Navigation Fundamentals
Navigation Fundamentals
Geometry of the Earth
The Geoid
Mean Sea Level (MSL) – Reference surface for altitude
Gravitational Equipotential surface
Navigation Fundamentals
Geometry of the Earth
The Geoid is a very irregular shape.
Need something mathematically simpler to use for
navigation (and surveying).
Use an ellipsoid (an ellipse rotated about the z axis)
defined by:
semi major axis (a)
eccentricity(e) or flattening(f)
coordinates of centre (x,y,z)
Navigation Fundamentals
Geometry of the Earth
Surveyers wanted ellipsoids which closely matched the geoid in
their part of the world so they generated “best fitting” ellipsoids
which minimized the “root square” differences in altitude between
the ellipsoids and the geoid
Navigation Fundamentals
Geometry of the Earth
There are hundreds
of Geodetic Systems
around the world
These are a few:
Navigation Fundamentals
Geometry of the Earth
With the arrival of GPS, a world-wide ellipsoid was
developed. This was called WGS84 (World Geodetic
System 1984)
Its main characteristics are:
a=6378137m
f=1/298.257
e2 = 2f-f2
gravity g = 9.78049(1+0.00529 sin2Φ) m/s2
Navigation Fundamentals
Geometry of the Earth
The difference between the WGS84 Ellipsoid and the
geoid is shown below
Navigation Fundamentals
Geometry of the Earth
Latitude:
Geocentric/Geodetic
Navigation Fundamentals
Geometry of the Earth
Radii of Curvature:
Required to convert linear to angular measurements (displacement
and speed)
Prime radius is the radius of a circle which best fits the vertical
east-west section through the point in question
Navigation Fundamentals
Geometry of the Earth
Meridian Radius of Curvature:
is the radius of a circle which best fits the vertical north-south
(meridian) section through the point in question
Navigation Fundamentals
Geometry of the Earth
These two give us conversions in two orthogonal directions and are
adequate for most applications
Sometimes a more general relation ship is required:
Gaussian Radius of Curvature:
is the radius of a sphere which best fits the ellipsoid at the
point in question
Navigation Fundamentals
Rates of Change
The rates of change of latitude and longitude are therefore:
Navigation Fundamentals
Coordinate Systems
Several coordinate systems have been devised to meet
particular requirements of navigation: The most important
of these are:
Earth-Centred, Earth-Fixed (ECEF) - Cartesian
We saw this in the GPS section:
z axis: Earth’ rotation axis
x axis: joins Earth centre and Greenwich meridian
origin: Earth centre of mass
Navigation Fundamentals
Coordinate Systems
Geodetic Spherical:
z1 = longitude (degrees)
z2 = latitude (degrees)
z3 = height above reference ellipsoid
Used for most long range navigation except that height
is normally height above geoid (MSL)
Navigation Fundamentals
Coordinate Systems
Generalized Spherical :
z = local vertical at origin note: origin could be in
motion
x, y tangent to earth’s surface at origin,
orientation of x axis depends on situation. e.g.
orientation of INS platform
Navigation Fundamentals
Coordinate Systems
Locally Level: (specialized case of Generalized Spherical)
z = local vertical at origin (origin is fixed)
x, y tangent to earth’s surface at origin,
orientation of x axis depends on requirements
e.g. centre line of a runway
Useful over a limited area (to where the error in
elevation becomes critical)
Navigation Fundamentals
Coordinate Transformations
The basic rotational coordinate transform was given in
the section on GPS. To convert from ECEF to Locally
Level (or Generalized Spherical) requires a minimum of
3 rotations
Navigation Fundamentals
Coordinate Transformations
TOP
VIEW
y’
y
X
x’
90-λ
x
Navigation Fundamentals
Coordinate Transformations
y’’
SIDE
VIEW
z
90-Φ
y’
x’
z’
Navigation Fundamentals
Coordinate Transformations
y’
y
N
x’
α
E
x
Navigation Fundamentals
Coordinate Transformations
The computations required for this transformation
are:
E
Navigation Fundamentals
Coordinate Transformations
Multiplied out this is:
Note: Given the values in this matrix, one can find
Φ = asin(C33)
λ = atan(C32/ C31)
α = atan(C13/ C23)
Navigation Fundamentals
Coordinate Transformations
Example:
Φ = asin(C33) = asin(0.707) = 45º
λ = atan(C32/ C31) = atan (0.683/0.184) =atan(3.71) = 74.9º
α = atan(C13/ C23) = atan(0/0.707) = atan (0) = 0º
Navigation Fundamentals
Coordinate Transformations
It is sometimes required to transform from ECEF to
Spherical Geodetic Coordinates and vice versa
Navigation Fundamentals
Coordinate Transformations
It is sometimes required to transform from ECEF to
Spherical Geodetic Coordinates and vice versa
Navigation Fundamentals
Dead Reckoning
Dead Reckoning (or DR) is a procedure for determining
position based on the knowledge of
•True Heading (best available true heading - BATH)
•Mag heading + Variation or Inertial
•True Airspeed (TAS)
•Wind Velocity
It is a predictive technique used in conjunction with
Position Fixing
Navigation Fundamentals
Dead Reckoning
Definitions:
Note:
Wind Direction is
the direction the
wind is coming
FROM
Navigation Fundamentals
Dead Reckoning
Example:
Heading: 135 (T)
TAS: 480kts
Wind Velocity: 50kts at 270(T)
Position: 50ºN 50ºW
Altitude:36000 Ft.
What will be the position of the aircraft in 20 minutes?
Navigation Fundamentals
Dead Reckoning
VE=390
Example:
340
-340
VN
50
Navigation Fundamentals
Dead Reckoning
Example:
ρM=3440.959NM
h=5.925NM
ρP=3451.168NM
cos(50)=0.643
dλ/dt = VE/(ρP+h)cos(Φ) = 390/(3451.168+5.925)·0.643
= 0.176 rad/hour or 10º/hour
dΦ/dt = VN/(ρM+h) = -340/(3440.959+5.925)
= -0.099 rad/hour or -5.6º/hour
Navigation Fundamentals
Best Estimate of Position
Modern aircraft usually have several position sensors.
It is desirable to use all of the information available to
get an estimate of position.
We would like to have a method for combining this
information in the best possible way.
Usually, the information from different sources has
different accuracies and we would like to make sure
that the most accurate source has the greatest influence
on the final result
Navigation Fundamentals
Best Estimate of Position
This is done by weighting the input values with factors
derived from their variances (E((x-m)2) or σ2
Assume we have 3 sources of x position, x1, x2 and x3
whose variances are σ12, σ22 and σ32 respectively.
We want to find x
First we form D = σ12 σ32 + σ22 σ32 + σ22 σ32
Navigation Fundamentals
Best Estimate of Position
The three weighting factors w1,w2, and w3 are formed
as follows
w1= (σ22· σ32)/D
w2= (σ12· σ32)/D
w3= (σ12· σ22)/D
and finally
x = w1x1 + w2x2 + w3x3
Navigation Fundamentals
Best Estimate of Position
Example
Navigation Fundamentals
Best Estimate of Position
Example
Navigation Fundamentals
Best Estimate of Position
Deterministically Biased Sensors (e.g. INS)
In systems like INS, some of the errors are a function of
initial conditions and can be considered deterministic
during the flight. e.g. the INS error due to gyro bias:
Navigation Fundamentals
Best Estimate of Position
This has a known shape:
and thus by measuring a few points, its future
values can be forecast
Navigation Fundamentals
Best Estimate of Position
Kalman Filters
Navigation Fundamentals
Bearing and Distance Calculation
The position calculation give latitude and longitude but
the pilot usually wants to know the direction and
distance to the the next waypoint.
The method used depends on the distances involved.
Short Distances: Assume Flat Earth Model
x=x0 + VEdt
y=y0 + VNdt
where x0 and y0 are the coordinates of the Starting Point
Navigation Fundamentals
Bearing and Distance Calculation
The True bearing and distance to the next waypoint are
then:
BT = tan-1 (x-x1)/(y-y1)
D = (x-x1)2 + (y- y1)2
where x1 and y1 are the coordinates of the waypoint
The pilot usually wants relative bearing to the waypoint
This is BT - ψ where ψ is the heading of the aircraft
Navigation Fundamentals
Bearing and Distance Calculation
x0,y0
ψ
BT
x,y
BR
x1,y1
Navigation Fundamentals
Bearing and Distance Calculation
For longer distances or where better accuracy is
required, spherical trigonometry is used.
Spherical Trigonometry deals with triangles on the
surface of a sphere.
The sides of all triangles are segments of Great
Circles
A Great Circle is the intersection of a sphere with
a plane passing through it’s the sphere’s centre
Navigation Fundamentals
Bearing and Distance Calculation
All sides and angles in spherical trigonometry are
give in terms of angles.
This is because the relationships are true for
spheres of any size.
For a particular sphere, the relationship between
the linear length of the side of a triangle and the
angular length is the radius of the sphere.
s = R·θ
Navigation Fundamentals
Bearing and Distance Calculation
To calculate bearing and distance we have to generate a
triangle.
Two sides are always drawn from the Pole to the two end
points in question.
Their lengths are 90º minus the latitude of the given point
The included angle is always the difference between the
longitudes of the two points
The two sides are always meridians and thus run true North
and South
Navigation Fundamentals
Bearing and Distance Calculation
Δλ
90-Φ2
90-Φ1
B2
B1
L
P1
Note: Bearing from P1 to P2 is B1
Bearing from P2 to P1 is 360º - B2
P2
Navigation Fundamentals
Bearing and Distance Calculation
The distance is converted to linear measure by multiplying the
angle of the side by the Gaussian radius of curvature