APPENDIX A

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.

Image

Image

Image

Image

Image

Image

Image

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 Image. 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,

Image

This means that Image

(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

Image

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 se(3) in Section 7.1, we notice that

Image

Then

Image

where we define

Image

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

Image

Taking the trace of (7.4), together with tr(X) = 0, we obtain

Image

which is equivalent to (7.6).

By (7.4), we obtain

Image

which is equivalent to (7.7).

Finally, we come to (7.5) and (7.8). What we should do is to derive

Image

Image

We will obtain these formulas by noticing Image. For (A.10),

Image

An alternative explanation of (A.10) will be the following. We assume that Y is of the form Image. We compute XY in two ways:

Image

By comparing the coefficients, we obtain Image and Image, which proves (A.10).

For (A.11), we put Image and Image, and write the equation

Image

Suppose Image and Image. This requirement is reduced to the system of linear equations in unknown variables b1 and b2 as

Image

which can be solved as

Image

If we put the explicit value of a1 and a2, then we obtain

Image

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)

Image

First note that the set of all matrices of the form sRδ is a vector space of skew-symmetric matrices:

Image

Here we denote

Image

So the left-hand side of the problem is rewritten as

Image

Recall the definition of Frobenius norm:

Image

where tr denotes the trace of a square matrix. Then

Image

Here, for the computation (*), we have used

Image

Image

Image

Now we will use

Image

since JT J = −J2 = I. This shows

Image

Here in the last equality, we have used the following identity

Image

This identity is equivalent, if we put Image, to the following identity

Image

which will be examined by expanding the left-hand side.

Now we see that the minimum is taken at

Image

and its minimum value is given by

Image

which is the desired result. This is the end of the proof.

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