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在移动机器人诸多技术的研究中，路径规划是技术研究中的重要部分之一，是机器人研究领域的热点，目的是在有障碍物的环境中按照某一种评价指标寻找一条从起始点位置到目标点位置的最优无碰撞路径[1-2]。
栅格法环境建模的经典方法，通过利用栅格对环境信息进行表示，栅格的大小决定了环境信息储存量的大小以机器人进行路径规划的时间长短[3-4]。
A*算法[5]是路径规划算法中经典的启发式搜索算法。
A*算法由Hart等[6]提出，结合Dijkstra算法和最佳优先搜索算法的优点。
文献[7]改进A*算法，在启发函数中加入余弦相似性和方向信息，做归一化处理。
文献[8]提出改进A*算法的扩展搜索邻域的思路，利用最小二叉堆优化A*算法OPEN列表存储结构提高搜索效率，但无法完成动态避障。
文献[9]成功地扩大搜索邻域，通过重新定义中心节点的位置，在每个节点的周围扩大无限可搜索邻域，较为快速实现无碰撞路径规划。
文献[10]基于A*算法，设计了优化的启发搜索函数，选取策略是使用关键点，剔除多余路径点以及非必要的转折点。
文献[11]通过引入二次A*算法，减少路径轨迹冗余点，缩短路径长度并且采用自适应圆弧优化算法与加权障碍物步长调改进A*算法与动态窗口法的机器人动态路径规划槐创锋，郭龙，贾雪艳，张子昊华东交通大学机电与车辆工程学院，南昌330013摘要：针对传统A*算法自身节点搜索策略存在路径转折点多、转折角度大、可行路径不是理论上的最优路径等缺点，将传统A*算法3×3的搜索邻域扩展为7×7，同时去除扩展邻域同方向的多余子节点，改进为7×7的A*算法，消除了传统A*算法的3×3邻域搜索和节点移动方向仅为0.25π的整数倍的限制，优化了搜索角度。
其次，针对移动机器人在复杂环境下动态路径规划问题，将改进7×7的A*算法与动态窗口算法进行融合，设计了一种全局最优路径的动态窗口评价函数，综合考虑移动速度、转角平滑度、安全性等因素，将改进7×7的A*算法与动态窗口法的融合算法与多种算法仿真比较，结果表明：改进7×7的A*算法与动态窗口法的融合算法更具有高效性和可行性。
2020，56（17）1引言近年来，无人机航迹规划问题成为国内外学者研究的热点，尤其在军事、抢险救灾和测绘等领域尤为重要。
为了提高无人机的生存机率并充分发挥其作战优基于改进蚁群算法的三维航迹规划魏江1，王建军1，王健1，2，秦春霞1，梅少辉11.西北工业大学电子信息学院，西安7101292.西北工业大学第365研究所，西安710129摘要：针对蚁群算法在无人机（UAV ）三维航迹规划中存在的收敛速度慢、空间复杂度高的缺点，提出了一种基于改进蚁群算法的无人机（UAV ）三维航迹规划方法。
该方法改进了局部搜索策略、初始信息素调整因子并在启发函数中加入了路径偏移因子，从而降低了航迹搜索空间的复杂度，提高了算法的搜索效率和收敛速度。
在利用DEM 数字高程数据建立的搜索空间中，该算法与现有算法相比，规划航迹缩短约24.08%，运行时间减少约11.56%，表明改进蚁群算法在无人机（UAV ）三维航迹规划中的可行性和有效性。
关键词：三维航迹规划；信息素调整因子；路径偏移因子；蚁群算法文献标志码：A 中图分类号：TP273doi ：10.3778/j.issn.1002-8331.2001-0134魏江，王建军，王健，等.基于改进蚁群算法的三维航迹规划.计算机工程与应用，2020，56（17）：217-223.WEI Jiang,WANG Jianjun,WANG Jian,et al.3D path planning based on improved ant colony puter Engi-neering and Applications,2020,56（17）：217-223.3D Path Planning Based on Improved Ant Colony AlgorithmWEI Jiang 1,WANG Jianjun 1,WANG Jian 1，2,QIN Chunxia 1,MEI Shaohui 11.College of Electronic and Information,Northwestern Polytechnical University,Xi ’an 710129,China2.No.365Institute,Northwestern Polytechnical University,Xi ’an 710129,ChinaAbstract ：For overcoming the disadvantages of traditional ant colony algorithm in UAV 3D path planning,such as slow convergence speed and high space complexity,this paper proposes a 3D path planning method for UAV based on improved ant colony algorithm.In this method,the local search strategy and the initial pheromone adjustment factor are improved,and the path offset factor is added to the heuristic function,which reduces the complexity of the path search space and improves the search efficiency and convergence speed of the algorithm.In the search space established using DEM digital elevation data,compared with existing algorithms,the algorithm in this paper shortens the planning path by 24.08%,and the running time of the algorithm is reduced by about 11.56%,which show the feasibility and effectiveness of the improved ant colony algorithm in the 3D path planning of UAVs.Key words ：3D path planning;pheromone adjusting factor;path offset factor;ant colony algorithm基金项目：陕西省重点产业创新链项目（No.2019ZDLGY14-02-02）。
Xiaojin Zhu ZHUXJ@ Zoubin Ghahramani ZOUBIN@ John Lafferty LAFFERTY@ School of Computer Science,Carnegie Mellon University,Pittsburgh PA15213,USAGatsby Computational Neuroscience Unit,University College London,London WC1N3AR,UKAbstractAn approach to semi-supervised learning is pro-posed that is based on a Gaussian randomfieldbeled and unlabeled data are rep-resented as vertices in a weighted graph,withedge weights encoding the similarity between in-stances.The learning problem is then formulatedin terms of a Gaussian randomfield on this graph,where the mean of thefield is characterized interms of harmonic functions,and is efficientlyobtained using matrix methods or belief propa-gation.The resulting learning algorithms haveintimate connections with random walks,elec-tric networks,and spectral graph theory.We dis-cuss methods to incorporate class priors and thepredictions of classifiers obtained by supervisedlearning.We also propose a method of parameterlearning by entropy minimization,and show thealgorithm’s ability to perform feature selection.Promising experimental results are presented forsynthetic data,digit classification,and text clas-sification tasks.1.IntroductionIn many traditional approaches to machine learning,a tar-get function is estimated using labeled data,which can be thought of as examples given by a“teacher”to a“student.”Labeled examples are often,however,very time consum-ing and expensive to obtain,as they require the efforts of human annotators,who must often be quite skilled.For in-stance,obtaining a single labeled example for protein shape classification,which is one of the grand challenges of bio-logical and computational science,requires months of ex-pensive analysis by expert crystallographers.The problem of effectively combining unlabeled data with labeled data is therefore of central importance in machine learning.The semi-supervised learning problem has attracted an in-creasing amount of interest recently,and several novel ap-proaches have been proposed;we refer to(Seeger,2001) for an overview.Among these methods is a promising fam-ily of techniques that exploit the“manifold structure”of the data;such methods are generally based upon an assumption that similar unlabeled examples should be given the same classification.In this paper we introduce a new approach to semi-supervised learning that is based on a randomfield model defined on a weighted graph over the unlabeled and labeled data,where the weights are given in terms of a sim-ilarity function between instances.Unlike other recent work based on energy minimization and randomfields in machine learning(Blum&Chawla, 2001)and image processing(Boykov et al.,2001),we adopt Gaussianfields over a continuous state space rather than randomfields over the discrete label set.This“re-laxation”to a continuous rather than discrete sample space results in many attractive properties.In particular,the most probable configuration of thefield is unique,is character-ized in terms of harmonic functions,and has a closed form solution that can be computed using matrix methods or loopy belief propagation(Weiss et al.,2001).In contrast, for multi-label discrete randomfields,computing the low-est energy configuration is typically NP-hard,and approxi-mation algorithms or other heuristics must be used(Boykov et al.,2001).The resulting classification algorithms for Gaussianfields can be viewed as a form of nearest neigh-bor approach,where the nearest labeled examples are com-puted in terms of a random walk on the graph.The learning methods introduced here have intimate connections with random walks,electric networks,and spectral graph the-ory,in particular heat kernels and normalized cuts.In our basic approach the solution is solely based on the structure of the data manifold,which is derived from data features.In practice,however,this derived manifold struc-ture may be insufficient for accurate classification.WeProceedings of the Twentieth International Conference on Machine Learning(ICML-2003),Washington DC,2003.Figure1.The randomfields used in this work are constructed on labeled and unlabeled examples.We form a graph with weighted edges between instances(in this case scanned digits),with labeled data items appearing as special“boundary”points,and unlabeled points as“interior”points.We consider Gaussian randomfields on this graph.show how the extra evidence of class priors can help classi-fication in Section4.Alternatively,we may combine exter-nal classifiers using vertex weights or“assignment costs,”as described in Section5.Encouraging experimental re-sults for synthetic data,digit classification,and text clas-sification tasks are presented in Section7.One difficulty with the randomfield approach is that the right choice of graph is often not entirely clear,and it may be desirable to learn it from data.In Section6we propose a method for learning these weights by entropy minimization,and show the algorithm’s ability to perform feature selection to better characterize the data manifold.2.Basic FrameworkWe suppose there are labeled points, and unlabeled points;typically. Let be the total number of data points.To be-gin,we assume the labels are binary:.Consider a connected graph with nodes correspond-ing to the data points,with nodes corre-sponding to the labeled points with labels,and nodes corresponding to the unla-beled points.Our task is to assign labels to nodes.We assume an symmetric weight matrix on the edges of the graph is given.For example,when,the weight matrix can be(2)To assign a probability distribution on functions,we form the Gaussianfieldfor(3) which is consistent with our prior notion of smoothness of with respect to the graph.Expressed slightly differently, ,where.Because of the maximum principle of harmonic functions(Doyle&Snell,1984),is unique and is either a constant or it satisfiesfor.To compute the harmonic solution explicitly in terms of matrix operations,we split the weight matrix(and sim-ilarly)into4blocks after the th row and column:(4) Letting where denotes the values on the un-labeled data points,the harmonic solution subject to is given by(5)Figure2.Demonstration of harmonic energy minimization on twosynthetic rge symbols indicate labeled data,otherpoints are unlabeled.In this paper we focus on the above harmonic function as abasis for semi-supervised classification.However,we em-phasize that the Gaussian randomfield model from which this function is derived provides the learning frameworkwith a consistent probabilistic semantics.In the following,we refer to the procedure described aboveas harmonic energy minimization,to underscore the har-monic property(3)as well as the objective function being minimized.Figure2demonstrates the use of harmonic en-ergy minimization on two synthetic datasets.The leftfigure shows that the data has three bands,with,, and;the rightfigure shows two spirals,with,,and.Here we see harmonic energy minimization clearly follows the structure of data, while obviously methods such as kNN would fail to do so.3.Interpretation and ConnectionsAs outlined briefly in this section,the basic framework pre-sented in the previous section can be viewed in several fun-damentally different ways,and these different viewpoints provide a rich and complementary set of techniques for rea-soning about this approach to the semi-supervised learning problem.3.1.Random Walks and Electric NetworksImagine a particle walking along the graph.Starting from an unlabeled node,it moves to a node with proba-bility after one step.The walk continues until the par-ticle hits a labeled node.Then is the probability that the particle,starting from node,hits a labeled node with label1.Here the labeled data is viewed as an“absorbing boundary”for the random walk.This view of the harmonic solution indicates that it is closely related to the random walk approach of Szummer and Jaakkola(2001),however there are two major differ-ences.First,wefix the value of on the labeled points, and second,our solution is an equilibrium state,expressed in terms of a hitting time,while in(Szummer&Jaakkola,2001)the walk crucially depends on the time parameter. We will return to this point when discussing heat kernels. An electrical network interpretation is given in(Doyle& Snell,1984).Imagine the edges of to be resistors with conductance.We connect nodes labeled to a positive voltage source,and points labeled to ground.Thenis the voltage in the resulting electric network on each of the unlabeled nodes.Furthermore minimizes the energy dissipation of the electric network for the given.The harmonic property here follows from Kirchoff’s and Ohm’s laws,and the maximum principle then shows that this is precisely the same solution obtained in(5).3.2.Graph KernelsThe solution can be viewed from the viewpoint of spec-tral graph theory.The heat kernel with time parameter on the graph is defined as.Here is the solution to the heat equation on the graph with initial conditions being a point source at at time.Kondor and Lafferty(2002)propose this as an appropriate kernel for machine learning with categorical data.When used in a kernel method such as a support vector machine,the kernel classifier can be viewed as a solution to the heat equation with initial heat sourceson the labeled data.The time parameter must,however, be chosen using an auxiliary technique,for example cross-validation.Our algorithm uses a different approach which is indepen-dent of,the diffusion time.Let be the lower right submatrix of.Since,it is the Laplacian restricted to the unlabeled nodes in.Consider the heat kernel on this submatrix:.Then describes heat diffusion on the unlabeled subgraph with Dirichlet boundary conditions on the labeled nodes.The Green’s function is the inverse operator of the restricted Laplacian,,which can be expressed in terms of the integral over time of the heat kernel:(6) The harmonic solution(5)can then be written asor(7)Expression(7)shows that this approach can be viewed as a kernel classifier with the kernel and a specific form of kernel machine.(See also(Chung&Yau,2000),where a normalized Laplacian is used instead of the combinatorial Laplacian.)From(6)we also see that the spectrum of is ,where is the spectrum of.This indicates a connection to the work of Chapelle et al.(2002),who ma-nipulate the eigenvalues of the Laplacian to create variouskernels.A related approach is given by Belkin and Niyogi (2002),who propose to regularize functions on by select-ing the top normalized eigenvectors of corresponding to the smallest eigenvalues,thus obtaining the bestfit toin the least squares sense.We remark that ourfits the labeled data exactly,while the order approximation may not.3.3.Spectral Clustering and Graph MincutsThe normalized cut approach of Shi and Malik(2000)has as its objective function the minimization of the Raleigh quotient(8)subject to the constraint.The solution is the second smallest eigenvector of the generalized eigenvalue problem .Yu and Shi(2001)add a grouping bias to the normalized cut to specify which points should be in the same group.Since labeled data can be encoded into such pairwise grouping constraints,this technique can be applied to semi-supervised learning as well.In general, when is close to block diagonal,it can be shown that data points are tightly clustered in the eigenspace spanned by thefirst few eigenvectors of(Ng et al.,2001a;Meila &Shi,2001),leading to various spectral clustering algo-rithms.Perhaps the most interesting and substantial connection to the methods we propose here is the graph mincut approach proposed by Blum and Chawla(2001).The starting point for this work is also a weighted graph,but the semi-supervised learning problem is cast as one offinding a minimum-cut,where negative labeled data is connected (with large weight)to a special source node,and positive labeled data is connected to a special sink node.A mini-mum-cut,which is not necessarily unique,minimizes the objective function,and label0other-wise.We call this rule the harmonic threshold(abbreviated “thresh”below).In terms of the random walk interpreta-tion,ifmakes sense.If there is reason to doubt this assumption,it would be reasonable to attach dongles to labeled nodes as well,and to move the labels to these new nodes.6.Learning the Weight MatrixPreviously we assumed that the weight matrix is given andfixed.In this section,we investigate learning weight functions of the form given by equation(1).We will learn the’s from both labeled and unlabeled data;this will be shown to be useful as a feature selection mechanism which better aligns the graph structure with the data.The usual parameter learning criterion is to maximize the likelihood of labeled data.However,the likelihood crite-rion is not appropriate in this case because the values for labeled data arefixed during training,and moreover likeli-hood doesn’t make sense for the unlabeled data because we do not have a generative model.We propose instead to use average label entropy as a heuristic criterion for parameter learning.The average label entropy of thefield is defined as(13) using the fact that.Both and are sub-matrices of.In the above derivation we use as label probabilities di-rectly;that is,class.If we incorpo-rate class prior information,or combine harmonic energy minimization with other classifiers,it makes sense to min-imize entropy on the combined probabilities.For instance, if we incorporate a class prior using CMN,the probability is given bylabeled set size a c c u r a c yFigure 3.Harmonic energy minimization on digits “1”vs.“2”(left)and on all 10digits (middle)and combining voted-perceptron with harmonic energy minimization on odd vs.even digits (right)Figure 4.Harmonic energy minimization on PC vs.MAC (left),baseball vs.hockey (middle),and MS-Windows vs.MAC (right)10trials.In each trial we randomly sample labeled data from the entire dataset,and use the rest of the images as unlabeled data.If any class is absent from the sampled la-beled set,we redo the sampling.For methods that incorpo-rate class priors ,we estimate from the labeled set with Laplace (“add one”)smoothing.We consider the binary problem of classifying digits “1”vs.“2,”with 1100images in each class.We report aver-age accuracy of the following methods on unlabeled data:thresh,CMN,1NN,and a radial basis function classifier (RBF)which classifies to class 1iff .RBF and 1NN are used simply as baselines.The results are shown in Figure 3.Clearly thresh performs poorly,because the values of are generally close to 1,so the major-ity of examples are classified as digit “1”.This shows the inadequacy of the weight function (1)based on pixel-wise Euclidean distance.However the relative rankings ofare useful,and when coupled with class prior information significantly improved accuracy is obtained.The greatest improvement is achieved by the simple method CMN.We could also have adjusted the decision threshold on thresh’s solution ,so that the class proportion fits the prior .This method is inferior to CMN due to the error in estimating ,and it is not shown in the plot.These same observations are also true for the experiments we performed on several other binary digit classification problems.We also consider the 10-way problem of classifying digits “0”through ’9’.We report the results on a dataset with in-tentionally unbalanced class sizes,with 455,213,129,100,754,970,275,585,166,353examples per class,respec-tively (noting that the results on a balanced dataset are sim-ilar).We report the average accuracy of thresh,CMN,RBF,and 1NN.These methods can handle multi-way classifica-tion directly,or with slight modification in a one-against-all fashion.As the results in Figure 3show,CMN again im-proves performance by incorporating class priors.Next we report the results of document categorization ex-periments using the 20newsgroups dataset.We pick three binary problems:PC (number of documents:982)vs.MAC (961),MS-Windows (958)vs.MAC,and base-ball (994)vs.hockey (999).Each document is minimally processed into a “tf.idf”vector,without applying header re-moval,frequency cutoff,stemming,or a stopword list.Two documents are connected by an edge if is among ’s 10nearest neighbors or if is among ’s 10nearest neigh-bors,as measured by cosine similarity.We use the follow-ing weight function on the edges:(16)We use one-nearest neighbor and the voted perceptron al-gorithm (Freund &Schapire,1999)(10epochs with a lin-ear kernel)as baselines–our results with support vector ma-chines are comparable.The results are shown in Figure 4.As before,each point is the average of10random tri-als.For this data,harmonic energy minimization performsmuch better than the baselines.The improvement from the class prior,however,is less significant.An explanation for why this approach to semi-supervised learning is so effec-tive on the newsgroups data may lie in the common use of quotations within a topic thread:document quotes partof document,quotes part of,and so on.Thus, although documents far apart in the thread may be quite different,they are linked by edges in the graphical repre-sentation of the data,and these links are exploited by the learning algorithm.7.1.Incorporating External ClassifiersWe use the voted-perceptron as our external classifier.For each random trial,we train a voted-perceptron on the la-beled set,and apply it to the unlabeled set.We then use the 0/1hard labels for dongle values,and perform harmonic energy minimization with(10).We use.We evaluate on the artificial but difficult binary problem of classifying odd digits vs.even digits;that is,we group “1,3,5,7,9”and“2,4,6,8,0”into two classes.There are400 images per digit.We use second order polynomial kernel in the voted-perceptron,and train for10epochs.Figure3 shows the results.The accuracy of the voted-perceptron on unlabeled data,averaged over trials,is marked VP in the plot.Independently,we run thresh and CMN.Next we combine thresh with the voted-perceptron,and the result is marked thresh+VP.Finally,we perform class mass nor-malization on the combined result and get CMN+VP.The combination results in higher accuracy than either method alone,suggesting there is complementary information used by each.7.2.Learning the Weight MatrixTo demonstrate the effects of estimating,results on a toy dataset are shown in Figure5.The upper grid is slightly tighter than the lower grid,and they are connected by a few data points.There are two labeled examples,marked with large symbols.We learn the optimal length scales for this dataset by minimizing entropy on unlabeled data.To simplify the problem,wefirst tie the length scales in the two dimensions,so there is only a single parameter to learn.As noted earlier,without smoothing,the entropy approaches the minimum at0as.Under such con-ditions,the results of harmonic energy minimization are usually undesirable,and for this dataset the tighter grid “invades”the sparser one as shown in Figure5(a).With smoothing,the“nuisance minimum”at0gradually disap-pears as the smoothing factor grows,as shown in FigureFigure5.The effect of parameter on harmonic energy mini-mization.(a)If unsmoothed,as,and the algorithm performs poorly.(b)Result at optimal,smoothed with(c)Smoothing helps to remove the entropy minimum. 5(c).When we set,the minimum entropy is0.898 bits at.Harmonic energy minimization under this length scale is shown in Figure5(b),which is able to dis-tinguish the structure of the two grids.If we allow a separate for each dimension,parameter learning is more dramatic.With the same smoothing of ,keeps growing towards infinity(we usefor computation)while stabilizes at0.65, and we reach a minimum entropy of0.619bits.In this case is legitimate;it means that the learning al-gorithm has identified the-direction as irrelevant,based on both the labeled and unlabeled data.Harmonic energy minimization under these parameters gives the same clas-sification as shown in Figure5(b).Next we learn’s for all256dimensions on the“1”vs.“2”digits dataset.For this problem we minimize the entropy with CMN probabilities(15).We randomly pick a split of 92labeled and2108unlabeled examples,and start with all dimensions sharing the same as in previous ex-periments.Then we compute the derivatives of for each dimension separately,and perform gradient descent to min-imize the entropy.The result is shown in Table1.As entropy decreases,the accuracy of CMN and thresh both increase.The learned’s shown in the rightmost plot of Figure6range from181(black)to465(white).A small (black)indicates that the weight is more sensitive to varia-tions in that dimension,while the opposite is true for large (white).We can discern the shapes of a black“1”and a white“2”in thisfigure;that is,the learned parametersCMNstart97.250.73%0.654298.020.39%Table1.Entropy of CMN and accuracies before and after learning ’s on the“1”vs.“2”dataset.Figure6.Learned’s for“1”vs.“2”dataset.From left to right: average“1”,average“2”,initial’s,learned’s.exaggerate variations within class“1”while suppressing variations within class“2”.We have observed that with the default parameters,class“1”has much less variation than class“2”;thus,the learned parameters are,in effect, compensating for the relative tightness of the two classes in feature space.8.ConclusionWe have introduced an approach to semi-supervised learn-ing based on a Gaussian randomfield model defined with respect to a weighted graph representing labeled and unla-beled data.Promising experimental results have been pre-sented for text and digit classification,demonstrating that the framework has the potential to effectively exploit the structure of unlabeled data to improve classification accu-racy.The underlying randomfield gives a coherent proba-bilistic semantics to our approach,but this paper has con-centrated on the use of only the mean of thefield,which is characterized in terms of harmonic functions and spectral graph theory.The fully probabilistic framework is closely related to Gaussian process classification,and this connec-tion suggests principled ways of incorporating class priors and learning hyperparameters;in 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Visual Complex Analysis —Applications ofHarmonic FunctionsAbstract:In this paper talk about Harmonic Functions in applications of mathematical physics and the theory of stochastic processes, which are the real or imaginary part of an analytic function.The standard applications are two dimensional steady state temperatures, electrostatics, fluid flow and complex potentials. We show 3 methods to solve Dirichlet problem.Noteworthy methods include Poisson's integral formulae, the Joukowski transformation, and so on. Modern computer software is capable of implemeting these complex analysis methods. Keywords: harmonic functions; physics; Dirichlet problem; Poisson integral formula ；stochastic processes1.Harmonic function [1]In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of R n ) which satisfies Laplace's equation, i.e.everywhere on U . This is usually written as or . ()(,)(,)f z x y i x y φψ=+which is an analytic function only and if only (,)x y φand (,)x y ψ are harmonic functions. It has many physical interpretations, some of which are listed below2.Application: The Joukowski Airfoil [2]Following functions are harmonic ：and . A carefulinspection shows that this is the polar form of , which is analytic for all z≠0. Hence that and are harmonic functions for all z≠0. The Russian scientist Nikolai Egorovich Joukowsky studied the functionHe showed that the image of a circle passing through and containing the point is mapped onto a curve shaped like the cross section of an airplane wing. We call this curve the Joukowski airfoil. If the streamlines for a flow around the circle are known, then their images under the mappingwill be streamlines for a flow around the Joukowski airfoil, as shown in FigureImage of a fluid flow under Joukowski mappingThis example illustrate how analytic function thery can be used to solve Laplace’s equation in region whose boundaries are identiable as level curves[3].2.Application ：Steady-State Temperature as a Harmonic Function[3]3.Physical meaning：Maximum principle and Second law of thermodynamics ——Heat cannot spontaneously flow from a colder location to a hotter location[3].We have emphasized practical applications of harmonic functions to steady state temperatures, electrostatics, and fluid flow。
路径规划技术作为机器人技术的重要分支之一备受关注，其目的是按照一定的评价标准规划出一条从起点到目标点的最优无碰路径。
按照机器人感知环境信息的程度，路径规划可分为全局路径规划和局部路径规划。
全局路径规划在环境信息已知的情况下进行，是一种事前、离线规划，对实时计算能力要求不高，可以规划出最优路径。
目前的全局路径规划算法及应用已经比较成熟。
局部路径规划是在环境信息未知或部分未知的情况下进行的路径规划，依靠超声波、红外线、激光雷达等传感器检测环境信息，实时规划出一条从起点到目标点的无碰路径[1]。
由于缺乏全局环境信息，并且机器人所搭载传感器检测范围有限，所以规划的路径一般不基于模糊势场法的移动机器人局部路径规划王迪，李彩虹，郭娜，刘国名，高腾腾山东理工大学计算机科学与技术学院，山东淄博255049摘要：针对采用传统人工势场法进行移动机器人局部路径规划时存在的局部极小点和规划路径过长等问题，提出了一种基于虚拟目标点和有限状态机的模糊势场法。
构造基于人工势场的虚拟目标点法来解决局部极小点问题，在合适的位置设置虚拟目标点使机器人逃离局部极小点区域。
将虚拟目标点法与模糊控制相结合，对障碍物环境进行预测，及时避障，解决机器人在复杂环境中采用虚拟目标点法规划路径时存在的路径过长问题。
设计一个有限状态机来判断障碍物环境，执行算法转换策略，使改进算法适用于多种复杂环境。
所设计算法在MATLAB平台上进行了仿真验证。
结果表明，该算法能够使机器人逃出局部极小点、缩短规划路径。
算法不仅适用于简单、离散环境，在传统算法运行困难的、复杂的环境中，例如墙型、U型和多U型障碍物环境，也能规划出可行的优化路径。
关键词：移动机器人；局部路径规划；模糊势场法；虚拟目标点；局部极小点；有限状态机文献标志码：A中图分类号：TP242doi：10.3778/j.issn.1002-8331.2008-0373Local Path Planning of Mobile Robot Based on Fuzzy Potential Field MethodWANG Di,LI Caihong,GUO Na,LIU Guoming,GAO TengtengSchool of Computer Science and Technology,Shandong University of Technology,Zibo,Shandong255049,China Abstract：A fuzzy potential field method based on virtual target point and Finite State Machine（FSM）is proposed tosolve the problem of local minimum point and over-long planned path existed in the traditional artificial potential field method for local path planning of the mobile robot.The virtual target point method based on artificial potential field is constructed to solve the local minimum problem,and it is set in an appropriate position to make the robot escape from the local minimum point area.The virtual target point method is combined with fuzzy control to predict the obstacle environ-ment and avoid obstacles in time,so as to solve the problem of over-long path when the robot adopts the virtual target point method to plan the path in complex environment.A model of the FSM is designed to judge the obstacle environ-ment,and the algorithm conversion strategy is implemented to make the improved algorithm suitable for various environ-ments.The designed method is tested on the MATLAB platform.The simulation results show that the algorithm can make the robot escape from the local minimum point and shorten the planned path.It is not only applicable to the simple and discrete environment,but also can plan the feasible optimization path in the complex environment where the traditional algorithm is difficult to operate,such as the wall-like,U-shaped and multiple U-shaped obstacles environments.Key words：mobile robot;local path planning;fuzzy potential field method;virtual target point;local minimum point; Finite State Machine（FSM）基金项目：国家自然科学基金（61473179，61973184）。
---------------------------------------------------------------范文最新推荐------------------------------------------------------ 调和函数的性质与应用+文献综述摘要：本论文首先讨论了调和函数的最值原理及次调和函数导出的Perron族；然后通过Perron函数将Dirichlet问题的判定转化为了闸函数的存在问题，从而给出了有解性的充分条件；接着通过Green函数法给出了Dirichlet问题的一般解，并讨论了一些其他的边值问题；另外，本文还对调和测度做了一些讨论，并讨论了Phragmén-Lindelöf定理。
论文的主要结构如下：第一章（绪论）介绍了调和函数的一些背景知识。
第二章讨论了调和函数自身的一些性质，主要根据次调和函数的性质引出了Perron族，以此来解决Dirichlet问题。
第三章主要运用Green函数法来讨论Dirichlet问题，1 / 18并讨论了其他边值问题。
第四章主要讨论调和测度和讨论了Phragmén-Lindelöf定理。
关键词调和函数边值问题Green函数极值原理5600毕业设计（论文）外文摘要TitleProperties and Application of Harmonic FunctionsAbstractIn this paper,we firstly discussed the maximum principle of harmonic functions and the Perron family deduced by subharmonic functions.Then we transformed the judgment of the Dirichlet problem to the existence of barrier function by using Perron family,thus we got the sufficient condition of the solution of the Dirichlet problem.Then we utilized the Green's function method to obtain the general solution of the Dirichlet problem,and we also discussed some other boundary value problems.In the---------------------------------------------------------------范文最新推荐------------------------------------------------------end,we discussed the Harmonic Measure and discussed the Phragmén-Lindelöf theorem.The paper mainly consists of the following parts:Chapter one introduced some background knowledge of the harmonic function;Chapter two discusses some properties of harmonic functions,and deduced Perron family by the nature of subharmonic functions,which we could use to solve the Dirichlet problem.偏微分方程包括弦振动方程，热传导方程，拉普拉斯方程，特里谷米方程等。
2019，55（13）基于深度强化学习的移动机器人路径规划董瑶1，2，葛莹莹1，2，郭鸿湧1，3，董永峰1，2，杨琛1，21.河北工业大学人工智能与数据科学学院，天津3004012.河北工业大学河北省大数据计算重点实验室，天津3004013.河北工程大学，河北邯郸056038摘要：为解决传统的深度Q 网络模型下机器人探索复杂未知环境时收敛速度慢的问题，提出了基于竞争网络结构的改进深度双Q 网络方法（Improved Dueling Deep Double Q -Network ，IDDDQN ）。
移动机器人通过改进的DDQN 网络结构对其三个动作的值函数进行估计，并更新网络参数，通过训练网络得到相应的Q 值。
移动机器人采用玻尔兹曼分布与ε-greedy 相结合的探索策略，选择一个最优动作，到达下一个观察。
机器人将通过学习收集到的数据采用改进的重采样优选机制存储到缓存记忆单元中，并利用小批量数据训练网络。
实验结果显示，与基本DDQN 算法比，IDDDQN 训练的机器人能够更快地适应未知环境，网络的收敛速度也得到提高，到达目标点的成功率增加了3倍多，在未知的复杂环境中可以更好地获取最优路径。
关键词：深度双Q 网络（DDQN ）；竞争网络结构；重采样优选机制；玻尔兹曼分布；ε-greedy 策略文献标志码：A 中图分类号：TP399doi ：10.3778/j.issn.1002-8331.1812-0321董瑶，葛莹莹，郭鸿湧，等.基于深度强化学习的移动机器人路径规划.计算机工程与应用，2019，55（13）：15-19.DONG Yao,GE Yingying,GUO Hongyong,et al.Path planning for mobile robot based on deep reinforcement puter Engineering and Applications,2019,55（13）：15-19.Path Planning for Mobile Robot Based on Deep Reinforcement LearningDONG Yao 1，2,GE Yingying 1，2,GUO Hongyong 1，3,DONG Yongfeng 1，2,YANG Chen 1，21.School of Artificial Intelligence,Hebei University of Technology,Tianjin 300401,China2.Hebei Provincial Key Laboratory of Big Data Computing,Hebei University of Technology,Tianjin 300401,China3.Hebei University of Engineering,Handan,Hebei 056038,ChinaAbstract ：To solve the problem of slow convergence under the basic deep Q -Network with which the robot explores the complex and unknown environment,an improved deep double Q network algorithm （Improved Dueling Deep Double Q -Network,IDDDQN ）based on dueling network structure is put forward.The mobile robot can estimate the state-action value function of its three actions through the improved DDQN network,update the network parameters and get the corresponding Q value through the training.With the combination of Boltzmann and ε-greedy adopted,the mobile robot chooses an optimal action,and reaches the next observation.It can also store the data into experience replay memory through network learning,and train the network with mini-batch data.According to the experiment results,the mobile robot using IDDDQN can quickly adapt to the unknown environment,the convergence speed of IDDDQN is improved,the success rate of reaching the target position adds up to more than three times,and the optimal path can also be gained in an unknown complex environment.Key words ：Deep Double Q -Network （DDQN ）;dueling network;resample experience replay memory;Boltzmann distribution;ε-greedy policy基金项目：天津市科技计划项目（No.14ZCDGSF00124）；天津市自然科学基金（No.16JCYBJC15600）。
