as
(2.50)
The above axis-angle representation characterizes a given rotation by four quantities, namely the three components of the equivalent axis k and the equivalent angle θ. However, since the equivalent axis k is given as a unit vector only two of its components are independent. The third is constrained by the condition that k is of unit length. Therefore, only three independent quantities are required in this representation of a rotation
(2.51)
Note, since k is a unit vector, that the length of the vector r is the equivalent angle θ and the direction of r is the equivalent axis k.
One should be careful to note that the representation in Equation (2.51) does not mean that two axis-angle representations may be combined using standard rules of vector algebra, as doing so would imply that rotations commute which, as we have seen, is not true in general.
2.5.4 Exponential Coordinates
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 Rk, θ using Equation (2.44). The components of the vector
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
(2.53)
and let eS(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 eS(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 eS(k)θ is in SO(3) we need to show that eS(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 eS(kθ) is an orthogonal matrix. Also
(2.56)
since the trace of a skew-symmetric matrix is zero. Thus eS(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.
Rodrigues’ Formula
Given the skew-symmetric matrix S(k) it is easy to show that S3(k) = −S(k), from which it follows that S4(k) = −S2(k), etc. Thus the series expansion for eS(k)θ reduces to
the latter equality following from the series expansion of the sine and cosine functions. The expression
(2.57)
is known as Rodrigues’ formula. It can be shown by direct calculation that the angle-axis representation for Rk, θ given by Equation (2.44) and Rodrigues’ formula in Equation (2.57) are identical.
Remark 2.1.
The above results show that the matrix exponential function defines a one-to-one mapping from so(3) onto SO(3). Mathematically, so(3) is a Lie algebra and SO(3) is a Lie group.
2.6 Rigid Motions
We have now seen how to represent both positions and orientations. We combine these two concepts in this section to define a rigid motion and, in the next section, we derive an efficient matrix representation for rigid motions using the notion of homogeneous transformation.
Definition 2.2.
A rigid motion is an ordered pair (d, R) where
A rigid motion is a pure translation together with a pure rotation.3 Let