The angular rate, and magnetic field sensors are relatively free
of dynamic perturbations. However, the accelerometer triad has fairly
complex dynamic perturbations. The attitude estimation algorithm
assumes that the values reported by the accelerometer triad represent
the gravity vector, or "down". Unfortunately, _{ }
does not consist only of the gravitational acceleration. It also
includes accelerations produced by vehicle accelerations and motions
sensed at the location of the accelerometer triad.
The acceleration measured at the sensor due to the motion of the
vehicle can be described by letting _{ }
represent the location of the accelerometer triad with respect to the
vehicle center of rotation. Additionally, let _{ }
represent the angular velocity of the vehicle, and _{ }
represent the linear velocity of the vehicle center of rotation. It
then follows that the linear velocity of the accelerometer triad
is

_{}. 
(4.30) 
where the first term describes the motion due to the vehicle linear velocity, and the second term is the velocity of the accelerometer triad due to the angular rotation of the vehicle. To find the acceleration at the location of the sensor, take the first derivative of (4.30).

_{} 
(4.31) 
Since the location _{ } of the accelerometer is constant (the accelerometer is fixed with respect to the vehicle),

_{}. 
(4.32) 
This finally gives

_{} 
(4.33) 
as the acceleration due to the motion of the vehicle measured at
the location of the sensor. The first term _{ }
describes the linear acceleration of the vehicle expressed in vehicle
coordinates. The next term _{ }
represents the tangential acceleration caused by the angular
acceleration of the vehicle. The final term _{ }
represents the centripetal acceleration, and is directed from the
position of the sensor, towards and perpendicular to the axis of
rotation. There is no Coriolis acceleration term since the
accelerometer triad is fixed with respect to the vehicle.
The state estimator assumes that these vehicle accelerations are
small compared to the gravity vector, so that the total measured
acceleration may be assumed to be due solely to gravity. This vector
is then assumed to be "down" by the state estimator. For slow
accelerations and rates, this produces reasonable results. However
for faster accelerations and rates the perturbations can clearly
affect the accuracy of the vehicle attitude estimate. For example if
the vehicle is hanging from a crane and is swung back and forth, the
acceleration vector will continuously point straight out of the
bottom of the vehicle, and the state estimator will not realize that
the vehicle is tilting back and forth as it swings.