![]() ![]() Forward is in the x direction, down is in its z direction and so on. We can also describe its motion with respect to this particular coordinate frame. Once we have a coordinate frame attached to the vehicle, we can then describe its orientation with respect to another coordinate frame. Now here is an example of a helicopter in flight and where we can attach a right-handed coordinate frame x-axis pointing forward, a y-axis out to the right, and a z-axis going straight down. In robotics, as in many other areas of engineering, it's really useful and important to attach coordinate frames to objects. So what we defined as a positive rotation for a 2-dimensional coordinate frame is actually a positive rotation around the z-axis, if you could imagine a z-axis being there. ![]() Here's our 2-dimensional coordinate frame again, here is our x and y-axis lying on top. When we looked previously at the 2-dimensional coordinate frame we also had a rule for the direction of positive rotation in two dimensions. Similarly a rotation around the x-axis - this a positive rotation around the x-axis and this is a positive rotation around the y-axis. So this is a positive rotation around the z-axis. So I take my right hand and I point my thumb along the axis - in this case it's the z-axis, and I curl my fingers around, and the direction of the curl, the direction of the tip of my fingers is the positive direction. When we are talking about rotations around axes we define the direction of positive rotation again using a right hand rule. So this positive angle convention in two dimensions corresponds to a positive rotation about the z-axis which is coming at us out of the screen. If we consider a 2-dimensional coordinate frame, then if we grow a z-axis out of the screen, that is the equivalent right handed 3-dimensional coordinate frame. The direction of the positive rotation is an important concept, and in a 2-dimensional case rotation was defined as being positive in this direction. So now when it comes to describing the pose of this aircraft it's got a translational component, that's the position of the origin of this coordinate frame with respect to the origin of this coordinate frame, and then there is the orientation of this coordinate frame with respect to this coordinate frame. So the way that we are going to formalise this is again similar to what we did for the 2-dimensional case, and again we attach a coordinate frame to the aircraft.Īnother coordinate frame here, a blue one and I'm going to fix it to the aircraft with the x-axis pointing forward and that means that the y-axis points over the wing and the z-axis points upward. So there are number of parameters that describe the orientation of this body in space. It can pitch up and down, it can roll to the right or it can roll to the left, and it can yaw to the right or to the left. There are a number of ways in which it can move. In 3-dimensions it is not as simple to describe the orientation of an object. And remembering that pose has got two components, a translational component that's the distance between the origins of frame A and frame B, and a rotational component how do I rotate coordinate frame A so that its axes are parallel to the axes of coordinate frame B. The notation is the same as it is for the 2-dimensional case and we read this symbol here as being the pose of coordinate frame B with respect to coordinate frame A. Later in this lecture we’ll talk about concrete implementations of ksi, what it actually means in terms of things that you can compute with, but for now just consider that as an abstract symbol. Things are not much different in 3-dimensions, once again we use this abstract notion of pose represented by Greek letter ksi, and now we represent the translation and orientation of one three-dimensional coordinate frame with respect to another. So we read this symbol as the pose of B with respect to A. And the notation we used is based on the Greek letter ksi and we used that to represent pose and we used the letter B the subscript to indicate that we're talking about the pose of the coordinate frame B and the leading index, in this case A, to indicate that the pose is with respect to coordinate frame A. In the previous lecture we introduced the notion of pose and poses a way of describing both the translation and the orientation of one coordinate frame with respect to another.
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