[MUSIC] Welcome to the first week of our special robotic course in mobility. This week we are going to be looking at the background materials, and we'll be motivating the effort. Our first segment will involve motivation and such questions as why, how, where to move. We'll spend a fair bit of time thinking about the animals, and we'll talk a fair bit about bioinspiration. And finally, we'll move toward what it means for robots to take bioinspiration and turn that inspiration into engineering. At that end of that first segment, we'll be looking at animals that don't use their limbs, and realizing that now, most of the animals do use their limbs. And we'll be spending the rest of our MOOC time on limbed and tailed mobility. The second half hour of this week will be background material. Those of you who are very strong on ordinary differential equations and linear algebra will race through this segment. And those of you who are weak are going to find it important to take some time to go through and think about the ideas being introduced, and possibly to do some background reading as well, in order to follow the development of the ideas. Really, it revolves around this simple second-order differential equation that most college students have in year two or year three in engineering, and even in other disciplines as well. We'll be thinking about this second-order mechanical differential equation as a dynamical system. And that change in thinking takes a little bit of getting used to, as exemplified by these figures where on the left-hand side of each panel you see a time trajectory. And on the right hand side, you see the position and the velocity on a plane. And time is collapsed down onto that plane so that the trajectories in time become orbits, so-called curves on the plane. That idea is crucial. We're not really concerned with linear ordinary differential equations, which is what you all learned in your sophomore or junior year. No, no, everything that's interesting about mobility in robotics has to do with nonlinear ordinary differential equations. And we'll spend some time reviewing what's known about nonlinear dynamical systems. The most important ideas that we'll need later have to with energy and basins. Welcome to week one. We'll be spending the rest of the three weeks using these ideas and turning them into marvelous robots, and thinking about how our marvelous robots might become even better.