Читать книгу Robot Modeling and Control - Mark W. Spong - Страница 67

In this section we introduce the so-called exponential coordinates and give an alternate description of the axis-angle transformation (2.44). We showed above in Section 2.5.3 that any rotation matrix R ∈ SO(3) can be expressed as an axis-angle matrix R_{k, θ} using Equation (2.44). The components of the vector are called exponential coordinates of R.

To see why this terminology is used, we first recall from Appendix B the definition of so(3) as the set of 3 × 3 skew-symmetric matrices S satisfying

(2.52)

For let S(k) be the skew-symmetric matrix

(2.53)

and let e^S(k)θ be the matrix exponential as defined in Appendix B

(2.54)

Then we have the following proposition, which gives an important relationship between SO(3) and so(3).

Proposition 2.1

The matrix e^S(k)θ is an element of SO(3) for any S(k) ∈ so(3) and, conversely, every element of SO(3) can be expressed as the exponential of an element of so(3).

Proof: To show that the matrix e^S(k)θ is in SO(3) we need to show that e^S(k)θ is an orthogonal matrix with determinant equal to + 1. To show this we rely on the following properties that hold for any n × n matrices A and B

1 

2 If the n × n matrices A and B commute, i.e., AB = BA, then eAeB = e(A + B)

3 The determinant , where tr(A) is the trace of A.

The first two properties above can be shown by direct calculation using the series expansion (2.54) for eA. The third property follows from the Jacobi Identity (Appendix B). Now, since ST = −S, if S is skew-symmetric, then S and ST clearly commute. Therefore, with S = S(kθ) ∈ so(3), we have

(2.55)

which shows that e^S(kθ) is an orthogonal matrix. Also

(2.56)

since the trace of a skew-symmetric matrix is zero. Thus e^S(kθ) ∈ SO(3) for S(kθ) ∈ so(3).

The converse, namely, that every element of SO(3) is the exponential of an element of so(3), follows from the axis-angle representation of R and Rodrigues’ formula, which we derive next.