M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

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Guide to kinematic theory for the analysis of spatial mechanisms and manipulators Kinematics of General Spatial Mechanical Systems Comprehensive in scope, the book covers topics ranging from rather elementary subjects such as spatial mechanisms with single degree of freedom to more advanced topics such as serial manipulators including redundant and deficient ones, parallel manipulators, and non-holonomic spatial cam mechanisms that involve rolling without slipping motions. The author presents an effective and accessible symbolic manipulation method making it possible to obtain neat and transparent expressions that describe the systems showing all the kinematic details. Such expressions readily lead to analytical or semi-analytical solutions. They also facilitate the identification and analysis of the multiplicities and singularities. 
This all-time beneficial book:
Provides an easy-to-use systematic formulation method that is applicable to all sorts of spatial machanisms and manipulators Introduces a symbolic manipulation method, which is effective and straightforward to use, so that kinematic relationships can be simplified by using all the special geometric features of the system Offers an accessible format that uses a systematic and easy-to-conceive notation which has proven successful Presents content written by an author who is a renowned expert in the field Includes an accompanying website Written for academicians, students, engineers, computer scientists and any other people working in the area of spatial mechanisms and manipulators,
provides a clear notation, formulation, and a logical approach to the topic and offers a fresh presentation of challenging material.

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Figure 34Two positions of a cube It is required to express the HTM between - фото 1589

Figure 3.4Two positions of a cube.

It is required to express the HTM картинка 1590between the two positions of the cube.

The translation vector can be expressed in as follows 3215 - фото 1591as follows:

3215 On the other hand - фото 1592 3215 On the other hand is oriented with respect to - фото 1593

(3.215) On the other hand is oriented with respect to so that 3216 - фото 1594

On the other hand, is oriented with respect to so that 3216 Hence in - фото 1595is oriented with respect to so that 3216 Hence in the expression of the translation v - фото 1596so that

(3.216) Hence in the expression of the translation vector becomes 3217 - фото 1597

Hence, in the expression of the translation vector becomes 3217 Then the column - фото 1598, the expression of the translation vector becomes

(3.217) Then the column matrix representation of in is obtained as 3 - фото 1599

Then, the column matrix representation of in is obtained as 3218 As for the rotation of the cube Figu - фото 1600in is obtained as 3218 As for the rotation of the cube Figure 34implies - фото 1601is obtained as

(3.218) As for the rotation of the cube Figure 34implies that 3219 Note that - фото 1602

As for the rotation of the cube, Figure 3.4implies that

(3.219) Note that Description 3219describes an IFB rotation sequence Therefore - фото 1603

Note that Description (3.219)describes an IFB rotation sequence. Therefore, referring to Section 3.7, the relevant transformation matrices can be obtained as shown below.

(3.220) 3221 3222 Hence 3223 - фото 1604

(3.221) 3222 Hence 3223 Having found the - фото 1605

(3.222) Hence 3223 Having found the rotational and translational displacement - фото 1606

Hence,

(3.223) Having found the rotational and translational displacement matrices ie and - фото 1607

Having found the rotational and translational displacement matrices, i.e. картинка 1608and the HTM can then be constructed as follows 3224 - фото 1609, the HTM can then be constructed as follows 3224 In order to ha - фото 1610can then be constructed as follows:

3224 In order to have a detailed expression the rotational partition - фото 1611

(3.224) In order to have a detailed expression the rotational partition can be written - фото 1612

In order to have a detailed expression, the rotational partition can be written as shown below 3225 Hence 3226 - фото 1613can be written as shown below.

(3.225) Hence 3226 As a verification of the expression of - фото 1614

Hence,

(3.226) As a verification of the expression of obtained above the coordinates of the - фото 1615

As a verification of the expression of картинка 1616obtained above, the coordinates of the points A 2and G 2are obtained as follows in 3227 - фото 1617:

3227 - фото 1618 3227 - фото 1619

(3.227) 3228 - фото 1620

3228 Note that by referring to Figure 34 both Eqs - фото 1621 3228 Note that by referring to Figure 34 both Eqs 3227and - фото 1622

(3.228) Note that by referring to Figure 34 both Eqs 3227and 3228can be - фото 1623

Note that, by referring to Figure 3.4, both Eqs. (3.227)and (3.228)can be verified by inspection, too. Moreover, Eq. (3.227)matches with Eq. (3.217), as it is supposed to do.

As an additional manipulation with the homogeneous transformation matrices, the coordinates of the points A 2and G 2can also be obtained in картинка 1624as explained below.

The HTM between and can be expressed as follows 3229 Recall that - фото 1625and can be expressed as follows 3229 Recall that has already be - фото 1626can be expressed as follows:

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