2D Geometry

Let's have a look at a slightly more realistic example. Here I have a 2-dimensional car and a 2-dimensional coordinate frame. I've labeled this coordinate frame W and that stands for the world coordinate frame and this is quite a common thing in robotics to have one reference frame that all other reference frames are described with respect to. And to indicate that very, very clearly, I've labeled the x and y axis with the subscript W, so it's very, very clear that this the x-axis of the W frame and the y-axis of the W frame. Now let's indicate some points of interest in this car, the left seat: the right seat, one of the mirrors, door handles, and perhaps where we refuel it. I can describe each of these points by a vector, with respect to the world coordinate frame.

Now let's imagine that a car has moved, now it's over here, I could now describe all these points again in terms of vectors with respect to the origin of the world coordinate frame. But it's a little bit tedious because the points haven't actually moved with respect to the car, the whole car has moved and now I have to redefine all the vectors again. Perhaps, there's a better way. And there it is, we are going to use the coordinate frame idea again and we are going to create a new coordinate frame and I'm going to attach it to the car. I labeled this frame C - C for car. And just to be clear, I labeled the axis XC and YC: X and Y axis are the coordinate frame that are attached to the car. Now it's a simple matter to describe all these points of interest on the car in terms of vectors with respect to the car's coordinate frame. Now if I moved the car, the vectors have changed with respect to the world but they haven't changed with respect to the car's coordinate frame.

Let's simplify things a bit and introduce a pose here. So introduce another one of these fat arrows which represents a pose, and this is the pose from the world coordinate frame to the car's coordinate frame so it's W ksi Z. Now let's look at just a single point, one of the points on the door of the car. I'm going to label that point d and we created vector from frame C to this point d, and we denote that in the way we've already discussed we use a symbol P to denote that we're talking about a vector. C is the reference frame so this is a vector with respect to frame C, so it begins at the origin of frame C and it goes to the point d. Now let's consider that I want to know what is this vector: the point D with respect to the world coordinate frame. In some sort of simplistic way, what I want to do is to add these 2 arrows. I want to move along here, and I want to move along here. But it's not quite that simple, we can't add a pose and a vector. We can't add a fat arrow to a thin arrow. Poses and vectors are different mathematical objects so we need to do this a little bit carefully.

The way we do it is to introduce a new operator, and it's the dot. In this formalism, we have a pose and it operates on a vector in order to transform it to a different coordinate frame. That's what this operator dot does: it transforms this vector which describes the point d with respect to the coordinate frame C, to a vector that describes the point d with respect to the coordinate frame W. It performs a transformation between coordinate frames. All of this is rather abstract. I’ve introduced this operator dot, but I haven't told you actually what it is - how would you code it? And, we introduced this abstract pose thing described by the Greek letter ksi - and I haven't said exactly what that is or how you could code that or implement that. These are things would get to in the following sections in this lecture.