The paper demonstrates the application of this online approach in collision avoidance, with a geometry-based method to decide new target points when obstacles are detected in the original path. These new target points are fed to the table-lookup online path planning algorithm to generate a collision-free path with minimum curvature variation.

In this paper, we propose a quintic non-polynomial spline approach coupled with $L1$ formula for the numerical treatment of time-fractional nonlinear Schrodinger equation. The unconditional stability is proved by Fourier method for $0 , where $\gamma$ is the order of the Caputo time-fractional derivative. The efficiency of the proposed numerical scheme is illustrated by numerical experiments, where the simulation results indicate better performance over previous work in the literature.

In medieval Arabic, no symbols are used to express mathematical problems and solutions. Instead, specific Arabic terms are used to describe mathematical processes. Nevertheless, a close examination of the text styles and how information is encoded into natural language phrases reveals that the notations contain no ambiguity, which makes it possible to formulate the problem and solve it. In this paper, we explore the idea of automatically generating modern symbolic mathematical equations from natural language medieval Arabic Algebra texts. We will describe a simple machine translation system using Rule-based Machine Translation (MT) with Dictionary Approach in order to translate medieval verbal equation to a modern equation. We construct a new dataset from scratch to facilitate our study, as there is no previously published similar dataset. Our system was able to achieve correct translation results for cubic equations with 100% accuracy, and quartic, quintic equations with 96.66% and 94.01%, respectively.

To realize the automatic measurement and improve the efficiency of the body measurement in the garment customization, we propose a body measurement method based on the ultrasonic sensor. The novel measurement structure and method are proposed to achieve the accurate bust, waist, and hip (BWH) information. First, the depth information is measured by the ultrasonic sensor arrays. The sensors should capture the front and back depth of the key body parts, the depth information and the certain position of sensors can generate the space coordinates. Second, the space coordinates are fitted with the quintic function to generate the curves. Third, the front curve, back curve, and the estimated parameter constitute the finial BWH data. Experimental results indicate that the proposed method can achieve accuracy of95.6%.

In this paper a trajectory tracking controller for a Mixed conventional/braking Actuation Mobile Robot (MAMR) is presented. The MAMR was designed to study the use of brakes in steering mobile robots that suffer from primary actuation failure. The MAMR design may also solve some common drawbacks of the conventional mobile robots such as control complexity, weight, and cost. This paper proposes a model predictive control approach to solve the trajectory tracking problem for the MAMR. The desired trajectories between two points are pre-generated using a quintic polynomial under curvature constraints. The dynamical model of the robot is used to solve the MPC problem. The effectiveness of the proposed control algorithm is tested through simulation.

Maximizing the endurance of an aircraft is a problem with several applications. This paper proposes an optimal control framework to solve the maximum endurance problem of a turbofan in cruise, and takes into account the presence of head or tail-winds. Since the optimal solution of this problem involves finding the root of a quintic polynomial, an approximate state-feedback analytical solution is proposed using the first iteration of Newton's method. This is the main contribution of the paper. Upper error bounds are also derived for the proposed algorithm in order to validate its accuracy. Simulation results are shown and compared with the maximum endurance performance mode of a Boeing 737 FMS. The simulation results show that the proposed speed is closer to the optimal solution than the speed of a commercial FMS.

In this paper, the optical transmission equation of fiber considering fourth-order dispersion and quintic nonlinearity is proposed. Simulation software of ultra-long haul transmission is developed by discretizing the equation with symmetric spilt-step Fourier method. Unrepeatered optical transmission system of 40Gb/s signal centred at 1550nm is designed. The results of numerical analysis show that the quintic nonlinearity with coefficient of 1W∼2/km will result in significant transmission impairments, and Q factor of the system is reduced by 1.9 at distance of 120km. When the optical transmitter power increases from 0.01 W to 0.5W, the average Q factor is declined by 32.5. Therefore, the influence of quintic nonlinearity needs to be considered in the system with high power.

In order to meet the requirement of rapid attitude maneuver control and vibration suppression of flexible spacecraft with disturbance and uncertainty, this paper presents a robust control scheme based on path planning and adaptive backstepping control. Firstly, a novel path planning scheme is proposed based on quintic polynomial transition. The continuous and smooth maneuver path is designed to solve the conflict between swiftness and overshoot in traditional path planning methods as well as to suppress the vibration of flexible appendages. By planning the desired angular acceleration of the maneuver path, appendages' vibration induced by attitude maneuver is decreased effectively. Furthermore, an adaptive robust controller is derived using the backstepping technique, including an adaptation mechanism to estimate the unknown disturbances and uncertainties so that the prior knowledge of the upper bounds on the total disturbance is not necessary for the controller design. Finally, the illustrative simulation results demonstrate the effectiveness and robustness of the proposed control scheme.

For a 5-DOF hybrid robot, its kinematics of position is analyzed, and it is modeled by using the virtual prototype software ADAMS. Kinematic DOF of the robot is verified via self-checking function. For an example in joint space, the trajectory of the robot is planned by using quantic polynomial algorithm, and STEP5 function is used for motion function, and the robot is simulated via kinematic simulation module in ADAMS. The result indicates that there are all smooth kinematic curves of end-effector and five branched chains, which using quintic polynomial algorithm could ensure that the hybrid robot runs smoothly in a simulation process.

Pythagorean Hodograph (PH) quintic transition curves are normally constructed because it is polynomial and has attractive properties, which is its arc-length is a polynomial of its parameter, and the formula for its offset is a rational algebraic expression. Pythagorean Hodograph is very important in formulating efficient real-time interpolator algorithm for CNC machines, suitable for rounding corners and blending smooth transition between two circles. By reinstating quintic Bezier curve with quintic trigonometric Bezier curve with two shape parameters, it will provide more flexibility to construct curves and surfaces. This new type of Pythagorean Hodograph will be more practical for curve designing especially in the computer aided geometric design of consumer products and useful for other computer graphics for designing highways.

This article presents a path planning concept for automated driving. It builds upon existing approaches using quintic polynomials for path planning in structured environments. The contribution of this work are analytical solutions to evaluate selected properties of quintic polynomial based paths such as maximum acceleration and jerk or overshooting behavior. In addition, a concept for generating asymmetrical lane change trajectories is proposed. The presented concepts are evaluated in terms of computational costs and applicability in a real test vehicle.

Automated ground vehicles are becoming a reality for future deployment due to their potential improvement of safety, comfort or emission reductions. However, some challenges remain unsolved such as navigation in dynamic urban environments, where safety and comfort are paramount. In this paper, a novel speed profile generator, based on quintic Bezier curves, is presented. This approach permits to improve the comfort in automated vehicles by constraining the global acceleration in the whole ride. Moreover, a comparison with a Jerk limitation algorithm for continuous acceleration profile and a smooth speed profile is performed. The quintic Bezier curves assure continuity of velocity curves and therefore a smooth Jerk and acceleration profiles. The ISO 2631-1 standard, for evaluation of human exposure to whole-body vibration, is considered in both methods for lateral and longitudinal comfort constraints. Simulations show that this approach improves comfort by smoothing the acceleration and jerk profiles, while also constraining the total acceleration perceived by the passengers.

A simple and efficient higher order finite element scheme is presented for obtaining highly accurate numerical solution for the two-dimensional Helmholtz equation in waveguides of arbitrary cross-section subjected to dirichlet boundary conditions. The above approach makes use of the Quintic order (5th order) parabolic arcs for accurately mapping the irregular cross section of the waveguide and then transforming the entire waveguide geometry to a standard isosceles triangle. In case of waveguides with regular geometry the transformation is done by straight sided quintic order finite elements. A unique and accurate point transformation technique is developed that ensures high accuracy of mapping by this quintic order curved triangular elements. This point transformation procedure gives a simple interpolating polynomial that defines the transformation from the global coordinate system to the local coordinate system. The above higher order finite element method is found to be highly optimal and accurate considering the various computational parameters like the number of triangular elements, degrees of freedom, nodal point distribution on the entire geometry, etc.

A new solution is proposed for the Real-time control of a 7-DOF (Degree Of Freedom) redundant manipulator by taking advantage of its inverse kinematics. The iterative optimization method is used to reduce the amount of calculating while ensuring its precision. In order to complete the manipulator's trajectory planning in Cartesian space, the quintic polynomial interpolation algorithm is established. Finally, the hardware platform as well as the control system for the manipulator is designed. The effectiveness of the trajectory planning method is verified by the analyzing the simulation results. It shows that the algorithm improves the efficiency of the inverse calculating obviously, which also enhance real-time performance of the system. Meanwhile, it meets the requirements for the speed and interpolation precision of the manipulator's performance.

In this paper an interpolation method based on quintic non-uniform rational B-spline(NURBS) is proposed to construct curves when to plan the trajectory of manipulators with respect to three objectives, time optimal, energy optimal and smoothness optimal. The mathematical model of quintic NURBS curve is set up to gain high-order continuous trajectories with endpoint configuration parameters can be specified, and a fast and elitist multi-objective genetic algorithm (NSGA-II) is adopted to optimize the trajectory of manipulators aims to get a series of Pareto optimal solutions under the three objectives. Through the simulation with six-degree of freedom robot shows that quintic NURBS curve can get high-order continuous trajectories and NSGA-II algorithm can provide an effective approach to get the perfect Pareto solutions for quintic NURBS curve. By constructing a average fuzzy membership function, a potential optimal solution can be selected from Pareto optimal set, then the high-order continuous optimal trajectory can be obtained.

This paper presents the use of Quartic and Quintic order finite elements for computing cutoff wavenumbers of arbitrary shaped waveguides. These finite elements are used for mapping the boundaries of waveguides with the highest accuracy. In the case of waveguides with curve geometries, the mapping is done by quartic and quintic order parabolic arcs. The domain of a particular waveguide is transformed to a suitable isosceles triangle with the help of these finite elements. The above method is found to be highly computationally efficient as compared to other methods found in literature.

This paper proposes a longitudinal motion planning method to reduce the velocity of an unmanned ground vehicle safely when it encounters stopped vehicles or obstacles on the road. The proposed method can generate the smooth velocity and acceleration profiles using the quintic polynomial curve. The optimal quintic polynomial curve can be determined by a numerical optimization approach. The performance of the proposed method is verified by simulations.

CQNLSE dark pulse is demonstrated in EDFL with NPR technique. Pulse repetition rate, pulse width, and highest pulse energy of the CQNLSE dark pulse are obtained at 1.52 MHz, 219 ns, and 0.59 nJ respectively.

This paper proposes a separate quintic polynomial trajectory for each sub phase of gait cycle to achieve smooth human-like walking. A biped robot with active hip, knee and ankle joints is considered here for human legs like appearance and walking performance. Trajectory planning is done on the basis of experimental flexion angle, velocity and acceleration data of three joint movements i.e. hip, knee and ankle in sagittal plane. The results are compared with cubic spline trajectory and other results available for gait generation. The result shows that the proposed trajectory can be used for walking of biped robot just like human walking.

Stability and collisions of moving solitons in a cubic-quintic nonlinear Bragg grating with dispersive reflectivity are investigated. Two disjoint soliton families are found and the outcomes of soliton-soliton collisions are discussed.

Stability and collisions of moving Bragg grating solitons in a cubic-quintic medium are investigated. The effect of solitons' initial velocity on the outcome of the collisions is analyzed.

Stationary Inflection Points (SIPs) around a frequency ω 0 are of the form ω-ω 0 ∝ km for any positive odd integer. We show theoretically that SIPs of order m can be created in optical waveguides such as optical fiber gratings or periodic nanobeam waveguides by coupling between m modes. Remarkably, we find that at the SIP frequency significant incident energy can be coupled into the waveguide even though the group velocity of the associated mode is zero. Here we demonstrate this frozen light for cubic (k3) and quintic (k5) SIPs and relate it to the existence of evanescent modes at the interface and demonstrate it.

We use the fixed point method to study the Hyers-Ulam stability of the quintic and sextic functional equations in non-Archimedean fuzzy normed spaces. In addition, we establish some results of approximately quintic and sextic mappings in non-Archimedean fuzzy normed spaces. Some applications of our result will be illustrated.

A set of quintic polynomial basis functions with two parameters are presented. Based on these basis functions, the quintic Bezier curve with two parameters which is called λμ -Bezier curves is defined. λμ -Bezier curves produces a closer fit to the guiding polygon than does the Bezier curves. We can attain local shape control of quintic λμ − B spline curve by modifying the shape parameters exactly to ensure two quintic λμ -Bezier curves segment satisfying C1-continuity at the common endpoint without affecting other parts. Finally,we present a example to illustrate the validity of the modification methods.

In this paper, an automatic assembly platform based on an equivalent parallel mechanism that includes four parallel precise locators is designed to adjust the position of a large unit during the assembly process. After the kinematics analysis of the platform, a posture calculation method based on a laser tracker is proposed. Additionally, a quintic polynomial is used to determine the part movements. The system uses twelve-axis synchronous control. Finally, experiments show that the platform is valid and efficient and that the final positioning accuracy is 0.1mm.

For a function f(t) defined on the discrete interval N[a, b + 2] = {a, a + 1, …, 6 + 2}, we develop a class of quintic discrete Hermite interpolate H ρ f(t) that involves only differences. Further, explicit error bounds are offered in the form of the inequality ||f-H ρ f||≤c j max/t∊N[a, b+2-j] | Δj f(t)|, 2≤j≤6 where the constants C j , 2 ≤ 6 are explicitly provided. We also establish the two-variable Hermite interpolate as well as the related error analysis for a function f(t, u) defined on N[a, b+2] × N[c, d + 2]. Four numerical examples are presented to illustrate the actual construction of the discrete Hermite interpolates, the actual errors are also computed to compare with the error bounds obtained.

In this paper, a perturbing nonlinear quintic Schrodinger equation is studied under limited time interval, complex initial conditions and zero Neumann conditions. The perturbation and Picard approximation methods together with the eigenfunction expansion and variational of parameters methods are used to introduce an approximate solution for the perturbative nonlinear case for which a power series solution is proved to exist. Using Mathematica, the solution algorithm is tested through computing the possible orders of approximations. The method of solution is illustrated through case studies and figures.

For a function f(t) defined on the discrete interval N[a, b + 2] = {a, a + 1, …, b + 2}, we develop a class of quintic discrete spline interpolate S ρ f(t) that involves only differences. Further, explicit error bounds are offered in the form of the inequality ||f-H ρ f||≤d j max/t∊N[a, b+2-j] |Δj f(t)|, 2≤j≤6 where the constants d j , 2 ≤ j ≤ 6 are explicitly provided. Three numerical examples are presented to illustrate the actual construction of the discrete spline interpolates, the actual errors are also computed to compare with the error bounds obtained.

Bifurcation of limit cycles for a quintic system is investigated using both qualitative analysis and numerical exploration. The investigation is based on detection functions which are particularly effective for the quintic system. The study reveals that the quintic system has 8 limit cycles using detection function approach, and two different distributed orderliness of 8 limit cycles for the quintic system are shown. By using method of numerical simulation, these limit cycles are observed and their nicety places are determined. The study also indicates that each of these limit cycles passes the corresponding nicety point. The results presented here are helpful for further investigating the Hilbert's 16th problem.

This paper finds a spiral segment matching G2 Hermite conditions for a single Pythagorean hodograph quintic polynomial. We have significantly simplified the existing methods with computationally stable spiral conditions and discovers more spiral regions for the existence of unsolved problems in the past. A spiral is free of local curvature extrema, making spiral design an interesting mathematical problem with importance for both physical and aesthetic applications. In the construction of highways or railway routes and in the path planning of non-holonomic mobile robots, it is often desirable to have a spiral segment with given positions, tangents, and curvatures at the end points.

In this paper, a new difference scheme by using quintic splines for solving a singlularly-perturbed boundary-value problem is derived. The proposed scheme is sixth order accurate at the interior nodal points and fourth order accurate at the end points, which is better than previous published results. Finally, the numerical results are given to illustrate the efficiency of our method.

This paper presents an extension of motion planning in consideration of travel time and velocity during the trajectory generation. In this study, cubic and quintic polynomials were adopted to obtain a smooth trajectory for a single nonholonomic mobile robot. In addition, a new strategy in generation of the trajectory has been introduced, which consists of offline and online planning. Offline planning is executed at the early stage of planning, in order to plan a trajectory within the known static obstacles in the presented environment. Whilst, online planning deals with the unknown static obstacles during the mobile robot navigates in the environment and to catch-up the time loss, due to avoiding the obstacles, in order to reach the final point at the specified time. The presented simulation results demonstrate the effectiveness of this new strategy in motion planning for the mobile robot.

This paper studies the relationship between amounts of new stockholders and share index in Chinese stock market. We analyze behavior characters of investors, construct a model to forecast the amounts based on polynomial fitness such as quintic, nine times’, eleven times’ polynomial fitness ordinally. Comparing with factual data, we apply polynomial simulation method to review the fitness of model, and confirm sensitivities of thedata. The result proves the validity of the simulation method.

5-axis machines are important in a number of industries ranging from aerospace to consumer-mold machining. The conventional interpolation algorithms for 5-axis CNC machining are focused on studying techniques to design position, orientation, and feedrate curves in such a way using isoparametric correspondence as to avoid excessive angular velocities. However, it is difficult to guarantee such correspondence. To help alleviate such difficulties, a research on interpolation algorithm for 5-axis tool paths generated using quaternion-quintic spherical Bezier spline for tool positioning and orientation was presented in this paper. The proposed algorithm was verified by Matlab platform. The analysis of results was also provided in this paper.

We elaborate in this article on a quadratic algorithm for a particular case of the 2-cyclic robotic scheduling problem, namely the case when at most two parts go through the production line at all times; or, what is the same, when the lifetime of the part is shorter than the duration of the cycle. This study is relevant since for the general case the best known algorithms are quintic.

This paper presents the algorithms to obtain the local minimum-time trajectory planning for the five-axis machining with or without the tool deflection. The tool and workpiece are first combined as a closed-chain system of rigid bodies. The tool path and shape are simplified to the tool-workpiece system curve and tool line. The forward and inverse kinematics are applied to obtain the kinematic relation between the system path and the position of the five motors. Based on the kinematic equations, the velocity, acceleration, and jerk of each motor can be derived. The motion trajectory of the end-effector of this system can be described through a sequence of intervals. In order to guarantee the dimension accuracy, the quintic polynomial was applied to fit the curve on each individual interval. The genetic algorithm (GA) with one generation is applied to find the minimum time period at every interval on the system curve in the case that the system has no deflection on the reference path. When there is the deflection on the system path, an algorithm is developed to find its projection at the ith interval on the reference path without deflection through the potential field method, and then the local minimum time periods at two new intervals composed with deflection, its projection, and the (i+1)th reference point, which is on the curve without deflection, respectively. The derived kinematic equations and the proposed algorithms are verified through the simulation.