Describing rotation and translation in 3D


So, as we should well know by now, pose has got 2 components; a translational component and a rotational component. And there are several ways that we can represent a pose. We could represent the translational component as a vector, and then we could represent the rotational component by a set of 3 Euler angles. Alternatively, we could represent it by a vector plus a set of 3 roll, pitch, yaw angles. Or we could represent it as a vector plus a quaternion. The fourth option is to represent it as a homogeneous transformation and this is what we did for the 2-dimensional case. As we've done several times already in this lecture we've lifted concepts from 2-dimensions in to 3-dimensions and that's what we're going to do next. We are going to work out how we describe pose as a homogeneous transformation in 3-dimensions.

Consider two 3-dimensional coordinate frames and a point P. Now, I'm going to introduce a frame {V}. It has the same origin as coordinate frame B but its axes are parallel to coordinate frame A. I can describe the point P with respect to the coordinate frame {V} by using a rotation, rotation from coordinate frame {V} to coordinate frame B. I can describe the origin of frame {V} by a vector with respect to frame A and I can describe the point P in terms of a vector with respect to coordinate frame A. I can write a simple vector relationship. I can add these 2 vectors because coordinate frame {V} has its axes parallel to those of coordinate frame A.

Now, I can substitute the first relationship and I have now an expression for a point with respect to frame A in terms of a vector with respect to the original coordinate frame B and I can substitute A for {V} because the axis of frames {V} and A are parallel to each other.

I can substitute in the elements of the rotation matrix and expand the vectors in terms of their elements and then, I can augment the vector by appending a 1 to it and move the translation between the origins of the coordinate frames into the matrix. And to keep things tidy, I'm going to augment the other vector as well which means I have to pack the matrix out with some zeroes and ones. So, what we have now is a homogeneous representation. We have a homogeneous transformation matrix and two homogeneous vectors. We can write it in compact form like this with the note that the homogeneous vectors by putting a tilde above them. We can see that this matrix comprises a rotation component, a translational component, 3 zeroes and a one.

So, this single 4 x 4 matrix encapsulates rotation and translation and allows us to transform a vector describing a point from coordinate frame B to coordinate frame A. This 4 x 4 matrix here, we refer to as a homogeneous transformation and these 2 vectors here are homogeneous vectors. And just a reminder on homogeneous vectors, we denote them by tilde on top. We take the Cartesian vector, in this case because it's 3-dimensional space, it's got 3 elements A, B and C. The homogeneous version of that has got 4 elements A, B, C and 1. We append a 1 to it. Here it is in a very compact form. We have a point denoted by a homogeneous vector with respect to frame B and we can transform it to a point described by a homogeneous vector with respect to frame A. Relative pose is described by single 4 x 4 matrix. A mathematician would say that this matrix belongs to the special Euclidean group of dimension 3 and the short hand is that T is an element of the set SE(3); 3 indicating the dimension in this case 3.

Instead of the abstract symbol ksi, we can use a homogeneous transformation matrix, a 4 x 4 matrix. Compounding or composition is simply a matrix-matrix product. To compound 2 poses, I simply multiply their representation in terms of homogeneous transformations. Note well that the inverse of a homogeneous transformation is not equal to its transpose, that property only belongs to the rotational component of a matrix which is an orthogonal matrix. The homogeneous transformation matrix is not an orthogonal matrix. It's inverse is not equal to its transpose. Finally, vector transformation is simply a matrix-vector product.

We learn how to describe the 3D pose of an object by a 4×4 homogeneous transformation matrix which has a special structure.

Professor Peter Corke

Professor of Robotic Vision at QUT and Director of the Australian Centre for Robotic Vision (ACRV). Peter is also a Fellow of the IEEE, a senior Fellow of the Higher Education Academy, and on the editorial board of several robotics research journals.

Skill level

This content assumes high school level mathematics and requires an understanding of undergraduate-level mathematics; for example, linear algebra - matrices, vectors, complex numbers, vector calculus and MATLAB programming.

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