Now back to the Qaudrotor dynamics. Equations at the bottom tell us the neck force and the neck moment. If we combine the neck force and the neck moment, with the Newton-Euler Equations we get these two sets of equations. On the right side, you see the total trust which is u1 and remember this thrust vector is known in the body fix frame. The matrix R is rotating this thrust vector to an inertia frame. At the bottom you see the net moment. Also known in the body fixed frame. But the equations as they've been written have components in the inertial frame on the top, and in a body fixed frame at the bottom. These are the Newton-Euler equations and these are the equations we've used to develop controllers and planners for our vehicles. One question you can ask is, how we actually calculate these parameters. Or in an online setting, how do we estimate these parameters. If you think carefully the parameters you really need to know are those corresponding to the geometry. The length L, for example, and those corresponding to the physical properties, the mass M and the inertia I. You can also see that these parameters appear linearly in these equations. So, if you had a systems that allowed to measured positions, velocities, and accelerations, it's actually not too hard to estimate the lengths, the masses and the inertias. You should also verify that it's quite easy to calculate the angular velocity in the body fixed frame. If you know the pitch, roll, and your angles, and you know the rate of change of the pitch, roll, and your angles on the right side, a simple transformation yields the angular velocity components p, q, and r along b1, b2, and b3. So this model is quite complicated. It involves three components of position, velocity, and acceleration, three components of rotations, angular velocities, and angular accelerations. To get a feel for the control problem I'd like to first look at the Planar version of this model so, we look at the equation of motion in the Y Z plan, we will resume that the robot cannot move out of this plan in other words there is no motion in the X direction and we'll also assume there is no yaw or pitch motions. As a result, we come up with the three equations that you see here, equations that describe the rates of change of velocity in the y and the z direction. And the rate of change of the angular velocity in the row direction. To describe these kinds of systems, it is useful to define a state vector. In the three dimensional case, we have six variables that describe the configuration of a robot. And a state vector includes the configuration and its derivator. So if Q is the sixth dimensional configuration vector, q and q dot constitute a 12 dimensional description off the state. The space of all such state vectors is called a state space. If you look at the equilibrium configuration, And that configuration is defined by the position x=0, y=0, z=0, the configuration by definition also has a zero roll angle and a zero pitch angle it could, of course, have any yaw angle. The equilibrium configuration must also correspond to zero velocity. If I write it down in terms of a state space vector, you have the equilibrium configuration q sub e and you have a 0 velocity vector. And that gives you a 12 dimensional vector where the bottom six elements are all 0. Similarly, for planar quadrotors, you have a three dimensional configuration space, a six dimensional state space, and an equilibrium configuration And an equilibrium state that can be similarly defined.