Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 56

In this example, the three independent parameters of , which is briefly denoted here as , are selected as its three elements, which are c₁₁, c₂₁, and c₁₂. The following values are specified for them.

(3.38)

It is required to determine by finding its remaining six elements.

The column‐by‐column expression of can be written as

(3.39)

By the given information, the first two columns of are determined partially as shown below.

(3.40)

Since , c₃₁ is found as follows with a sign ambiguity represented by σ₁ = ± 1:

(3.41)

The fact that leads to the following equation.

(3.42)

The fact that leads to the following additional equation.

(3.43)

When c₂₂ is eliminated between Eqs. (3.42) and (3.43), the following quadratic equation is obtained for c₃₂.

(3.44)

Hence, c₃₂ is found as follows with an additional sign ambiguity represented by σ₂ = ± 1:

(3.45)

Upon inserting c₃₂, Eq. (3.43) gives c₂₂ as

(3.46)

Finally, the third column is found as follows according to Eq. (3.34):

(3.47)

Note that the procedure described above provides four different outcomes for due to the independent sign variables σ₁ and σ₂. To pick up one of these solutions, let σ₁ = σ₂ = + 1. This particular choice of σ₁ and σ₂ leads to , which is shown below.

(3.48)

As a check for the validity of the above solution, it can be shown that .