1. Navigation Fundamentals

2. Navigation Fundamentals Geometry of the Earth
The Geoid
Mean Sea Level (MSL) – Reference surface for altitude
Gravitational Equipotential surface

3. 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)

4. 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

5. Navigation Fundamentals Geometry of the Earth
There are hundreds of Geodetic Systems around the world
These are a few:

6. 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= m
f=1/
e2 = 2f-f2
gravity g = ( sin2Φ) m/s2

7. Navigation Fundamentals Geometry of the Earth
The difference between the WGS84 Ellipsoid and the geoid is shown below

8. Navigation Fundamentals Geometry of the Earth
Latitude:
Geocentric/Geodetic

9. 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

10. 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

11. 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

12. Navigation Fundamentals Rates of Change
The rates of change of latitude and longitude are therefore:

13. 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

14. 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)

15. 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

16. 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)

17. 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

18. Navigation Fundamentals Coordinate Transformations
TOP
VIEW y y’ X x’
90-λ x

19. Navigation Fundamentals Coordinate Transformationsz
SIDE
VIEW
y’’ z’
90-Φ x’ y’

20. Navigation Fundamentals Coordinate Transformationsy’ N y x’ α E x

21. Navigation Fundamentals Coordinate Transformations
The computations required for this transformation are: E

22. 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)

23. 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

24. Navigation Fundamentals Coordinate Transformations
It is sometimes required to transform from ECEF to Spherical Geodetic Coordinates and vice versa

25. Navigation Fundamentals Coordinate Transformations
It is sometimes required to transform from ECEF to Spherical Geodetic Coordinates and vice versa

26. 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

27. Navigation Fundamentals Dead Reckoning
Definitions:
Note:
Wind Direction is the direction the wind is coming FROM

28. 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?

29. Navigation Fundamentals Dead Reckoning
VE=390
Example:
340
480 VN
-340 50

30. Navigation Fundamentals Dead Reckoning
Example:
ρM= NM h=5.925NM
ρP= NM cos(50)=0.643
dλ/dt = VE/(ρP+h)cos(Φ) = 390/( )·0.643
= rad/hour or 10º/hour
dΦ/dt = VN/(ρM+h) = -340/( )
= rad/hour or -5.6º/hour

31. 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

32. 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

33. 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

34. Navigation Fundamentals Best Estimate of Position
Example

35. Navigation Fundamentals Best Estimate of Position
Example

36. 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:

37. Navigation Fundamentals Best Estimate of Position
This has a known shape:
and thus by measuring a few points, its future values can be forecast

38. Navigation Fundamentals Best Estimate of Position
Kalman Filters

39. 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

40. 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

41. Navigation Fundamentals Bearing and Distance Calculation
x0,y0 ψ BT
x,y BR
x1,y1

42. 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

43. 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

44. 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

45. Navigation Fundamentals Bearing and Distance CalculationΔλ
90-Φ2
90-Φ1 B2 P2 B1 L P1
Note: Bearing from P1 to P2 is B1
Bearing from P2 to P1 is 360º - B2

46. 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