The infinite horizon line model has been based on some significant assumptions. Viola-
tion of these assumptions will affect the accuracy of the attitude determination process. The effects that these violations have on the attitude measurement are investigated in
the following subsections. The assumption violations which are investigated are:
1. Limited View Distance (Violation of the infinite view distance assumption)
2. Curved Horizon (Violation of the straight horizon line assumption)
3. Spherical Earth (Violation of the planar ground assumption)
4. Terrain Effects (Violation of the flat planar ground assumption)
5.4.1 Limited View Distance
One of the main assumptions with this model, is the assumption that the horizon is
infinitely far away and that the view distance is infinite. This assumption allows the
vanishing line of the ground plane to be calculated. If the view distance was fixed at
finite distance, then the attitude of the horizon would be a function of the altitude of the aircraft. The horizon would be effectively pitched down, sitting lower in the image
than it should be. For a forward facing camera this would bias the pitch attitude
h
θ
θ
hθ
measR
θmeas= θ + arctan ( h R ) ∴ lim R→∞θmeas= θ ∴ lim R→cθmeas= θ + arctan ( h c ) = θ + θhFigure 5.1: Horizon Pitch Bias
If the view range R is very much larger than the altitude h such that R ≫ h then the pitch bias effect will be negligible. For pitch bias of less than a 1◦ then the ratio must be (h/R) < 0.0175. This means that at an operating altitude of 3, 000 ft a minimum
view distance of 57 km is required.
The infinite horizon line method cannot account for this horizon pitch bias unless
fused with another attitude sensor or multiple cameras are used to resolve this horizon
pitch bias. This leads on to the investigation of a multiple camera attitude determina-
tion method in Chapter 6, which fits a horizon plane rather than a horizon line. This
method permits the identification of the horizon pitch biases.
A secondary effect of the non-infinite view distance is the curvature of the horizon. This is discussed in the next subsection.
5.4.2 Curved Horizon
The apparent horizon is not a straight line but it is curved. This can easily be seen
with the human eye, looking out over the ocean. This effect is due to a combination of altitude, finite view distance and the curvature of the Earth effects.
When a curve drawn on the ground plane (with a finite view radius) is projected
into the image plane, it forms a straight line in the image as long as the camera is
on the ground. When this ground curve is observed from a non-zero altitude, the
curvature of the line becomes apparent in the image. Figure 5.2 demonstrates this
altitude dependency effect.
Ground Plane h
Image Plane
Figure 5.2: Horizon Plane Curvature Effects
malised image coordinates as:
ˆ
u = tan(ψr)
ˆ
v =−h
Rsec(ψr)
The effect of this can be seen in Figure 5.3, where different curves for different alti-
tude/range ratios are plotted. The straight line corresponds to the infinite horizon
case, such that R→ ∞. As the altitude to range ratio starts to become apparent, the horizon drops inside the image and the amount of curvature increases.
−25 −20 −15 −10 −5 0 5 10 15 20 25 −25 −20 −15 −10 −5 0 5 10 15 20 25 Azimuth (deg) Declination (deg)
Horizon Curvature Effects
(h/R) = 0.05
(h/R) = 0.15
(h/R) = 0.25 (h/R) = 0
The effect of this horizon curvature is to introduce biases into the straight line
fit. This would feed through and bias the attitude determination process. The pitch
declination at the horizon point ψr = 0◦ follows the horizon pitch bias relationship as
described in the subsection above. The maximum declination deviation at the image
edges would introduce a slightly larger pitch bias than compared to the point ψr = 0◦.
The main effect that this horizon curvature could have on the straight line fit apart
from the pitch bias, is to effect the slope of the line fit as the line could be fitted
tangential to any point along the curved horizon. The slope of the horizon curvature at any point can be calculated with:
ϕh = arctan ( ∂ ˆv ∂ ˆu ) = arctan ( −h Rsin(ψr) )
For a 45◦ FOV camera and an altitude to range (h/R) ratio of 0.25, this maximum
possible horizon bank bias would correspond to 5.4649◦.
5.4.3 Spherical Ground Model
Infinite horizon line models assume that the ground is a flat plane. Since the Earth
is roughly spherical and not a flat plane, this will introduce errors into the attitude
measurements. Again the main effect of this will be a horizon pitch bias with the horizon sitting lower in the image because the Earth is sloping away from the camera
at large distances. This effect will be magnified at higher altitudes. The shape of the
Earth introduces a horizon pitch bias effect similar to the effects outlined in the above
subsections. Assuming the Earth is a sphere, then the introduced horizon pitch bias
h θh
Re
Re
O
Figure 5.4: Spherical Earth Pitch Bias
The induced horizon pitch bias can be calculated by the equation:
θh = arccos ( Re Re+ h ) (5.41)
where Re is the radius of the Earth (6371 km) and h is the altitude.
For a 1◦error induced by the spherical world effect, then the corresponding altitude
h would have to be 970 m or 3, 184 ft. While this effect would be negligible for small
platforms (such as UAVs or MAVs), it can become apparent for medium to large civilian
aircraft. For example, for a very high altitude of 10, 000 ft for a general aviation light
aircraft, this horizon pitch bias would be 1.77◦. If the altitude and the horizon profile
is known, then this can be accounted for, otherwise again this horizon pitch bias must
be estimated at high altitudes by fusion with another attitude sensor or by using a multiple camera horizon method.
5.4.4 Terrain Effects
As it has been stated before, the ground has previously been approximated as a flat
plane. If the ground is not flat, then the terrain can effect the shape of the horizon.
If the terrain variations are large compared to the altitude of the camera, then terrain
occlusions can occur, obscuring the distant horizon. Since the apparent horizon would
no longer be straight in this situations, any line fitted will be influenced by the terrain shape. Hence this would introduce biases in the attitude measurements. No account
or approximation of these effects can be undertaken without a valid terrain model.
eliminating any attitude biases. This terrain-aided approached is developed later on in
Chapter 8 and it can account for all of the previously identified sources of error. This
makes the terrain-aided model the most accurate horizon based attitude determination
method.
An estimate of the maximum horizon pitch bias which could be introduced at low
altitudes can be calculated from the VFR operating conditions [1]. At low altitudes, the
minimum flight visibility is 5 km while the minimum terrain clearance or Lowest Safe
Altitude (LSALT) is approximately 1, 500 ft. The corresponding maximum horizon pitch bias for this condition is 5.2246◦.