The foundation of the model lies in a system of five differential equations that describe the forces and moments acting on the vehicle. While a more accurate model would use a system of six equations, accounting for all three moments, we have simplified this model by making a few assumptions and arbitrary coordinate choices. First let us note that because we are using rotation matrices to describe the orientation of the vehicle in three-space, we must specify the Euler angles in an abitrary sequence (recall that matrix multiplication is not commutative). Also note that the general geometry of an airplane is symmetric about the roll axis, with the exception of stabilizers, and thus is not as susceptible to moments induced by relative wind in comparison to similar induced moments about the pitch and heading axes. Therefore, we make the simplification that we only need consider moments about the roll axis produced by pilot input. We also make the assumptions that we will be working with fairly realistic input and that the moment of inertia about the roll axis is small in comparison to its pitch and heading counterparts, such that a pilot has nearly explicit control over the roll of the vehicle, again assuming realistic input. Thus, we can furthur our previous simplification by specifying the roll angle as the last angle in the Euler angle sequence and allowing said roll angle to be explicitly determined by our pilot. This reduces the number of moments produced by relative winds that we must calculate from three moments (pitch,heading,roll) to two moments (pitch,heading). Again, this may be an over-simplification of the model for more advanced maneuvers, but it appears to produce realistic results for typical input. It also appears to produce fairly realistic results for some more advanced pilot input, including full loops and barrel rolls.

We also make some corrections for the dynamic drag and lift as proportionally determined by the inner product of the vehicle's motion vector and its thrust vector.

Once the initial conditions of the system (position,velocity) and the pilot input functions (thrust,heading,pitch,roll) have been defined, Mathematica will numerically solve the system and produce a plot of the calculated trajectory much like the plot shown below.



This is a plot of the vehicle performing a simple rolling maneuver. While the plot may appear to look a bit unrealistic, it is just not to scale, notice the values along the bounding box edges.

Of course, now that we have the trajectory information we can grab the airplane shape (from a previous notebook), apply it to the trajectory, and then animate the system to create a visual representation of the plane traversing the model path.



This notebook also provides custom functions for exporting the images of the animations directly to the file system.