Polynomial models represent the resultant aerodynamic forces and moments at the CG as polynomial functions of state variables and control variables, or values derived from these. The coefficients of the polynomials are constant parameters, defined to fit the aerodynamics as closely as possible for the flight conditions of significance. The polynomials can have terms in more than one variable.
One of the amazing things about airplane aerodynamics is that, for all of its inherent complexity, a mostly linear polynomial model can be fairly accurate for a wide range of flight conditions. Certain phenomena cause the relative linearity of the airplane to break down; accurate prediction of aerodynamic forces and moments in these situations require a higher-order model. The two most important phenomena that require the higher-order models are stall (high angles of attack) and drag divergence (high Mach numbers). But in flight regimes not involving stall or drag divergence, a mostly linear model can be quite accurate.
In order for a polynomial model to work well, it must do three things. First, the model must calculate aerodynamic moments about the reference point (or some other convenient fixed point). Calculating the moment about the non-fixed CG requires the polynomial model to incorporate CG location as a variable, an unnecessary complication. After the moments about the fixed point have been calculated, the moments are transfered to the CG for use in the equations of motion.
Second, the model must calculate forces and moments in wind axes.
After the forces are determined in wind axes, they are rotated to body
axes using Equation 43.
Third, the model must use nondimensionalized forces and moments.