外 文 翻 译
Introduction to Robotics
Mechanics and Control
机器人学入门
力学与控制
系
别： 机械与汽车工程系 专学业生
名姓
称： 机械设计制造及其自动化 名： 郭仕杰
学
号：
06101315 指导教师姓名、职称： 贺秋伟 副教授
完成日期 2014 年2 月28日 Introduction to Robotics
Mechanics and Control
Abstract This book introduces the science and engineering of mechanical manipulation.This branch of the robot has been in several claical field based.The main related fields such as mechanics, control theory, computer science.In this book, Chapter 1 through 8 topics ranging from mechanical engineering and mathematics, Chapter 9 through 11 cover control theory of material, and twelfth and 13 may be claified as computer science materials.In addition, this book emphasizes the computational aspects of the problem; for example, each chapter it mainly mechanical has a brief section calculation.This book is used to teach the cla notes introduction to robotics, Stanford University in the fall of 1983 to 1985.The first and second versions have been through 2002 in use from 1986 institutions.Using the third version can also benefit from the revised and improved due to feedback from many sources.Thanks to all those who modified the author\'s friends.This book is suitable for advanced undergraduates the first grade curriculum.If students have contributed to the dynamics and linear algebra course in advanced language program in a basic course of statics.In addition, it is helpful, but not absolutely neceary, let the students finish the course control theory.The purpose of this book is a simple introduction to the material, intuitive way.Specifically, does not need the audience mechanical engineer strict, although much of the material is from the field.At the Stanford University, many electrical engineers, computer scientists, mathematicians find this book very readable.Here we only on the important part to extract.
The main content
1、Background
The historical characteristics of industrial automation is popular during the period of rapid change.Either as a cause or an effect of automation technology, period of this change is closely linked to the world economy.Use of industrial robots, can be identified in a unique device 1960\'s, with the development of computer aided design (CAD) system and computer aided manufacturing (CAM) system, the latest trends, automated manufacturing proce.The technology is the leading industrial automation through another transition, its scope is still unknown.In
1 the northern America, machinery and equipment used in early 80\'s of the 20th century, the late 80\'s of the 20th century a short pull.Since then, the market more and more (Figure 1.1), although it is affected by economic fluctuations, all the market.Figure 1.2 shows the robots were installed in a large number of annual world industrial zone.Notably, the number of Japan\'s report is different from other areas: they count the number of machine of robot in other parts of the world are not considered robot (instead, they would simply be considered \"factory machines\").Therefore, the reported figures for the Japanese exaggerated.One of the main reason for the growth in the use of industrial robots is that they are falling costs.Fig.1.3 shows that, in the last century 90\'s ten years, robot prices dropped although human labor costs.At the same time, the robot is not only cheaper, they become more effective and faster, more accurate, more flexible.If we factor these quality adjusted to the number, the use of robots to decrease the cost of even than their price tag faster.More cost-effective in the robot they become, as human labor to become more expensive, more and more industrial work become robot automation candidate.This is the most important trend to promote the industrial robot market growth.The second trend is, in addition to the economic, as robots become more can become more tasks they can do, may have on human workers engaged in dangerous or impoible.Industrial robots perform gradually get more complex, but it is still, in 2000, about 78% installation welding or material handling robot in USA robot.A more challenging field, industrial robots, accounted for 10% unit.This book focuses on the dynamics and control of the most important forms of industrial robot, manipulator.What is the industrial robot is sometimes debate.Equipment, as shown in Figure 1.4 is always included, and CNC milling machine (NC) is usually not.The difference lies in the programmable complex place if a mechanical device can be programmed to perform a variety of applications, it may be an industrial robot.This is the part of a limited cla of tasks are considered fixed automation.For the purpose of this difference, do not need to be discued; the basic properties of most materials suitable for various programmable machine.In general, the mechanical and control research of the mechanical hand is not a new science, but a collection of the theme from the \"claic\" field.Mechanical engineering helps to machine learning methods for static and dynamic conditions.The mathematical description of movement of the tool manipulator space supply and other attributes.Provide design evaluation tool to realize the motion and force the desired algorithm control theory.Electrical engineering technology applied in the design of electrical engineering technology for sensor applied in design and industrial robot interface sensor, are programmed to perform the required task of basic computer science and the equipment.
Figures：
2
FIGURE 1.1: Shipments of industrial robots in North America in millions of US
dollars
FIGURE 1.2: Yearly installations of multipurpose industrial robots for 1995-2000 and
forecasts for 2001-2004
3
FIGURE 1.3: Robot prices compared with human labor costs in the 1990s
FIGURE 1.4:The Adept 6 manipulator has six rotational joints and is popular in many applications.Courtesy of Adept Technology, Inc.
2、Control of mechanical arm In the study of robots, 3D spatial position we constantly to the object of interest.These objects are all manipulator links, parts and tools, it deals, and other objects in the robot\'s environment.In a coarse and important level, these objects are described by two attributes: the position and direction.Of course, a direct interest in the topic is the attitude in which we represent these
4 quantities and manipulate their mathematics.In order to describe the human body position in space and direction, we will always highly coordinate system, or frame, rigid object.Then we continue to describe the position and orientation of the reference frame of the coordinate system.Any framework can be used as a reference system in the expreion of a body position and direction, so we often think of conversion or transformation of the body of these properties from one frame to another description.The 2 chapter discues the Convention methods of dealing with job descriptions discued method of treating and post convention described positioning and manipulation of coordinate system the quantity and mathematics different.Well developed skills relevant to the position and rotation of the description and is very useful in the field of rigid robot.Kinematics is the science of sports, the movement does not consider the force which resulted in it.In the scientific research of kinematics, a position, velocity, acceleration, and the location variable high order derivative (with respect to time of all or any of the other variables (S)).Therefore, the kinematics of manipulator is refers to the geometric and temporal characteristics of all movement.The manipulator comprises nearly rigid connection, which is the relative movement of the joint connection of adjacent links.These nodes are usually instrument position sensor, so that adjacent link is a relative position measurement.In the case of rotating or rotary joint, the displacement is called the joint angle.Some robots including sliding (or prism) connection, in which the connection between the relative displacement is a translation, sometimes called the joint offset.The manipulator has a number of independent position variables are specified as the mechanism to all parts of the.This is a very general term, any mechanism.For example, a four connecting rod mechanism has only one degree of freedom (even with three members of the movement).In the case of the typical industrial robots, because the robots is usually an open kinematic chain, because each joint position usually define a variable, the node is equal to the number of degrees of freedom.The free end of the link chain consisting of the manipulator end effector.According to the application of robot, the end effector can be a starting point, the torch, electromagnet, or other device.We usually by mechanical hand position description framework description tool, which is connected to the end effector, relative to the base, the base of the mobile manipulator.In the study of mechanical operation of a very basic problem is the kinematics.This is to compute the position
5 of mechanical static geometric problems in hand terminal positioning.Specifically, given a set of joint angles, the forward kinematics problem is to compute the position and orientation relative to the base of the tool holder.Sometimes, we think this is a change from the joint space is described as a manipulator position that Cartesian space description.\"This problem will be discued in the 3 chapter.In the 4 chapter, we will consider the inverse kinematics problem.The problems are as follows: the end effector position and direction of the manipulator, computing all poible joint angle, can be used to achieve the position and direction of a given.(see Figure 1.7.) This is a practical problem of manipulator is fundamental.This is quite a complex geometry problem, the conventional solution in tens of thousands of humans and other biological systems time every day.In a case like a robot simulation system, we need to create computer control algorithm can make the calculation.In some ways, the solution to this problem is the most important element in the operating system.This is quite a complex geometry problem, the conventional solution in tens of thousands of humans and other biological systems time every day.In a case like a robot simulation system, we need to create computer control algorithm can make the calculation.In some ways, the solution to this problem is the most important element in the operating system.We can use this problem as a mapping on 3D Descartes \"position\" space \"position\" in the robot joint space.This need will occur when the 3D spatial objects outside the specified coordinates.Lack of this kind of algorithm some early robot, they just transfer (sometimes by hand) required for the position, and then be recorded as a common set of values (i.e., as a position in joint space for later playback).Obviously, if the playback position and motion pattern recording and joint of the purely robot in Cartesian space, no algorithm for the joint space is neceary.However, the industrial robot is rare, the lack of basic inverse kinematics algorithm.The inverse kinematics problem is not a simple forward kinematics of A.The equation of motion is nonlinear, their solution is not always easy (or even poible in a closed form).At the same time, the existing problems of solutions and multiple solutions occur.The study of these problems provides an appreciation of what the human mind nervous system is achieved when we, there seems to be no conscious thought, object movement and our arms and hands operation.Manipulator is a solution of the presence or absence of a given definition of work area.A solution for the lack of means of mechanical hands can not reach the desired position and orientation, because it is in the manipulator working area.
6 In addition to static positioning problem, we can analyze the robot motion.Usually, the analysis in the actuator velocity, it is convenient to define a matrix called the Jacobi matrix of the manipulator.The speed of Jacobi matrix specified in Descartes from the velocity mapping space and joint space.(see Figure 1.8.) This mapping configuration of the manipulator changes the natural changes.At some point, called a singularity, this mapping is not to make the transformation.This phenomenon are important to the understanding of the mechanical hand designers and users.
Figures：
FIGURE 1.5: Coordinate systems or \"frames\" are attached to the manipulator and to
objects in the environment.
7
FIGURE 1.6: Kinematic equations describe the tool frame relative to the base frame
as a function of the joint variables.
8
FIGURE 1.7: For a given position and orientation of the tool frame, values for the joint variables can be calculated via the inverse kinematics.
9
FIGURE 1.8: The geometrical relationship between joint rates and velocity of the end-effector can be described in a matrix called the Jacobian.
3、Symbol Symbol is always the problems in science and engineering.In this book, we use the following convention: First: Usually, uppercase variables vector or matrix.Scalar lowercase variables.Second：Tail buoy use (such as the widely accepted) indicating inverse or transposed matrix.Third：Tail buoy not subject to strict conventions, but may be that the vector components (for example, X, Y, Z) or can be used to describe the PBO / P in a position of the bolt.Fourth：We will use a lot of trigonometric function, we as a cosine symbol angle E1 can adopt the following methods: because the E1 = CE1 = C1.In the vector sign note general: many mechanics textbook treatment number of vector at a very abstract level and often used vector is defined relative to expreion in different coordinate systems.The most obvious example is, in addition to vector is relative to a given or known a different frame of reference.This is usually very convenient, resulting in compact structure, elegant formula.
10 For example, consider the angular velocity, connected in series with the last body ° W4 \'four rigid body (such as the manipulator links) relative to the fixed seat chain.Due to the angular velocity vector addition, angular velocity equation at last link we can write a very simple vector:
However, unle the information is relative to a common coordinate system, they cannot be concluded, therefore, although elegant, equation (1.1) calculation.Most of the \"work\".A case study of the manipulator, such statements, (1.1) work coordinate system hidden bookkeeping, which is often we need to practice.Therefore, in this book, we put the symbol reference frame vectors, we don\'t and carrier, unle they are in the same coordinate system.In this way, we derive expreions for computing numerical solution, \"bookkeeping\" problem can be directly applied to the actual.
Summary The robot is a typical electromechanical integration device, it uses the latest research results of machinery and precision machinery, microelectronics and computer, automation control and drive, sensor and information proceing and artificial intelligence and other disciplines, with the development of economy and all walks of life to the automation degree requirements increase, the robot technology has been developing rapidly, the emergence of a variety of robotic products.The utility of robot products, not only can solve many practical problems difficult to solve by manpower, and the promotion of industrial automation proce.At present, the research and development of robot relates to many aspects of the technology, the complexity of system structure, development and development cost is generally high, limiting the application of the technology, to some extent, therefore, the development of economic, practical, high reliability of robot system with a wide range of social significance and economic value.Based on the design of mechanical structure and drive system, the kinematics and dynamics of the cleaning robot is analyzed.Kinematics analysis is the basis of path planning and trajectory control of the manipulator, the kinematics analysis, inverse problem can complete the operation of space position and velocity mapping to drive space, using the homogeneous coordinate transformation method has been the end of manipulator position and arthrosis transform relations between the angle, geometric analysis method to solve the inverse kinematics problem of manipulator, provides a theoretical basis for control system design.The robot dynamics is to study the relationship between the motion and force of science, the purpose of the study is to meet the need of real-time control, this paper use straightaway language introduced the related mechanical industrial robots and control knowledge
11 for us, pointing the way for our future research direction.Robot is a very complicated learning, in order to go into it, you need to constantly learn, the road ahead is long, I shall search.
机器人学入门
力学与控制
摘要
本书介绍了科学与工程机械操纵。这一分支学科的机器人已经在几个经典的领域为基础的。主要的相关的领域是力学，控制理论，计算机科学。在这本书中，第1章通过8个主题涵盖机械工程和数学，第9章通过11个盖控制理论材料，第12和13章可能被归类为计算机科学材料。此外，这本书强调在计算方面的问题；例如，每章这方面主要以力学有一个简短的章节计算考虑。这本书是从课堂笔记用来教机器人学导论，斯坦福大学在1983的秋天到1985。第一和第二版本已经通过2002在从1986个机构使用。第三版也可以从中受益的使用和采用的修正和改进由于许多来源的反馈。感谢所有那些谁修正了作者的朋友们。这本书是适合高年级本科生一年级的课程。如果学生已经在静力学的一门基础课程有助于动力学和线性代数课程可以在高级语言程序。此外，它是有帮助的，但不是绝对必要的，让学生完成入门课程控制理论。本书的目的是在一个简单的介绍材料，直观的方式。具体地说，观众不需要严格的机械工程师，虽然大部分材料是从那场。在斯坦福大学，许多电气工程师，计算机科学家，数学家发现这本书很易读。在这里我们仅对其中重要部分做出摘录。
主要内容
1、背景
工业自动化的历史特点是快速变化的时期流行的方法。无论是作为一个原因或一个效果，这种变化的时期自动化技术是紧密联系在一起的世界经济。利用工业机器人，成为可识别在1960年代的一个独特的装置，随着计算机辅助设计（CAD）系统和计算机辅助制造（CAM）系统的特点，最新的趋势，制造业的自动化过程。这些技术是领先的工业自动化
12 通过另一个过渡，其范围仍然是未知的。在美国北部，在早期有机器设备多采用世纪80年代，其次是上世纪80年代后期一个简短的拉。自那时起，市场越来越多的（图1.1），虽然它是受经济波动，是所有市场。图1.2显示的机器人被安装在大数每年世界各国的工业区。值得注意的是，日本的报告数量有所不同从其他地区一样：他们算一些机器的机器人在世界的其他地方都没有考虑机器人（而不是，他们会简单地认为是“工厂的机器”）。因此，该报告的数字为日本有些夸大。
在工业机器人的使用增长的一个主要原因是他们正在下降成本。图1.3表明，在上世纪90年代的十年中，机器人的价格下降了虽然人类的劳动成本增加。同时，机器人不只是越来越便宜，他们变得更有效更快，更准确，更灵活的。如果我们的因素这些质量调整成数，使用机器人的成本下降甚至比他们的价格标签更快。在他们的工作机器人变得更具成本效益的，作为人类劳动继续变得更加昂贵，越来越多的工业工作成为机器人自动化的候选人。这是最重要的趋势推动了工业机器人的市场增长。第二个趋势是，除了经济，随着机器人变得更能成为他们能够做的更多以上的任务，可能对人类工人从事危险的或不可能的。工业机器人执行逐步得到更多的应用复杂的，但它仍然是，在2000年，大约78%安装在美国进行焊接或材料搬运机器人的机器人。
一个更具挑战性的领域，工业机器人，占10%装置。这本书着重于力学和最重要的形式控制的工业机器人，机械手。到底什么是工业机器人是有时辩论。设备，如图1.4所示是总是包括在内，而数控（NC）铣床通常不。区别在于的可编程的复杂的地方如果一个设备机械设备可以被编程为执行各种应用程序，它可能是一个工业机器人。这是最机部分有限的一类的任务被认为是固定的自动化。为目的本文的区别，不需要讨论；大多数材料的基本性质适用于各种可编程机。
总的来说，其力学和控制机械手的研究不是一个新的科学，而只是一个收集的主题从“经典”的领域。机械工程有助于机器学习方法静态和动态的情况下。数学描述空间供应工具机械手的运动和其他属性。控制理论提供了工具以实现所期望的运动和力的应用评价算法设计。电气工程技术施加在传感器的设计电气工程技术施加在传感器的设计和工业机器人接口，与计算机科学的基础这些设备进行编程以执行所需任务。
附图：
13
图1.1在数以百万计的人在美国北部的工业机器人的出货量美元
图1.2 年安装的多用途的工业机器人1995-2000年和2001年至2004年预测
图1.3 机器人的价格与上世纪90年代的人类劳动成本的比较
14
图1.4 娴熟的6臂有六个转动关节（流行于众多制造行业）
2、力学和机械臂的控制
机器人的研究中，我们不断的关注对象的位置三维空间。这些对象是机械手的链接，零件和工具，它的交易，并在机器人的环境的其他对象。在一个粗而重要的水平，这些对象是由两个属性描述：位置和方向。当然，一个直接感兴趣的话题是态度在我们所代表的这些量和操纵他们的数学。
为了描述人体在空间中的位置和方向，我们将始终高度坐标系统，或框架，严格的对
15 象。然后我们继续相对于一些参考描述该帧的位置和方向坐标系统。任何框架可以作为一个参考系统内的表达一个身体的位置和方向，所以我们经常认为转化或改变身体的这些属性从一帧到另一个的描述。2章讨论了公约的方法处理与职位描述讨论了公约的方法处理与职位描述定位和操纵这些量与数学不同的坐标系统。发展良好的技能有关的位置和旋转的描述甚至在刚体机器人领域是非常有用的。
运动学是科学的运动，对运动不考虑力这导致它。在运动学的科学研究，一个位置，速度，加速度，和所有的高阶导数的位置变量（相对于时间或任何其他变量（S））。因此，机械手的运动学研究是指所有的运动的几何和时间特性。机械手包括近刚性连接，这是由关节连接允许相邻链接的相对运动。这些节点通常仪表有位置传感器，使邻近的链接是相对位置测量。在旋转或旋转接头的情况下，这些位移被称为关节角度。一些机器人包含滑动（或棱镜）连接，其中之间的联系相对位移是一个翻译，有时也被称为联合偏移量。机械手具有数独立的位置的变量会被指定为定位该机制的所有部分。这是一个总称，任何机制。为例如，一个四连杆机构只有一个自由度（即使有三运动的成员）。在典型的工业机器人的情况下，因为机器人通常是一个开放的运动链，因为每个关节的位置通常定义一个变量，节点的数目等于自由度。
在链接组成的机械手的末端执行器的自由端链。根据机器人的应用，末端执行器可以是一个抓手，焊枪，电磁铁，或其他装置。我们一般通过描述工具的框架描述的机械手的位置，这是连接到端部执行器，相对于底座，所对移动机械手的基础。在机械操作的研究一个非常基本的问题就是了运动学。这是计算的位置的静态几何问题机械手的末端定位。具体而言，给定一组关节角，正向运动学问题是计算位置和方向工具架相对于底座。有时，我们认为这是改变从关节空间描述为一个机械手位置的表示笛卡尔空间的描述。“这个问题将在3章探讨。在4章中，我们将考虑的逆运动学问题。这个问题提出了如下：给出了末端执行器的位置和方向机械手，计算所有可能的关节角度，可以用来实现这个给定的位置和方向。（见图1.7。）这是一个根本性的问题机械手的实际应用。这是一个相当复杂的几何问题，常规的解决在人类和其他生物系统时间每天成千上万。在一个案例像一个机器人仿真系统，我们需要创建的控制算法计算机可以使这个计算。在某些方面，这个问题的解决方案是在操作系统中最重要的元素。
这是一个相当复杂的几何问题，常规的解决在人类和其他生物系统时间每天成千上万。在一个案例像一个机器人仿真系统，我们需要创建的控制算法计算机可以使这个计算。在某些方面，这个问题的解决方案是在操作系统中最重要的元素。
我们可以把这个问题作为一个映射在三维笛卡尔的“位置”空间的“位置”在机器人的关节内的空间。这需要自然会出现每当目标外部三维空间指定的坐标。一些早期的机器人缺乏这种算法，他们只是转移（有时用手）所需的的位置，然后被记录为一组共同的值（即，作为一个位置关节空间）用于以后回放。显然，如果机器人用纯粹的模式记录和关节的位置和运动的播放，没有算法有关的关节空间的笛卡尔空间是必要的。然而，是罕见的工业机器人，缺乏基本的逆运动学算法。逆运动学问题不是简单的正向运动学一个。由
16 于运动方程是非线性的，他们的解决方案并不总是容易（甚至可能在一个封闭的形式）。同时，对存在的问题解和多解的出现。这些问题的研究提供了一个欣赏什么人的心灵神经系统是实现当我们，似乎没有有意识的思考，移动和我们的双臂和双手操作的对象。一个解的存在或不存在的定义工作区一个给定的机械手。一个解决方案的缺乏意味着机械手不能达到所需的位置和方向，因为它在机械手的外工作区。
除了处理静态定位问题，我们不妨分析机器人的运动。通常，在执行机构的速度分析，它是方便的定义一个矩阵的数量称为机械手的雅可比矩阵.指定的速度雅可比矩阵在笛卡尔从关节空间的速度映射空间。（见图1.8。）这种映射配置的自然变化机械手的变化。在某些点，称为奇点，这映射是不使转化。这一现象的理解是设计师和用户的重要机械手。
附图：
图1.5 坐标系统或“帧”连接到机械手环境中的物体
17
图1.6运动学方程描述刀具架相对于底座作为一个联合变量的函数
18
图1.7 对于一个给定的位置和方向的工具框架，值为关节变量可以通过逆运动学计算
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图1.8 联合率和速度之间几何关系端部执行器可以在一个矩阵描述了所谓的雅可比矩阵
3、标识符号
符号一直是科学和工程问题。在这本书中，我们使用以下公约： 第
一、通常，大写变量表示的向量或矩阵。小写的变量的标量。 第
二、尾标使用（如被广泛接受的）指示逆或转置矩阵。
第三、尾标不受严格的公约，但可能表明向量的组件（例如，X，Y，Z）或可用于述在PBO / P一个螺栓的位置。
第四、我们将使用许多三角函数，我们为一个余弦符号角E1可以采用下列方式：因
为E1 = CE1 = C1。
在一般的矢量符号注：许多力学教材处理矢量在一个非常抽象的层次上的数量和经常使用向量定义相对于在表达不同的坐标系统。最明显的例子是，除了向量是给定的或已知的相对于不同的参考系。这是通常很方便，导致结构紧凑，有优雅的公式。为例如，考虑角速度，在串联连接的最后一次身体°W4 \'四刚体（如机械手的链接）相对的固定座链。由于角速度矢量相加，我们可以写一个非常简单的向量的最后环节的角速度方程：
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然而，除非这些量是相对于一个共同的坐标表示系统，他们不能总结，所以，虽然优雅，方程（1.1）隐藏大部分的“工作”的计算。为研究个案机械手，这样的陈述，（1.1）隐藏簿记的工作坐标系统，这往往是我们需要实践的想法。因此，在这本书中，我们把符号参考框架向量，我们不要和载体，除非他们在同一坐标系统。在这种方式中，我们推导出的表达式，解决“记账”问题可直接应用于实际的数值计算。
总结
机器人是典型的机电一体化装置，它综合运用了机械与精密机械、微电子与计算机、自动控制与驱动、传感器与信息处理以及人工智能等多学科的最新研究成果，随着经济的发展和各行各业对自动化程度要求的提高，机器人技术得到了迅速发展，出现了各种各样的机器人产品。机器人产品的实用化，既解决了许多单靠人力难以解决的实际问题，又促进了工业自动化的进程。目前，由于机器人的研制和开发涉及多方面的技术，系统结构复杂，开发和研制的成本普遍较高，在某种程度上限制了该项技术的广泛应用，因此，研制经济型、实用化、高可靠性机器人系统具有广泛的社会现实意义和经济价值。在完成机械结构和驱动系统设计的基础上，对物料抓取机械手运动学和动力学进行了分析。运动学分析是路径规划和轨迹控制的基础，对操作臂进行了运动学正、逆问题的分析可以完成操作空间位置和速度向驱动空间的映射，采用齐次坐标变换法得到了操作臂末端位置和姿态随关节夹角之间的变换关系，采用几何法分析了操作臂的逆向运动学方程求解问题，对控制系统设计提供了理论依据。机器人动力学是研究物体的运动和作用力之间的关系的科学，研究的目的是为了满足是实时性控制的需要，本文用通俗易懂的语言为我们介绍了工业机器人的相关力学与控制的知识，为我们以后的研究方向指明了道路。机器人的研究是一门非常复杂的学问，为了深入去探究它的方方面面，就需要不断的去学习，正所谓路漫漫其修远兮，吾将上下而求索。
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