October 2000, Issue 123

Navigating With GPS

WAYPOINT NAVIGATION

Waypoint navigation is based on great circle navigation. Great circle navigation is general and good for planes, boats, and cars. Waypoint navigation systems navigate via latitude and longitude pairs. The navigation computer accesses a list of latitude/longitude pairs and calculates the distance and bearing from one point to another. Information presented is usually the current distance and bearing from your present position to the next waypoint. Often, a dynamic directional pointer is displayed, which you follow to the next waypoint.

Before navigating, the data from the GPS receiver must be converted to a form acceptable to the great circle navigation algorithms (i.e., the distance and bearing formulae). First and foremost, all of the data must be in radians. This seems straightforward, but there’s a complication. The latitude and longitude data emitted by most receivers is in a form that cannot be directly converted to radians. So, an intermediate latitude and longitude conversion sequence must take place.

All NMEA data is emitted as ASCII data. Latitude and longitude data received from a GPS receiver in NMEA-0183 format is in units ddmm.mmmm, where dd equals degrees, mm equals minutes, and .mmmm is decimal minutes. These units are not appropriate for the distance and course calculations; they must be converted to degrees and decimal degrees, then to radians.

The first step is converting the latitude and longitude data from the form ddmm.mmmm to dd.dddd. This is a straightforward algorithm, but it still takes a substantial amount of code. To determine the algorithm, first separate and save dd from the incoming latitude and longitude string. Then, divide mm.mmmm by 60, resulting in an exponent of zero and a new mantissa, 0.dddd. Third, add the saved dd to the result, yielding dd.dddd.

After you finish converting both latitude and longitude, radian conversion is possible. The formula to convert from dd.mmmm to radians is:

After performing the distance and bearing calculations, the data needs to be converted back to dd.mm.mmmm. This is done by following the formula degrees = radians × 57.2957795. To convert back to the form ddmm.mmmm, save dd, multiply .dddd by 60, and add the exponent to the result, yielding mm.mmmm. Then concatenate the saved dd, resulting in ddmm.mmmm.

That takes care of the latitude and longitude conversions. Now, you can tackle the knots and nautical miles (NM) conversions. All speed and distance data contained in NMEA messages is in terms of knots and nautical miles. One NM corresponds to the traversal of 1 s of arc. One knot is 1 NMph. So, if you’re traveling at five knots (5 NMph), you’ll traverse 5 s of arc in 1 h.

Now, convert knots to miles per hour and nautical miles to statute (land) miles (see Table 2). Remember that the output of the navigation calculations is in radians. The conversion to nautical miles is NM = radians × 3437.7387. Next, you can convert nautical miles to land miles (MI) using MI = NM × 1.150779. Converting from land miles to feet (FT), the formula is FT = MI × 5280.