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For critical systems, redundancy is built into the system. It is common knowledge among designers/architects to have three different inputs so in case one is faulty, input from the remaining two can be used to find (and shut off possibly) the faulty one (two against one).

Why is not every airplane equipped with 3 angle-of-attack sensors and triple modular voting when these devices are critical for the safety?

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    $\begingroup$ The million dollar question $\endgroup$ – Ben Mar 14 at 6:25
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    $\begingroup$ Do you know any aircraft with 3 AoA sensors? $\endgroup$ – DeepSpace Mar 14 at 8:41
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    $\begingroup$ @DeepSpace: A320 has three, then again when two fail, the system thinks the third (good) one has failed. $\endgroup$ – ymb1 Mar 14 at 9:28
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    $\begingroup$ Not only A320, all airbus aircraft have 3 AoA sensors. A350 even has additional 4th AoA sensor. $\endgroup$ – Mad Max Mar 14 at 10:49
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    $\begingroup$ @Sean The question isn't about the crashes (especially as edited) so I don't think this is off-topic. $\endgroup$ – David Richerby Mar 15 at 10:40
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Triple redundancy is necessary to detect a fault and exclude it. The system then continues to operate through the fault. Double redundancy is used to detect a fault but cannot exclude it, so the system stops operating. The important fact is that the faults they actually detect are identical.

Stall events are rare and are normally not expected in flight. There is no immediate hazard if handling augmentation or stall warnings are disabled. Therefore, there is no need for triple redundancy.

Simply put, if the system detects an AoA discrepancy, it can simply trip off and stay off until it is repaired on the ground.

If the double redundant system is ideally designed, then only a simultaneous fault will escape detection. Note too that if the same simultaneous fault occurs to two sensors in a triple redundant system, then it will also escape detection because it will outvote the correctly operating sensor. Therefore, both systems share the exact same failure mode.

Double and triple simultaneous faults can and do occur with common causes including environmental factors (AF 447), maintenance errors (XL 888), and birdstrikes (US 1549). It also allows faults in the voting logic (QF 72). Both recent AF and XL fatal accidents are signs of an overreliance of buying 3 of the same box and then calling it "safe".

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    $\begingroup$ "only a simultaneous fault will escape detection" Only if they fail in the same way. Suppose two sensors, one reading -1 and the other reading +1; you can know they are reporting different values, but without additional knowledge, you cannot know which one is correct. In the case of three sensors, if the value set is (+1, +1, -1) you can reasonably (see XL888T) conclude that the -1 is in error; however, if the value set is (+1, 0, -1) you cannot conclude which, if any, of the values are right or wrong. $\endgroup$ – a CVn Mar 14 at 15:37
  • $\begingroup$ Whether faults are truly independent is another matter. Multiple AOA sensors are likely to have been made by the same factory (and probably around the same time), maintained by the same mechanics and are flying through the same air. That radically increases the odds of common mode failures, which will defeat any redundancy system. $\endgroup$ – StephenS Apr 3 at 15:06
  • $\begingroup$ @aCVn While true, AoA sensors failing simultaneously will almost certainly fail in much the same way: the only way they can realistically fail is through blockage (eg ice or debris). Ice buildup, for example, is likely to impact similar sensors in a similar way (see: AF447) $\endgroup$ – Jon Story Apr 3 at 15:06
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Two AoA sensors are more reliable than three!

Let's have a look at probability calculation, and assume the fault probability of one sensor to be p = 0.1 % (per flight, or whatever you like to choose). The probability of the same sensor to work as expected is q = 1 − p = 99.9 %.

Two Sensors

The probability for

  • no fault: q2 ≈ 99.8 %
  • a discrepancy (1 fault): 2 p q ≈ 0.2 %
  • an undetected double fault: p2 = 10-6

Three Sensors

The probability for

  • no fault: q3 ≈ 99.7 %
  • 1 recovered fault: 3 p q2 ≈ 0.3 %
  • undetected faults: 1 − q3 − 3 p q2 ≈ 3 · 10-6

Which solution is preferable?

Autonomous system

If we were talking about an autonomous system, like a drone or maybe a satellite, we would be looking at the ability of the system to take a decision on its own.

A decision cannot be taken with

  • 2 sensors if a discrepancy or a double fault occurs. The probability for that is 0.2 %.

  • 3 sensors if more than 1 fault occurs. The probability for that is 3 · 10-6.

3 · 10-6 is 667 times better than 0.2 %. The autonomous system is better off with three sensors and TMR voting.

Aircraft with pilots

The situation is different if the system is monitored by a pilot, who can intervene in the case of a discrepancy. A false positive alarm is acceptable. Undetected faults are not acceptable. The likelihood for an undetected fault is 1 · 10-6 with 2 sensors, and 3 · 10-6 with 3 sensors. The 2 sensor system is 3 times more reliable under this premise!

In addition, a single fault is more obtrusive in the case of the 2 sensor configuration. A single fault with three sensors - if noticed at all - is more easily ignored instead of being eliminated.

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    $\begingroup$ Ah good ol' Lusser's law. $\endgroup$ – ymb1 Mar 14 at 10:10
  • $\begingroup$ Logic in answer seems little odd to me. Why probability of sensor working in 3 sensor setup is q^3 and not 1-p^3? Isn't it that all 3 need not to be working for getting AoA information, only 1 is sufficient to do the job? $\endgroup$ – Mad Max Mar 14 at 10:40
  • $\begingroup$ @MadMax: q^3 is (by definition) the probability for all 3 sensors working = no fault. 1-p^3 is the probability for 0, 1, or 2 faults combined. In a TMR configuration, 2 sensors are required for a correct results. 1 is not sufficient. $\endgroup$ – bogl Mar 14 at 11:49
  • $\begingroup$ Originally gave +1 but after few days thinking realised that this is actually wrong. The three sensor case does not work like that. The system does not aggregate the ok/fail state, if it has that data it will just ignore failed sensors. What it aggregates is the actual measurement. So the relevant statistic for undetected is actually the probability of two sensors failing the same way and giving the same false measurement. Usually this also needs to happen at the same time as the system would generally ignore the sensor that it thinks is giving false data. (cont.) $\endgroup$ – Ville Niemi Mar 21 at 2:43
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    $\begingroup$ Dear @VilleNiemi, you are right that my answer does not reflect the situation in the B737M. Instead it covers one theoretical aspect. To describe the scenario, it is not sufficient to count the sensors, but it matters how they are evaluated. The subject question keeps changing and adapting to the incomplete knowledge we have about an ongoing accident investigation. This should be avoided. Do you have reliable information about the B737M design? I do not. I consider moving this answer to a different question (to be created). $\endgroup$ – bogl Mar 21 at 8:23

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