Next, we'd like to start thinking about how to create agile robots. Robots that can start from a position of rest, accelerate pretty quickly to a maximum speed, stop when it sees obstacles, and then accelerate again to a maximum speed. You see this in the video clip that you're seeing. This is a manually piloted robot and you can see this is an expert pilot that's able to drive the vehicle really quickly through fairly complex environments, and we'd like to create autonomous robots that can do exactly this. Let's see what it means to stop from a configuration where the robot is going at maximum forward speed. First, the robot is pitching forward when it's going at maximum speed. When you decide to bring it to a position of rest, you must pitch it backward, reversing the direction of thrust so that you get deceleration. This means that the robot must be pitching back at an aggressive angle. This will cause the robot to slow down. But as a result, the thrust factor which now points in a direction other than the vertical direction will also cause the robot to loose height because the component of thrust in the vertical direction is now less than the weight. As we maximize agility, we really want to be thinking about minimizing the stopping distance. The other thing worth exploring is the robot's ability to turn quickly, now I want you to think of this robot flying forward at maximum speed and then turning as quickly as possible. What we'd like to do here is to minimize this turning radius which you see denoted as ro. In both these examples, stopping from maximum speed and turning at maximum speed, is actually sufficient to consider a fairly simple model of a quadrotor. What you see here is a diagram of a vehicle in the vertical plane. The propellers apply a thrust and the sum of these two thrusts, actually four thrusts for a quadrotor, is the vector you want. That vector you want now has two components, one in the horizontal direction and one in the vertical direction. The difference of the thrust contributes to the moments and that's u2. If I write down the equations of motion in the plane, essentially, I get three equations of motion that describe how the components of the thrust, u1, and how the turning moment, u2, accelerate the robot in the yz plane, and also turn the robot in the direction of the pitch angled feet. Again, you have two accelerations, linear, denoted by a, with components in the y and the z direction and angular, denoted by alpha, which obviously has only one component and this is the rotation in the plane. The two key ideas are that you wanna accelerate quickly and you wanna roll and pitch quickly. To accelerate quickly, you wanna maximize acceleration, I denote that by a sub-max. And in order to roll and pitch quickly, you wanna maximize alpha sub-max. The first quantity is the linear acceleration. The second quantity is the angular acceleration. To maximize the first quantity, you want to maximize the ratio of u1 to W. In other words, take the maximum thrust, divide that by the weight, and maximize that ratio. If you think about the second quantity, that you can maximize by taking u2, which is the turning moment, maximize that divided by the moment of inertia along the x-axis. We've developed a very simple simulation that illustrates this. You will see the robot starting from a maximum forward speed and effectively slamming on the brakes but again, the brakes are slammed on by creating a reverse pitch which generates a reverse thrust, and that slows the vehicle down. One of the things you can do is then calculate the stopping distance for different acceleration rates. Here we show two curves. One at 5 meters per second squared, again, roughly half the acceleration due to gravity. The second is 10 meters per second squared, again, roughly equal to the acceleration due to gravity. In both cases, we've essentially used a dynamic simulation to create a graph of the stopping distance with respect to the maximum velocity the robot starts off with. As the robot travels with a larger velocity, the stopping distance increases, and clearly, the higher the ability of the robot to accelerate or decelerate, the smaller the stopping distance. These are two curves we have generated that gives you a flavor of what it means to maximize the agility. You want to be able to stop quickly if the vehicle sees an unexpected obstacle, and this is quite critical to maneuverability. To explore this, we've designed a very simple math lab exercise where you're given a simulator, and you're gonna use this simulator to explore this design space of operating speeds, the maximum acceleration and the inertia and the mass of the vehicle. The larger the mass, or larger the inertia, clearly, the larger the stopping distance. Likewise, your ability to accelerate fast is also gonna decrease the stopping distance. The larger the velocity you're going with, the greater the stopping distance and ultimately, you wanna decrease the stopping distance for the same operating speed.