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
(2.58)
Now consider three coordinate frames o0x0y0z0, o1x1y1z1, and o2x2y2z2. Let d1 be the vector from the origin of o0x0y0z0 to the origin of o1x1y1z1 and d2 be the vector from the origin of o1x1y1z1 to the origin of o2x2y2z2. If the point p is attached to frame o2x2y2z2 with local coordinates
(2.59)
and
(2.60)
The composition of these two equations defines a third rigid motion, which we can describe by substituting the expression for
(2.61)
Since the relationship between
(2.62)
Comparing Equations (2.61) and (2.62) we have the relationships
(2.63)
(2.64)
