In lecture, we claimed that in the linearized equations of motion for the quadrotor, the second derivative of position is proportional to U1, and the fourth derivative of position is proportional to U2. In this segment, we'll derive these relationships explicitly. Recall the quadrotors equations of motion, which come from the linear and angular momentum balances. We can use U1 to represent the total thrust applied and U2 to represent the moment vector. We could then rewrite the equations of motion in terms of U1 and the components of U2. At the equilibrium hover configuration, the position and yaw angle of the quadrotor can be at some arbitrary value r0 and sin0 respectively. However, the angles theta and phi, as well as r dot, theta dot, phi dot and sy dot are all zero. We want to derive expressions for the equations of motion when the quadrotor is near this equilibrium configuration. First, let's consider the value of cosine theta near the equilibrium configuration when theta equals zero. Around the theta equals 0, the function cosine of theta can be approximated using the Taylor expansion which is shown here. We can consider all the terms after the first two terms in this series to be negligibly small. The value of cosine at theta equals 0 is 1. The derivative of cosine is negative sin theta. Since sin theta, a theta equals 0 is 0. The approximation of cosine theta near theta equals 0 is simply 1. Can you confirm this qualitatively? By looking at the value of the cosine function, near theta equals 0. We can plot the cosine function for angles negative 30 to 30 degrees. We can see that the cosine values are indeed close to 1. We can repeat this process to approximate the function sin theta near theta equals 0. Again, we use the Taylor series. For sin theta we arrive at the following expression. Since sin theta, a theta equals 0 is 0, and the derivative of sin theta is cosine theta, we can simplify the expression as follows. Since the value of cos(theta) at theta=0 is 1, the value of sin(theta) near theta=0 is approximately theta. This suggests that around theta = 0, we expect the sin function to look linear. If we again plot the values of sin theta for angles from -30 to 30 degrees, we see that the sin function indeed looks linear. We will use these two approximations to linearize the equations of motion of the quadrotor. Again, we want to use this approximations to help direct simplified versions of the equations of motion that apply when the quadrotor is near the equilibrium hover configuration. First, consider the linear momentum equation. We can explicitly write the rotation matrix in terms of the Euler angles. We can then perform the matrix multiplication to arrive at the following second order differential, which at equilibrium, the pitch angle theta and the roll angle phi are both approximately 0. Therefore we can use the equation we found earlier to approximate the sins and cosines of these angles. Substituting in these approximations reduces the differential equations to the ones shown here. You can clearly see that the second derivative of position is proportional to the input, u1. Now consider the relationship between the angular velocity components p, q, r and the first derivatives of the Euler's angles. Again, we carry out the matrix multiplication to arrive at three equations to relating p, q ad r to phi dot, theta dot and psi dot. We can then substitute in the approximations for sins and cosines of theta and phi. This will resolves in the equation shown here. Next, we can approximate all terms that are the product of an angle and an angles derivative as 0. Near Hover, theta and phi in all angular derivatives are close to 0. The product of two terms near 0 will be very small. And as a result you can approximate these terms as 0 itself. Substituting these approximations into the equations, tells us that around hover, the angular velocity components are approximately the time derivatives of the Euler angles. Finally consider the angular momentum equation. First, we approximate the off diagonal inertia terms as close to 0. This allows us to simplify the inertia matrix. Performing the matrix multiplication gives us the following set of equations. We saw earlier that around Hover, p, q and r are approximately phi dot, theta dot and sin dot respectively and are therefore also close to 0. The pattern of any two angular velocity components can then be approximated as 0. This gives us the following set of equations. Using again the approximation of the angular velocity components as the Euler angle derivatives, we arrive at this set of second order differential equations. Now let's go back to the linear momentum equation in the x direction. We can differentiate this equation twice. We can then substitute the approximations of the angular momentum equations of motion into the expression for the fourth derivative of x. Omitting terms that do not contain the second derivative of an Euler angle, we arrive at the following expression for the fourth derivative of x. Carrying out this procedure in y and z directions yields similar equations. You can see that the fourth derivative of position is indeed proportional to u2.
