[MUSIC] Welcome to week four, the last installment of our robotics MOOC. In this week, we will bring you from what we understand today about composition of behavior to the horizons of research. Where we can begin to achieve behaviors that we imperfectly understand but have some hope of understanding well enough to become truly compositional as if we have a language of programing work. In the first segment, we'll be thinking about the abstract idea of dynamical composition, we'll come back to this ideas of sequential uncomposition, sequential and parallel composition that were introduced in the very original first week and we'll treat them a little bit more carefully both from the point of view of mathematics and of energetic design. Here's an example of a RHex like hexapod seen from the side and the sagittal plane, kinematically behaving just as if it were the pogo stick on the left. And here's an instantiated version of RHex in the physical world, running through the grass. Acting on the grass just as if it were a pogo stick, even though it doesn't look anything like one. You can see in slow motion when RHex is properly tuned, it does begin to look like a pogo stick from the point of view of its mass center. We'll be talking about these ideas of anchoring the templates in the bodies in 4.1. In 4.2, we'll introduce you to a family of compositions that we have been practicing for a while, but only recently do we begin to think we have the mathematics to underlie them formally. We'll start by talking about why parallel composition is hard. We'll then discuss the minotaur robot that you've seen before and the jerboa robot that you've seen before. As increasingly complicated compositions of simple pieces. You remember the Raibert hopper from before, and you remember the rimless wheel from before. We're going to put these two together to build a two dimensional hopper, and then we're going to put them together with that inertial tail reorienting style behavior that we showed you earlier to get the minotaur. We're going to do the same thing to the more complicated jerboa body. And we can begin to get mathematical proofs of the empirical behaviors that you see revealed in these videos. The last segment of this course will bring you to the horizons of research of what we understand. Here, we actually get the minotaur to bound around in a stable manner using simple compositions of vertical Raibert style hoppers in ways that we can almost begin to prove are correct. We can use the same compositions in different bodies. We can use different compositions in the same body. The last piece of this week will be spent thinking about the jagged edge of what we understand, which are these transitional behaviors such as Aaron Johnson's leaping Rhex. And Aaron Johnson's scrambling Rhex, where you can see if we could get this down right, we would be able to hugely increase the operating capacity and the rescue abilities of machines like Rhex. Welcome to the final installment of our mobility component of this robotics MOOC. In this week four of the mobility effort, we will be talking about the beginnings of programing. What we do and don't understand, and what sorts of behaviors we can get in the physical world with what sorts of mathematical guarantees.