I am interested in the methods implemented by Flight Management Systems in order to calculate a trajectory path of vertical navigation guidance (VNAV), which takes into consideration the legs of the instrument procedures used for a flight plan, among other things such as aircraft's performance characteristics, wind etc.

In other words, a manual or a guide from official sources (for example, FAA, ICAO or Eurocontrol), but other reliable third-party solutions would also suffice.

I am mainly interested in a way to determine the vertical profile of an instrument flight procedure (SID, STAR, IAP) based on its legs and any crossing altitude or speed constraints that are assigned to any of their waypoints.

Can someone point me in the right direction? I am aware that there is not one way of determining the vertical profile of a route, however my guess is that some universally accepted guidelines must exist.

EDIT: I will provide some more information about what I have in mind, based on Noah's contribution. Let's take for example FINNZ2 and HHERO3 departure procedures of KSNA airport. We can see that some of the waypoints have crossing altitudes assigned.

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Can an acceptable implementation of vertical navigation be based on the aforementioned crossing altitudes, by considering them as mandatory altitudes that an aircraft must have while passing from them? This would mean that every such crossing altitude would be considered as "CROSS X AT Y ALTITUDE" restriction, even if it is At or Above, At or Below or Block.

Something like the following vertical profiles:

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In other words, I am trying to figure out if this would be a useful or realistic information that an FMS would also provide.

My other idea would be to take a predefined climb gradient for the aircraft, let's say 500 feet per nautical miles and adjust the vertical path if any waypoint crossing altitude restrictions are violated.

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    $\begingroup$ The equipment onboard the aircraft knows the altitude and position it is located plus the position of the proximate waypoint. Along with the distance to waypoint and groundspeed, it is simple math to calculate the rate of descent/climb to arrive properly. $\endgroup$ – Mike Brass Oct 22 '19 at 23:47
  • $\begingroup$ That sounds like an answer to me... $\endgroup$ – Michael Hall Oct 22 '19 at 23:57

For this answer:

  • Hard altitude requirement - a fixed altitide which an airplane must be at. (eg. 7000ft)
  • Soft altitude requirement - a window of altitudes for the aircraft to be at (eg. abv. 3500ft, between 9000ft and 12000ft)
  • Using a 757 for all examples unless specified otherwise.

A simple FMS may simply assume and use the minimum of a soft requirement as the desired altitude, as you correctly stated.

However, in airliners, it is much more common to use what is called a "Cost Index" (CI) to calculate speeds, climb rates and the like. Cost index can be calculated as so:

CI = (Time cost/hr)/(Fuel cost/lb)

A CI of 9999 means that the FMC will aim for Velocity Maximum Operating (VMO) in every stage of flight. Whereas a CI of 0 will result in the FMS aiming for the following: +=================================+===============+===============+ | Climb | Cruise | Descent | +=================================+===============+===============+ | Minimum fuel to cruise altitude | Maximum Range | Max L/D ratio |

So that will lead to the following speeds: +============+=======+========+========+ | Cost Index | Climb | Cruise | Descent| +============+=======+========+========+ | 0 | 290kn | .778m | 250kn | +------------+-------+--------+--------+ | 9999 | 345kn | .847m | 334kn | +------------+-------+--------+--------+ Now we have climb speeds, we can work backwards to calculate the best climb or descent rate for this speed (I can't find a reliable source for this, so that's where the practical examples end). From here; TOD, TOC and TTC. As long as an FMS can remain at its best CI speed to meet an altitude requirement, it will. Otherwise it will alter best CI speed by the smallest amount possible to meet the requirements.

Some good (debatable) reading is this report by MITRE, an information security organisation. They found that most FMS' tested had very little variation in the time they began descents and climbs, as well as altitudes for a fixed CI. They also found that the RTCA standards for RNP - called RTCA DO-236 where generally adhered to in every tested FMS. That means the following was met:

... tolerances for a flight along a specified vertical path is 160’ for 0’- 5000’, 210’ for 5000’-29000’, and 260’ for 29000’- 41,000

MITRE also says:

If a preceding waypoint is encountered before the altitude of the constraint specified for that waypoint has been reached, then the constraint altitude of the waypoint is adopted for the reference path altitude at the waypoint.

Now, armed with our new CI based climb rates and speeds, my other answer can be used to finish our calculations and work out where (in the Z dimension) the aircraft will reach a soft requirement.

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    $\begingroup$ Apologies for the rambling. If you need anything clarified LMK. $\endgroup$ – Noah Nov 1 '19 at 2:22
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    $\begingroup$ Sorry, that's 77.8% of mach. Mach 0.778. TOD - Top of descent, TOC - Top of climb, Time to climb $\endgroup$ – Noah Nov 2 '19 at 5:09
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    $\begingroup$ My mistake, I should have realised that by myself. If I could accept both of your answers as correct then I would definitely do! :-) $\endgroup$ – Vector Zita Nov 2 '19 at 21:01
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    $\begingroup$ By the way, how did you calculate those speed values? Is there any source/database that can help me calculate speed values based on Cost Index, as you did? In addition, you stated that "As long as an FMS can remain at its best CI speed to meet an altitude requirement, it will". This seems to me to imply that climb/descent rate affects climb/descent gradient (feet per nautical mile). Is there any source that makes this connection (eg for X type of aircraft, optimal climb gradient is this value based on that speed range (climb/descent rates). Apologies if I'm asking something too obvious! $\endgroup$ – Vector Zita Nov 3 '19 at 3:02
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    $\begingroup$ Boeing: boeing.com/commercial/aeromagazine/articles/qtr_02_10/pdfs/… $\endgroup$ – Noah Nov 4 '19 at 22:04

I programmed a virtual FMS for testing of new airways and STARs, and I completely agree with the comment left by @MikeBrass. Ultimately, there is no standard for how FMS systems complete this task as long as they come to the same result.

It is very simple trigonometry and other, high school level, math concepts. The FMS knows the 3D coordinates for the waypoint (We'll call these X2,Y2,Z2) as well as the current GPS coordinates, altitude, true heading, and a myriad of other sensors that it uses to calculate the aircraft's current position in space (X1,Y1,Z1). It also uses the differences in GPS GS and CAS to work out the forward (Or rearward) component of the wind, and the differences in True Heading and TMG to calculate the sideways component of wind. We now have all the information required to complete calculations for nearly any scenario.

Required Crossing Altitude Scenario / Speed Restraints:

In this instance, the first calculation would be one of Altitude to gain (OR loose). This is as simple as Z2-Z1. Now, assuming a fixed (Pilot set speed or speed restrictions based on procedure) speed we can easily calculate a climb rate in FPM. This is done by creating a 2D line that travels between points X2,Y2 and X1,Y1 and calculating distance between those 2 points, along the line. Dividing this by GPS ground speed, and computing heading and GS changes along this leg similar to how a student pilot might do with a "Whiz Wheel". If the pilot changes speed, or the forward component of the wind changes during the climb or decent, you just need to run this calculation again but with the new position of the aircraft in 3D space.

Curved approach segments complicate this slightly but interpreting these as a 2D curve will greatly simplify your calculations.

If this doesn't answer your question, you may want to reword it.

  • $\begingroup$ @Noah thank you very much for your thorough answer. What troubles me is that a waypoint does not necessarily have an assigned crossing altitude, which means that an FMS knows the 2D (Χ, Υ), rather than the 3D (X, Y, Z) coordinates of this waypoint. How do you fill those missing altitude values? In addition, I can grasp the relatively easiness of these calculations but do you have in mind any official documentation that describes the necessary equations? Because other sources mention more complex calculations that also take aircraft performance (eg weight, max CG etc) into consideration. $\endgroup$ – Vector Zita Oct 23 '19 at 12:12
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    $\begingroup$ @VectorZita, If there is no crossing ALT, then the climb is simply considered as a part of the original climb segment. To either the next required crossing ALT or the final cruise ALT. It's just one climb with a turn at some point along it. You certainly are correct, most modern FMS systems to take into account all the available Performance Data. Aircraft performance is mostly used to calculate details that the Autopilot (If coupled) will use to complete the climb, EG, Engine power settings, trim stab settings, ETC. This data really has very little relevance to actual 3D navigation. $\endgroup$ – Noah Oct 23 '19 at 22:30
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    $\begingroup$ I'd think that any information like what you're requesting would be trade secret and wouldn't be available outside an FMS manufacturer or OEM like Boeing. I'd have a look at the ARNIC 429 data format, in addition to DO-178C. ARNIC will tell you what data an FMS will have about a procedure, and DO-178C will tell you all about what an FMS must be resilient against. You might be able to answer some of your questions with these Docs. $\endgroup$ – Noah Oct 23 '19 at 22:37
  • $\begingroup$ Noah thanks again for your feedback, really appreciated. I edited my question in order to clarify one last thing about an idea that I have, based on your contribution. Could you check it out and offer some advice about the demonstared examples too? $\endgroup$ – Vector Zita Nov 1 '19 at 0:46
  • $\begingroup$ Ah okay, I understand more of what you're asking now. I might put it as a separate answer as it really has little relevance to this answer. $\endgroup$ – Noah Nov 1 '19 at 1:10

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