PREDICTION |
Initial State Vector
The state vector contains some of the parameters needed to describe the
system. For kinematic applications its main components are usually position
(and perhaps velocity). Additional parameters can be included, depending
on the task and data at hand. At time t0, the state vector and
its variance-covariance matrix are represented by 
and Q.
Knowledge of the initial state vector may have been derived from measurements,
or it is the result of a previous (filter) adjustment.
Kinematic Model
To express the kinematic behaviour of the system between two epochs, a kinematic model is defined by a linear relation between the system parameters at two different epochs:
 |
(7.4-1) |
where:
 |
is the state vector at time t0, |
| x | is the state vector at time t, |
| F | is the transition matrix, and |
| w | is the system noise. |
The noise term w expresses the possible deviations from the defined trajectory of the vehicle between two epochs and is assumed to have mean zero and variance-covariance Qww. The term "dynamic" is also commonly used, but "kinematic" is preferred here, because the underlying forces governing the behaviour of the system do not appear explicitly in the model.
Parametric Expression
Implicit in the kinematic model is the capability to estimate the state vector at future epochs. If x has been estimated at time t0, as w has mean zero, the predicted value at time t is simply:
 |
(7.4-2) |
According to the Law of Propagation of Variances, and assuming that and
w are independent, the variance-covariance matrix of the
predicted estimate is:
 |
(7.4-3) |
The predicted estimate of the state vector can be regarded as an observation
of its true value, with random error vx and weight P
= Q-1:
 |
(7.4-4a) |
It is tempting, and usually possible, to choose 
= , thus
reducing eqn (7.4-4) to 
= - vx, however
such a substitution can be considered as a particular case of a more general
result. Nevertheless, eqn (7.4-4) can be slightly modified:
 |
(7.4-4b) |
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© Chris Rizos, SNAP-UNSW, 1999