06 WB-Mech_Ch05_Modal
TBM减速器两级行星齿轮传动系统动力学特性研究
2024年第48卷第3期Journal of Mechanical TransmissionTBM减速器两级行星齿轮传动系统动力学特性研究徐尤南1李明钦1刘汕娟1刘志强1,2（1 华东交通大学机电与车辆工程学院，江西南昌330013）（2 江西水利职业学院机电工程系，江西南昌330013）摘要隧道掘进机（Tunnel Boring Machine，TBM）减速器为TBM刀盘驱动系统重要部件。
为了揭示TBM减速器两级行星齿轮传动系统的动力学特性，考虑TBM减速器两级行星齿轮传动系统的齿面摩擦、时变啮合刚度、齿侧间隙、传递误差等影响因素，运用集中参数法建立了TBM减速器两级行星齿轮传动系统的扭转动力学模型，求解分析了动力学特性。
固有特性分析结果表明了行星轮系的3种振动模态形式：刚体模态、扭转振动模态、行星轮振动模态。
动态响应分析得到各齿轮的振动位移及齿轮副间的动态啮合力。
分析结果为该行星齿轮传动系统动态优化设计奠定了基础。
关键词TBM 两级行星轮系动力学动态响应固有特性Research on Dynamic Characteristics of the Two-stage Planetary GearTransmission System of TBM ReducersXu Younan1Li Mingqin1Liu Shanjuan1Liu Zhiqiang1,2（1 School of Mechatronics and Vehicle Engineering, East China Jiaotong University, Nanchang 330013, China）（2 Department of Mechanical and Electrical Engineering, Jiangxi Water Resources Institute, Nanchang 330013, China）Abstract Tunnel boring machine (TBM) reducers are important components of TBM's cutter-head drive system. In order to reveal the dynamic characteristics of the two-stage planetary gear transmission system of the TBM reducer, the influence factors such as tooth surface friction, the time-varying meshing stiffness, the back‑lash and meshing error of the two-stage planetary gear transmission system of the TBM reducer are considered. The torsional dynamics model of the two-stage planetary gear transmission system of the TBM reducer is estab‑lished by the lumped parameter method, and its dynamic characteristics are solved and analyzed. The inherent characteristic analysis results show three vibration modes of the planetary gear train: rigid body vibration mode, torsional vibration mode and the planetary gear vibration mode; the vibration displacement of each gear and the dynamic meshing force between the gear pairs are obtained from the dynamic response analysis. The result lays the foundation for the dynamic optimization design of planetary gear trains.Key words TBM Two-stage planetary gear train Dynamics Dynamic response Inherent character‑istic0 引言全断面硬岩隧道掘进机（Tunnel Boring Machine，TBM）[1]是集机、电、液及传感与信息技术于一体的国家重点工程建设、军事与国防工程所急需的重大技术装备。
Explicit_Dynamics_Chapter 2_Intro_to_WB
• ANSYS Workbench is comprised of various applications (some examples):
– Mechanical for performing structural and thermal analyses using the ANSYS solver – Explicit Dynamics for performing non-linear dynamic analyses using the AUTODYN solver. – Geometry (DesignModeler) for creating and modifying CAD geometry to prepare the solid model for use in Mechanical. – Meshing for generating Structural and CFD meshes – Advanced Meshing for generating Structural and CFD meshes using ICEM CFD – Design Exploration for optimization – Finite Element Modeler (FE Modeler) for translating a NASTRAN and ABAQUS mesh for use in ANSYS – AUTODYN for explicit dynamics simulations featuring modeling of nonlinear dynamics
• Native applications are launched and run entirely in the Workbench window.
Matlab与VB结合在机械振动模态分析中的应用
Matlab与VB结合在机械振动模态分析中的应用
孙鲁涌;周异;赵文礼
【期刊名称】《现代机械》
【年(卷),期】2003(000)004
【摘要】阐述了应用MATLAB与VB编程工具相结合的手段开发机械振动特性研究软件的方法,以二自由度可解耦系统的振动模态分析为例编制了相应程序,可快速有效地进行不同物理常数时模态数据计算和波形表达.
【总页数】3页(P39-41)
【作者】孙鲁涌;周异;赵文礼
【作者单位】浙江大学生产工程研究所,310027;杭州电子工业学院机电分
院,310037;杭州电子工业学院机电分院,310037;杭州电子工业学院机电分
院,310037
【正文语种】中文
【中图分类】TH132
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1.MATLAB与VB结合在开采沉陷中的应用 [J], 祁慧敏;彭程
2.利用MatrixVB插件实现在VB中的MATLAB编程及其在结构动力分析中的应用 [J], 陈辉国;石少卿
3.MATLAB与VB结合在水电机组测试分析中的应用 [J], 唐拥军;潘罗平
4.机械振动与噪声控制中的MATLAB应用 [J], 徐红金
5.机械振动与噪声控制中的MATLAB应用 [J], 徐红金
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离散S曲线算法控制的机械手臂运动特性研究
China Forest Products Industry林产工业，2021，58（04）：32-36离散S曲线算法控制的机械手臂运动特性研究∗刘晓飞 高兴华 郭庆东 崔金鹏 （北华大学机械工程学院，吉林省吉林市 132021）摘 要：针对机械手装配应用中对作业精度及响应速度的要求，研究了步进电机驱动的正交两指机械手臂动作高精度、快响应的平滑控制方法。
研究基于连续S曲线函数离散处理的算法实现步进电机平滑控制调速的机理，分析了每步脉冲离散处理算法对步进电机升频和降频的作用。
通过理论计算确定优化离散S曲线，并应用于机械臂中进行仿真，结果表明：输出转矩曲线与理论曲线拟合度基本一致。
将离散优化的S曲线算法在还原魔方机器人正交两指机械手臂上进行试验，试验表明：在斜率常数为a=0.44时，步进电机可实现快速精确控制，能有效解决其在高速精确运动中的失步、过冲等问题。
关键词：正交两指机械手臂； 离散化S曲线； 步进电机调速控制； 运动特性； 仿真建模中图分类号：TS64文献标识码：A文章编号：1001-5299 （2021） 04-0032-05DOI:10.19531/j.issn1001-5299.202104007Research on Motion Characteristics of Robot Arm Controlled by Discrete S-curve AlgorithmLIU Xiao-fei GAO Xing-hua GUO Qing-dong CUI Jin-peng(College of Mechanical Engineering, Beihua University, Jilin132021, Jilin,P.R.China) Abstract:Aiming at the requirements of operation accuracy and response speed in robotic assembly applications, the high-precision, fast-response smooth control method of the motion of the orthogonal two-finger manipulator driven by the stepper motor were studied in this paper. The mechanism of smooth speed control of stepper motor based on discrete processing algorithm of continuous S-curve function was studied, and the effect of each step of the discrete processing algorithm on the up-frequency and down-frequency of the stepper motor was analyzed as well. The optimized discrete S-curve was determined by theoretical calculation and applied to the robot arm for simulation. The moment curve was basically consistent with the theoretical curve. The discrete optimized S-curve algorithm was tested on the orthogonal two-finger mechanical arm of the restored Rubik's cube robot. The experimental results showed that when the slope constant a of the motor was 0.44, the stepper motor can realize fast and accurate control, and can effectively solve the problems of out-of-step and overshoot in high-speed and accurate motion.Key words: Orthogonal two-finger manipulator; Discrete S-curve; Stepping motor speed control; Motion characteristics; Simulation modeling机械手臂作为机器人的末端执行机构[1-4]已在一些流水线生产中得到广泛应用[5]。
Workbench12.0modal分析
本节所述的功能，一般适用于ANSYS DesignSpace Entra及以上版本的许 可。
5-2
Vibration Analysis
模态系统分析基础
对于模态分析，振动频率ωi和模态φi 是根据下面的方程计算的出的：
Training Manual
假设:
([K ] ω [M ]){φ } = 0
2 i i
– [K] 和 [M] 不变:
假设材料特性为线弹性的 利用小位移理论，并且不包括非线性的 不存在[C] ,因此无阻尼 无{F} , 因此无激振力 结构可以强迫振动也可以不强迫振动
– 模态 {φ} 是相对值，不是绝对值
5-3
Vibration Analysis
A.模态系统分析步骤
Training Manual
Workbench - Mechanical Introduction
第五章 模态分析
5-1
Vibration Analysis
简介
– 假设用户已学习了第4章线性静力结构分析部分。
Training Manual
在这一章中，将介绍模态分析。进行模态分析类似线性静力分析。 本章内容:
– 模态分析步骤 – 有预应力的模态分析步骤
Vibration Analysis
… 检查结果
模态:
– 由于在结构上没有激励作用，因此振型只是与自由振动相关的相对值。 – 在详细列表里可以看到每个结果的频率值. – 应用图形窗口下方的时间标签的动画工具栏来查看振型
Training Manual
5-10
Vibration Analysis
B. Workshop 5.1 – 模态
[K ]{xo } = {F }
空间滚动轴承MoS2_薄膜三体接触摩擦分析
第52卷第12期表面技术2023年12月SURFACE TECHNOLOGY·343·空间滚动轴承MoS2薄膜三体接触摩擦分析董绍江1，程伟伦1，胡小林2，徐向阳1，潘雪娇1（1.重庆交通大学 机电与车辆工程学院，重庆 400074；2.重庆工业大数据创新中心有限公司，重庆 400707）摘要：目的针对太空尘导致的固体润滑轴承失效机理难以揭示的问题，提出采用分子动力模拟的方法，从原子尺度对空间滚动轴承二硫化钼（MoS2）润滑薄膜的摩擦特性进行研究。
方法首先建立MoS2薄膜的原子模型，在此基础上，根据实际场景，将MoS2薄膜的摩擦接触条件分为两体和三体接触。
其次，通过对比两体和三体接触的摩擦结果，探究三体接触条件下MoS2薄膜的摩擦磨损机理。
最后，通过改变基底结构，探究粗糙基底对MoS2薄膜三体接触磨损的影响。
结果三体接触对薄膜的磨损和摩擦系数均小于两体接触。
在相同的滑移速度下，磨料的自旋速度越快，摩擦系数越低，但薄膜上出现的损伤越多。
三体接触条件下，基底的粗糙表面不利于薄膜吸附，并且加快了薄膜的磨损。
结论三体与两体接触的摩擦磨损机理不同。
三体接触摩擦过程中，薄膜表面不会出现犁削现象进而造成薄膜的撕裂。
原子尺度下，摩擦过程中磨料的自旋作用削弱了薄膜表面切向力的积累，从而有效减小了摩擦力，减少了薄膜磨损。
在保持磨料滑移速度不变的情况下，磨料的自旋速度会对薄膜的摩擦和磨损产生影响。
在三体接触摩擦过程中，粗糙基底上的微凸结构阻碍薄膜运动，增大了摩擦力，薄膜与基底接触面积的减少削弱了基底与薄膜的吸附性。
关键词：二硫化钼；三体接触摩擦；两体接触摩擦；分子动力学仿真；磨损；摩擦中图分类号：TH133 文献标识码：A 文章编号：1001-3660(2023)12-0343-08DOI：10.16490/ki.issn.1001-3660.2023.12.028Three-body Contact Friction of MoS2 Thin Films inSpace Rolling BearingsDONG Shao-jiang1, CHENG Wei-lun1, HU Xiao-lin2, XU Xiang-yang1, PAN Xue-jiao1(1. School of Mechatronics and Vehicle Engineering, Chongqing Jiaotong University,Chongqing 400074, China;2. Chongqing Industrial Big Data Innovation Center Co., Ltd., Chongqing 400707, China)ABSTRACT: Because of the weightless environment of space, space dust can easily enter the spacecraft. In the long time running, the friction pair will produce debris. Space dust and debris can affect mechanical lubrication. It is difficult to reveal the failure mechanism of solid lubricated bearings caused by space dust and debris. The friction characteristics of MoS2 (molybdenum disulfide) lubricated film of space rolling bearing were studied by molecular dynamic simulation. Fe-Ni-Cr alloy收稿日期：2022-10-12；修订日期：2023-04-12Received：2022-10-12；Revised：2023-04-12基金项目：国家自然科学基金（51775072）；重庆市科技创新领军人才支持计划项目（CSTCCCXLJRC201920）；重庆市高校创新研究群体（CXQT20019）；重庆市北碚区科学技术局技术创新与应用示范项目（2020-5）；重庆交通大学研究生科研创新项目（CYS22414）Fund：The National Natural Science Foundation of China (51775072); Chongqing Science and Technology Innovation Leading Talents Support Program (CSTCCCXLJRC201920); Chongqing University Innovation Research Group (CXQT20019); Technology Innovation and Application Demonstration Project of Chongqing Beibei Science and Technology Bureau(2020-5); Graduate Research and Innovation Project of Chongqing Jiaotong University(CYS22414)引文格式：董绍江, 程伟伦, 胡小林, 等. 空间滚动轴承MoS2薄膜三体接触摩擦分析[J]. 表面技术, 2023, 52(12): 343-350.DONG Shao-jiang, CHENG Wei-lun, HU Xiao-lin, et al. Three-body Contact Friction of MoS2 Thin Films in Space Rolling Bearings[J]. Surface Technology, 2023, 52(12): 343-350.·344·表面技术 2023年12月was used as the reference bearing material in the simulation to explore the interaction between the film and the substrate. Firstly, the atomic model of MoS2 film and Fe-Ni-Cr alloy was established. Based on this, the friction contact conditions of MoS2 film were divided into two-body contact and three-body contact according to the actual scene. Secondly, the friction and wear mechanism of MoS2 film under three-body contact condition was explored by comparing the friction results of two-body contact and three-body contact. Finally, the effect of rough substrate on three-body contact wear of MoS2 film was explored by changing the substrate structure. For the surface,under the condition of two-body contact, the abrasive particles will tear the film, and the ploughing phenomenon will appear under the action of load. Under the condition of three-body contact, there is no tearing or ploughing phenomenon on the surface of the film. The wear of three-body contact is less than that of two-body contact. The friction coefficient of three-body contact is less than that of two-body contact. Under the condition of three-body contact, the rough surface of the substrate is not conducive to the adsorption of thin films and accelerates the wear of thin films. The effect of load on the adsorption of rough substrate and thin film is obvious. When the load exceeds 300 nN, the adsorption energy of rough substrate and thin film will change abruptly. The friction and wear mechanism of three-body contact is different from that of two-body contact. Under the condition of two-body contact, the abrasive particles lead to sliding friction. During the process of motion, the abrasive particles accumulate and release tangential force on the surface of the film (stick-slip phenomenon).When the accumulated tangential force exceeds the strain limit of the film, the film will be destroyed. Under the condition of three-body contact, the abrasive particles have both rolling friction and sliding friction. When the abrasive slip velocity is kept constant, the abrasive spin velocity will affect the friction and wear of the film. In the process of three-body contact friction, there is no ploughing phenomenon on the surface of the film, which leads to the tear of the film. At atomic scale, the spin effect of abrasive particles weakens the accumulation of tangential force on the surface of the film, thus effectively reducing the friction force and the wear of the film. In the process of three-body contact friction, the slight convexity on the rough substrate obstructs the motion of the film and increases the friction force, and the reduction of the contact area between the film and the substrate weakens the absorbability of the substrate and the film.KEY WORDS: molybdenum disulfide; three-body contact friction; two-body contact friction; molecular dynamics simulation;wear; friction随着我国航天事业的发展[1]，对于航天器零部件失效机理的研究就显得尤其重要。
a fast and elitist multiobjective genetic algorithm NSGA-II
A Fast and Elitist Multiobjective Genetic Algorithm:NSGA-IIKalyanmoy Deb ,Associate Member,IEEE ,Amrit Pratap,Sameer Agarwal,and T.MeyarivanAbstract—Multiobjective evolutionary algorithms (EAs)that use nondominated sorting and sharing have been criti-cizedmainly for their:1)is thenumber of objectives and is the populationsize);2)nonelitism approach;and 3)the need for specifying a sharing parameter.In this paper,we suggest a nondominated sorting-based multiobjective EA (MOEA),called nondominated sorting genetic algorithm II (NSGA-II),which alleviates all the above three difficulties.Specifically,a fast nondominated sorting approach withsolutions.Simulation results on difficult test problems show that the proposed NSGA-II,in most problems,is able to find much better spread of solutions and better convergence near the true Pareto-optimal front compared to Pareto-archived evolution strategy and strength-Pareto EA—two other elitist MOEAs that pay special attention to creating a diverse Pareto-optimal front.Moreover,we modify the definition of dominance in order to solve constrained multiobjective problems efficiently.Simulation results of the constrained NSGA-II on a number of test problems,including a five-objective seven-constraint nonlinear problem,are compared with another constrained multiobjective optimizer and much better performance of NSGA-II is observed.Index Terms—Constraint handling,elitism,genetic algorithms,multicriterion decision making,multiobjective optimization,Pareto-optimal solutions.I.I NTRODUCTIONTHE PRESENCE of multiple objectives in a problem,in principle,gives rise to a set of optimal solutions (largely known as Pareto-optimal solutions),instead of a single optimal solution.In the absence of any further information,one of these multiple solutions,it has to be applied many times,hopefully finding a different solution at each simulation run.of multiobjective evolu-been suggested [1],[7],[13],revised February 5,2001andSeptember 7,2001.The work of K.Deb was supported by the Ministry of Human Resources and Development,India,under the Research and Development Scheme.The authors are with the Kanpur Genetic Algorithms Laboratory,Indian In-stitute of Technology,Kanpur PIN 208016,India (e-mail:deb@iitk.ac.in).Publisher Item Identifier S 1089-778X(02)04101-2.[20],[26].The primary reason for this is their ability to find multiple Pareto-optimal solutions in one single simulation run.Since evolutionary algorithms (EAs)work with a population of solutions,a simple EA can be extended to maintain a diverse set of solutions.With an emphasis for moving toward the true Pareto-optimal region,an EA can be used to find multiple Pareto-optimal solutions in one single simulation run.The nondominated sorting genetic algorithm (NSGA)pro-posed in [20]was one of the first such EAs.Over the years,the main criticisms of the NSGA approach have been as follows.1)High computational complexity of nondominated sorting:The currently-used nondominated sorting algorithm has acomputational complexity of(where is the population size).Thismakes NSGA computationally expensive for large popu-lation sizes.This large complexity arises because of the complexity involved in the nondominated sorting proce-dure in every generation.2)Lack of elitism:Recent results [25],[18]show that elitism can speed up the performance of the GA significantly,which also can help preventing the loss of good solutions once they are found.3)Need for specifying the sharing parameterwe describe the proposed NSGA-II algorithm in details.Sec-tion IV presents simulation results of NSGA-II and compares them with two other elitist MOEAs(PAES and SPEA).In Sec-tion V,we highlight the issue of parameter interactions,a matter that is important in evolutionary computation research.The next section extends NSGA-II for handling constraints and compares the results with another recently proposed constraint-handling method.Finally,we outline the conclusions of this paper.II.E LITIST M ULTIOBJECTIVE E VOLUTIONARY A LGORITHMS During1993–1995,a number of different EAs were sug-gested to solve multiobjective optimization problems.Of them,Fonseca and Fleming’s MOGA[7],Srinivas and Deb’s NSGA[20],and Horn et al.’s NPGA[13]enjoyed more attention.These algorithms demonstrated the necessary additional oper-ators for converting a simple EA to a MOEA.Two commonfeatures on all three operators were the following:i)assigningfitness to population members based on nondominated sortingand ii)preserving diversity among solutions of the samenondominated front.Although they have been shown to findmultiple nondominated solutions on many test problems and anumber of engineering design problems,researchers realizedthe need of introducing more useful operators(which havebeen found useful in single-objective EA’s)so as to solvemultiobjective optimization problems better.Particularly,the interest has been to introduce elitism to enhance theconvergence properties of a MOEA.Reference[25]showedthat elitism helps in achieving better convergence in MOEAs.Among the existing elitist MOEAs,Zitzler and Thiele’s SPEA[26],Knowles and Corne’s Pareto-archived PAES[14],andRudolph’s elitist GA[18]are well studied.We describe theseapproaches in brief.For details,readers are encouraged to referto the original studies.Zitzler and Thiele[26]suggested an elitist multicriterion EAwith the concept of nondomination in their SPEA.They sug-gested maintaining an external population at every generationstoring all nondominated solutions discovered so far beginningfrom the initial population.This external population partici-pates in all genetic operations.At each generation,a combinedpopulation with the external and the current population is firstconstructed.All nondominated solutions in the combined pop-ulation are assigned a fitness based on the number of solutionsthey dominate and dominated solutions are assigned fitnessworse than the worst fitness of any nondominated solution.This assignment of fitness makes sure that the search is directedtoward the nondominated solutions.A deterministic clusteringtechnique is used to ensure diversity among nondominatedsolutions.Although the implementation suggested in[26]is,with proper bookkeeping the complexity of SPEAcan be reduced to1)-evolutionstrategy.Instead of using real parameters,binary strings wereused and bitwise mutations were employed to create offsprings.In their PAES,with one parent and one offspring,the offspringis compared with respect to the parent.If the offspring domi-nates the parent,the offspring is accepted as the next parent andthe iteration continues.On the other hand,if the parent dom-inates the offspring,the offspring is discarded and a new mu-tated solution(a new offspring)is found.However,if the off-spring and the parent do not dominate each other,the choice be-tween the offspring and the parent is made by comparing themwith an archive of best solutions found so far.The offspring iscompared with the archive to check if it dominates any memberof the archive.If it does,the offspring is accepted as the newparent and all the dominated solutions are eliminated from thearchive.If the offspring does not dominate any member of thearchive,both parent and offspring are checked for their near-ness with the solutions of the archive.If the offspring resides ina least crowded region in the objective space among the mem-bers of the archive,it is accepted as a parent and a copy of addedto the archive.Crowding is maintained by dividing the entiresearch space deterministically in is thedepth parameter andevaluationsas,where,theoverall complexity of the algorithm is,each solutioncan be compared with every other solution in the population tofind if it is dominated.This requiresis the number of objectives.When thisprocess is continued to find all members of the first nondomi-nated level in the population,the total complexity isfront,the solutions of the first front are discounted temporarily and the above procedure is repeated.In the worst case,the task of finding the second front alsorequiresfronts and there exists onlyone solution in each front.This requires anoverall computations.Notethat,and 2)dom-inates.This requireswith,we visit each member (the domi-nation count becomes zero,we put it in a separate list and thethird front is identified.This process continues until all fronts are identified.For each solution.Thus,each solutiontimes before its domination count becomes zero.At this point,the solution is assigned a nondomination level and will never be visited again.Since there are at most.Thus,the overall complexity of the procedureistimes as each individual can be the memberof at most one front and the second inner loop (for eachtimes for each individual[each individual dominatescomparisons]resultsin the overall.B.Diversity PreservationWe mentioned earlier that,along with convergence to the Pareto-optimal set,it is also desired that an EA maintains a good spread of solutions in the obtained set of solutions.The original NSGA used the well-known sharing function approach,which has been found to maintain sustainable diversity in a popula-tion with appropriate setting of its associated parameters.The sharing function method involves a sharing parameter--for each if thenAddelse if thenifthenUsed to store the members of the next frontfor eachif then比pFig.1.Crowding-distance calculation.Points marked in filled circles are solutions of the same nondominated front.2)Since each solution must be compared with all other so-lutions in the population,the overall complexity of thesharing function approachis.In the proposed NSGA-II,we replace the sharing function approach with a crowded-comparison approach that eliminates both the above difficulties to some extent.The new approach does not require any user-defined parameter for maintaining diversity among population members.Also,the suggested ap-proach has a better computational complexity.To describe this approach,we first define a density-estimation metric and then present the crowded-comparison operator.1)Density Estimation:To get an estimate of the density of solutions surrounding a particular solution in the population,we calculate the average distance of two points on either side of this point along each of the objectives.Thisquantityth solution inits front (marked with solid circles)is the average side length of the cuboid (shown with a dashed box).The crowding-distance computation requires sorting the pop-ulation according to each objective function value in ascending order of magnitude.Thereafter,for each objective function,the boundary solutions (solutions with smallest and largest function values)are assigned an infinite distance value.All other inter-mediate solutions are assigned a distance value equal to the ab-solute normalized difference in the function values of two adja-cent solutions.This calculation is continued with other objective functions.The overall crowding-distance value is calculated as the sum of individual distance values corresponding to each ob-jective.Each objective function is normalized before calculating the crowding distance.The algorithm as shown at the bottom of the page outlines the crowding-distance computation procedure of all solutions in an nondominatedsetth objective function value of the and theparametersand arethe maximum and minimum values oftheindependent sortings of atmost)are in-volved,the above algorithmhasare assigned adistance metric,we can compare two solutions for their extent of proximity with other solutions.A solution with a smaller value of this distance measure is,in some sense,more crowded by other solutions.This is exactly what we compare in the proposed crowded-comparison operator,described below.Although Fig.1illustrates the crowding-distance computation for two objectives,the procedure is applicable to more than two objectives as well.2)Crowded-Comparison Operator:The crowded-compar-ison operator(in the populationhas two attributes:1)nondomination rank();2)crowding distance(ifandThat is,between two solutions with differing nondomination ranks,we prefer the solution with the lower (better)rank.Other-wise,if both solutions belong to the same front,then we prefer the solution that is located in a lesser crowded region.With these three new innovations—a fast nondominated sorting procedure,a fast crowded distance estimation proce-dure,and a simple crowded comparison operator,we are now ready to describe the NSGA-II algorithm.C.Main LoopInitially,a random parentpopulation.Sinceelitism--foreachtois introduced by comparing current population with previously found best nondominated solutions,the procedure is different after the initial generation.We first describethe,we definitely choose all members of thesetpopulation members,we sort the solutionsof the lastfrontis now used for se-lection,crossover,and mutation to create a newpopulationofsize;2)crowding-distance assignmentis;3)sortingon.The overall complexity of the algorithmismem-bersin----th nondominated front in the parentpopsort in descending orderusing-use selection,crossover and mutation to createa newpopulationTABLE IT EST P ROBLEMS U SED IN T HIS STUDYAll objective functions are to be minimized.A.Test ProblemsWe first describe the test problems used to compare different MOEAs.Test problems are chosen from a number of signifi-cant past studies in this area.Veldhuizen [22]cited a number of test problems that have been used in the past.Of them,we choose four problems:Schaffer’s study (SCH)[19],Fonseca and Fleming’s study (FON)[10],Poloni’s study (POL)[16],and Kursawe’s study (KUR)[15].In 1999,the first author suggested a systematic way of developing test problems for multiobjec-tive optimization [3].Zitzler et al.[25]followed those guide-lines and suggested six test problems.We choose five of those six problems here and call them ZDT1,ZDT2,ZDT3,ZDT4,and ZDT6.All problems have two objective functions.None of these problems have any constraint.We describe these prob-lems in Table I.The table also shows the number of variables,their bounds,the Pareto-optimal solutions,and the nature of the Pareto-optimal front for each problem.All approaches are run for a maximum of 25000function evaluations.We use the single-point crossover and bitwisemutation for binary-coded GAs and the simulated binary crossover (SBX)operator and polynomial mutation [6]forreal-coded GAs.The crossover probabilityofand a mutation probabilityofis the number of decision variables for real-coded GAsandequalto four and an archivesizeFig.3.Distance metric7.nondominated solutions of the combined GA and external populations at the final generation to calculate the performance metrics used in this study.For PAES,SPEA,and binary-coded NSGA-II,we have used30bits to code each decision variable.B.Performance MeasuresUnlike in single-objective optimization,there are two goals in a multiobjective optimization:1)convergence to the Pareto-op-timal set and2)maintenance of diversity in solutions of the Pareto-optimal set.These two tasks cannot be measured ade-quately with one performance metric.Many performance met-rics have been suggested[1],[8],[24].Here,we define two per-formance metrics that are more direct in evaluating each of the above two goals in a solution set obtained by a multiobjective optimization algorithm.The firstmetricchosen solutions on the Pareto-optimal front.The averageof these distances is used as the firstmetricchosen solutions on the Pareto-op-timal front for the calculation of the convergence metric and so-lutions marked with dark circles are solutions obtained by analgorithm.The smaller the value of this metric,the better theconvergence toward the Pareto-optimal front.When all obtainedsolutions lie exactlyonandvariance of this metric calculated for solutionsets obtained in multiple runs.Even when all solutions converge to the Pareto-optimal front,the above convergence metric does not have a value of zero.Themetric will yield zero only when each obtained solution lies ex-actly on each of the chosen solutions.Although this metricaloneFig.4.Diversity metric1.can provide some information about the spread in obtained so-lutions,we define an different metric to measure the spread insolutions obtained by an algorithm directly.The secondmetricof these distances.Thereafter,from the obtained set of non-dominated solutions,we first calculate the extreme solutions(inthe objective space)by fitting a curve parallel to that of the truePareto-optimal front.Then,we use the following metric to cal-culate the nonuniformity in thedistribution:is the average of alldistances,assuming that therearesolutions,there aresolutions lie on one solution.It is interesting tonote that this is not the worst case spread of solutions possible.We can have a scenario in which there is a large varianceinand wouldmakewould be zero,making the metric to take a valuezero.For any other distribution,the value of the metric would begreater than zero.For two distributions having identical valuesof takes a higher value with worse distri-butions of solutions within the extreme solutions.Note that theabove diversity metric can be used on any nondominated set ofsolutions,including one that is not the Pareto-optimal ingTABLE IIM EAN(F IRST R OWS)AND V ARIANCE(S ECOND R OWS)OF THE C ONVERGENCE M ETRIC7TABLE IIIM EAN(F IRST R OWS)AND V ARIANCE(S ECOND R OWS)OF THE D IVERSITY M ETRIC1a triangularization technique or a V oronoi diagram approach[1]to calculateobtained using four algorithms NSGA-II(real-coded),NSGA-II(binary-coded),SPEA,and PAES.NSGA-II(real coded or binary coded)is able to convergebetter in all problems except in ZDT3and ZDT6,where PAESfound better convergence.In all cases with NSGA-II,the vari-ance in ten runs is also small,except in ZDT4with NSGA-II(binary coded).The fixed archive strategy of PAES allows betterconvergence to be achieved in two out of nine problems.Table III shows the mean and variance of the diversity metricFig.7.Nondominated solutions with SPEA onKUR.Fig.8.Nondominated solutions with NSGA-II (binary-coded)on ZDT2.In both aspects of convergence and distribution of solutions,NSGA-II performed better than SPEA in this problem.Since SPEA could not maintain enough nondominated solutions in the final GA population,the overall number of nondominated solutions is much less compared to that obtained in the final population of NSGA-II.Next,we show the nondominated solutions on the problem ZDT2in Figs.8and 9.This problem has a nonconvex Pareto-op-timal front.We show the performance of binary-coded NSGA-II and SPEA on this function.Although the convergence is not a difficulty here with both of these algorithms,both real-and binary-coded NSGA-II have found a better spread and more solutions in the entire Pareto-optimal region than SPEA (the next-best algorithm observed for this problem).The problem ZDT4has21here],that study clearly showed that a population of size of about at least 500is needed for single-objective binary-coded GAs (with tournament selection,single-point crossover and bitwise mutation)to find the global optimum solution in more than 50%of the simulation runs.Since we have used a population of size 100,it is not expected that a multiobjective GA would find the global Pareto-optimal solution,but NSGA-II is able to find a good spread of solutions even at a local Pareto-optimal front.Since SPEA converges poorly on this problem (see Table II),we do not show SPEA results on this figure.Finally,Fig.11shows that SPEA finds a better converged set of nondominated solutions in ZDT6compared to any other algorithm.However,the distribution in solutions is better with real-coded NSGA-II.D.Different Parameter SettingsIn this study,we do not make any serious attempt to find the best parameter setting for NSGA-II.But in this section,we per-Fig.11.Real-coded NSGA-II finds better spread of solutions than SPEA on ZDT6,but SPEA has a better convergence.TABLE IVM EAN AND V ARIANCE OF THE C ONVERGENCE AND D IVERSITY M ETRICSUP TO 500GENERATIONSform additional experiments to show the effect of a couple of different parameter settings on the performance of NSGA-II.First,we keep all other parameters as before,but increase the number of maximum generations to 500(instead of 250used before).Table IV shows the convergence and diversity metrics for problems POL,KUR,ZDT3,ZDT4,and ZDT6.Now,we achieve a convergence very close to the true Pareto-optimal front and with a much better distribution.The table shows that in all these difficult problems,the real-coded NSGA-II has converged very close to the true optimal front,except in ZDT6,which prob-ably requires a different parameter setting with NSGA-II.Par-ticularly,the results on ZDT3and ZDT4improve with genera-tion number.The problem ZDT4has a number of local Pareto-optimalfronts,each corresponding to particular valueof.A large change in the decision vector is needed to get out of a local optimum.Unless mutation or crossover operators are capable of creating solutions in the basin of another better attractor,the improvement in the convergence toward the true Pareto-op-timal front is not possible.We use NSGA-II (real-coded)with a smaller distributionindexand diversitymeasureFig.12.Obtained nondominated solutions with NSGA-II on problem ZDT4.These results are much better than PAES and SPEA,as shown in Table II.To demonstrate the convergence and spread of so-lutions,we plot the nondominated solutions of one of the runs after 250generations in Fig.12.The figure shows that NSGA-II is able to find solutions on the true Pareto-optimal frontwith.V .R OTATED P ROBLEMSIt has been discussed in an earlier study [3]that interactions among decision variables can introduce another level of dif-ficulty to any multiobjective optimization algorithm including EAs.In this section,we create one such problem and investi-gate the working of previously three MOEAs on the following epistatic problem:minimizewhere,but the aboveobjective functions are defined in terms of the variablevector by a fixed rotationmatrixFig.13.Obtained nondominated solutions with NSGA-II,PAES,and SPEA on the rotated problem.within the prescribed variable bounds,we discourage solutionswithresulting].This example problem demonstrates that one of the known dif-ficulties(the linkage problem[11],[12])of single-objective op-timization algorithm can also cause difficulties in a multiobjec-tive problem.However,more systematic studies are needed toamply address the linkage issue in multiobjective optimization.VI.C ONSTRAINT H ANDLINGIn the past,the first author and his students implemented apenalty-parameterless constraint-handling approach for single-objective optimization.Those studies[2],[6]have shown howa tournament selection based algorithm can be used to handleconstraints in a population approach much better than a numberof other existing constraint-handling approaches.A similar ap-proach can be introduced with the above NSGA-II for solvingconstrained multiobjective optimization problems.A.Proposed Constraint-Handling Approach(ConstrainedNSGA-II)This constraint-handling method uses the binary tournamentselection,where two solutions are picked from the populationand the better solution is chosen.In the presence of constraints,each solution can be either feasible or infeasible.Thus,theremay be at most three situations:1)both solutions are feasible;2)one is feasible and other is not;and3)both are infeasible.For single objective optimization,we used a simple rule for eachcase.Case1)Choose the solution with better objective functionvalue.Case2)Choose the feasible solution.Case3)Choose the solution with smaller overall constraintviolation.Since in no case constraints and objective function values arecompared with each other,there is no need of having any penaltyparameter,a matter that makes the proposed constraint-handlingapproach useful and attractive.In the context of multiobjective optimization,the latter twocases can be used as they are and the first case can be resolved byusing the crowded-comparison operator as before.To maintainthe modularity in the procedures of NSGA-II,we simply modifythe definition of domination between two solutions.Definition1:A solution,if any of the following conditions is true.1)Solution is not.2)Solutions are both infeasible,but solutionand dominatessolutionTABLE VC ONSTRAINED T EST P ROBLEMS U SED IN T HIS S TUDYAll objective functions are to be minimized.Three different nondominated rankings of the population arefirst performed.The first ranking is performed using-di-mensional vector.The second ranking is performedusing only the constraint violation values of all(Fig.14.Obtained nondominated solutions with NSGA-II on the constrained problemCONSTR.Fig.15.Obtained nondominated solutions with Ray-Tai-Seow’s algorithm on the constrained problem CONSTR.can be maintained for a large number of generations.However,in each case,we obtain a reasonably good spread of solutions as early as 200generations.Crossover and mutation probabilities are the same as before.Fig.14shows the obtained set of 100nondominated solu-tions after 500generations using NSGA-II.The figure shows that NSGA-II is able to uniformly maintain solutions in both Pareto-optimal region.It is important to note that in order to maintain a spread of solutions on the constraint boundary,the solutions must have to be modified in a particular manner dic-tated by the constraint function.This becomes a difficult task of any search operator.Fig.15shows the obtained solutions using Ray-Tai-Seow’s algorithm after 500generations.It is clear that NSGA-II performs better than Ray–Tai–Seow’s algorithm in terms of converging to the true Pareto-optimal front and also in terms of maintaining a diverse population of nondominated solutions.Next,we consider the test problem SRN.Fig.16shows the nondominated solutions after 500generations usingNSGA-II.Fig.16.Obtained nondominated solutions with NSGA-II on the constrained problemSRN.Fig.17.Obtained nondominated solutions with Ray–Tai–Seow’s algorithm on the constrained problem SRN.The figure shows how NSGA-II can bring a random population on the Pareto-optimal front.Ray–Tai–Seow’s algorithm is also able to come close to the front on this test problem (Fig.17).Figs.18and 19show the feasible objective space and the obtained nondominated solutions with NSGA-II and Ray–Tai–Seow’s algorithm.Here,the Pareto-optimal region is discontinuous and NSGA-II does not have any difficulty in finding a wide spread of solutions over the true Pareto-optimal region.Although Ray–Tai–Seow’s algorithm found a number of solutions on the Pareto-optimal front,there exist many infeasible solutions even after 500generations.In order to demonstrate the working of Fonseca–Fleming’s constraint-han-dling strategy,we implement it with NSGA-II and apply on TNK.Fig.20shows 100population members at the end of 500generations and with identical parameter setting as used in Fig.18.Both these figures demonstrate that the proposed and Fonseca–Fleming’s constraint-handling strategies work well with NSGA-II.。
基于改进的RRT^()-connect算法机械臂路径规划
随着时代的飞速发展，高度自主化的机器人在人类社会中的地位与作用越来越大。
而机械臂作为机器人的一个最主要操作部件，其运动规划问题，例如准确抓取物体，在运动中躲避障碍物等，是现在研究的热点，对其运动规划的不断深入研究是非常必要的。
机械臂的运动规划主要在高维空间中进行。
RRT （Rapidly-exploring Random Tree）算法[1]基于随机采样的规划方式，无需对构型空间的障碍物进行精确描述，同时不需要预处理，因此在高维空间被广为使用。
近些年人们对于RRT算法的研究很多，2000年Kuffner等提出RRT-connect算法[2]，通过在起点与终点同时生成两棵随机树，加快了算法的收敛速度，但存在搜索路径步长较长的情况。
2002年Bruce等提出了ERRT（Extend RRT）算法[3]。
2006年Ferguson等提出DRRT （Dynamic RRT）算法[4]。
2011年Karaman和Frazzoli提出改进的RRT*算法[5]，在继承传统RRT算法概率完备性的基础上，同时具备了渐进最优性，保证路径较优，但是会增加搜索时间。
2012年Islam等提出快速收敛的RRT*-smart算法[6]，利用智能采样和路径优化来迫近最优解，但是路径采样点较少，使得路径棱角较大，不利于实际运用。
2013年Jordan等通过将RRT*算法进行双向搜索，提出B-RRT*算法[7]，加快了搜索速度。
同年Salzman等提出在下界树LBT-RRT中连续插值的渐进优化算法[8]。
2015年Qureshi等提出在B-RRT*算法中插入智能函数提高搜索速度的IB-RRT*算法[9]。
同年Klemm等结合RRT*的渐进最优和RRT-connect的双向搜基于改进的RRT*-connect算法机械臂路径规划刘建宇，范平清上海工程技术大学机械与汽车工程学院，上海201620摘要：基于双向渐进最优的RRT*-connect算法，对高维的机械臂运动规划进行分析，从而使规划过程中的搜索路径更短，效率更高。
基于VB的伸缩臂式擦窗机工作臂有限元分析
基于VB的伸缩臂式擦窗机工作臂有限元分析郭海军，贾莉（长安大学工程机械学院，陕西西安710064）［摘要］运用ANSYS软件对擦窗机伸缩臂进行三维建模、装配及有限元分析，得到最大应力和最大位移等数据；再使用VB对有限元分析过程进行封装，即用VB界面实现可视化，用户只需在VB界面中选择尺寸、材料性能和载荷参数等，系统自动调用ANSYS计算程序，进行网格划分、加载以及求解，最终将计算结果显示在VB界面中。
［关键词］擦窗机；伸缩臂；有限元分析；VB［中国分类号］TU976+.42 ［文献标识码］A ［文章编号］1001-554X（2016）06-0057-04Finite element analysis for working arm of BMU based on VBGUO Hai-jun，JIA Li以VB（Visual Basic）可视化、面向对象的编程语言为开发工具，可以设计用户界面友好的擦窗机设计计算系统，提高了设计质量和效率，缩短产品开发周期，降低生产成本，提高产品的市场竞争力。
本文运用ANSYS对擦窗机伸缩臂进行应力应变分析以及对可靠性和安全性进行验证，使用VB 通过其内部shell函数实现对Windows应用程序的调用，利用VB向后台ANSYS程序写入APDL文件，控制ANSYS实现对擦窗机工作臂结构的交互式、参数化设计。
1 伸缩臂式擦窗机结构分析1.1 结构分析擦窗机是用于建筑物或构筑物窗户和外墙清洗、维修等作业的常设悬吊接近设备，是一种将作业人员、工具和材料等通过作业平台运送到空中指定位置进行各种安装、维修等作业的高空作业设备。
主要由悬挂吊船、悬挂装置、起升机构等组成，如图1所示。
按照臂的形式可分为直臂式、俯仰式、伸缩臂式等。
1.2 计算载荷工况（1）载荷工况1。
工作状态的载荷考虑：①总悬挂载荷（包括钢丝绳和电缆总重）；②各部分重量；③悬挂装置、吊篮和物料承受的风载；④运动部件的惯性力。
WB-Mech_120_WS_08
ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.ຫໍສະໝຸດ b.WS8.1-12
Workshop Supplement
February 23, 2009 Inventory #002594
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February 23, 2009 Inventory #002594
WS8.1: Results Processing
Workshop 8.1 – Project Schematic
Workshop Supplement
• Open the Project page.
• From the Units menu verify:
– See the contact description on the next page.
• The inlet pipe and bracket are modeled using structural steel while the expansion chamber is Polyethylene.
a.
ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.
WS8.1-9
b.
February 23, 2009 Inventory #002594
WS8.1: Results Processing
Workshop 8.1 - Environment
10. Apply the pressure load to the pipe:
a. Highlight the “Static Structural (A5) branch.
Mastercam9.1后处理-FANUC-数控铣
# --------------------------------------------------------------------------
# This post supports Generic Fanuc code output for 3 and 4 axis milling.
# Post Name : MPFAN
# Product : MILL
# Machine Name : GENERIC FANUC
# Control Name : GENERIC FANUC
# Description : GENERIC FANUC MILL POST
# the position of the rotary axis. This is the default setting.
#
#3 Axis Rotary (Polar)
# Polar positioning is offered in Mastercam 3 axis toolpaths through the
# following functions to enable/disable.
# Cantext value:
# 1 = Stop = output the "M00" stop code
# 2 = Ostop = output the "M01" optional stop code
# CNC 07/02/01 - Add cantext to cancel drill and tool retract, jph
# CNC 01/09/02 - Initial post update for V9.0, rjm
09 WB-Mech_120_NL_WS05B
WS5B-9
Workbench Mechanical – Structural Nonlinearities
…作业 5B: 局部p Supplement
• 双击 或 RMB=>Edit…) Model 打开 FE模型 双击(或 模型(Mechanical Session) 模型
WS5B-15
Workbench Mechanical – Structural Nonlinearities
…作业 5B: 局部屈服 作业
• 对等效应力进行后处理并和作业 的进行比较. 对等效应力进行后处理并和作业4A的进行比较 的进行比较
Workshop Supplement
WS5B-16
Tip
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模型描述 • 注射成型机械喷嘴尖端装配 注射成型机械喷嘴尖端装配2D 轴对称模型 • 材料 材料:
– 机架为钢 机架为钢(760Mpa 屈服强度 屈服强度) – 尖端铜合金 尖端铜合金(280Mpa屈服强度 屈服强度) 屈服强度 – 绝热钛
Insulator
Housing
WS5B-2
Workbench Mechanical – Structural Nonlinearities
作业 5B: 局部屈服
模型描述 (续) 续 • 接触对 接触对:
A. 机架对喷嘴尖端的无摩擦接触 B. 喷嘴尖端对绝热器的绑定接触 C. 机架和喷嘴尖端间螺纹连接处(螺 机架和喷嘴尖端间螺纹连接处 螺 纹细节没有建模) 纹细节没有建模 的绑定接触
Workbench- Mechanical 结构非线性
WS5B-1
Workbench Mechanical – Structural Nonlinearities
马波斯P7WB安装手册-中文
马波斯担负社会责任去不断支持每一种形式的改革创新，从而降低或最小化任何对人类 健康和环境有负面影响的风险。
因此马波斯自愿实行从我们的产品中进一步去除受管制物质。在马波斯这样做的同时， 还一直秉承着一个清晰的理念：任何在保护健康和环境上取得的改善不应该为了产品质量和可 靠性而妥协。
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2. 技术规格 ................................................................................................. 错误!未定义书签。 2.1. 存放和工作的环境条件 ...................................................................... 错误!未定义书签。 2.1.1. 污染介质 ............................................................................................... 错误!未定义书签。 2.1.2. 高度....................................................................................................... 错误!未定义书签。 2.2. 日常保养 ........................................................................................... 错误!未定义书签。 2.2.1. 电子装置的清洁 ..................................................................................... 错误!未定义书签。 2.3. 数据备份电池 .................................................................................... 错误!未定义书签。
Freescale Kinetis E 系列 KE06 微控制器开发板用户指南说明书
Freescale Semiconductor User’s Guide1FRDM-KE06 hardware overviewThe FRDM-KE06 Freedom development platform microcontroller board is assembled with the following features:•Kinetis E series KE06 MCU in an 80-pin LQFP package•On-board serial and debug adapter (OpenSDA)•I/O headers for easy access to MCU I/O pins •Freescale inertial sensor, MMA8451Q •Reset push button •RGB LED•Infrared communication •One thermistor•Motor control function for simple BLDC motor control on APMOTOR56F8000E •CAN communication Figure 1 shows a block diagram of the FRDM-KE06 board.Document Number:FRDMKE06UGRev. 0, 03/2014Contents1.FRDM-KE06 hardware overview . . . . . . . . . . . . . . . . 12.FRDM-KE06 hardware description . . . . . . . . . . . . . . 23.Serial and Debug Adapter (OpenSDA) . . . . . . . . . . . 4Freedom KE06 User’s Guideby: Freescale Semiconductor, Inc.FRDM-KE06 hardware descriptionFigure1. FRDM-KE06 block diagramThe FRDM-KE06 features two microcontrollers, the target MCU and a serial and debug adapter (OpenSDA) MCU. The target MCU is a Kinetis E series KE06 family device. The OpenSDA MCU is a Kinetis K series K20 family device, the K20DX128VFM5.2FRDM-KE06 hardware description2.1Power supplyThe FRDM-KE06 offers a design with multiple power supply options. It can be powered from the USB connector, the V IN pin on the I/O header, an off-board 1.71-3.6V supply from the 3.3V pin on the I/O header or 3.3V from motor control board. The USB and V IN supplies are regulated on-board using a 3.3V linear regulator to produce the main power supply. The other two sources are not regulated on-board. Note that KE06 on FRDM board can be powered by 5V or 3.3V.The following figure shows the schematic drawing for the power supply inputs and the on-board voltage regulator.FRDM-KE06 hardware descriptionFigure 2. FRDM-KE06 power supplyTable 1 provides the operational details and requirements for the power supplies.Note that the OpenSDA circuit is only operational when a USB cable is connected and supplying power to J6. However, the protection circuitry is in place to allow multiple sources to be powered at once.Table 1. Power supply requirementsSupply Source Valid RangeOpenSDA Operational?Regulated on-board?OpenSDA USB (J6)5VYes Yes V IN Pin on I/O header 4.3-9V Yes Yes 3.3V Pin on I/O header 1.71-3.6V Yes No 3.3V Pin on motor control header3.3VYesNoTable 2. Power suppliesPower Supply Name DescriptionVDD_PERIPHPeripheral power supply, including RGB LED, Key buttons, infrared, thermistor, reset circuit.J14 Pin1&2 connected, 3.3V power supply;J14 Pin2&3 connected, 5V power supply.Serial and Debug Adapter (OpenSDA)Notes:1.J9 and J10 are not populated by default on the production version. The two pins of these headers are shorted together by 0 ohm resistor R12 and R26 on the PCB. To measure the energyconsumption of either the KE06 or the OpenSDA MCU, the 0 ohm resistor between these pins must first be cut. A current probe or a shunt resistor and voltage meter can then be applied to measure the energy consumption on these rails. When the MCU current measurement is done, this 0 ohm resistor can be soldered on again.2.To better get ADC accuracy on KE06, it is recommended that a 0 ohm resistor R13 be soldered on. Ensure there is no power supply from P3V3_MOTOR and P3V3 sourced from I/O headers.3Serial and Debug Adapter (OpenSDA)OpenSDA is an open-standard serial and debug adapter. It bridges serial and debug communications between a USB host and an embedded target processor as shown in Figure 3.VDD_KE06KE06 MCU power supply.Header J9 provides a convenient means for KE06 energy consumption measurements.Header J14 for KE06 power supply selection: 3.3V or 5V.J14 Pin1&2 connected, 3.3V power supply,J14 Pin2&3 connected, 5V power supply.P3V3_SDAOpenSDA circuit power supply. Can be 3.3V only.Header J10 provides a convenient means for K20 energy consumption measurements.P5V_SDA INPUT [J6 Pin1] Input 5V Power supplied from the OpenSDA USB connector. P5-9V_VIN INPUT [J4 Pin16] Power supplied from the V IN pin of the I/O headers.P3V3_MOTOR INPUT [J2 Pin13] Input 3.3V power supplied from motor control header.P5V_USB OUTPUT[J4 Pin10] Output 5V to the I/O headers.Sourced from J6 USB (P5V_SDA) supply through a back drive protection Schottky diode.Table 2. Power suppliesSerial and Debug Adapter (OpenSDA)Figure3. OpenSDA block diagramOpenSDA is managed by a Kinetis K20 MCU built on the ARM® Cortex™-M4 core. The OpenSDA circuit includes a status LED (D4) and a reset pushbutton (SW1). The pushbutton asserts the reset signal to the KE06 target MCU. It can also be used to place the OpenSDA circuit into Bootloader mode by holding down the reset pushbutton while plugging the USB cable to USB connector J6. Once the OpenSDA enters bootloader mode, other OpenSDA applications such as debug app can be programmed. SPI and GPIO signals provide an interface to the SWD debug port of the KE06. Additionally, signal connections are available to implement a UART serial channel. The OpenSDA circuit receives power when the USB connector J6 is plugged into a USB host.3.1Debugging InterfaceSignals with SPI and GPIO capability are used to connect directly to the SWD of the KE06. These signals are also brought out to a standard 10-pin (0.05”) Cortex Debug connector (J7) as shown in Figure 4. It is possible to isolate the KE06 MCU from the OpenSDA circuit and use J7 to connect to an off-board MCU. To accomplish this, cut the 0 ohm resistor R58. This will disconnect the SWD_CLK pin to the KE06 so that it will not interfere with the communications to an off-board MCU connected to J7.When KE06 on FRDM board is 5V powered, and the OpenSDA is power off, there need to connect an external debugger to debug KE06 on board.Serial and Debug Adapter (OpenSDA)Figure4. SWD debug connector to KE063.2Virtual Serial PortA serial port connection is available between the OpenSDA MCU and UART1 pin PTC7 (TXD1) and PTC6 (RXD1) of KE06. Several of the default OpenSDA applications provided by Freescale, including the MSD Flash Programmer and the P&E Debug Application, provide a USB Communications Device Class (CDC) interface that bridges serial communications between the USB host and this serial interface on the KE06.3.3KE06 Microcontroller3.3.1Clock SourceThe Kinetis KE06 microcontrollers feature an on-chip oscillator compatible with two ranges of input crystal or resonator frequencies: 32 kHz (low frequency mode), 4-20 MHz (high frequency mode).The KE06 on the FRDM-KE06 is clocked from an 8 MHz crystal.3.3.2Serial PortThe serial port interface signals used with OpenSDA are UART1 pin PTC7 (TXD1) and PTC6 (RXD1). These signals are also connected to I/O header J1.3.3.3ResetThe PTA5/RESET signal on the KE06 is connected externally to a pushbutton SW1. The reset button can be used to force an external reset event in the target MCU. The reset button can also be used to force the OpenSDA circuit into bootloader mode when plugging the USB cable to J6. See Section3, “Serial and Debug Adapter (OpenSDA)” section for more details.Serial and Debug Adapter (OpenSDA)3.3.4DebugThe sole debug interface on all Kinetis E Series devices is a Serial Wire Debug (SWD) port. The primary controller of this interface on the FRDM-KE06 is the onboard OpenSDA circuit. However, a 2x5-pin (0.05”) Cortex Debug connector, J7, provides access to the SWD signals for the KE06 MCU. The following table shows SWD connector signals description for KE06:3.4ThermistorOne thermistor (RT1) is connected to two ADC inputs (PTF4/ADP12, PTF5/ADP13) of KE06 for evaluating the ADC module.Figure 5. Thermistor connectionTable 3. ARM JTAG/SWD mini Connector DescriptionPin FunctionConnection to KE061VTref 3.3V or 5V KE06 power supply (VDD_KE06)2SWDIO/TMS PTA4/SWD_DIO 3GNDGND4SWDCLK/TCK PTC4/SWD_CLK 5GND GND 6SWO/TDO NC 7NC NC 8TDI NC 9NC NC10RESETPTA5/RESETSerial and Debug Adapter (OpenSDA)3.5Infrared PortOne infrared Rx port and one Tx port (as shown in the following figure) are connected to ACMP0 input pin (ACMP0_IN1) and UART0 TXD0 pin of KE06 to demonstrate the capability of SCI0 modulated by a flextimer to generate infrared signals and use ACMP0 as a filter to receive the SCI data via infrared signal.Figure6. Infrared connection3.6Key buttonsTwo key buttons are connected to PTH3/4 to demonstrate KBI function of KE06, which can capture both falling edge and rising edge of key button input, as shown in the following figure.Serial and Debug Adapter (OpenSDA)Figure7. Key buttons connection3.7Three-axis accelerometerA Freescale MMA8451Q low power, three-axis accelerometer is interfaced through an I2C bus and two GPIO signals as shown in the following table. By default, the I2C address is 0x1D (SA0 pulled high).Table4. Accelerometer signal connectionsMMA8451Q KE06SCL PTA3SDA PTA2INT1 PTD4INT2 PTD3Serial and Debug Adapter (OpenSDA)Figure8. Accelerometer connection3.8RGB LEDThree PWM-capable pins are connected to a red, green, blue LED. The signal connections are shown in the table below.Table5. RGB LED Signal ConnectionsRGB LED KE06Red Cathode PTG5/FTM2CH3Green Cathode PTG6/FTM2CH4Blue Cathode PTG7/FTM2CH5Serial and Debug Adapter (OpenSDA)Figure9. GB LED connection3.9CANThe CAN phy on KE06 FRDM board is 3.3V powered.Table6. CAN signal connectionCAN Phy KE06D CAN_TXR CAN_RXSerial and Debug Adapter (OpenSDA)Figure10. CAN3.10Input/Output HeadersThe KE06 microcontroller is packaged in an 80-pin LQFP. Some pins are utilized in on-board circuitry, but many are directly connected to one of four I/O headers (J1, J2, J3, J4 and J5). J1 and J2 also function as motor control headers to provide access to a motor control board such as simple BLDC motor driving board APMOTOR56F8000E.Serial and Debug Adapter (OpenSDA)Figure11. I/O headers3.11Arduino CompatibilityThe I/O headers on the FRDM-KE06 are arranged to allow compatibility with peripheral boards (known as shields) that connect to Arduino and Arduino-compatible microcontroller boards. The pins on the headers share the same mechanical spacing and placement as the I/O headers on the Arduino Uno Revision 3 board design. See Figure11 for compatible signals.Document Number:FRDMKE06UG Rev. 003/2014Information in this document is provided solely to enable system and software implementers to use Freescale products. There are no express or implied copyright licenses granted hereunder to design or fabricate any integrated circuits based on the information in this document.Freescale reserves the right to make changes without further notice to any products herein. Freescale makes no warranty, representation, or guarantee regarding the suitability of its products for any particular purpose, nor does Freescale assume any liability arising out of the application or use of any product or circuit, and specifically disclaims any and all liability, including without limitation consequential or incidental damages. “Typical” parameters that may be provided in Freescale data sheets and/or specifications can and do vary in different applications, and actual performance may vary over time. All operating parameters, including “typicals,” must be validated for each customer application by customer’s technical experts. Freescale does not convey any license under its patent rights nor the rights of others. Freescale sells products pursuant to standard terms and conditions of sale, which can be found at the following address: /SalesTermsandConditions.How to Reach Us:Home Page:Web Support:/supportFreescale, the Freescale logo, and Kinetis, are trademarks of FreescaleSemiconductor, Inc., Reg. U.S. Pat. & Tm. Off. All other product or service names arethe property of their respective owners. ARM and Cortex are the registered trademarksof ARM Limited. ARMCortex-M4 is the trademark of ARM Limited.© 2014 Freescale Semiconductor, Inc.。
Journal of Electronic Imaging 13(3), 411–417 (July 2004).
Superquadric representation of automotive parts applying part decompositionYan ZhangAndreas KoschanMongi A.AbidiUniversity of TennesseeDepartment of Electrical and Computer Engineering334Ferris HallKnoxville,Tennessee37996-2100E-mail:yzhang@Abstract.Superquadrics are able to represent a large variety of objects with only a few parameters and a single equation.We present a superquadric representation strategy for automotive parts composed of3-D triangle meshes.Our strategy consists of two ma-jor steps of part decomposition and superquadricfitting.The origi-nalities of this approach include the following two features.First,our approach can represent multipart objects with superquadrics suc-cessfully by applying part decomposition.Second,superquadrics re-covered from our approach have the highest confidence and accu-racy due to the3-D watertight surfaces utilized.A novel,generic3-D part decomposition algorithm based on curvature analysis is also proposed.Experimental results demonstrate that the proposed part decomposition algorithm is able to segment multipart objects into meaningful single parts efficiently.The proposed superquadric rep-resentation strategy can then represent each individual part of the original objects with a superquadric model successfully.©2004 SPIE and IS&T.[DOI:10.1117/1.1762516]1IntroductionObject representation denotes representing real-world ob-jects with known graphic or mathematical primitives that can be recognized by computers.This research has numer-ous applications in areas including computer vision,com-puter graphics,and reverse engineering.An object can be represented by three levels of primitives in terms of the dimensional complexity:volumetric primitives,surface el-ements,and contours.The primitive selected to describe the object depends on the complexity of the object and the tasks involved.As the highest level primitives,volumetric primitives can better represent global features of an object with a significantly reduced amount of information com-pared with surface elements and contours.In addition, volumetric primitives have the ability to achieve the highest data compression ratio without losing the accuracy of the original data.The primarily used volumetric primitives in-clude generalized cylinders,geons,and superquadrics.1Su-perquadrics are a generalization of basic quadric surfaces and they can represent a large variety of shapes with only a few parameters and a single equation.An object initially represented by thousands of triangle meshes can be repre-sented by only a small set of superquadrics.This compact representation can be applied to object recognition to aid, for example,automated depalletizing of industrial parts or robot-guided bin picking of mixed nuclear waste in a haz-ardous environment.The quality control of both tasks can be enhanced by employing superquadrics.Furthermore,the registration of multiview data is indispensable to measure the size of partially occluded objects or their distances from each other in several image-based quality control tasks.Su-perquadrics can be used to efficiently register multiview range data of scenes with small overlap.2Most early research on superquadric representation con-centrated on representing single-part objects from single-view intensity or range images by assuming that the objects have been appropriately segmented.3–13This category of research focused on the data-fitting process,including ob-jective function selection,fitting criteria measurements,andPaper ORNL-007received Jul.30,2003;accepted for publication Feb.23,2004. 1017-9909/2004/$15.00©2004SPIE andIS&T.Fig.1Real range image of a multipart object obtained from Ref.18. Journal of Electronic Imaging13(3),411–417(July2004).Journal of Electronic Imaging/July2004/Vol.13(3)/411convergence analysis.For complex,multipart objects or scenes,there are two major types of approaches in the re-search literature.The first type of method incorporates an image segmentation step prior to the superquadric fitting.11–15The other type of method directly recovers su-perquadrics from a range image withoutpresegmentation.16–19Compared with superquadric repre-sentation of single-part object,these two types of methods can represent more complex objects and have wider appli-cations in related tasks including robotic navigation,object recognition,and virtual reality.However,existing super-quadric representation methods have several weaknesses.First,existing methods cannot handle arbitrary shapes or significant occlusions in the scene.Figure 1shows an ex-ample of the most complicated object that can be repre-sented by superquadrics appeared in the research literature.18We observe that the range image shown in Fig.1con-tains very few occlusions due to the simplicity of the ob-ject.In this case,an optimal viewpoint can easily be found from which each part of the object is visible.When an automotive part,i.e.,a complex,multipart object such as shown in Fig.2,is of interest,no existing methods can represent this object correctly because heavy occlusions are inevitable from any single viewpoint due to the complexity of the object.The second weakness of existing methods is that they utilize only single-view images.Again,for the automotive part shown in Fig.2͑a ͒,it is too difficult to find an optimal viewpoint from which all the parts are visible due to self-occlusions and occlusions,as shown in Fig.2͑b ͒.In addi-tion,the confidence of recovered superquadrics is low due to incomplete single-view data utilized and the accuracy of the recovered models highly depends on the viewpoint used to acquire the data.How complicated,multipart objects canbe represented by superquadrics with high confidence and accuracy remains unknown from the literature.In this paper,we propose an efficient strategy to repre-sent multipart objects with superquadrics.We also present a novel 3-D part decomposition algorithm based on curvature analysis to facilitate our superquadric representation strat-egy.Experiments are shown for automotive parts composed of 3-D triangulated surfaces.The remainder of this paper proceeds as follows.Section 2presents a superquadric representation approach for mul-tipart objects.Section 3proposes the 3-D part decomposi-tion algorithm for triangle meshes.The experimental results are presented in Sec.4and Sec.5concludes the paper.2Superquadric Representation of Multipart Objects Utilizing Part DecompositionA diagram for the proposed superquadric representation al-gorithm is illustrated in Fig.3.Beginning with a multipart object composed of triangle meshes,we propose a part de-composition algorithm to segment the meshes into single parts.Next,each single part is fitted with a superquadric model.Utilizing part decomposition,the difficult represen-tation problem of complicated objects is solved.We use 3-D triangulated surfaces reconstructed from multiview range images as input so that the recovered superquadrics have significantly higher confidence than those recovered from single-view images.In addition,our proposed algo-rithms are generic and flexible in the sense of triangle mesh handling ability since triangle meshes have been the stan-dard surface representation elements in many computer-related areas.A triangulation step is required only if un-structured 3-D point clouds areprovided.Fig.2Distributor cap:(a)photograph of the object,(b)rendering of 3-D triangulated surfaces scanned from view 1,and (c)rendering of 3-D triangulated surfaces scanned from view2.Fig.3Diagram of the proposed superquadric representation strategy utilizing part decomposition.Zhang,Koschan,and Abidi412/Journal of Electronic Imaging /July 2004/Vol.13(3)2.1Introduction to SuperquadricsA set of superquadrics with various shape factors is shown in Fig.4.The implicit definition of superquadrics is ex-pressed as 18F ͑x ,y ,z ͒ϭͫͩx a 1ͪ2/2ϩͩy a 2ͪ2/2ͬ2/1ϩͩz a 3ͪ2/1ϭ1,1,2͑0,2͒,͑1͒where (x ,y ,z )represents a surface point of the superquad-ric,(a 1,a 2,a 3)represent sizes in the (x ,y ,z )directions,and (1,2)represent shape factors.To represent a super-quadric model with global deformations in the world coor-dinate system,15parameters are needed.They are summa-rized as 18∧ϭ͑a 1,a 2,a 3,1,2,,,,p x ,p y ,p z ,k x ,k y ,␣,k ͒,͑2͒where the first 11parameters define a regular superquadric.Parameters k x and k y define the tapering deformations and ␣and k define the bending deformations.Most approaches define an objective function and find the superquadric pa-rameters through minimizing this objective function.The objective function used in this paper is 1G ͑∧͒ϭa 1a 2a 3͚i ϭ1N͓F 1͑x c ,y c ,z c ͒Ϫ1͔2.͑3͒The Levenberg-Marquardt method 20was implemented tominimize the objective function due to its stability and ef-ficiency.In addition,our superquadric fitting algorithm is able to recover superquadrics with global deformations from unstructured 3-D data points.3Curvature-Based 3-D Part DecompositionMany tasks in computer vision,computer graphics,and re-verse engineering involve objects or models.These tasks become extremely difficult when the object of interest is complicated,e.g.,it contains multiple parts.Part decompo-sition can simplify the original task performed on multipart objects into several subtasks each dealing with their con-stituent single,much simpler parts.21,22While a significant amount of research for part decomposition of 2-D intensity or 2.5-D range images has been conducted over the last 2decades,23–25little effort has been made on part segmenta-tion of 3-D data.26,27Therefore,a novel 3-D part decompo-sition algorithm is proposed in this paper.Figure 5illus-trates the difference between region segmentation and part decomposition.A scene consisting of a barrel on the floor is segmented into three surfaces by a region segmentation al-gorithm and two single-part objects by a part decomposi-tion algorithm.We can observe that the scene can be rep-resented by two superquadrics,which is consistent with the part decomposition result.Therefore,we conclude that part decomposition is more appropriate for high-level tasks such as volumetric primitives-based object representation and recognition.A diagram of the proposed part decomposition algorithm is shown in Fig.6.The proposed part decomposition consists of four major steps:Gaussian curvature estimation,boundary detection,region growing,and postprocessing.Boundaries between two articulated parts are composed of points with highly negative curvature according to the transversality regularity.21,22These boundaries are therefore detected by thresholding estimated curvatures for each vertex.A component-labeling operation is then performed to grow nonboundary vertices into parts.Finally,a postprocessing step is performed to assign nonlabeled vertices,including boundary vertices,to one of the parts and merge parts con-taining fewer vertices than a prespecified threshold into their neighbor parts.This part decomposition algorithm is summarized as follows.ˆAlgorithm 1…3-D part decomposition of triangle meshes …‰ˆInput:‰Triangulated surfaces.ˆStep 1.‰Compute Gaussian curvature for each vertex on the surface.ˆStep 2.‰Label vertices of highly negative curvatureasFig.4Superquadrics with various shapeparameters.Fig.6Diagram of the proposed 3-D part decomposition algorithm.Superquadric representation of automotive parts ...Journal of Electronic Imaging /July 2004/Vol.13(3)/413boundaries and the remaining vertices as seeds.ˆStep 3.‰Perform iterative region growing on each seed vertex.ˆStep 4.‰Assign nonlabeled vertices to parts and merge parts having less than a prespecified number of vertices into their neighboring parts.ˆOutput:‰Decomposed single parts.The major steps of this part decomposition algorithm are described respectively in the following sections.3.1Gaussian Curvature Estimation and BoundaryDetectionThe Gaussian curvature for each vertex on a triangulated surface is estimated by K ͑p ͒ϭ3͑2Ϫ͚i ϭ1N i ͒i ϭ1NA i␦2͑p Ϫp i ͒,͑4͒using the method proposed in Ref.28.Variable p representsthe point of interest,p i represents a neighboring vertex of the point p ,and A i represents the area of the triangle con-taining the point p .Variable i represents the interior angle of the triangle at p ,and ␦is the Dirac delta function.The triangles sharing the vertex p are illustrated in Fig.7.After Gaussian curvature is obtained for each vertex on the surface,a prespecified threshold is applied to label ver-tices as boundary or seed.Vertices of highly negative cur-vature are labeled as boundaries between two parts accord-ing to the transversality regularity,21while the rest are labeled as seeds.The threshold is critical and affects the performance of region growing.This threshold is deter-mined in a heuristic manner and depends on mesh resolu-tion.Two types of isolated vertices defined in this work according to their labels include:͑1͒a point that is labeled as boundary while all of its neighbors are labeled as seeds and ͑2͒a point that is labeled as a seed while all of its neighbors are labeled as boundary.The isolated vertices are removed by changing their labels to be the same as those of their neighbors.3.2Region Growing and PostprocessingAfter the vertices are labeled,a region-growing step is per-formed on each vertex labeled as seed.Figure 8showstriangle meshes around the point p .To illustrate the region growing process,a set of two-ring neighbor meshes around point p is shown in this figure.Region growing is performed as follows.Starting from a seed vertex p ,the region number 1is first assigned to the vertex.Second,all the neighbors p i initially labeled as seeds are then labeled with the same region number as the point p .The same labeling process is performed for each neighbor p i to label vertices p i j .This process terminates when the grown region is surrounded by boundary vertices,i.e.,the neighbors of the edge vertices of the region are all labeled as boundaries (Ϫ1).This process is repeated for every other vertex labeled as seed ͑0͒,but not for a vertex that has been grown and labeled with one of the region numbers (1,2,...,N ).After all the seed vertices are assigned new labels,a postprocessing is performed for each bound-ary vertex.Given a seed point x ,all its neighbors x i are first sorted in an ascending order based on their Euclidean distance to the point x .Next,a neighboring vertex x i ,which is the first point labeled with a region number ͑Ͼ0͒,is picked up.The boundary vertex x is then labeled the same as the vertex x i ,i.e.,the label of x is changed from Ϫ1to a region number (Ͼ0).Finally,with the exception of a few missing vertices,each vertex is labeled as 1,2,3,...,N ,the number of the parts.Missing vertices are usually located around boundaries between two articulated parts,and they are further assigned to parts during the post-processing step.Finally,a postprocessing step is performed to assign the nonlabeled vertices to parts.For example,the vertex p is an unlabeled vertex and needs further postprocessing.Assum-ing that p i (i ϭ1,2,...,N )represents a neighboring vertex of the point p ,the neighboring vertices are first selected if they have the same sign of curvature as that of the vertex p and belong to one of the segmented parts.Next,among those neighbor vertices,the vertex that has the smallest Euclidean distance to the vertex p is selected as a target vertex.For example,the vertex p 1is assumed to be the target vertex of the vertex p .Vertex p is assigned thesameFig.7Curvature estimation for the vertex p utilizing triangle meshinformation.Fig.8Region growing process for the vertex p .Zhang,Koschan,and Abidi414/Journal of Electronic Imaging /July 2004/Vol.13(3)Superquadric representation of automotive parts...Fig.5Region and part segmentation of a synthetic scene:(a)rendering of a synthetic scene consist-ing of a barrel on thefloor,(b)three segmented regions rendered in different colors,and(c)twocolors.decomposed parts rendered in differentview range images from the IVP Ranger system29and consists of37,171vertices and73,394tri-angles.The part decomposition results consist of two parts:(a)photograph of the original object,(b)rendering of the reconstructed mesh,(c)decomposed parts rendered in different colors,and(d)twocolors.recovered superquadrics rendered in differentmultiview range images from the IVP Ranger system29and consists of58,975vertices and117,036triangles.The part decomposition results consist of13parts:(a)photograph of the original object,(b)rendering of the reconstructed mesh,(c)decomposed parts rendered in different colors,and(d)colors.recovered superquadrics rendered in differentmultiview range images from the IVP Ranger system29and consists of58,784vertices and117,564triangles.The part decomposition results consist of nine parts:(a)photograph of the original object,(b)rendering of the reconstructed mesh,(c)decomposed parts labeled in different colors,and(d)recov-ered superquadrics rendered in different colors.Journal of Electronic Imaging/July2004/Vol.13(3)/415label as vertex p1,i.e.,the same segmented part.Further-more,parts composed of fewer vertices than a specified threshold are merged with adjacent regions.4Experimental ResultsExperimental results on superquadric representation for multipart,automotive objects including a disk brake,a dis-tributor cap,and a water neck are shown in this section. The meshes were reconstructed from multiview range im-ages scanned from the IVP Ranger System.29The recovered superquadrics were rendered in three dimensions using quad meshes.30Figure9shows the disk brake and its part decomposition and superquadric representation results.The reconstructed3-D triangulated surface shown in Fig.9͑b͒consists of37,171vertices and73,394triangles.Starting from this reconstructed mesh,our part decomposition algo-rithmfirst decomposed the disk brake into two single parts, as shown in Fig.9͑c͒.Each decomposed part was nextfit-ted to a superquadric model,as shown in Fig.9͑d͒.The decomposed parts and recovered superquadrics are ren-dered in different colors.We observe that our part decom-position algorithm successfully decomposed the disk brake into its constituent parts and the superquadric representa-tion strategy recovered correct superquadrics in terms of their size,shape,and pared to the original triangle mesh representation consisting of37,171vertices and73,394triangles,the recovered superquadrics describe the disk brake with only22parameters͑11parameters for each superquadric without global deformations͒.This low representation cost of superquadric representation can sig-nificantly benefit tasks including virtual reality,object rec-ognition,and robotic navigation.However,the hole at the center of the disk brake failed to be represented since su-perquadrics can only represent objects with genus of zero.19 Figure10shows the distributor cap and its part decom-position and superquadric representation results.The recon-structed mesh shown in Fig.10͑b͒consists of58,975ver-tices and117,036triangles and was decomposed into13 single parts,as shown in Fig.10͑c͒.We observe that this decomposition result is consistent with human perception. The recovered superquadrics shown in Fig.10͑d͒correctly represent the distributor cap.The recovered superquadric parameters and the ground truths for one of the small cyl-inders on top of the distributor cap are shown in Table1. We can observe that the recovered superquadric parameters for this cylinder have the correct size and shape informa-tion when compared with the ground truth parameters of the object.In addition,superquadrics represent the distribu-tor cap with only143floating numbers,while the original triangle mesh consists of58,975vertices and117,036tri-angles.Figure11shows the water neck and its part decomposi-tion and superquadric representation results.The recon-structed mesh shown in Fig.11͑b͒consists of58,784ver-tices and117,564triangles and was decomposed into nine single parts,as shown in Fig.11͑c͒.We observe that the decomposed parts are consistent with human perception. The recovered superquadrics shown in Fig.11͑d͒correctly represent the water neck.The recovered superquadric pa-rameters and the ground truths for the handle,the ball,and the small cylinder next to the handle of the water neck are shown in Table2.From this table,we observe that the recovered superquadric parameters have the correct size and shape information when compared with the ground truth parameters of the objects.Again,superquadrics repre-sent the water neck in a desirable accuracy with only99 parameters while the original triangle mesh consists of 58,784vertices and117,564triangles.5ConclusionsThis paper proposed a superquadric representation ap-proach for multipart objects.Superquadrics can represent objects in an acceptable accuracy with only a few param-eters,while other surface primitives and contours usually require thousands of representation elements.Such a com-pactness and low representation cost can significantly ben-efit tasks including virtual reality,object recognition,and robot navigation,e.g.,it enables these tasks to run in a real-time manner.The advantages of the proposed super-quadric representation approach include:͑1͒it can success-fully represent complicated,multipart objects byfirst de-composing them into single-part objects,and͑2͒the recovered superquadrics have the highest confidence and accuracy since the input we use are3-D triangulated sur-faces reconstructed from multiview range images.The in-completeness and ambiguities contained in single-view im-ages were eliminated during the multiview surface reconstruction process.We also proposed a3-D part de-composition algorithm to decompose compound objects represented by triangle meshes into their constituent single parts based on curvature analysis.Considering the fact that the triangle mesh has been a standard surface representation element in computer vision and computer graphics,the pro-posed part decomposition algorithm is generic,flexible,and can facilitate computer vision tasks such as shape descrip-tion and object recognition.Furthermore,the part decom-position algorithm can segment a large number of triangle meshes͑over100,000͒in only seconds on an SGI Octane workstation.Table1Recovered superquadric parameters and ground truths for one of the small cylinders shown in Fig.10(d)where the unit is millimeters.Parameters a1a2a312 Ground truths15.215.620.10.1 1.0 Superquadric parameters16.4515.6720.420.120.96Table2Recovered superquadric parameters and ground truths for the water neck shown in Fig.11(d)where the unit is millimeters. Object Parameters a1a2a312 Handle Ground truths39.739.417.60.1 1.0 Superquadric parameters40.2340.5866.830.130.98 Ball Ground Truths50.047.656.0 1.0 1.0 Superquadric parameters51.6247.5654.28 1.020.95 Cylinder Ground truths16.517.844.20.1 1.0 Superquadric parameters17.5617.9443.380.110.95Zhang,Koschan,and Abidi 416/Journal of Electronic Imaging/July2004/Vol.13(3)AcknowledgmentsThis work was supported by the University Research Pro-gram in Robotics under Grant No.DOE-DE-FG02-86NE37968,by the Department of Defense/U.S.Army Tank-automotive and Armaments Command/National Au-tomotive Center/Automotive Research Center Program R01-1344-18,and by the Federal Aviation Administration National Safe Skies Alliance Program R01-1344-48/49. 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Documentation:MAPP2500Ranger PCI System,Version1.6,Integrated Vision Products,Sweden͑2000͒.30.J.Wernecke,The Inventor Mentor:Programming Object-oriented3DGraphics with Open Inventor,Addison-Wesley,Reading,MA͑1994͒.Yan Zhang received her BS and MS de-grees in electrical engineering from Hua-zhong University of Science and Technol-ogy,China,in1994and1997,respectively,and her PhD degree in electrical engineer-ing from the University of Tennessee,Knoxville,in2003.Her research interestsinclude3-D image processing,computervision,and patternrecognition.Andreas Koschan received his MS de-gree in computer science and his PhD incomputer engineering from the TechnicalUniversity Berlin,Germany,in1985and1991,respectively.He is currently a re-search associate professor with the De-partment of Electrical and Computer Engi-neering,the University of Tennessee,Knoxville.His work has primarily focusedon color image processing and3-D com-puter vision including stereo vision and la-ser rangefinding techniques.He is a coauthor of two textbooks on 3-D image processing and a member of IS&T andIEEE.Mongi A.Abidi is a W.Fulton Professorwith the Department of Electrical and Com-puter Engineering,the University of Ten-nessee,Knoxville,which he joined in1986.Dr.Abidi received his MS and PhD degreesin electrical engineering in1985and1987,both from the University of Tennessee,Knoxville.His interests include image pro-cessing,multisensor processing,3-D imag-ing,and robotics.He has published over120papers in these areas and coedited the book Data Fusion in Robotics and Machine Intelligence(Academic Press,1992).He is the recipient of the1994to1995Chancellor’s Award for Excellence in Research and Creative Achievement and the2001Brooks Distinguished Professor Award.He is a member of the IEEE,the Computer Society,the Pattern Recognition Society, SPIE and the Tau Beta Pi,Phi Kappa Phi,and Eta Kappa Nu honor societies.He also received the First Presidential Principal Engineer Award prior to joining the University of Tennessee.Superquadric representation of automotive parts...Journal of Electronic Imaging/July2004/Vol.13(3)/417。
微型滚珠丝杠副摩擦力矩模型的建立与实验验证
微型滚珠丝杠副摩擦力矩模型的建立与实验验证陈勇将;汤文成【摘要】Being capable of predicting the friction torque of micro-type ball screw accurately allows the designer to correctly determine the energy loss and heat generation rate. Since the calculation of traction force is complex and requires much computer time, the forces and moments equilibrium are used for deriving the rolling element-race sliding force. Take into account the influence of spin sliding and helix angle, the final friction torque of micro-type ball screw can be calculated. The micro-type ball screw tests were conducted on a special machine for verification. It is found that good a-greement is held between the predictions of the new analytic torque model and the test results obtained with the micro-type ball screw torque rig. Analysis indicates that the performance parameters of lubricant and friction coefficient in ball-races contacts are the main factor to influence the friction torque of micro-type ball screws.%为了可以快速且准确地预测微型滚珠丝杠副的能量损耗与温升速率,建立了新的摩擦力矩模型.首先基于微型滚珠丝杠副的摩擦机理,利用力与力矩平衡的方法求得滚珠与滚道接触界面间的摩擦力,该方法比通过对接触区域上的切应力积分求解摩擦力的方法更简单方便,易于编程实现;其次考虑自旋滑动摩擦及螺旋升角的影响,推导出微型滚珠丝杠副的摩擦力矩模型;最后在微型滚珠丝杠副综合测试实验台上进行了实验验证.分析结果发现,在不同转速下,微型滚珠丝杠副摩擦力矩模型的预测值与实验值都有较好的吻合,表明摩擦力矩主要受润滑介质的性能参数及滚珠与滚道接触的滑动摩擦系数的影响.【期刊名称】《东南大学学报（自然科学版）》【年(卷),期】2011(041)005【总页数】5页(P982-986)【关键词】微型滚珠丝杠副;摩擦力矩;摩擦机理;滑动摩擦【作者】陈勇将;汤文成【作者单位】东南大学机械工程学院,南京,210096;东南大学机械工程学院,南京,210096【正文语种】中文【中图分类】TH132.1;TH117微型滚珠丝杠副的摩擦力矩是指各种摩擦因素对微型滚珠丝杠副运动构成的阻力矩，其带来的能量损耗降低了传动效率，引起的温升产生热变形影响了定位精度［1-2］.对于广泛应用于高精度小型机电一体化设备的微型滚珠丝杠副来说［3］，摩擦力矩模型的建立十分必要，可以快速且准确地预测能量损耗与温升速率.文献［4］建立了一个估计滚珠丝杠副摩擦力矩的模型，主要参考了滚动轴承建立摩擦力矩模型的方法，没有考虑自旋滑动摩擦及滚珠丝杠副独有的螺旋结构对摩擦力矩的影响，且没有进行实验验证.文献［5］将文献［4］建立的滚珠丝杠副摩擦力矩模型计算结果与文献［6］测量到的滚珠丝杠副摩擦特性数据进行了比较，发现文献［4］建立的摩擦力矩模型计算结果与实验数据存在较大差异.本文从微型滚珠丝杠副的摩擦机理出发，得出影响摩擦力矩的各种阻力与阻力矩;考虑自旋滑动摩擦及螺旋升角的影响，推导出适用于微型滚珠丝杠副的摩擦力矩模型;通过实验对模型进行验证.1 微型滚珠丝杠副的摩擦机理微型滚珠丝杠副中接触面虽为滚动界面，但滚动接触所引起的摩擦损耗很少，损耗主要来自于滚动过程中发生的滑动行为.这些滑动产生于滚珠与丝杠滚道接触界面间、滚珠与螺母滚道接触界面间、滚珠与返向器滚道接触界面间和滚珠与滚珠接触界面间(见图1)，其中滚珠与丝杠滚道、滚珠与螺母滚道及滚珠与返向器滚道接触界面间的滑动摩擦为损耗的主要来源.返向器是螺母的一部分，所以滚珠与返向器滚道接触可看作滚珠与螺母滚道接触的特例，将这3个接触界面间的滑动摩擦统称为滚珠与滚道接触界面间的滑动摩擦.图1 微型滚珠丝杠副的摩擦机理分解图1.1 滚珠与滚道接触产生的摩擦微型滚珠丝杠副中滚珠与滚道间的纯滚动与滑动一样也会发生能量损耗，主要是接触表面的应力循环引起的黏着损失和弹性滞后损耗.虽然滚动接触发生在离散的微凸体上，但由于滚动表面的微凸体沿着法向接近和分离，不像滑动接触那样沿着切线方向，因而滚动接触面积变化不大;滚动接触区的黏着力主要来自范德华力，黏着力将在滚动接触区末端以拉伸形式被分离，而不像滑动接触那样以剪切形式产生分离，所以滚动接触所引起的摩擦损耗较滑动摩擦相比很少.滚动摩擦的弹性滞后损耗源于消耗功与复原功之差，单个接触点处由弹性滞后所形成的摩擦力矩为［7］式中，Dw为滚珠直径;Qi为滚珠与滚道间的法向载荷;k为滚珠与滚道接触的椭圆率.滑动摩擦主要包括楔紧滑动、差动滑动和自旋滑动.楔紧滑动源于滚珠丝杠副的螺旋结构，是滚珠丝杠副特有的运动，而滚珠与滚道间的接触变形引起的差动滑动，仅在2条窄带上不发生滑动，同时由于接触角的存在，滚珠丝杠副中还将引起绕纯滚动点的自旋滑动.当滚珠与滚道接触区域上存在2条纯滚动线时，差动滑动摩擦力矩MCi和自旋滑动摩擦力矩MPi的计算公式为［8］式中，μ为滑动摩擦系数;l为接触椭圆的长半径;ri为滚珠与滚道间接触变形后的半径;fc，fp分别为差动滑动MCi和自旋滑动MPi计算系数.1.2 滚珠在返向器中产生的摩擦滚珠在返向器中的摩擦主要为滚珠进出返向器时碰撞力产生的摩擦和滚珠在返向器滚道内的滑动摩擦［9］.由于滚珠在返向器中不承受载荷，所以滚珠中心的线速度不高，相对于滚珠与丝杠滚道及滚珠与螺母滚道产生的滑动摩擦而言，这一部分摩擦很小，通常情况下可以忽略.碰撞力产生的摩擦与丝杠转速成正比，高速下这一部分摩擦较大［10-11］，但微型滚珠丝杠副的应用领域对转速要求不高，因此这一部分摩擦也不作考虑.1.3 润滑剂产生的阻力及滚珠间的滑动摩擦在流体润滑微型滚珠丝杠副中，滚珠必须克服润滑剂产生的黏性摩擦力.润滑剂产生的阻力取决于散布在滚珠丝杠副内部润滑剂的量.运用弹流润滑理论求得润滑剂引起的滚动阻力为［7］式中，E为等效弹性模量;Li为滚珠与滚道间接触点法向平面内的综合曲率半径;G 为材料参数;U无纲量速度参数;W为载荷参数.微型滚珠丝杠副的螺旋传动难以实现在滚珠间加上类似于轴承的保持架，所以滚珠间的滑动摩擦力无法避免，滚珠间摩擦产生的摩擦力矩可表示为式中，μb为滚珠间摩擦系数;FB为滚珠间摩擦力.2 摩擦力矩模型的建立2.1 滚珠与滚道接触的摩擦力微型滚珠丝杠副中滚珠与滚道接触的摩擦力FSi可以通过滚珠上力与力矩的平衡关系式求得.由微型滚珠丝杠副摩擦机理可知，作用于滚珠滚动方向的阻力除了滚珠与滚道间摩擦力FSi外，还有滚珠间接触力FB及润滑剂引起的滚动阻力FRi和压力FPi;而作用于滚珠上沿运动方向绕滚珠球心的阻力矩主要为差动滑动摩擦力矩MCi、弹性滞后阻力矩Mi及滚珠间摩擦力矩MB(见图2)，由力和力矩平衡可得式中，FSS为滚珠与丝杠滚道间的摩擦力;FSN为滚珠与螺母滚道间的摩擦力;FPN 为滚珠与螺母滚道间的润滑剂引起的压力;FPS为滚珠与丝杠滚道间的润滑剂引起的压力;MCN为滚珠与螺母滚道间的差动滑动摩擦力矩;MCS为滚珠与丝杠滚道间的差动滑动摩擦力矩;MN为滚珠与螺母滚道间的弹性滞后阻力矩;MS为滚珠与丝杠滚道间的弹性滞后阻力矩.图2 单个滚珠所受力与力矩平衡关系图由式(6)和(7)解得FSS和FSN的表达式为润滑剂作用于滚珠球心的压力FP可以表示成以FR为变量的函数，即式中，d0为微型滚珠丝杠副的公称直径;aS为滚珠与丝杠滚道的接触角;aN为滚珠与螺母滚道的接触角.将式(10)和(11)代入式(8)和(9)可得滚珠与滚道接触界面间摩擦力的最终表达式为2.2 滚珠与滚道接触的摩擦力矩模型对于微型滚珠丝杠副中单个滚珠的摩擦力矩，除了包含滚珠与滚道间摩擦力与力臂之积外，还应将自旋滑动摩擦力矩MPi投影至旋转轴上(见图2)，其分量应为MPisinaicosβ，因此单个滚珠与滚道接触的摩擦力矩表达式为式中，RS为滚珠与丝杠滚道接触点至丝杠轴心的距离;RN为滚珠与螺母滚道接触点至丝杠轴心的距离;β为螺旋升角.因此，对于滚珠数为Z的微型滚珠丝杠副，其摩擦力矩最终表达式为3 摩擦力矩模型的实验验证3.1 试验设备及方法微型滚珠丝杠副摩擦力矩的测量是在微型丝杠综合测试实验台上进行的，测量实验台结构如图3所示.由安装在工作台上的驱动电机与滚珠丝杠副的一端连接带动旋转，另一端用尾座顶尖限制，通过摩擦力矩测量装置的悬臂与安装在螺母上连接杆相配合，来约束滚珠丝杠副的周向与轴向运动.当丝杠旋转时，螺母旋转运动被限制只能带动摩擦力矩测量装置在滑动导轨上作轴向运动，在此方向上由气压缸施加轴向载荷给滚珠丝杠副，此轴向力的大小由拉压式力传感器测得.而限制周向运动的阻力由压力传感器测得，转换成数字信号后，乘以力臂因子即可得到所测量的微型滚珠丝杠副的摩擦力矩.图3 微型滚珠丝杠副综合测试实验台结构示意图试验产品为FF型微型滚珠丝杠副(南京工艺装备制造有限公司)，具体结构参数及性能参数如表1所示.被测试的微型滚珠丝杠副首先以400 r/min的转速跑合5 min后停转，然后保持轴向载荷恒定，再调节驱动电机转速从零增加到800r/min.在这期间测量螺母移动方向与轴向载荷方向相反时的摩擦力矩，为验证试验结果的准确性和可重复性，同一转速下重复测量多次.试验时测量参数的采集频率设置为100，力臂直径为38 mm.表1 FF1605型微型滚珠丝杠副结构与性能参数16公称导程/mm 5滚珠直径/mm 3.5弹性模量/GPa 210泊松比 0.3滚珠数 40动力黏度/GPa 0.04黏压系数/(nm·N－1)参数数值公称直径/mm 223.2 结果与讨论图4所示为FF1605型微型滚珠丝杠副在不同转速下测得的摩擦力矩值及其变化范围，摩擦力矩是在同一转速下螺母移动方向与轴向载荷方向相反时重复试验多次所取的平均值.由图4可知，在较低转速(n=400 r/min)下，摩擦力矩的变化幅度较大，随着转速的增加，摩擦力矩的变化速度有所放缓;微型滚珠丝杠副的摩擦力矩极不平稳，存在较大的波动，且波动范围的大小与转速没有明显关联，这主要是由于影响微型滚珠丝杠副的因素较多，因而摩擦力矩特性具有随机性和复杂性.图4 不同转速下微型滚珠丝杠副的摩擦力矩实验值及变化范围由不同转速下摩擦力矩实验值与模型值对比可知(见图5)，不同转速下微型滚珠丝杠副的摩擦力矩模型值与实验值都有较好的吻合.随转速的提高，摩擦力矩模型值一直以较大的速度上升，而实验值在n=400 r/min以下与模型值上升速度相近，尤其当转速为400 r/min时两值误差几乎为零.但随着转速进一步升高，实验值的上升速度慢于模型值，但上升趋势不变.图5 不同转速下微型滚珠丝杠副的摩擦力矩模型值与实验值对比图4 结论在拟静力学分析基础上，考虑了微型滚珠丝杠副螺旋升角及自旋滑动摩擦的影响，利用力与力矩平衡的方法，推导出了微型珠丝杠副摩擦力矩模型，实验证明所建立的新摩擦力矩模型基本符合实际情况.由依据微型滚珠丝杠副摩擦机理建立的新模型可知，润滑介质的性能参数及滚珠与滚道接触的滑动摩擦系数对摩擦力矩影响较大，实验也表明摩擦力矩的变化主要是转速影响了滚珠所受的润滑剂黏性阻力，因此要减小微型珠丝杠副摩擦力矩、提高摩擦力矩的平稳性，需要特别重视润滑的研究.参考文献(References)［1］ Brecher C，Witt S，Yagmur T.Infuence of oil additives on the wear behavior of bal screws［J］.Production Engineering，2009，3(3):323-327. ［2］ Wei C W，Lin J F.Kinematic analysis of the ball screw mechanism considering variable contact angles and elastic deformations［J］.Journalof Mechanical Design，2003，125(4):717-733.［3］严虎勤，闫笃政，田茂林.HJ微型滚珠丝杠副的设计与制造［J］.制造技术与机床，2006，4(9):109-110.Yan Huqin，Yan Duzheng，Tian Maolin.Design and manufacture of micro-type ball screw［J］.Manufacturing Technology ＆ Machine Tool，2006，4(9):109-110.(in Chinese)［4］ Olaru D，Puiu G C，Balan L C，et al.A new model to estimate friction torque in a ball screw system［C］//Advanced Computer Aided Design of Ecological Products.Brasov，Romania，2004:333-346.［5］苏益德.滚珠导螺杠之摩擦力分析研究［D］.中国台湾:中兴大学机械工程系，2009.［6］ Kamalzadeh A.Precision control of high speed ball screw drives ［D］.Waterloo，Canada:Department of Mechanical Engineering，University of Waterloo，2008.［7］ Houpert L.Ball bearing and tapered roller bearing torque:analytical，numerical and experimental results［C］//Proc of STLE AnnualMeeting.Houston，USA，2002:19-23.［8］ Houpert L.Numerical and analtical calculations in ball bearing［C］//Proc of Congres Roulements.Toulouse，France，1999:1-15.［9］Braccesi C，Landi L.A general elastic-plastic approach to impactanalysis for stress state limit evaluation in ball screw bearing return system ［J］.International Journal of Impact Engineering，2007，34(7):1272-1285. ［10］ Hung J P，Wu J S S，Chiu J Y.Impact failure analysis of re-circulating mechanism in ball screw ［J］.Engineering Failure Analysis，2004，11(4):561-573.［11］张佐营，宋现春，姜洪奎，等.滚珠丝杠副中滚珠与返向器的碰撞研究［J］.中国机械工程，2008，19(7):850-881.Zhang Zuoyin，Song Xianchun，Jiang Hongkui，et al.Research on the impact between balls and re-circulating mechanism in ball screw ［J］.China Mechanical Engineering，2008，19(7):850-881.(in Chinese)。
基于修正的G-K公式的组合机构自由度的计算
2023年第47卷第5期Journal of Mechanical Transmission基于修正的G-K公式的组合机构自由度的计算许兆棠（三江学院机械与电气工程学院，江苏南京210012）摘要用修正的Grübler-Kutzbach的机构自由度计算公式，计算组合机构的自由度，有时会产生错误的结果。
为了正确计算组合机构的自由度，并考虑计算组合机构自由度的方法便于工程技术人员学习和应用，根据修正的Grübler-Kutzbach的机构自由度计算公式，导出组合机构自由度的计算公式，得到机构自由度的一般计算公式；提出计算组合机构自由度的注意事项，给出了组合机构划分为基本机构的方法，并给出了算例。
修正的Grübler-Kutzbach的机构自由度计算公式只能用于1个基本机构的自由度计算及各个基本机构的公共约束相同的机构自由度计算；对于一般的组合机构，要用组合机构自由度的公式计算其自由度。
组合机构自由度等于各个基本机构自由度的和。
关键词组合机构自由度计算基本机构公共约束Calculation of Mobility on Combined Mechanisms Based on theModified G-K's FormulaXu Zhaotang（School of Mechanical and Electrical Engineering， Sanjiang University， Nanjing 210012， China）Abstract Using the modified Grübler-Kutzbach's formula of mechanism mobility, the calculation of the mobility of combined mechanism may produce wrong results. In order to correctly calculate the mobility of com⁃bined mechanism, and to consider that the calculation method of the mobility of combined mechanism is conve⁃nient for engineers for study and application, according to the modified Grübler-Kutzbach's formula of the mech⁃anism mobility, the calculation formula of mobility of combined mechanism is deduced. The general calculation formula of mobility of mechanism is obtained. The attentions of calculating mobility of combined mechanism are proposed. The method of dividing the combined mechanism into basic mechanisms is given. Numerical examples are given. The modified Grübler-Kutzbach's formula of mechanism mobility can only be used to calculate mobil⁃ity of a basic mechanism, and to calculate mobility of combined mechanism in that each basic mechanism has the same common constraint. The mobility of general combined mechanism should be calculated by the calcula⁃tion formula of mobility of combined mechanism. The mobility of combined mechanism is equal to the sum of the mobility of each basic mechanism.Key words Combined mechanism Mobility Calculation Basic mechanism Common constraint0 引言修正的Grübler-Kutzbach的机构自由度计算公式（简称修正的G-K公式）早已编入中外许多教科书中[1]115-116[2]42-48，并已为众多工程师所熟悉和运用。
基于ESO_的水下机器人机械臂系统鲁棒模型预测控制
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
0 0 0 0 0
λ =
Jλθ˙200 0001(4)
考虑建模误差引起的不确定性因素, 将参数矩
阵分成已知与未知部分, 即
Mγ = Mˆ γ + ∆M Cγ = Cˆ γ + ∆C Dγ = Dˆ γ + ∆ D
(5)
由式(3)与式(5)可以得到模型
因此, 文中旨在提出一种基于扩张状态观测器 (extended state observer, ESO)的 RMPC 方法, 以同 时解决条件约束与抗干扰问题。首先, 定义忽略 不确定因素和干扰的名义系统, 在名义系统的基 础上设计 RMPC 算法。 然后, 使用 ESO 来估计总 干扰并提前进行补偿, 以提高 UVMS的系统鲁棒 性和抗干扰能力。接着利用反馈线性化方法求出 作用于真实系统的控制器, 并进行稳定性分析。 最后, 通过仿真实验验证了其有效性。
ESO 的 RMPC 具有很好的轨迹跟踪性能和抗扰动能力。
关键词: 水下机器人机械臂系统; 鲁棒模型预测控制; 扩张状态观测器; 轨迹跟踪
中图分类号: TJ630.33; U664.82
文献标识码: A
文章编号: 2096-3920(2023)06-0827-12
DOI: 10.11993/j.issn.2096-3920.2022-0074
gv(qv) ]
gm(qm)
] ]
(2)
式中: q = [qv, qm]T包含水下机器人本体的位姿向 量 qv = [x, y, z, ψ]T和 2 个 机 械 臂 关 节 的 角 度 qm = [θ1, θ2]T; ψ为机器人的偏航角; M(q)为系统的惯性
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Workbench -Mechanical Introduction
第五章
模态分析
简介
Training Manual •在这一章中，将介绍模态分析。
进行模态分析类似线性静力分析。
–假设用户已学习了第4章线性静力结构分析部分。
•本章内容:
–模态分析步骤
–有预应力的模态分析步骤
•本节所述的功能，一般适用于ANSYS DesignSpace Entra及以上版本的许可。
Training Manual
模态系统分析基础
•对于模态分析，振动频率ωi 和模态φi 是根据下面的方程计算的出的：
2•假设:
[][](){}0
=−i
i
M K φω–[K] 和[M] 不变:
•假设材料特性为线弹性的
•利用小位移理论，并且不包括非线性的•不存在[C] ,因此无阻尼•无{F} , 因此无激振力
•
结构可以强迫振动也可以不强迫振动
–模态{φ} 是相对值，不是绝对值
A.模态系统分析步骤
Training Manual •模态分析与线性静态分析的过程非常相似，因此不对所有的步骤做详细介绍。
用蓝色斜体字的步骤是针对模态分析的。
–附加几何模型
–设置材料属性
–定义接触区域(如果有的话)
–定义网格控制(可选择)
–定义分析类型
–加支撑(如果有的话)
–求解频率测试结果
–设置频率测试选项
–求解
–查看结果
…几何体和质点
Training Manual •模态分析支持各种几何体：
实体, 表面体和线体
–,
•可以使用质量点:
•质点在模态分析中只有质量（无硬度）。
质点在模态分析中只有质量（无硬度）
•质量点的存在会降低结构自由振动的频率。
•材料属性: 杨氏模量,泊松比, 和密度是必需的。
密度是必需的
Training Manual
… 分析类型
Training Manual •从Workbench的工具栏中选择“Modal”指定模型的分析类型。
•在Analysis Settings中:
–提取的模态阶数：1到200（默认的是6）。
–指定频率变化的范围（默认的是0到1e+08Hz）。
… 载荷和约束
Training Manual •结构和热载荷无法在模态中存在。
•约束:
–假如没有或者只存在部分的约束, 刚体模态将被检测. 这些模态将处于0HZ附近. 与静态结构分析不同, 模态分析并不要求禁止刚体运动.
–边界条件对于模态分析来说，是很重要的。
因为他们能影响零件的振型和固有频率.
边界条件对于模态分析来说是很重要的因为他们能影响零件的振型和固有频率
因此需要仔细考虑模型是如何被约束的.
–压缩约束是非线性的，因此在此分析中不被使用.
… 求解
Training Manual •求解模型(没有要求的结果).
•求解结束后，求解分支会显示一个图标，显示频率和模态阶数.
•可以从图表或者图形中选择需要振型或者全部振型进行显示.
•嵌入“Total Deformation”结果中需要求解模态。
… 检查结果
Training Manual •模态:
–由于在结构上没有激励作用，因此振型只是与自由振动相关的相对值。
–在详细列表里可以看到每个结果的频率值.
–应用图形窗口下方的时间标签的动画工具栏来查看振型
B. Workshop 5.1 –模态
Training Manual •Workshop 5.1 –模态分析
•目标:
–检查用18号钢制造的马达盖子（如下所示）的振动特性。
Training Manual
C. 有预应力的模态
•在某些情况下，在执行模态分析时可能需要考虑预应力影响。
–处于不变载荷作用的应力作用下的结构可能会影响固有频率，例如吉他调弦。
[]{}{}
F x K o =[][]
S o →σ进行线性静力分析
应力刚度矩阵用于计算结
构分析
2[][](){}0
=−+i i
M S K φω
原来的模态方程被修改为包括[S]阵
…没有/有预应力作用的例子
Training Manual •在项目图表里建立一个静力结构分析与模态分析相结合的并有预应力存在的分析模型。
•注意在模型分支里，结构分析结果变成开始状态
的。
…没有/有预应力作用的例子
Training Manual •对一边固支的薄板的结构，简单对比一下两种分析
–进行两个分析，对比模态和有预应力的模态分析的不同。
模态分析有预应力的模态分析
Training Manual
…没有/有预应力作用的例子
•在这个例子中，提供了在有外力存在的拉伸状态下增加固有频率的方法。
模态
第一模态频率: 83.587 Hz
有预应力的模态第一模态频率: 99.679 Hz
D. Workshop 5.2 –预应力模态
Training Manual •Workshop 5.2 –预应力模态分析
•目标:模拟拉杆在有应力和无应力状态下的模态响应（如下所示）。