If the inputs to the vehicle attitude and angular velocity
estimators are truly a vector pointing down, a vector in the north
down plane (between north and down), and a vector describing the
angular velocity, then the results produced by the state estimation
equations are completely accurate. Unfortunately, several factors
affect the accuracy of the data provided to the state estimator.
Among these factors are noise, sensor bias, and dynamics that cause
the sensors to measure phenomenon other than the effect that you are
trying to isolate.

When estimating angular velocity, errors in angular rate measurement
add directly to the actual value. When measuring attitude, the effect
of sensor error is somewhat more complex. Error in the measurement of
the gravity vector directly affects _{ },
and then later causes errors in the determination of _{ }
(4.13). These errors are then propagated through the calculation of
_{ }
(4.14). Errors_{ }
in the determination of the gravity vector that are tangential to a
circle surrounding _{ }
(Figure 4-1) perturb the attitude angle _{ }
calculated about that axis by

where _{ }
is the length of the gravity vector.

Figure 4-1 Errors in determination of gravity vector tangential to
circle about _{ }

Since the acceleration and magnetic field vectors are not
orthogonal, errors in the measurement of the gravity vector that are
tangential to a circle surrounding the magnetic field vector perturb
the angle calculated about the magnetic field vector by

where _{ }
is the angle between the magnetic field vector, and the gravity
vector. Notice that the effect is amplified due to the small angle
between the two vectors. The noise has the smallest perturbing effect
when _{ }.

Figure 4-2 Errors in determination of gravity vector tangential to circle about the magnetic field vector

The magnetic field measured by Ranger NBV is only about 18°
from the acceleration vector. This causes the resulting angular
errors to be amplified from the orthogonal case by a factor of

Errors in the determination of the earth's magnetic field vector also
produce errors, but due to the structure of the state estimation
equations, their effect is limited to only one axis. Errors in the
determination of the gravity vector_{ }
that are tangential to a circle surrounding _{ }
(Figure 4-3) do not perturb the attitude angle calculated about that
axis since they do not modify the direction of the unit vector _{
}
calculated in (4.13).

Figure 4-3 Errors in determination of magnetic field vector
tangential to circle about _{ }

Again, since the acceleration and magnetic field vectors are not
orthogonal, errors in the measurement of the magnetic field vector
that are tangential to a circle surrounding the gravity vector
perturb the angle calculated about the gravity vector by

This is the same equation as above however d is now the length of the
magnetic field vector. As before, _{ }
is the angle between the magnetic field vector and the gravity
vector.

Figure 4-4 Errors in determination of magnetic field vector tangential to circle about the gravity vector

The following three subsections detail several of the known measurement errors that lead to the estimation inaccuracies described above, and the methods employed to attempt to eliminate them.