In the previous segment, we looked at control using a very simple idealized model. What's wrong with that idealized model? The biggest thing that's wrong is that we assumed that the motors were capable of producing whatever thrust the controller required. So in reality, if you look at this model, the motor thrusts are limited, because the motors have a limited capacity. So if I write down this equation again, and look at forces in the vertical direction, clearly the thrusts have to compensate for the weight, and the thrust that exceeds the weight, will produce a quadratic acceleration. But the thrust that the motor can produce is limited by the peak torque. So let's assume that this peak torque is known to us, and that in turn determines the maximum thrust we can produce, T sub-max. This in turn determines the maximum acceleration. If you look at that model, u, the control input, is now determined by the sum of the motor thrust and the weight. And of course, this has to be a vector sum. You have to remember that the thrust points in the opposite direction to the weight. Assuming you know what Tmax is, you can calculate umax by simply taking the maximum thrust and adding it to the weight. Again, remember that the thrust is in the vertical direction pointing up. The weight is in the vertical direction pointing down. And you really need to take the vector sum to get umax. So now when we do PD control, u(t) is determined not just by the proportion of the derivative control law. But also the maximum thrust that can be applied. So if you take the minimum of these two functions, that'll give you the true value of the controlled input that can be applied. The same for the PID control. I'm gonna now show you two videos. The only thing that's different in the two videos is the assumed value of maximum thrust. On the left side, the maximum thrust to weight ratio is 2. And on the right side the maximum thrust to weight ratio is 1.2. So these two videos or simulations illustrate the differences between using two different motors. Or you could ask the question, what happens if you keep the motors the same, but change the payload? Again, it's only the thrust to weight ratio that changes. And you qualitatively get different performances, as you can see in these simulations. Now, what I want you to do is to use the same simulator, using the control law we had before. And study how changing the thrust to weight ratio effects the response of the quadrotor. Change the mass or the payload of the robot and see how the response changes. Using this simulation, you should also be able to determine the maximum payload that the robot can carry. Such that the performance is within acceptable levels.