of wt is denoted by Qt, and the noise terms in different time steps are assumed to be uncorrelated to each other (such as,

E[wi wj]=0 if i≠j) and to x0.

For estimation, we have a random variable x0|0 that captures our belief about the likelihood of different states at time t=0, and two random variables xt|t− 1 and xt|t at each time step t = 1, 2, … that capture our beliefs about the likelihood of different states at time t before and after fusion with the measurement, respectively. The mean and covariance matrix of a random variable xi|j are denoted by and ∑i|j, respectively. We assume (no bias).

Prediction essentially uses xt− 1|t− 1 as a proxy for xt− 1 in Equation 32 to determine xt|t− 1 as shown in Equation 33.

a priori estimate and denoted by . The a priori estimate is then fused with zt, the state estimate obtained from the measurement at time t, and the result is the a posteriori state estimate at time t, denoted by . This a posteriori estimate is used by the model to produce the a priori estimate for the next time step and so on. As described here, the a priori and a posteriori estimates are the means of certain random variables; the covariance matrices of these random variables are shown within parentheses each estimate in Figure 6b, and these are used to weight estimates when fusing them.

We first present the state evolution model and a priori state estimation. Then we discuss how state estimates are fused if an estimate of the entire state can be obtained by measurement.

Finally, we discuss how to address this
problem when only a portion of the
state can be measured directly.

State evolution model and prediction.
The evolution of the state over time is
described by a series of random vari-
ables x0, x1, x2,…
• The random variable x0 captures
the likelihood of different initial
states.

• The random variables at succes-
sive time steps are related by the
following linear model:
( 32)

Here, ut is the control input, which is assumed to be deterministic, and wt is a zero-mean noise term that models all the uncertainty in the system. The covariance matrix

Figure 8. Computation of a posteriori estimate.