Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 27

Consider the following 3 × 3 matrix equation, which is to be solved for .

(1.81)

The matrix can be expressed as follows in terms of its columns.

(1.82)

Along with , can be expressed as follows in terms of its elements.

(1.83)

Hence, Eq. (1.81) can be written in a more detailed form as

(1.84)

Equation (1.84) leads to the following scalar equations with the indicated premultiplications.

(1.85)

(1.86)

(1.87)

Note that, for i ∈ {1, 2, 3} and j ∈ {1, 2, 3},

(1.88)

Thus, Eqs. (1.85)–(1.87) reduce to the following equations.

(1.89)

(1.90)

(1.91)

Equations (1.89)–(1.91) imply that

(1.92)

Therefore, if , Eqs. (1.89)–(1.91) give the elements of as follows:

(1.93)

(1.94)

(1.95)

Note that the solution obtained above is the same as the solution provided by Cramer's rule.