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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For the sake of comparing Eqs. (1.29)and (1.38)from the viewpoint of the notational logic , Eq. (1.29)is written again below.

(1.39) Here it is instructive to pay attention to the interchanged location of the - фото 435

Here, it is instructive to pay attention to the interchanged location of the superscript ( a ) in Eqs. (1.38)and (1.39). In Eq. (1.39), картинка 436must not bear ( a ) because it is a vector that is specified without necessarily knowing anything about the observation frame картинка 437, whereas картинка 438must necessarily bear ( a ) because it is one of the basis vectors of картинка 439. In Eq. (1.38), on the other hand, картинка 440must necessarily bear ( a ) because it represents the appearance of картинка 441as observed in картинка 442, whereas картинка 443must not bear ( a ) because it is not tied up to any reference frame as explained above and expressed by Eq. (1.36).

1.6 Matrix Operations Corresponding to Vector Operations

1.6.1 Dot Product

Consider two vectors картинка 444and Kinematics of General Spatial Mechanical Systems - изображение 445, which are resolved as follows in a reference frame Kinematics of General Spatial Mechanical Systems - изображение 446:

(1.40) Kinematics of General Spatial Mechanical Systems - изображение 447

(1.41) Kinematics of General Spatial Mechanical Systems - изображение 448

The dot product of and can be expressed as 142 On the other hand according to - фото 449and can be expressed as 142 On the other hand according to Eq 124 143 - фото 450can be expressed as

(1.42) On the other hand according to Eq 124 143 Hence Eq 142becomes - фото 451

On the other hand, according to Eq. (1.24),

(1.43) Hence Eq 142becomes 144 Owing to the definition of δ ij Eq - фото 452

Hence, Eq. (1.42)becomes

(1.44) Owing to the definition of δ ij Eq 144becomes simplified to 145 - фото 453

Owing to the definition of δ ij, Eq. (1.44)becomes simplified to

(1.45) Equation 145can also be written as follows in terms of and which - фото 454

Equation (1.45)can also be written as follows in terms of картинка 455and картинка 456, which are the column matrix representations of картинка 457and in 146 - фото 458in Kinematics of General Spatial Mechanical Systems - изображение 459:

Kinematics of General Spatial Mechanical Systems - изображение 460

(1.46) Kinematics of General Spatial Mechanical Systems - изображение 461

Equation (1.46)shows that the dot product of two vectors is equivalent to the inner product of their column matrix representations in a reference frame such as картинка 462.

1.6.2 Cross Product and Skew Symmetric Cross Product Matrices

Consider the same two vectors картинка 463and картинка 464, which are expressed by Eqs. (1.40)and (1.41)as resolved in the reference frame Their cross product can be expressed as 147 On the other hand according - фото 465. Their cross product can be expressed as

(1.47) On the other hand according to Eq 125 148 Hence Eq 147becomes - фото 466

On the other hand, according to Eq. (1.25),

(1.48) Hence Eq 147becomes 149 Equation 149implies that 150 - фото 467

Hence, Eq. (1.47)becomes

(1.49) Equation 149implies that 150 By using the definition of ε ijkgiven by - фото 468

Equation (1.49)implies that

(1.50) By using the definition of ε ijkgiven by Eq 126 Eq 149can be worked - фото 469

By using the definition of ε ijkgiven by Eq. (1.26), Eq. (1.49)can be worked out to what follows:

(1.51) Upon comparing the coefficients of the basis vectors of on each side of Eq - фото 470

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