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# 数学代写|Matlab代考|CSC113 Kinematic Chains

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## 数学代写|Matlab代考|Kinematic Chains

Bodies linked by joints form a kinematic chain as shown in Fig. 1.5. A contour or loop is a configuration described by a closed polygonal chain consisting of links connected by joints, Fig. 1.5a. The closed kinematic chains have each link and each joint incorporated in at least one loop. The closed loop kinematic chain in Fig. 1.5a is defined by the links $0,1,2,3$, and 0 . The open kinematic chain in Fig. 1.5b is defined by the links $0,1,2$, and 3 . A mechanism is a closed kinematic chain. A robot is an open kinematic chain. The mixed kinematic chains are a combination of closed and open kinematic chains. The crank is a link that has a complete revolution about a fixed pivot. Link 1 in Fig. 1.5a is a crank. The rocker is a link with oscillatory rotation and is fixed to the ground. The coupler or connecting rod is a link that has complex motion and is not fixed to the ground. Link 2 in Fig. 1.5a is a coupler. The ground or the fixed frame is a link that is fixed (non-moving) with respect to the reference frame. The ground is denoted with 0 .

A planar mechanism is shown in Fig. 1.6a. The mechanism has five moving links $1,2,3,4,5$, and a fixed link, the ground 0 . The translation along the $i$ axis is denoted by $\mathrm{T}_i$, and the rotation about the $i$ axis is denoted by $\mathrm{R}_i$, where $i=x, y, z$. The motion of each link in the mechanism is analyzed in terms of its translation and rotation about the fixed reference frame $x y z$. The link 0 (ground) has no translations and no rotations. The link 1 has a rotation motion about the $z$ axis, $\mathrm{R}_z$. The link 2 has a planar motion ( $x y$ is the plane of motion) with a translation along the $x$ axis, $\mathrm{T}_x$, a translation along the $y$ axis, $\mathrm{T}_y$, and a rotation about the $z$ axis, $\mathrm{R}_z$. The link 3 (slider) has a rotation motion about the $z$ axis, $\mathrm{R}_z$. The link 4 has a planar motion ( $x y$ the plane of motion) with a translation along $x, \mathrm{~T}_x$, a translation along $y, \mathrm{~T}_y$, and a rotation about $z, \mathrm{R}_z$. The link 5 has a rotation about the $z$ axis, $\mathrm{R}_z$.

A graphical construction for the mechanism connectivity is the contour diagram $[3,18]$. The numbered links are the nodes of the diagram and are represented by circles, and the joints are represented by lines that connect the nodes. Figure $1.6 \mathrm{~b}$ is the contour diagram of the planar mechanism. The link 1 is connected to ground 0 at $A$ and to link 2 at $B$ with revolute joints. The link 2 is connected to link 3 at $C$ with a slider joint. The link 3 is connected to ground 0 at $C$ with a revolute joint. Next, the link 2 is connected to link 4 at $D$ with a revolute joint. Link 2 is a ternary link because it is connected to three links. Link 4 is connected to link 5 at $D$ with a slider joint. Link 5 is connected to ground 0 at $E$ with a revolute joint. The independent contour is the contour with at least one link that is not included in any other contours of the chain. The number of independent contours, $N$, of a kinematic chain is computed as
$$N=c-n,$$
where $c$ is the number of joints, and $n$ is the number of moving links.
For the mechanism shown in Fig. 1.6a the independent contours are $N=c-n=$ $7-5=2$, where $c=7$ is the number of joints and $n=5$ is the number of moving links. Some contours of the mechanisms can be selected as: 0-1-2-3-0, 0-1-2-4-5-0, and 0-3-2-4-5-0. Only two contours are independent contours.

The dyad (binary group) is a fundamental kinematic chain with two links $(n=2)$ and three one degree of freedom joints $\left(c_5=3\right.$ ). Figure $1.7$ depicts different types of dyads: rotation rotation rotation (dyad RRR) or dyad of type one, Fig. 1.7a; rotation rotation translation (dyad RRT) or dyad of type two, Fig. 1.7b; rotation translation rotation (dyad RTR) or dyad of type three, Fig. 1.7c; translation rotation translation (dyad TRT) or dyad of type four, Fig.1.7d; rotation translation translation (dyad RTT) or dyad of type five, Fig. 1.7e. The advantage of the group classification of a mechanical system is in its simplicity. The solution of the whole mechanical system can be obtained by adding partial solutions of different fundamental kinematic chains [55-57].

The number of DOF for the mechanism in Fig. $1.8$ is $M=1$. If $M=1$, there is one driver link (one actuator). The rotational link 1 can be selected as the driver link. If the driver link is separated from the mechanism the remaining moving kinematic chain (links $2,3,4,5$ ) has the number of DOF equal to zero. The dyad is the simplest system group with two links and three joints. On the contour diagram, the links 2 and 3 form a dyad and the links 4 and 5 represent another dyad. The mechanism has been decomposed into a driver link (link 1) and two dyads (links 2 and 3, and links 4 and A two degrees of freedom joint can be substituted with one link $(n=1)$ and two one degree of freedom joints $\left(c_5=2\right)$.

Figure $1.9$ shows a cam and follower mechanism. There is a two degrees of freedom joint at the contact point $C$ between the links 1 and 2. The two degrees of freedom joint at $C$, is replaced with one link, link 3, and two one degree of freedom joints at $C$ and $D$ as shown in Fig. 1.9b. To have the same relative motion, the length of link 3 has to be equal to the radius of curvature $\rho$ of the cam at the contact point $C$.

In this way the two degrees of freedom joint at the contact point can be substituted with two one degree of freedom joints, $C$ and $D$, and an extra link 3 , between links 1 and 2 . The new mechanism still has one degree of freedom, and the cam and follower system $(0,1$, and 2$)$ is a R-RRT mechanism $(0,1,2$, and 3 ) in a different aspect.

# Matlab代写

## 数学代写|Matlab代考|Kinematic Chains

$$N=c-n,$$
，其中$c$是关节的数量，$n$是移动链接的数量。对于图1.6a所示的机构，独立轮廓为$N=c-n=$$7-5=2$，其中$c=7$为关节的数量，$n=5$为移动链接的数量。一些机构的轮廓可以选择为:0-1-2-3-0,0-1-2-4-5-0，和0- 3-2-2 -5-0。只有两条轮廓是独立的轮廓

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。