# Understanding and breaking down pointing requirements

“< 1 deg, 3σ total angle pointing stability” - What does the 3sigma indicate? Does this mean that 99.7% of the total pointing error (which I assume is computed via a sim), is within 1 deg? If it was 1-sigma, would that mean only 68% of the pointing errors need to be within 1 deg? What does total angle mean vs single-axis? Does total angle mean you RSS the x,y,z pointing errors?

• Can you edit your post to provide some context? Commented Nov 7, 2023 at 18:46

In my professional experience in writing spaceflight software requirements, reading them, implementing them, validating them, and verifying them, these "what does that mean" questions come up way more often than anyone ever expects. It turns out that requirements from you-in-the-past are just as mysterious as requirements from other people.

That said:

• yes, your understanding of sigma as being used for the standard deviation percentages is correct in both manufacturing and for other errors like pointing errors (though I really should get an authoritative source. I like that source's phrasing though

This means that 99.7% of all outcomes are within this range of accuracy

So if we do, for example, a Monte Carlo simulation, only 0.3% of our cases should exhibit pointing errors beyond the stated limit.

THAT SAID, if someone said to me "no no, over the cumulative time of this set of simulation runs, we're only pointing outside the limits 0.3% of that time even though all of our cases do violate that limit" ... I would be wary, because the simulation parameters themselves are going to be defined per-case with their own means and sigmas, generally (aside from some variables not assumed to be normally distributed). I don't know if that's a fight I can win on a contract level to say that the requirements weren't met. Hence, it might be a good idea to codify exactly what you mean in the requirement (99.7% of simulation cases do not exceed this pointing error, to be verified by Monte Carlo simulation) instead of relying on the commonly-understood meaning of the words.

Have I been bitten by someone doing this kind of thing before? I can't say, but I have a headache.

• I would expect total angle to mean you're comparing two vectors (e.g. the boresight vector of an antenna and the vector from the a particular part of that antenna to its nominal target) and taking the subtended angle between them, e.g. by solving the dot product $$\vec{a} \cdot \vec{b} = ab\cos{\theta}$$, or but I have explicitly seen it defined as the RSS of three Euler angles. (I agree with David Hammen's comment that this isn't the right thing to do, and there are extra ways to find a single total angle between two arbitrary orientations because of Euler's Rotation Theorem including by representing them as quaternions). Single axis would mean one axis. Again, explicitly defining it in your requirements can save you a headache later.

Note that if you're using Euler angles to specify rotations you should also specify exactly what they mean: these are rotation applied in this order around these axes of this defined coordinate system to end up at this transformed coordinate system. I have seen people try to define them much more casually.

I've just realized that I've been writing this assuming that you are writing requirements instead of interpreting them. If your job is to interpret ambiguous requirements: start asking questions, maybe even push for a revision in the requirements themselves if you can. If you get pushback along the lines of "you should know what this means" a) that's a bad engineering culture and b) try to get agreement right there about what it means and write that down, including who and when is defining it.

• Love this " It turns out that requirements from you-in-the-past are just as mysterious as requirements from other people." Commented Nov 7, 2023 at 22:56
• Minor nitpick: I would never use the RSS of three Euler angles. Calculating the single axis rotation angle error using quaternions is trivial. Given two unit quaternions $Q_1$ and $Q_2$, the angular difference between them is the arcsine of the magnitude of the vector part of $Q_1 \times Q_2^*$ (or $Q_1^* \times Q_2$, or any similar combination). Commented Nov 7, 2023 at 23:58
• @DavidHammen neither would I, but I've seen it in practice at NASA at least once :( Commented Nov 8, 2023 at 0:08
• @uhoh The problem here is that angular error as a single axis rotation is not Gaussian. Similarly, the distance error for hitting the center of the docking port is not Gaussian. Neither the angular error nor the distance error can be negative, and the probability of the error being exactly zero typically is zero (zero is a sample space of measure zero). Some people at NASA have actually read The Handbook and have started writing more specific requirements. (continued) Commented Nov 8, 2023 at 13:45
• @uhoh I ran across the "3σ" requirement multiple times during my career. Such requirements typically do not assume a Gaussian distribution. They simply mean 99.73% of the test cases must pass, most likely not adjusted for a finite sample size. Commented Nov 9, 2023 at 6:13