How many significant digits should I use for simulating aircraft equations of motion?

Question

I have coded the equations of motion in Matlab, and I am using the ode45 function to simulate the nonlinear dynamics of F/A-18 Hornet without any controller (just the airframe). I have given initial conditions as trim conditions.

d2r = pi/180;           % degree to rad

V       =  438.653328;  % Airspeed , ft/s
beta    =  0*d2r;       % Sideslip Angle, rad
alpha   =  10*d2r;      % Angle-of-attack, rad

p       =  0*d2r;       % Roll rate, rad/s
q       =  0*d2r;       % Pitch rate, rad/s
r       =  0*d2r;       % Yaw rate, rad/s

phi     =  0*d2r;       % Roll Angle, rad
theta   =  10*d2r;      % Pitch Angle, rad
psi     =  0*d2r;       % Yaw Angle, rad

pN = 0;                 % X position in Earth Frame, ft
pE = 0;                 % Y position in Earth Frame, ft
h = -25000;             % Z position in Earth Frame, ft

During the first iteration, C_m has a very positive small value (1e-10), which leads to some pitching moment when multiplied by S, c, and qbar (even though it's small). But, as time keeps on increasing, this value of C_m goes on increasing and as a result, a large moment is created. Due to the high precision of Matlab, the pitching moment coefficient is not for the given trim condition.

My question is: should I round off the values of C_m to 3 or 4 digits? By rounding off, the C_m will be zero and I get M as zero.

The below figure shows how velocity is varying with time when the aforementioned initial conditions are given.

This figure is obtained after rounding off C_m to 4 significant digits.

This figure is obtained without rounding off C_m to 4 significant digits.

Alpha and theta values without rounding off C_m.

Answer 1

From your description, it's not an issue with the aerodynamic coefficients. If as you have described previously that you're modeling F-18, then the airframe should be unstable longitudinally. A 1e-10 deviation (assuming it applies to body rates, body speeds, etc. as well) is well-within typical trim error bound; in fact, a deviation is unavoidable when dealing with computational floating point. But if it's longitudinally unstable, any deviation will magnify over time exponentially.

Answer 2

Computers in these days do not care about "significant decimal digits" for computation as they use binary IEEE-754 representation internally. The typical number sizes are float (most often 32 bits in these days) and double (most often 64 bit). C offers 80-bit extended precision (long double) and sometimes 128 bit double is available. You can read about sizes here. In addition, some programming languages and packages can do arbitrary precision but it is usually slow.

There are often "corner cases" in model that may require unusually high precision. For instance, the expression (a-b)/(c-d) may be very tough to compute when a gets close to b while c gets close to d. Because of such cases, I usually prefer to use the highest precision available (least long double) in simulations. It may even be corner cases that require complex type arithmetic, even if both input and result are of the real type (sqrt(-1)**2).

Smaller sizes are often not even faster. They sometimes make sense if you work with data matrices so large that memory becomes a problem, of if you use a gamer GPU card for computing that does not feature double precision.

For human readable reports, it is common to leave enough decimal digits to represent the value with known accuracy, or one more, but not longer. For instance, if the value is known with the 0.1 accuracy, it could be printed as 0.6 or 0.66 for instance, but not as 0.6666666666666666. But if this text output is not for humans and will be read back by computer, there is no reason to limit the number of digits. Also, if you know more digits, rounding can only do the same or worse.