Homing cross track error
updated 9/24/2004
Disclaimer
I hacked some of this out rather quickly. I think that it is correct, but I have not checked it thoroughly.
What is homing?
After corresponding with a reader, Peter Bell, I got curious about the magnitude of the effects of homing. Interestingly enough, the discussion was not about flying, but about kayaking. Although currents that a kayaker has to deal with are much smaller than winds a pilot deals with in an absolute sense, the problems can be similar when the relative speeds of the vehicles are taken into account.
Homing is a problem for boating or flying. The problem occurs when you aim for a point and cross currents and/or winds push you off course. As you are pushed off course, you continually must adjust your heading so that you are aimed at the point. The good news is that you will eventually reach the aim point. The problem is that you will travel along a curved path to get there. Depending on circumstances, this varies between bad style and serious mis-navigation.
It is always interesting to find that it is possible to misapply new technologies to commit old errors. I have heard of people committing this navigational error using GPS for guidance, which I will get to. Some of the traditional ways in which people end up homing instead of tracking.
- Aviation: Following the ADF needle.
- Aviation: Centering the OBS needle on the VOR, flying the heading from the OBS and repeating the process as the aircraft homes to the station.
- Aviation or boating: Visually flying directly to a point by heading directly for it in a crosswind. I have never navigated a boat with radar, but I would imagine the same principal applies for navigating with radar.
There are a variety of ways to properly maintain a track as opposed to homing depending on the method of navigation, most of which I will leave for other discussions. However, one of the points that I make in my two books, Basic GPS Navigation and Cockpit GPS, is that by matching TRACK to BEARING you will travel a straight course to a waypoint and avoid this problem. Unfortunately, many people end up using their GPS incorrectly to home instead of track. This will occur if you take the GPS BEARING and steer it as a heading. This can also happen in a boating environment if you are using a GPS with a built in electronic compass. With such a GPS, the GPS will switch to using the compass if you are below a certain speed, usually about 10 knots. This threshold speed can be adjusted in the setup menu so that the GPS will give an indication of TRACK from the GPS receiver rather than HEADING from the compass sensor.
What are the effects of homing?
I think that it is uncontroversial to say that homing is not the best way to navigate. What I have not seen is a discussion of just how much of an effect it has -- some examples with numbers. Thus, I have wrote a spreadsheet to play with the problem.
I had wanted to solve the problem analytically as it properly should be done. Not being able to do so, I resorted to an incremental approximation of the phenomenon using an Excel spreadsheet. In the mean time, Ed Williams responded to a newsgroup posting with some formulas that I attempted to derive but was unable to do so. If I get the time and interest, I may get around to redesigning the sheet and this page in light of Mr. William's work. In the mean time, I have hastily updated my sheet to more accurately calculate the coordinates using Mr. William's forumulas.
The spreadsheet is available in Microsoft Excel format at:
If you get a request for a password, you should be able to press CANCEL and load the sheet.
Inputs:
Inputs are in blue.
Distance is the total length of the leg being analyzed. The units are not important as long as you use units consistent with the speeds.
TAS is True Airspeed. If you are boating, this would be the speed relative to the water with no current considered.
Wind Speed is the speed of the wind or current.
Wind Direction is the direction relative to the nose. Thus, 0 is directly on the nose and 180 is directly on the tail, 90 is directly from the side. (This has been locked out since I updated the x,y formulas. It is locked at a 90 crosswind)
Results:
Tracking results for comparison:
These are the results that you would get if you would properly track a straight line from start to finish for comparison with the homing case.
As we start out from the beginning (x=0, y=0) we are aimed directly towards the finish point. We head in a direction so that the HEADING matches the bearing, BRG. As we travel along this incremental leg:
x is the distance from the start point along the original course.
y is the distance from the original course line. In GPS terms, this would be XTK or OFF COURSE.
BRG is in degrees and measured relative to the original course.
Vx is the velocity along the original course line.
Vy is the velocity perpendicular to the course line.
V is the speed along the path, this is groundspeed if you are a pilot.
Track is the angle in degrees of the incremental leg relative to the original course line.
Total time is the time in minutes from the start at the point.
Leg time is the time for the incremental leg beginning at this point.
Total distance is the distance traveled along the path. It is the sum of the previous legs up to this point.
Leg distance is the distance along the incremental leg from the point.
As we get to the next incremental point, we turn so that we are aimed towards the finish again and travel the next incremental leg.
Ed William's Formulas
Below are Ed William's formulas. Ed has a fantastic collection of Aviation Formulary on his site.
almost twenty years ago, was on exactly this subject, in response to a
discussion between Bob Dubner and Barry Schiff.
Suppose an airplane flying at unit speed starts homing on the origin
of the (x,y) plane, starting at (1,0) in a crosswind of u. The
equations of motion are
dx/dt = -x/sqrt(x^2 + y^2)
dy/dt = -y/sqrt(x^2 + y^2) + u
with x-1, y=0 at t=0
so
dy/dx = (y - u sqrt(x^2 - y^2))/x
You can verify the the solution of this ODE is:
y =(x/2) * (x^(-u) - x^u)
and that for the homing to succeed, we must have u < 1 (less
crosswind than airspeed!)
From this we can derive a couple of interesting results:
(1) The time to home is 1/(1-u^2), which we can compare to the time
to track, which is 1/sqrt(1-u^2). Reverting to dimensional units, we
can say that it takes longer by 1/sqrt(1 - (xwind/TAS)^2) to home than
to track in a direct crosswind.
(2) The maximum cross-track displacement (where dy/dx=0) is
y_max = (1/2) ( ((1-u)/(1+u))^(1/2u -1/2) - ((1-u)/(1+u))^(1/2u
+1/2) )
A great deal more numerical resolution is required to get accurate
results from your spreadsheet for other than small u, particularly
near the origin (homing point), where the track ends up coming in at
right angles to the course, however small u (but non-zero) may be.
Ed
http://williams.best.vwh.net
Derivations:
Here is my old explanation. The problem with this is that I was unable to solve the differential equation to get a direct formula for y=f(x). Thus, I resorted to an incremental approximation using the spreadsheet. Considering that the need for precision was not critical, it actually worked reasonably well.
These are not the full derivations. However, this should get you an idea of how I figured things.
Direction of travel is left to right.
Vx and Vy are just the velocity of the boat in the respective axis. Which is the sum of the component from the no current speed of the boat and the current along each axis.
Vx=dx/dt= no_current_speed * (d-x)/sqrt(y^2+(d-x)^2) – current * cos (current_angle)
Vy=dy/dt= no_current_speed * y/sqrt(y^2+(d-x)^2) – current * sin (current_angle)
dy/dx=Vy/Vx
As an approximation: Yn=Yn-1 + incrent_of_X * dy/dx
BRG=atan (y/x)
Feedback
I hope you find this interesting and useful. I welcome your feedback here: Feedback
John Bell 9/21/2004