Formula Derivation
In this appendix, we give a few remarks on Rodrigues formulas in Chapters 2 and 4 as well as the energy formula in Chapter 6.
A.1 SEVERAL VERSIONS OF RODRIGUES FORMULA
Rodrigues formulas vary in liturature. Many variation are known and used. For convenience, we summarize the relation between these formula.
Here u is assumed to be a unit vector in (A.1), (A.2), (A.3) and (A.4), while u in (A.6) may not be a unit vector. Note that u = (u1, u2, u3) in (A.4). Also, for (A.5) and (A.7), A = −AT should be ‘unit’, that is, it is assumed that 
. In all the cases, u shows the direction of the rotation axis.
For (A.6), the relation between A and u is given by Av = u × v. To be more explicit,
This means that 
(A.1) explains the meaning as the 2D rotation of angle θ in the plane orthgonal to the vector u. In a sense, (A.2) is most popular in the liturature. (A.2) is same as (2.14), and (A.5) is same as (2.15). The direct implications among these equivalent formulas are illustrated as
A.2 RODRIGUES TYPE FORMULA FOR MOTION GROUP
We explain the computation of formulas (7.3)–(7.8), where (7.5) and (7.8) might be less well-known, as compared with the others.
Having X̂ ∊ se(3) in Section 7.1, we notice that
Then
where we define
This shows R = exp(X) and d = Yl.
The relations (7.3),(7.4),(7.6) and (7.7) on X and R are known as Rodrigues formulas. Actually, (7.4) is essentially (A.5). (7.3) coincides with the requirement on the relation between A and u in (A.5).
(7.3) gives
Taking the trace of (7.4), together with tr(X) = 0, we obtain
which is equivalent to (7.6).
By (7.4), we obtain
which is equivalent to (7.7).
Finally, we come to (7.5) and (7.8). What we should do is to derive
We will obtain these formulas by noticing 
. For (A.10),
An alternative explanation of (A.10) will be the following. We assume that Y is of the form 
. We compute XY in two ways:
By comparing the coefficients, we obtain 
and 
, which proves (A.10).
For (A.11), we put and 
, and write the equation
Suppose 
and 
. This requirement is reduced to the system of linear equations in unknown variables b1 and b2 as
which can be solved as
If we put the explicit value of a1 and a2, then we obtain
This is the required formula (A.11) for the inverse of Y.
A.3 PROOF OF THE ENERGY FORMULA
We give a proof of the formula (6.5)
First note that the set of all matrices of the form sRδ is a vector space of skew-symmetric matrices:
Here we denote
So the left-hand side of the problem is rewritten as
Recall the definition of Frobenius norm:
where tr denotes the trace of a square matrix. Then
Here, for the computation (*), we have used
since JT J = −J2 = I. This shows
Here in the last equality, we have used the following identity
This identity is equivalent, if we put 
, to the following identity
which will be examined by expanding the left-hand side.
Now we see that the minimum is taken at
and its minimum value is given by
which is the desired result. This is the end of the proof.