Читать книгу Robot Modeling and Control - Mark W. Spong - Страница 71

Just as we represented rotation matrices as exponentials of skew-symmetric matrices, we can also represent homogeneous transformations as exponentials using so-called twists.

Definition 2.3.

Let v and k be vectors in with k a unit vector. A twist ξ defined by k and v is the 4 × 4 matrix

(2.75)

We define se(3) as

(2.76)

se(3) is the vector space of twists, and a similar argument as before in Section 2.5.4 can be used to show that, given any twist ξ ∈ se(3) and angle , the matrix exponential of ξθ is an element of SE(3) and, conversely, every homogeneous transformation (rigid motion) in SE(3) can be expressed as the exponential of a twist. We omit the details here.