Trajectory Method Research Articles

Biomimetic design of fibrous composite structures

The paper is devoted to the analysis of the use of Nature’s methods in the design of composite structures, in particular, with the use of optimal curvilinear paths for laying fibers. In the first part models of different structural levels in macro-, mini-, micro-mechanics of composites are noted. The second part includes analysis of the optimal elastic-strength properties of wood and composites for crack stopping by weak interfaces. The third part is devoted to methods for modeling curvilinear fibers trajectories “flowing around the holes”. In the fourth part the technologies for manufacturing joints are analyzed in which holes are formed using curved fiber trajectories. The fifth part describes the “bio-inspired” principles for the optimal design of composite pipe structures which are similar to multilink bamboo stems. In the sixth part examples of the effective use of fiber composites in elastic elements are considered. The seventh part is devoted to the intensively developing additive technologies of three-dimensional printing of parts made from fiber composites with laying the fibers along the calculated trajectories. The role of composite technologies in creating structures in open space is analyzed.

Prediction of shock standoff distance with modified rotational relaxation time of air mixture

The rotational relaxation time of an air mixture is modified as an approach to improve accuracy when predicting hypersonic shock standoff distance. A novel atomistic quasi-classical trajectory (QCT) method with a modified approach is devised to drastically reduce computational cost, and rigorously model the rotational relaxation time of N2 in N2–N and N2–N2 collisions. The calculated full sets of rotational state-to-state transition rates obtained by the QCT method are fed into the rotational state-resolved master equations to determine the rotational relaxation time of N2. Clear discrepancies are observed when the present rotational relaxation time is compared with existing empirical data for N2. The existing empirical model is utilized to determine the rotational relaxation time of other atmospheric gas species. Then the present set of rotational relaxation times for the air mixture is employed to predict the hypersonic shock standoff distance over a blunt body of the ground and flight experiments. Compared with the results from the two-temperature model, the rotational nonequilibrium enlarges the hypersonic shock standoff distance. This increase in shock standoff distance by the rotational nonequilibrium is attributed to the delay in chemical reactions inside the shock layers. The accuracy of the predicted measured shock standoff distance is improved by considering the present rotational relaxation time of the air mixture.

Transient particle acceleration by a dawn–dusk electric field in a current sheet

The influence of a dawn–dusk electric field Ey on transient particles in a 1D current sheet (CS), characterized by the normal (Bz) and tangential (Bx) components of the magnetic field, is studied. The motion and energization of particles injected at the edges of a CS are investigated within the framework of the trajectory method. The analytical treatment reveals that in the case of uniform Bz and Ey, the dynamics of transient particles are described by magnetic flux conservation on specific segments of the trajectory, which allows prediction of some specific properties of the velocity space inside the CS. Verification of the analytical treatment by means of test-particle numerical modeling demonstrates good agreement. In particular, it is shown that the CS can play the role of a converging lens that focuses particles to pitch-angle values close to θ∼π. At the same time, the analysis reveals that the particle energy gain stays within the range of ΔW∈2m[(Ey/Bz)2,(Ey/Bz)(v0+Ey/Bz)], where m is the particle mass and v0 is the initial particle speed (i.e., v0=v·v). The limits of the range depend only weakly on the CS half-thickness. The analysis reveals that for the typical parameters of Ey and Bz in the stationary terrestrial magnetotail, protons with v0≃450 km/s (before CS crossing) can be accelerated along the CS up to vx≃1800 km/s.

Trajectory planning for mobile manipulators with vibration reduction

A new method of online trajectory planning for mobile manipulators using the extended Jacobian approach, taking into account state and control constraints, allowing the mobile manipulator to reduce the arm vibrations is presented. The proposed approach is based on the concept of virtual control which scales the trajectory preventing robot controls from changing rapidly and ensuring that the control constraints are fulfilled. The simulation results presenting the features of the developed algorithm have been shown. The effectiveness of the method in vibration reduction is confirmed by an experiment involving a KUKA youBot mobile robot.

Energy-Saving Trajectory Planning for Robotic High-Speed Milling of Sculptured Surfaces

In the field of sculptured surfaces machining, the robot trajectory planning, under high-order complex constraints, aiming at minimizing energy or time, is always a challenge. The complexity of curvature characteristics of sculptured surfaces and the nonlinearity of relevant constraints are the main reasons. This article proposes an efficient planning method of minimum-energy robot trajectory, for high-speed machining of sculptured surfaces. First, the energy characteristic model of the robot machining system (RMS) is established, to acquire the energy-optimal feedrate, under velocity constraints, to use in the subsequent trajectory planning. Next, a trajectory planning model, with complex constraints, is developed. The proposed method transforms the original trajectory planning into a minimal modification of the initial objective-optimal B-spline feedrate curve (BFC). Furthermore, two main coupling problems, which influence the minimal change of curves, in the direct evolution-based BFC modification, are addressed. Based on the derived solutions, a novel modification algorithm of BFC, with a callback mechanism (CBM), is proposed. Finally, the performance of the proposed method and the specific algorithm is validated by two case studies. Results show that the proposed method can significantly improve the efficiency of robot trajectory planning, aiming at minimum energy, while exhibiting excellent performance. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This work was motivated by how to automatically and effectively acquire the energy-saving trajectory of a robotic system, executing sculptured surface machining. This article suggests a new approach that transforms the original trajectory optimization into a minimal modification of the initial energy-optimal BFC, obtained under velocity constraints. This approach calculates the complex energy formula only once. A CBM, for constraint-based BFC modification, is proposed, to revise the redundant reduction of the feedrate. The proposed modification algorithm significantly decreases the detrimental effect of two coupling problems, on the adjustment of the BFC. Results show that the computational efficiency of the proposed method is significantly superior to the one of the direct optimization method. The performance of the proposed modification algorithm is obviously advanced compared to the existing B-spline evolution algorithm. In blade machining, after optimizing the trajectory using the proposed method and algorithm, both energy and time decrease by more than 45% compared to the conservative feed. The proposed algorithm improves significantly the trajectory compared to the prior art algorithm. Technologists benefit from reduced planning time and improved machining performance. However, incorporation into a CAM or manufacturing system has not yet been realized and remains a significant work to be done in the future.

Long-term trends in particulate sulfate at Grand Canyon National Park

Visibility impairment, particularly at the iconic southwestern U.S. national parks, was an early and persistent concern that helped motivate parts of the Clean Air Act (CAA) and its 1977 and 1990 amendments. Grand Canyon National Park (GCNP), Arizona, in particular, was the site of many early air quality studies that found that before 1990 particulate sulfate was the largest contributor to light extinction at GCNP. Since a peak in the 1960s, sulfate concentrations at GCNP have declined dramatically as sulfur dioxide (SO2) emissions in the southwestern United States and northern Mexico have been reduced, often due to successful regulations. The relative importance of upwind source regions has also changed. During the 1960s and 1970s the predominant SO2 emitters were copper smelters in Arizona and surrounding states. As smelter emissions declined, in the 1980s and 1990s electric generating units (EGUs) became the largest upwind SO2 sources. More recently, EGU emissions have also been reduced, especially since about 2000. Three back trajectory methods are used to examine the sulfate concentrations at GCNP as a function of upwind source regions. 1) Residence time analysis shows that Arizona, southern California, southern Utah, and northern Mexico are most often upwind of GCNP. There were highly significant reductions in mean sulfate concentrations at GCNP when air arrived after transport through most source regions. 2) Concentration trends by source region show that the maximum rates of decline for 1980–2018 were about −2%/year when there was transport through Colorado and the Navajo Generating Station (NGS) source regions. 3) The Trajectory Mass Balance (TrMB) model was used to apportion sulfate seasonally for 3-year rolling averages during 1981–2017. As expected, relative source attributions have evolved through time. Largest mean contributors in summer are northern Mexico (25–70%), Baja (0–25%), Arizona smelters (0–35%), the Mohave Generating Station (<10–18%) and Southern California (1–15%). During winter the Navajo Generating Station (10–25%) and Cholla (3–12%) often contributed the largest fractions. Mean concentrations are much lower, and there are larger fractional declines in contributions from many source areas during winter as compared to summer. These analyses document the successful outcomes that can result from effective regulations.

Query the trajectory based on the precise track: a Bloom filter-based approach.

Fast and precise querying in a given set of trajectory points is an important issue of trajectory query. Typically, there are massive trajectory data in the database, yet the query sets only have a few points, which is a challenge for the superior performance of trajectory querying. The current trajectory query methods commonly use the tree-based index structure and the signature-based method to classify, simplify, and filter the trajectory to improve the performance. However, the unstructured essence and the spatiotemporal heterogeneity of the trajectory-sequence lead these methods to a high degree of spatial overlap, frequent I/O, and high memory occupation. Thus, they are not suitable for the time-critical tasks of trajectory big data. In this paper, a query method of trajectory is developed on the Bloom Filter. Based on the gridded space and geocoding, the spatial trajectory sequences (tracks) query is transformed into the query of the text string. The geospace was regularly divided by the geographic grid, and each cell was assigned an independent geocode, converting the high-dimensional irregular space trajectory query into a one-dimensional string query. The point in each cell is regarded as a signature, which forms a mapping to the bit-array of the Bloom Filter. This conversion effectively eliminates the high degree of overlap and instability of query performance. Meanwhile, the independent coding ensures the uniqueness of the whole tracks. In this method, there is no need for additional I/O on the raw trajectory data when the track is queried. Compared to the original data, the memory occupied by this method is negligible. Based on Beijing Taxi and Shenzhen bus trajectory data, an experiment using this method was constructed, and random queries under a variety of conditions boundaries were constructed. The results verified that the performance and stability of our method, compared to R*tree index, have been improved by 2000 to 4000 times, based on one million to tens of millions of trajectory data. And the Bloom Filter-based query method is hardly affected by grid size, original data size, and length of tracks. With such a time advantage, our method is suitable for time-critical spatial computation tasks, such as anti-terrorism, public safety, epidemic prevention, and control, etc.

Seasonal Characteristics of Air Pollutant Sources and Transport Pathways in Xining City

Based on the analysis of seasonal characteristics of PM10 and PM2.5 particle mass concentrations in Xining from 2016 to 2018, the daily 72 hour backward trajectories were calculated using the hybrid single-particle Lagrangian integrated trajectory (HYSPLIT) model and global data assimilation system (GDAS) data. The main transport pathways of PM10 and PM2.5 were identified and their characteristics were analyzed by clustering analysis for four seasons. The potential source regions and their contributions were defined using the potential source contribution function (PSCF) model and the concentration-weighted trajectory (CWT) method provided by TrajStat software. Results indicated that the sources were mostly distributed in the north-west and north-east regions and heights were low in the surrounding and adjacent areas of Xining. The transport pathways were mainly affected by airflows from the west, northwest, southwest, and east in Xining city. The trajectories with the highest probability of occurrence were characterized by short distance, low height, and slow-moving speed, originated from Qinghai in spring, summer and autumn, but from Xinjiang in winter, and was dominated by intra-provincial transportation. Different transport trajectories had different effects on PM10 and PM2.5 concentrations. Polluted airflows mainly originated from internal sources in Qinghai, external sources in Xinjiang, and foreign sources in the west of Xinjiang, with all the source regions located in deserts or Gobi areas. Obviously seasonal differences existed in the distribution and contribution of the potential source areas, with the widest and largest contribution in winter, followed by spring and autumn, and the smallest in summer. The most important potential source regions were located in northern, central, and eastern Qinghai, and southern, central, and eastern Xinjiang, while the surroundings were potential source regions for medium contribution.

Theoretical studies of carbon isotopic fractionation in reactions of C with C2: dynamics, kinetics, and isotopologue equilibria

Context. Our current understanding of interstellar carbon fractionation hinges on the interpretation of astrochemical kinetic models. Yet, the various reactions included carry large uncertainties in their (estimated) rate coefficients, notably those involving C with C2. Aims. We aim to supply theoretical thermal rate coefficients as a function of the temperature for the gas-phase isotope-exchange reactions 13C+12C2(X1Σg+,a3Πu)⇌13C12C(X1Σg+,a3Πu)+12C and 13C+13C12C(X1Σg+,a3Πu)⇌13C2(X1Σg+,a3Πu)+12C. Methods. By relying on the large masses of the atoms involved, we employ a variation of the quasi-classical trajectory method, with the previously obtained (mass-independent) potential energy surfaces of C3 dictating the forces between the colliding partners. Results. The calculated rate coefficients within the range of 25 ≤ T∕K ≤ 500 show a positive temperature dependence and are markedly different from previous theoretical estimates. While the forward reactions are fast and inherently exothermic owing to the lower zero-point energy content of the products, the reverse processes have temperature thresholds. For each reaction considered, analytic three-parameter Arrhenius-Kooij formulas are provided that readily interpolate and extrapolate the associated forward and backward rates. These forms can further be introduced in astrochemical networks. Apart from the proper kinetic attributes, we also provide equilibrium constants for these processes, confirming their prominence in the overall C fractionation chemistry. In this respect, the 13C+12C2(X1Σg+) and 13C+12C2(a3Πu) reactions are found to be particularly conspicuous, notably at the typical temperatures of dense molecular clouds. For these reactions and considering both equilibrium and time-dependent chemistry, theoretical 12C/13C ratios as a function of the gas kinetic temperature are also derived and shown to be consistent with available model chemistry and observational data on C2.

PERANCANGAN ULANG TRANSFER CHUTE 63/72 UNTUK MENAIKKAN UMUR LINER, STUDI KASUS DI PT FREEPORT INDONESIA

Transfer Chute is one of the important components in the transportation of ore originating from open pit to the stock pile. Damage or leakage in the transfer chute when there is no over haul will result in so that it can cause production losses for 10 hours . Damage that often occurs in chute transfers occurs after production increases which causes the conveyor speed to change from 2 m / s to 4 m / s so that the initial design of the transfer chute is not suitable for conveyor speeds to 4 m / s, ore originating from conveyor 63 is not form a rock box on the transfer chute 63/72 but directly hit the liner on the transfer chute. In this paper presents two material discharge trajectory methods and expain the differences between them, and then we choose a method that approaches the actual condition. From the results of the trajectory analysis, we can determine new dimensions of transfer chute. Based on the modification of the chute transfer, the life of the liner increased from 4 weeks to 6 months and there was never a down time due to an increase in the transfer chute 63/72.

Nanotwinned diamond cutting tool processed by femtosecond pulsed laser milling with trochoidal trajectory

Ultrahard nanotwinned diamond (nt-D) is an excellent diamond material with higher hardness, better fracture toughness and thermal stability than the commonly used single crystal diamond (SCD), thus it is considered as a promising tool material applied in ultra-precision cutting. However, extremely high hardness results in the fabrication of nt-D tool more difficult than SCD because of very low machining efficiency when using the conventional mechanical methods. Aiming to mass-produce of nt-D tool, the femtosecond pulsed laser milling technique was applied, and the related phase transition mechanism was explored during processing nt-D. Then, a trochoidal trajectory was proposed to acquire high forming accuracy and sharp edge in this study. It is found that the cubic diamond within nt-D was transformed into amorphization directly rather than graphitization under laser processing, which is attributed to the enhanced phase stability for nt-D. Furthermore, the nt-D surface integrity processed by laser was mainly restricted by diamond falling-off pits dewetting from grain boundary. Then, an optimized trochoidal trajectory method was proved to effectively decrease the depth of the falling-off pits and improve the surface integrity by the aid of laser pulse delay. Through the distribution calculation of pulse overlapping with the deduced theoretical equations and processing parameters optimization of galvanometer rotation, an nt-D cutting tool was successfully fabricated with smooth surface and sharp cutting edge. Finally, the wear mechanism, wear resistance and cutting performance of the as-fabricated nt-D tool were explored through cutting testing of hard polycrystalline MgF2. The results supported that nt-D tool exhibits better wear resistance and cutting performance than nanograined polycrystalline diamond (NPD) tool.

Measuring curvature of trajectory traced by coupler of an optimal four-link spherical mechanism

This research paper presents design of a gripper finger mechanism and a new curvature measurement method of fingertip trajectory. Firstly, the conditions of a grasp provided by gripper, which was designed from a spherical mechanism, was explained. Therefore, trajectory length of coupler point was modeled as objective function which is maximized. Ball-Burmester theory was used to form constraints which can provide a spherical curve with curvature close to zero. A plausible dimension range of mechanism, transmission angle, instantaneous invariants were arranged as constraint function. A meta-heuristic algorithm previously developed by one of authors was used to search the best instantaneous invariants based on the type of mechanism. The optimal solution was prototyped, and three-fingered gripper assembled. Secondly, an endoscopic camera peculiarly was used to measure curvature of the trajectory traced by coupler point of an optimal four-link spherical mechanism which was supposed to compose a finger of a robotic gripper. The coupler of the mechanism was expected to trace approximately a great circle arc to provide a grasp. Therefore, the technique to measure curvature was based on image processing and geodetic curvature theory. A calibrated endoscopic camera was used for acquisition of video record above the prototype which was centered in the center of the camera screen. The curvature of coupler point of optimal spherical mechanism design and prototype were compared. According to performed regression analysis, results of the measurement were %97 reliable for the useful part of the trajectory.

Extended virtual detector theory including quantum interferences

We extend an earlier “virtual detector” method [X. Wang, J. Tian, and J. H. Eberly, Phys. Rev. Lett. 110, 243001 (2013)], a hybrid quantum mechanical and classical trajectory method based on the concept of probability current, to include phases in the classical trajectories. Effects of quantum interferences, lost in the earlier method, are partially restored. The obtained photoelectron momentum distributions agree well with the corresponding numerical solutions of the time-dependent Schrödinger equation.

Development of a New Hybrid Stochastic/Trajectory Droplet Collision Model for Spray Simulations in Internal Combustion Engines

Focusing on the spray simulation in engines using the Lagrangian-Drop Eulerian-Fluid (LDEF) approach, a new hybrid stochastic/trajectory (HST) droplet collision model with the improved collision frequency, the redivided collision cell, and an updated method for selection of the candidate collision pairs was developed in this study. The stochastic method used in the HST model was based on the theoretical framework of the time-counter (TC) direct simulation Monte Carlo (DSMC) method. Besides the stochastic method, the trajectory method was also introduced for the cases with sparse spray conditions in order to improve the reliability of the model predictions. The factor of “every parcel represents a certain number of droplets” was well considered in the calculations of the droplet collision frequency in the present model, which has not been attracted enough attention in previous models. Moreover, an adaptive collision cell technique was used for reducing the dependence of the calculation results on the computational mesh. The formula for selecting the collision pairs was updated in order to accurately reproduce the spatial distribution of the spray droplets after collision. To assess the performance of the droplet collision model, the experimental data with the direct spray-to-spray impingement under wide operating conditions were used for model validations. In addition, the numerical results by the droplet collision models including O'Rourke model, Nordin model, and no-time-counter (NTC) model were presented for comparison. The results indicate that the HST model is capable of satisfactorily reproducing the spray morphology, collision frequency, and Sauter mean diameter (SMD) of the droplets during the spray process. Meanwhile, good independence on the computational grid size, time step, and parcel number can be achieved by the HST model.

On the Dragging Trajectory of Anchors in Clay for Merchant Ships

Anchor dragging is a major threat to the structural integrity of submarine pipelines. A mathematical model in which the mechanical model of chain and the bearing model of anchor were coupled together. Based on the associated flow rule, an incremental procedure was proposed to solve the spatial state of anchor until it reaches the ultimate embedding depth. With an indirect measurement method for the anchor trajectory, a model test system was established. The mathematical model was validated against some model tests, and the effects of two parameters were studied. It was found that both the ultimate embedding depth of a dragging anchor and the distance it takes to reach the ultimate depth increase with the shank-fluke pivot angle, but decrease as the undrained shear strength of clay increases. The proposed model is supposed to be useful for the embedding depth calculation and guiding the design of the pipeline burial depth.

Research on the Method and Application of MapReduce in Mobile Track Big Data Mining

Background: Mass movement trajectory data with real scenarios has been evolved with big data mining to solve the data redundancy problem. Methods: This paper proposes a parallel path based on the Map Reduce compression method, using two kinds of piecewise point mutual crisscross, the classified method of trajectory, and then segment trajectory distribution to multiple nodes to parallelize the compression. Results: Finally, the results based on both compression methods have been simulated for the different real-time data by merging both techniques. Conclusion: The performance test results show that the parallel trajectory compression method proposed in this paper can greatly improve the compression efficiency and completely eliminate the error caused by the failure of the correlation between the segments.

On the formation of van der Waals complexes through three-body recombination.

In this work, we show that van der Waals molecules X-RG (where RG is the rare gas atom) may be created through direct three-body recombination collisions, i.e., X + RG + RG → X-RG + RG. In particular, the three-body recombination rate at temperatures relevant for buffer gas cell experiments is calculated via a classical trajectory method in hyperspherical coordinates [Pérez-Ríos et al., J. Chem. Phys. 140, 044307 (2014)]. As a result, it is found that the formation of van der Waals molecules in buffer gas cells (1 K ≲ T ≲ 10 K) is dominated by the long-range tail (distances larger than the LeRoy radius) of the X-RG interaction. For higher temperatures, the short-range region of the potential becomes more significant. Moreover, we notice that the rate of formation of van der Walls molecules is of the same order of the magnitude independent of the chemical properties of X. As a consequence, almost any X-RG molecule may be created and observed in a buffer gas cell under proper conditions.

On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions

&lt;p style='text-indent:20px;'&gt;The objective of this paper is to study the fractal dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Inspired by the idea of the &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \ell $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-trajectory method, we prove the existence of a finite dimensional global attractor in an auxiliary normed space for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions and estimate the fractal dimension of the global attractor in the original phase space for this system by defining a Lipschitz mapping from the auxiliary normed space into the original phase space.&lt;/p&gt;

A Lightweight Processing Method for Hand-Drawn Pressure-Sensitive Trajectories Oriented Toward Web-Based 3D Modeling

Digital geometry processing of hand-drawn pressure-sensitive trajectories oriented toward Web-based three-dimensional (3D) modeling becomes attractive problem. Existing practices usually focus on converting raw trajectories into 3D stroke meshes with consistent line width, which the pressure information can not been fully utilized for denoising, simplification and reflecting realistic graphics of 3D stroke mesh. Also, there are little research on the high quality of trajectory with data transmission fluency and light weight in the Web environment. We propose a lightweight processing method for hand-drawn pressure-sensitive trajectories that is oriented toward Web-based 3D modeling. An angle-chord height united trajectory simplification (ACUTS) algorithm is proposed, in which the discrete points are simplified by comprehensively considering the angle deviation and chord height deviation with the sliding window mechanism. In addition, a lightweight reconstruction algorithm based on pressure -sensitivity for discrete points (LRPDP) is also proposed, in which the pressure value can be fully utilized for simplifying the redundant points while ensuring the reconstructed 3D stroke mesh more realistic. Besides, the experimental results show that compared to some other methods, we perform best in terms of data reduction using compression ratio and reduction percentages. We also perform best in terms of high accuracy of 3D stroke mesh using mean stochastic error distance, local maximum error and total length error. Although the insufficient utilization of the pressure-sensitive information limits our research to some extents, the significance of our research is to provide a new strategy to realize Web-based 3D modeling using hand drawing mode.

Trajectory Planning of Spray Gun With Variable Posture for Irregular Plane Based on Boundary Constraint

In order to solve the problem of low spraying efficiency and excessive paint waste caused by overspray when the robot automatically sprays irregular plane, a method of spray trajectory planning with boundary-constrained considering spray gun changing posture is proposed. The static spraying model of the spray gun with variable posture, and the dynamic spraying model of the spray gun along the arc path are established separately, an optimization method for the symmetry of coating distribution based on the variable spray angle when the gun spraying along the arc path is proposed. Orthogonal experiment method based on numerical simulation is used to clarify the influence of boundary curve shape and spray gun posture on boundary constraint distance, the prediction of the value range of the boundary constraint distance at the boundary of different curvature radius is realized. Within the range of the value of the boundary constraint, combined with the equal division method of the plane intercept line and the distance search method, a boundary constraint spray path generation algorithm is established by setting a uniform boundary constraint distance value. Finally, a multivariable spraying parameter optimization algorithm based on PSO is established, the spraying velocity, spraying height and spraying angle in the dynamic spraying process are solved, and the coating uniformity is optimized. The effectiveness and feasibility of the proposed spray trajectory planning method is verified by robot spraying experiment for any irregular plane.