In order to design behaviors or controllers for, for robots, we inevitably need models of how the robots actually behave. And we're going to start with one of the most common models out there, which is the model of a differential drive mobile robot. So, differential drive wheeled mobile robot has two wheels and the wheels can turn at different rates and by turning the, the wheels at different rates, you can make the robot move around. So, this is the robot we are going to start with and the reason for it is because it is extremely common. In fact, the Khepera 3, which is the robot that we are going to be using quiet a lot in this course is a differential drive wheeled mobile robot. But a lot of them out there are, in fact, differential drive robots. Typically, they have the two wheels and then a caster wheel in the back. and the way these robots work is you have the right wheel velocity that you can control and the left wheel with velocity that you can't control. So, for instance, if they're turning at the same rate, the robot is moving straight ahead. If one wheel is turning slower than another, then you're going to be turning towards the direction in which the slower wheel is. So, this a way of actually being able to, to make the robot more round. So, let's start with this kind of robot and see what does a robot model actually look like. Well, here's my cartoon of the robot. The circle is the robot and the black rectangles are supposed to be the wheels. The first thing we need to know is what are the dimensions of the robot. And I know I've said that a good controller shouldn't have to know exactly what particular parameters are because typically dont know what the friction coeficcient is. Well, in this case, you are going to need to know two parameters. And one parameter you need to know is the wheel base, meaning how far away are the wheels from each other? We're going to call that L. So, L is the wheel base of the robot. You're also going to need to know the radius of the wheel, m eaning how big are the wheels? We call that capital R. Now, luckily for us, these are parameters that are inherently easy to measure. You take out the ruler and you measure it on your robot. But these parameters will actually play a little bit of a role when we're trying to, to design controllers for these robots. Now, that's the cartoon of the robot. What is it about the robot that we want to be able to control? Well, we want to be able to control how the robot is moving. But, at the end of the day, the control signals that we have at our disposal are v sub r, which is the rate at which the right wheel is turning. And v sub l, which is the rate at which the left wheel is turning. And these are the two inputs to our system. So, these are the inputs, now, what are the states? Well, here's the robot. Now, I've drawn it as a triangle because I want to stress the fact that the things that we care about, typically, for a robot is, where is it, x and y. It's the position. And which direction is it heading in? So, phi is going to be the heading or the orientation of the robot. So, the things that we care about are where is the robot, and in which direction is it going? So, the robot model needs to connect the inputs, which is v sub l and v sub r, to the states, somehow. So, we need some way of doing this transition. Well, this is not a course on kinematics. So, instead of me spending 20 minutes deriving this, voila, here it is. This is the differential drive robot model. It tells me how vr and vl translates into x dot, which is, how does the x position of the robot change? Or to y dot, which is how is the y position, or phi dot, meaning how is the robot turning? So, this is a model that gives us what we need in terms of mapping control inputs onto states. The problem is, that it's very cumbersome and unnatural to think in terms of rates of various wheels. If I asked you, how should I drive to get to the door, you probably not going to tell me how what v sub l and v sub r are, your probably g oing to tell me don't drive too fast and turn in this direction. Meaning, you're giving me instructions that are not given in terms of v sub l and v sub r, which is why this model is not that commonly used when you're designing controllers. However, when you implement them, this is the model you're going to have to use. So, instead of using the differential drive model directly, we're going to move to something called the unicycle model. And the unicycle model overcomes this issue of dealing with unnatural or unintuitive terms, like wheel velocities. Instead, what it's doing is it's saying, you know what, I care about position. I care about heading, why don't I just control those directly? In the sense that, let's talk about the speed of the robot. How fast is it moving? And how quickly is it turning, meaning the angular velocity? So, translational velocity, speed, and angular velocity is how quickly is the robot turnings. If I have that my inputs are going to be v, which is speed, and omega, which is angular velocity. So, these are the two inputs. They're very natural in the sense that we can actually feel what they're doing which, we typically can't when we have vr and vl. So, if we have that, how do we map them on to the actual robot. Well, the unicycle dynamics looks as follows, x dot is v cosine phi. The reason this is right is, if you put phi equal to 0, then cosine phi is 1. In this case, x dot is equal to v, which means that your moving in a straight line, in the x-direction, which makes sense. Similarly for y, so y dot is v sine phi and phi dot is omega because I'm controlling the heading directly or the, the, the, the rate at which the heading is changing directly. So, this model is highly useful, we're going to be using it quite a lot which is why it deserves one of the patented sweethearts. Okay, there is a little bit of problem though because this is the model we're going to design our controllers for, the unicycle model. Now, this model is not the differential drive wheele d model, this is. So, we're going to have to implement it on this model and now, here we have v and omega. These are our, the, the control inputs we're going to design for. But here, v sub r and v sub l are the actual control parameters that we have. So, we somehow need to map them together. Well, the trick to doing that is to find out that this x dot, that's the same as this x dot, right? They're the same thing. This y dot is the same as the other y dot. So, if we just identify the two x dots together, then divide it by cosine 5, we actually get that the velocity v is simply r over 2, v sub r plus v sub l or 2v over r is vr plus vl. So, this is an equation that connects v, which is the translational velocity or the speed, to these real velocities. And we do the same thing for omega. We get this equation. So, only l over r is vr minus vl. Now, these are just two linear equations, we can actually solve these explicitly for v sub r and v sub l and if we do that, we get that v sub r is this thing and v sub l is this other thing. But the point now is, this is what I designed for, this is what I designed for. So, v and omega are design parameters. l and r are my known measured parameters for the robot, the base of the robot, meaning how far the wheels are apart, and the radius of the wheel. And with these parameters, you can map your designed inputs, v and omega, onto the actual inputs that are indeed running on the robot. So, this is step 1, meaning we have a model. Now, step 2 is, okay, how do we know anything about the world around us?
