In the following we'll zoom in on a little piece of magic that is built-in our HSIs. I call it the "lubber line method". In general I don't see a lot of advantages of an HSI over our "traditional" VOR "heads", but the lubber line method, which cannot be flown with a normal VOR head and which guarantees perfectly smooth intercepts, without ever over- or undershooting or causing us to yank the plane, is an extremely powerful reason for using an HSI -- at least to intercept radials.
This is in particular important if you are intercepting an ILS final approach course localizer, and not just a VOR at great distance at slow speed. As the ILS localizer is very narrow (and flight instructors try to impart in their students to never fly the ILS too slow, lest the wind will bustle us around and increase a potential proclivity for needle-chasing), it's very easy to yank the plane due to needle sensitivity and to overshoot or undershoot and then "wiggle around". Such oscillations are terrible for your passengers, and they impose additional strain on your plane (do you want to be a connecting rod or Cessna bungy in your plane when the pilot does heavy needle-chasing? Neither do they!). You aren't supposed to fly your plane in a manner that causes the needles to resemble windshield-wipers. And mind you, when flying IFR you are supposed to limit your bank angles to 30 degrees (25 with FMS), another reason to avoid sharp turns as they come too late. Use the lubber line method shown below, and your intercept will use only small bank angles and be smooth, steady, reliable, and exact.
If your plane doesn't have an HSI or glass cockpit and you would like a flexible low-cost solution, consider one of the many mobile/hand-held GPS-based solutions, such as Android tablets with appropriate apps (for example Garmin Pilot for Android tablets, such as the Nexus 10), or other GPS-based systems, such as Garmin's GPSMap 696, or several others. Until this date (20130720) ForeFlight on the iPad STILL does not have an HSI. On Android you have Garmin Pilot and Fly-Is-Fun (and probably others that I am not aware of) that all have HSIs, and Garmin (and others) have numerous aviation GPS units with HSIs. Here is what the HSI looks like on Garmin Pilot on Android:
Sorry, I don't need much glass cockpit beyond that, a simple and relatively inexpensive large Android tablet does it for me. Furthermore, a GPS-based HSI has the advantage that we can use this method on ANY waypoint/fix/airport, not just VORs. If we want to intercept a particular NDB bearing, for example on an NDB approach, we can use our GPS-based HSI. You can't do this with your physical HSI. You can approach an airport with a particular bearing, for example runway alignment, with this method using your glass HSI. All these are reasons to embrace either a real glass cockpit or a tablet-based HSI or a tablet-like aviation GPS, such as the GPSMap 696. But I don't want to get sidetracked on glass cockpit methods here, and this glass cockpit item should not mislead us into believing that the HSI lubber line method was anything new: it's not. This method has been used ever since the HSI has been around. I just find that many pilots, including young instructors, are not aware of it, and with webMathematica we have the tools to explain the math in detail, hence this article.
What's a lubber line? The wikipedia says:
A lubber line is a fixed line on a compass binnacle or radar plan position indicator display pointing towards the front of the ship or aircraft and corresponding to the craft's centerline (being the customary direction of movement).
The glossaries of the PHAK and IFH merely say:
Lubber line. The reference line used in a magnetic compass or heading indicator.
Here it is on the HSI:
Note the white line at the top of the HSI.
We get the same depiction on an electronic flight display / glass cockpit HSI, such as the G1000 system (it's the white marker at the top):
The cleverness of the lubber line method now is the following. As you approach the radial to be intercepted (inbound our outbound), you initially don't have a "live needle" yet (your position radial is more than 10 degrees off the target radial), so you fly a heading which corresponds to a somewhat more "aggressive" intercept angle: 30 degrees, 45 degrees, depending on how you fly it may even be 60 or 90 degrees (frankly, I see nothing wrong with 60 - 90 degree intercepts if you are sufficiently far away from the station and not too fast -- as long as they are smooth and steady and not yanky!). But let's assume you come with 45 degrees. Once you get a live needle, which means you are 10 degrees away from your target radial, the lubber line method comes into play. Now all you need to do is keep the upper end of the CDI constantly "at" or "below" the lubber line. Think of an imaginary extension of the lubber line towards the center of the HSI. All you need to do now is adjust your turn in a manner that just keeps the top of the CDI exactly on the lubber line (or its imaginary extension towards the center of the HSI), and you'll fly a perfectly smooth intercept that neither overshoots nor undershoots nor causes you to yank the plane. It's nicely "rounded out" for a perfect intercept that is safe and imposes minimal strain on your plane and your passengers. Yanking, overshooting, and undershooting are all very bad in my book, and again comes my reminder here that when flying IFR you are supposed to limit your bank angles to 30 degrees (25 with FMS). With the lubber line method your bank angles will be relatively shallow as you prevent sharp turns, and as you level out, you're dead on!
Let's look at the math, which will explain us why this works. Let's call the distance the CDI has from the centerline d. On the HSI it's a distance (as the CDI is horizontally offset from the centerline), but that distance does, in fact, represent degrees on the course deviation scale. Let's call the number of degrees we're off the target radial beta (note this is in degrees now). Let's call half the length of the CDI needle f. Let's call our intercept angle alpha. As additional overlays to an HSI picture would get too busy and I'm in the process of practising some hand-drawing skills, let's consider this scan of a hand-drawn figure I made:
Note that the top of the CDI needle is exactly on the lubber line (or its extension), I have marked it with a small circle. And we can clearly see that tan alpha = beta/f, or alpha = arctan (beta/f). As f is a constant number (a fixed length), we see that the intercept angle by using this method only depends on the inverse tangent of the angle between our target radial and our current position radial:
with +/-90 degrees being the asymptotes (red).
Now all we need to do is compute a new intercept angle alpha based on our new position radial, which we can compute with simple trig from our new position (beta = ArcTan(y/x)), with x and y being the x- and y-coordinates in cartesian (latitude/longitude) coordinates. We do this 50,000 times for a small step length of 1/10,000. We start with an initial intercept angle of 45 degrees, we assume that half the CDI needle length is roughly twice the distance of a full deflection of the CDI, we know that a full scale deflection of the CDI is 10 degrees, and we also assume, just to fix numbers, that our initial position where we first have a live needle (10 degrees) is 10 nm East and 1.76 nm North of the VOR (as that will give us 10 degrees).
This is what we get. Here is a plot of 50,000 steps (points) having a step length of 1/10,000. Thus, at a distance from the station of about 10 nm, one step roughly corresponds to 1/1000 of a mile, or 6 ft. A pilot cannot possibly readjust his heading every 6 ft., but I want to show how precise this method is (reducing all deviations to the inability of the pilot to constantly keep the top of the CDI on the lubber line) as the intercept angle reduces from the initial 45 degrees to 0 while the position radial (thin black line, 10 degrees off, =beta) converges to the target radial.
The following zooms in on the interesting intercept area (we drop the last 4 nautical miles to the VOR station):
Is that smooth and exact or what? As we near the target radial, we become more and more parallel with it, our "offness" decreases dramatically, and we never yanked, and we didn't overshoot nor undershoot. Note that the smooth intercept curve is totally independent of your actual speed, as the computed intercept angle only depends on your position radial. Of course, your actual ground speed DOES affect how fast / how well you can control the turn and command the plane so as to ensure the top end of the CDI is on the lubber line. If you are fast and close, you will most likely "lag" the plane a bit, but at a greater distance and slower ground speed, it shouldn't be too hard. If you can't master that, you'll do even worse with any other method you try as you yank the plane around without ever establishing yourself in a stable manner on the target radial. I also want to point out that the initial intercept angle (45 degrees in our example) is relatively unimportant, as alpha only depends on beta (and f, but that is constant). You could, in theory, come with an intercept angle of 90 degrees, but then you should turn pretty quickly to get the top of the CDI needle under the lubber line. Then you proceed as if you had originally come with 20, 30, 45, or 60 degrees. In fact, I claim, the lubber line method is the best way for you to cope if you were really coming with a 90 degree intercept angle (which would be pretty bad ATC vectors or poor math on your part).
At higher speeds and shorter distances to the station: compare that with the alternative: fly your intercept without using this method. You will end up making just guesses and will overshoot and/or undershoot and yank the plane around. If you are fast or close, you will end up yanking the plane, just to ensure you won't overshoot the target radial. To prevent that, you use your foresight and thus you may start yanking early, resulting in undershooting. Either way, you will do one or two or three of the following: yank, overshoot, undershoot, all three of which are, as mentioned before, pretty bad for multiple reasons. And just because you'll lag the plane a bit, doesn't mean this method is inexact. As your steering is a dynamic process, you will automatically and subconsciously adjust your steering, as you have constantly positive feedback on your HSI (to remind you, the HSI is a performance instrument in both the control/performance as well as the primary/secondary method of IFR basic attitude flying, with minimal lag/latency).
Speaking more mathematically, what we have here is what is called a differential system, and your brain (together with your visual perception and steering commands through your hands) acts as a resolver for this differential system. As pilots we do that all the time as we fly our planes, we just don't notice it. We have a HUGE number of differential systems in our planes and navigation strategies, for example in our avionics and in our pitot-static system, as well as many other mechanical parts. Whenever we navigate, our brains resolve differential systems, but we are used to that so much that we don't think about these things explicitly as differential systems. The moment you steer something "larger" or "more intensely" because some type of error or "offness" is getting too big, we have a differential system, and we resolve that with our brains. An autopilot is a wonderful example of an extremely powerful differential system resolver that is just more agile than we are. We use the same terminology here: intercept, track, cross-track, and gain component. And the lubber line method represents, at its core, also a differential system.
I suggest you do the following:
- Practise the lubber line method in a sim
- Practise radial intercepts without using the lubber line method in a sim (do something else, whatever you want)
- Compare the two
- Increase your speed and decrease your distance to the VOR and keep practising. Your intercepts will get better and better with the lubber line method and worse and worse without, as you increase the problem difficulty.
Wilhelm Thaller also has a very good description of the method with outstanding graphical depictions in his book "Never Get Lost". Scroll to page 4:
Lubber Line Method in: Wilhelm Thaller: Never Get Lost
You'll see that the top of the CDI is always kept "under" or "at" the lubber line. And the subsequent pictures show that by using this method your intercept angle becomes progressively smaller and smaller, as you are forced to turn the plane to a smaller intercept angle as you get nearer the radial.
In closing I point out that the ratio of the lengths
of the CDI needle and the maximum distance to the centerline shown on the HSI is a crucial figure here. This is a key figure in the computation of alpha. Unless there is a binding standard (that I am not aware of) for the manufacturers that produce HSIs (or the EFD software) they can make the CDI needles longer or shorter and/or use larger/shorter distances for maximum CDI deflection, the ratio of which will be relevant for alpha. The larger the ratio CDI needle length / max deflection distance is, the more sensitive this method is, and vice versa. I don't have several HSIs at my disposal that I could use to measure these distances/lengths myself with a ruler, but please be aware the sensitivity of this method highly depends on that ratio in the particular
HSI that is being used. On the HSI shown near the top of this page this ratio is about 0.8, but other instruments may have a different ratio. YOUR turn sensitivity depends on YOUR HSI!
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Copyright (c) 2013 Andreas Lauschke. All Rights Reserved.