Is there a manual which describes the algorithmic procedure on which the Vertical Navigation Guidance is based?

Question

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.

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:

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.

Answer 1

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.

Answer 2

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.