Hi, I'm Smriti Chopra. I am a graduate student at Dr. Egerstedt's lab, and I will be your instructor for the Glue Lectures, for the duration of this course, which is Control of Mobile Robots. And just a quick idea: what are these Glue Lectures? Basically, they're going to kind of connect, you know what you learn with Dr. Egerstedt to the quizzes we are going to give you. And give you helpful hints about the quizzes. Clarify, repeat certain concepts. By that I really mean we, you know, work out a few examples, understand the math that goes behind some of the lectures, etc. Just to help you guys move along the course. And the format is going to be one Glue Lecture every week. And the course is seven weeks, so seven Glue Lectures. And on a side note Amy LaViers is a former grad student in our lab, and she was last year's instructor for the Glue Lectures. So we will be reusing some of her lectures. And so in the coming weeks, you know she and I are going to share. Between the two of us, we're going to teach you guys, out of the seven lectures. But whatever questions, etc, you have, you direct them to me, because well she's not here, and I'll be answering those questions for you on the forums, etc. And with that, let's get into the lectures. The first one is Dynamical Models. This week with Dr. Egerstedt, you guys kind of, you know, went through this introductory notion of what a model is. And he tells you that really, a model is basically, you know, something that describes how your system changes or evolves with time. And your system could be a robot, for example, and a model is just going to describe, let's say, how the positions change with time, or how the angle, angles change with time, etc. And the whole idea behind controls is that we're going to kind of try and influence this change to make our system do something we want to do. And we're going to do this by injecting controls. But let's go really down to the basics. What really is a model? And before we do that, let's do a little exercise in derivatives. So here we have, let's say, position x, with the respect to time, is given by this exponential function. Now this is in its general form. I'm saying that let's say me, for example, my position is x of t, and I wake up at time t equal to 0. I'm at position 10. So and I'm saying that my position evolves with time with respect to this function here, and it's given by this graph here. That's okay. That's simple. And now I can take the derivative of this position with respect to time, right? And I get what, what is the velocity. And you'll see this is actually 2x of t, really. And then you can do this further on, and you can take more derivatives, and now we have the acceleration, which is 4x of t. One thing I want to point, point out really quick here is that x dot of t equal to 2x of t, for example. Now what is this? This is really nothing but a differential equation. Basically something that relates a variable x to its derivatives, second, third whatever. So in this case, x dot equal to 2x of t is a differential equation. Something he mentioned in class, right? Okay. So this was just an exercise in derivatives. You guys should be able to take exponentials and their derivatives, etc. No problem. We saw the graphs related to these, as well. But in action, what does it really mean? So I'm telling you that my position with respect to chi, time changes like this: 10e to the power 2t. What it really means is, let's say that's me, or that's a pink ball, whatever. And now I draw this line, which is the x axis. And I start at 10, which is where I wake up, because you guys see, if we're here in this equation, if you put x at time 0, you'll get 10, e to the power 0, which is 10. So at time 0, when I wake up, I'm at 10. Right? And now, the whole point is that let's say at time t equal to 0, like I said, we wake up. Now I say start, right? And then I start moving. And as my time increases, my position is going to keep changing with respect to this function here. And because it's exponential, let's say at time point 1, I'm here. And then I jump really high at point 2, and then even further. because exponentially my position is changing. So, so from graphs, all of a sudden we see how it maps onto, these equations map onto actual motion. That's all I wanted to really show in this. But now, instead, I tell you that, you know what, I'm not going to tell you how x changes with respect to time. I'm instead going to give you this set of equations, which is a differential equation, and then an initial condition where I wake up. And now I want to find what my x of t is going to be, given this, which is x dot of t and x of 0. So in this case what do you do? Well, what you do is, and this is what you did in your lectures as well, just to see how you evolve with res, with time, you discretize the world. And real quick, let's say this is your time axis. And I'm going to divide up this time axis into, you know, delta t chunks, where each chunk is 0.5 seconds, or whatever. So it's very simple. I just divide up the axis. And now, I'm going to say that my t is actually given by this k delta t. This is how you discretize, right? And k becomes a counter. Why? Because when k is equal to 0, my time is 0. And when k is 1, it's 0.5. When k is 2, I'm at one, etc. So k becomes kind of like a counter, right? All right. Now let's see how the ball is moving. So we know that we discretized time. What we want to see is now, as I move k from 0, 1, 2, 3, how does my ball move? And sort of extract x of t from it. Right? Okay. So again, this is our equation in continuous time. So remember I told you I have given you only two things. These are the two things that I've given you. Okay, so I know this guy here, that at time 0, I'm here. And now I have the equation, the differential equation. Well, you can discretize this differential equation through something called Taylor's, Taylor extension, which you guys did with Dr. Egerstedt, which basically is given by this equation here. And a quick note. What it really is doing is, it's kind of saying, x k plus 1 is nothing but x times k plus 1 times delta t. Right? And xk is nothing but x at time k delta t. So it's, by incrementing k, what we are doing is we are finding out at the next time instant, discrete dimension where should I be, depending on my previous time instant. That's all this equation is really doing. A good thing to note here, though, is that we will be using just the first two terms of the Taylor expansion, which is what you guys did in the lectures as well. This expansion is really an approximation, because you see these dot dot dot here it just goes on and on. You keep taking derivatives, and you keep going on, so obviously the more terms you use, the better your approximation of x of t will be. But as of now, we're just going to use the first two terms of the expansion. So let's see. Here we have x of t is 2x, delta t is 0.5, we know this, t is k delta t, and now let's put this entire thing into this equation of ours, the discrete time equation. And we get this guy here, x k plus 1 is equal to 2xk. You guys should do this yourselves. It's really simple. But just, you know, plug in values. You get this one equation. And this is now our discrete time update equation. And how do we use this? Well, we are simply going to say okay, so at k is equal to 0, x of 0 is 0, and now xk plus 1, which is x1, will be 2 times x of 0, which is actually 20, right? So we saw that at k is equal to 1, which is actually time point 0.5, I jumped from here to here. Okay? And now I keep incrementing k. So now I take k is equal to 1, and I calculate my x2, which is now 40. And then I take k is equal to 2, and I increment, and I get x3, which is 80. For all you guys, you guys can kind of see that this is really a linear approximation. It's not changing exponentially, and like we thought it should, because we know that it should change exponentially, but it's not. And this is because, like I said, the more terms you take in the Taylor series for your approximation, the better the approximation. So here, since we took just two terms, it's a kind of linear approximation. If I take more terms and added, you know, acceleration, etc, then your approximation would have been better. And so, what we actually just found out, was nothing more than a dynamical model, and how you kind of solve it to see how your system is evolving. So your differential equation and an initial condition, or just a condition, basically telling you this is where I am at a certain time. These two things are more than enough to find out how your system is evolving with respect to time, and what is your x of t. And in general, general form, this is, how you say, is your dynamical model. Basically, it's an equation x star of t. That is a function of its, you know, state or time. And later on, this is where you put in your control input, you as well, when you guys go later into the course. So you kind of, like I said, influencing the change x dot of t, to make x of t do whatever you want. Right now we're not dealing with control, so it's just simply x dot of t is given by something. And you have an equation here x t star is x star, which is basically saying that at some time, I'm going to tell you where you should be in space. And with just this information, you should be able to evolve x of t, and see how your state is changing with time. So that was pretty simple. Punchline. This is the following dynamical model, which we've been solving all this while. We know how x evolves. But we know how x evolves numerically, right? We discretize the world. And we saw how x should evolve as we move the counter from 1, 2, 3, 4, etc. But we didn't really get the equation out. You know, the mathematical expression for x of t, which we know is the exponential 2t, 10. We know that this was the actual solution, but we really didn't get this equation, pretty equation out. What we got was a list of basically times, and where x should land up, numerical solutions. So if you had to, let's say, get the equation out, what do you do? Well, you integrate. And again, like I said, we know already the solutions. That's our hint. All we are given is this. For all of those, this is not important for the course, but for all those people who want to see how you integrate and get, you know, x of t, we can go over it really quick here. It's really nothing but, of course, an integration. Let's say dx by dt is 2x. You kind of separate the variables real quick. So you say over x is equal to 2dt. Integrate both sides. This is actually your logx is equal to two t plus c, c is your constant. And now this is an equation you've got in which you're going to plug in your initial condition. So log10 is equal to c. And from here you say, okay, so let me put all this back into my main equation. Logx becomes two t plus log10. And now you kind of, you know, shift terms here and there, and you get x is equal to 10e to the power 2t, x of t. So now you guys even saw that, you know, if you don't want to do the whole discretizing the world and kind of simulating where you should be, instead you can integrate. And a word of caution here, that you cannot always integrate and find the solution. Sometimes you have to rely on numerical methods, because analytical solutions may not even exist, you know? You may not get pretty equations for dynamical models, etc. But here, just to kind of show you that, you know, you can do it. Just integrate, and you'll get the answer. Again, it's not important for the course, but woo, we got it. But what is important for the course is that even though maybe, okay, we won't ask you to integrate, what we'll do instead is we'll say, okay, here is your model. And I'm going to give you what I call a candidate solution, x of t should be 10e to the power 2t. And now what you need to do is figure out, is this indeed a solution to the model or not. And for that, there are just two simple steps that you do. First is called the initial condition check. What does that mean? It basically means that I'm given this candidate solution, and then within the solution, I'm going to kind of put in my initial condition x of 0 and see if I actually get 10 or not. And here I do get 10, so it satisfies this equation, my x of t. And then the second check is the differential equation check. And what is that? That is really nothing but okay, again I have my candidate solution, but I'm going to now take the derivative of that candidate solution. And 2e to the power of 2t, which is really my 2 times x, and see if it satisfies this equation. And it does. So this is another technique. So in case, you know, given a model, you don't want to find out through integration or numerical methods what your x of t should be, but instead I'm telling you what it should be, this is how you check or confirm that it is indeed a solution or not. Okay. So the take home message of this entire lecture is really first, dynamical models. What are dynamical models? I hope I've kind of given you guys an idea, and made you sort of comfortable with the idea of, you know, models. And for example, you have something like this, right? Okay, then how do we solve them? We solve, one we can do by numerical solving, which is numerically discretize the world and solve it. And then, another is analytically, through integration, right? And then the third thing, which is very important, is that I give you the model and then I say okay, this is the candidate solution. You check the solution through two checks, the first being initial condition, second being differential equations. And sort of try and verify that this candidate solution is indeed satisfying the model. And this is a homework assignment for you guys. For this model, see if you can numerically, analytically, and this whole checking thing, whether, you know, you can kind of solve it and see if you get this equation or not. And before we wrap up this lecture, there's just one concept more, which is the equilibrium point concept, which I want to introduce. Which is really nothing but basically this concept that if my state x wakes up here at this point, or at this position, or at this value, it stays there. As simple as that. So it's really not anything more than saying that my x dot of t is going to be 0. So to find an equilibrium point, you do simply x dot of t equal to 0, and find the value of x. And like I said, now if, if I was to wake up here, or if at, let's say, time t is equal to 0 I or 3, or whenever I wake up, I am at this equilibrium point. Because my x dot of t is 0 and there's no change, my change is basically 0. I will stay there. So it's really simple. So in our case, in the dynamical model case, x is equal to negative 1 by 3 is your equilibrium point. So if you're asked, you know, find an equilibrium point, simply just put x dot equal to 0, or whatever function you have here equal to 0, and find the values of your state, x in this case, and that will be your equilibrium point. And also, with that check the forums. Good luck with Quiz One. And bye bye.
