MATLAB在机构与机器人分析中的应用实例

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"《机构与机器人MATLAB分析》是一本由Dan B. Marghitu教授编著的教材，专注于机械工程领域中的机构与机器人分析。书中强调了MATLAB在这一领域的关键作用，因为MATLAB不仅提供了强大的数值计算能力，还支持计算机辅助的分析计算，显著提升了设计和控制机械系统，尤其是机构和机器人系统的效率。本书的核心内容围绕着利用MATLAB解决机制和机器人设计中的实际问题，作者通过丰富的示例展示了MATLAB语言的运用，使读者能够直观地理解并掌握如何通过编程实现复杂的运动学和动力学分析。书中包含大量使用MATLAB解决的问题，这些程序作为附录供读者参考和实践，以便他们能够在实际工作中灵活运用。 MATLAB对于理解机构和机器人链路的运动学、如速度、加速度和力的传递，以及动态行为，如平衡分析、轨迹规划和控制设计等方面具有显著优势。它简化了繁琐的数学计算，使得工程师和学生能够快速且准确地处理复杂的系统模型。此外，书中的教材也遵循版权法规，除非出于研究、个人学习或批评性评论目的，否则复制或传播内容需事先获得出版商书面许可。《机构与机器人MATLAB分析》是一本实用的教程，不仅教授基础理论，更侧重于通过MATLAB工具的实际操作，帮助读者提升在现代机械工程领域中分析和控制机构与机器人系统的能力。对于任何希望深入理解并应用MATLAB在该领域的人来说，这是一本不可或缺的参考资料。"

1.2 Kinematic Pairs 7

joint of order two (two-pin joints)

(multiple joint)

link 1 (crank)

link 4 (coupler or connecting rod)

link 3 (rocker)

(a)

loop

(b)

sphere

joint

fixed

base

moving

platform

(

)

link 0

(ground)

link 0

(ground)

link 0

(ground)

end-effector

Fig. 1.8 (a) Mechanism with ﬁve moving links, (b) parallel link robot, and (c) Stewart mechanism

• a rocker is a link that has oscillatory (back and forth) rotation and is ﬁxed to a

grounded pivot;

• a coupler or connecting rod is a link that has complex motion and is not ﬁxed to

ground.

Ground is deﬁned as any link or links that are ﬁxed (non-moving) with respect to

the reference frame. Note that the reference frame may in fact itself be in motion.

Figure 1.8b illustrates a ﬁve-bar linkage consisting of ﬁve links, including the

base link 0, connected by ﬁve joints. The mechanism can be viewed as two link

arms (1, 2 and 3, 4) connected at a point C. It is a closed kinematic chain formed

by the ﬁve links. The position of the end-effector is determined if two of the ﬁve

joint angles are given. Figure 1.8c shows the Stewart mechanism, which consists of

8 1 Introduction

a moving platform, a ﬁxed base, and six powered cylinders connecting the moving

platform to the base frame. The position and orientation of the moving platform are

determined by the six independent actuators. This mechanism has spherical joints

(three degrees of freedom joints).

The concept of number of degrees of freedom is fundamental to the analysis of

mechanisms. It is usually necessary to be able to determine quickly the number of

DOF of any collection of links and joints that may be used to solve a problem.

The number of degrees of freedom or the mobility of a system can be deﬁned as:

the number of inputs that need to be provided in order to create a predictable system

output, or the number of independent coordinates required to deﬁne the position of

the system.

The class f of a mechanism is the number of degrees of freedom that are elimi-

nated from all the links of the system.

Every free body in space has six degrees of freedom. A system of class f consist-

ing of n movable links has (6 − f ) n degrees of freedom. Each joint with j degrees of

constraint diminishes the freedom of motion of the system by j − f degrees of free-

dom. The number of joints with k degrees of constraint is denoted as c

.Adriver

link is that part of a mechanism that causes motion. An example is a crank. The

number of driver links is equal to the number of DOF of the mechanism. A driven

link or follower is that part of a mechanism whose motion is affected by the motion

of the driver.

1.3 Dyads

For the special case of planar mechanisms ( f =3) the number of degrees of freedom

of the particular system has the form

M = 3n −2c

−c

, (1.1)

where n is the number of moving links, c

is the number of one degree of freedom

joints, and c

is the number of two degrees of freedom joints.

There is a special signiﬁcance to kinematic chains that do not change their de-

grees of freedom after being connected to an arbitrary system. Kinematic chains

deﬁned in this way are called system groups or fundamental kinematic chains. Con-

necting them to or disconnecting them from a given system enables given systems to

be modiﬁed or structurally new systems to be created while maintaining the original

degrees of freedom. The term system group has been introduced for the classiﬁca-

tion of planar mechanisms used by Assur and further investigated by Artobolevski.

Limiting to planar systems from Eq. 1.1, it can be obtained as

3n −2 c

= 0, (1.2)

according to which the number of system group links n is always even. In Eq. 1.2

there are no two degrees of freedom joints because a c

joint (two degrees of free-

1.5 Planar Mechanism Decomposition 11

Fig. 1.11 Planar R-RTR-RTR mechanism

To ﬁnd the number of c

a connectivity table will be used, Fig. 1.12a. The links

are represented with bars (two node links) or triangles (three node links). The one

degree of freedom joints (rotational joint or translation joint) are represented with

a cross circle. The ﬁrst column has the number of the current link, the second col-

umn shows the links connected to the current link, and the last column contains the

graphical representation. The link 1 is connected to ground 0 at A and to link 2 at B,

Fig. 1.12a. The link 2 is connected to link 1 at B and to link 3 at B. Next, link 3 is

connected to link 2 at B, link 0 at C, and link 4 at D. Link 3 is a ternary link because

it is connected to three links. At B there is a joint between link 1 and link 2 and a

joint between link 2 and link 3. Link 4 is connected to link 3 at D and to link 5 at

D. The last link, 5, is connected to link 4 at D and to 0 at A. In this way the table in

Fig. 1.12a is obtained. At A there is a multiple joint, two rotational joints, one joint

between link 1 and link 0, and one joint between link 5 and link 0.

The structural diagram is obtained using the graphical representation of the table

connecting all the links Fig. 1.12b. The c

joints (with cross circles), all the links,

and the way the links are connected are represented on the structural diagram. The

number of one degree of freedom joints is given by the number of cross circles.

From Fig. 1.12b it results that c

= 7. The number of DOF for the mechanism is

M = 3 (5) −2 (7)=1. If M = 1, there is just one driver link. One can choose link

1 as the driver link of the mechanism. Once the driver link is taken away from the

mechanism the remaining kinematic chain (links 2, 3, 4, 5) has the mobility equal to

zero. The dyad is the simplest system group and has two links and three joints. On

the structural diagram one can notice that links 2 and 3 represent a dyad and 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 5).

Another graphical construction for the connectivity table, shown in Fig. 1.12a, is

the contour diagram, that can be used to represent the mechanism in the following

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