Velocity kinematics in 3D

Let’s do a quick introduction to skew symmetric matrices. These matrices are sometimes called anti-symmetric matrices and they have this interesting property that the transpose is equal to the negative of the original matrix. These matrices are always singular. Their determinant is always equal to zero.

Any matrix is the sum of a symmetric matrix and a skew symmetric matrix. In three dimensions, we can write the skew symmetric matrix which is a function of a three element vector in this fashion. So, we have a vector whose elements are X, Y, and Z. The skew symmetric matrix looks like this. The obvious features are a diagonal of zeros. There are two X's in there. Two Y's and two Z's. One of them has a positive sign and one of them has a negative sign.

We can also write a vector cross product as a matrix vector product. One of the matrices is a skew symmetric matrix computed from the first vector.

Here’s a skew symmetric matrix and I am going share with you some rules of thumb for how you work out which elements are where when you write this matrix down. To start with, there is a zero diagonal. So, that’s pretty easy. You write three zeros down the diagonal.

There is also what I think of as an exclusion rule. So, in row X and in column Y, that element will not be an extra Y. It will be the other thing. And in this case, it will be a Z.

If we consider a cyclic rotation sequence. So, we have an endless sequence - X, Y, Z, X, Y, Z and so on. If I consider the element in column Y and in row X, moving from Y to X, it is going backwards in that cyclic rotation sequence. Because we are going backwards in the sequence, I am going to put a minus sign on that particular element.

If we look at this one over here, we are going from Z to X. And, that’s a forward in the cyclic rotation sequence. So, we give this element a positive sign.