Navigation Basics and
Definitions
The navigation algorithms employed by the Navigator
system are based on the concept of spherical trigonometry which
is known as Great Circle Navigation. Here is a quick overview
of these concepts, and some useful definitions and conversions.
We all understand that the Earth is a sphere, and any point
on that sphere is described by a latitude (north-south) and a
longitude (east-west) coordinate. Confining our discussion to the
northern hemisphere, the range of latitude is 0 degrees at the
equator to 90 degrees at the North Pole. Zero degrees of
longitude by definition starts in England and proceeds westbound
around the Earth for 360 degrees. Navigational distance is
defined as a nautical mile (nm) which is 1/60 of a degree of
latitude in distance, known as a minute. A difference of one
minute of longitude at the equator also equates to one nm.
Here is where the fun starts!
Meridians of longitude begin to merge proceeding north and
converge at the pole. Thus, as we proceed north, the distance
between minutes of longitude decreases by an amount based upon
the cosine of one’s latitude. Envision a triangle on the Earth’s
surface joining any three cities in the world at differing latitudes
— however, our triangle has curved not straight sides due to the
curvature of the Earth — our favorite sphere! The interior angles
of this spherical triangle do not necessarily sum to 180 degrees as
in a plane triangle, and it does not have to be a right triangle.
Consequently, the shortest distance between two of those cities
follows an arc — not a straight line — and is known as a great
circle route; one which has a small, but continuous change of
heading between any two points on the arc.
So, standard trig functions such as sin, cosine, and tangent
do not accurately apply when measuring distance and heading
(degrees measured clockwise from north). We must employ
so-called spherical trig functions arc sin, arc cosine, and arc
tangent, which are supported on scientific calculators and by most
programming languages. These functions apply to calculations
involving spherical (curved) surfaces.
The standard unit of measurement in navigation work is the
radian which is defined as the angle subtended by a circular arc
of unit length and unit radius. All internal computations in my
programs are done using radians. Here are some useful
conversions between units we understand and radians:
1 radian = degrees (π/180) = 0.017453
1 degree = radians (180/π) = 57.29577
1 nm = 1,852 meters (along an arc!)
1 nm = ((180* 60)/π)*distance expressed in radians
1 radian = (π/(180* 60))*distance expressed in nm
1 minute = 1 nm
1 degree = 60 minutes or 60 nm
It should be noted that the great circle navigation formulae
are very accurate over short distances and small changes in
heading. The Navigator converts degrees and distance to
radians internally — so, don’t worry about having to deal with
this yourself!
location any time in
the future. It’s a free
download from
Trimble Navigation
at
www.trimble
.com/ mgis.shtml.
I have several
improvements planned
— it’s hard to avoid
“feature creep” when
there is still PIC
memory available!
The major area of
planned upgrade is the
steering module. I hope
to make the rudder
corrections linear to provide a smoother ride down a leg,
and there is a plan to incorporate a compass sensor module
(a PWM device) to allow the waypoint major course
changes to be anticipated and smoothed.
The hardware can also be consolidated — as
mentioned, I used what I had on hand. I plan to combine
the PWM functions with the GPS board, and possibly
reduce the entire project to a single board using a
PIC18F4620 in the TQFP package with a 16F876A
coprocessor.
Next time, we will take a look at how to construct and
connect the modules that comprise the Navigator, and take
a detailed look at each software module. SV
SERVO 07.2009
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