Instrumentation & Measurement Magazine 25-1 - 39

measurements from all of the sensors except one are fused into
one pseudo-measurement. Then, the method for calculating
CRLB for two sensors is used. Fig. 2 illustrates a comparison
between the predicted performance (CRLB), the actual covariance
obtained with a bias estimation algorithm, and the RMSE
of measurement bias estimates for a sensor. In this simulation,
measurements are range and angle, where both have biases.
Tracking With Uncertainties
Measurement Biases
In a target tracking system, estimating the measurement biases
is a key aspect of getting precise target location estimates. The
measurement equation with bias can be written as:
z hx b w
k kk k
Fig. 1. PCRLB over time for single target with sensor location uncertainties
(adapted from [4]).
compared to another algorithm. If the sensors are tracked
along with the targets, PCRLB says that we can get better
tracking accuracies of the targets in addition to reducing the
uncertainties in the sensor locations compared to incorporating
measurement bias with the measurement covariance.
Detailed discussion of tracking with sensor location uncertainty
and corresponding RMSE values is presented below.
When estimating measurement biases, usually multiple
sensors are used (a detailed discussion will follow below).
In the case of two sensors and multiple targets, the PCRLB
is calculated in a batch process [3]. For multiple sensors,
where bk
  k
(4)
is the unknown bias, which can vary over time, and wk
Fig. 2. CRLB, covariance and RMSE of bias estimation for a sensor with range and angle as the measurements (from [3],
©2016 IEEE).
February 2022
IEEE Instrumentation & Measurement Magazine
is a zero-mean Gaussian random noise.
Broadly, there are two ways to deal with the measurement
biases: 1) the bias is rectified from the measurement, and an
unbiased measurement is used in the fusion system to get the
estimated target trajectories; or 2) the biased measurements are
used in the tracker, and the bias is estimated with the trajectory
of the targets. For the first approach, the system needs to know
the exact bias information prior to the start of the operation. In
real life scenarios, knowing the complete information about
the sensor bias is impossible. Hence, the second approach is
much more widely used. In most cases, multiple sensors are
needed to estimate the measurement biases. The biases could
be unknown fixed values
or time-varying.
To estimate the measurement
biases, the classical
approach is to include the
bias vector in the state vector
and to use an augmented
state Kalman filter (ASKF).
With multiple targets, the
states of all the targets and
the measurement biases are
stacked in a single vector.
The computational infeasibility
makes it extremely
difficult to estimate the bias
in this process. A few algorithms
to estimate the
biases without stacking all
of the targets are presented
for fixed and dynamic measurement
biases. In most
of these algorithms, it is
assumed that in the multisensor
network, the sensors
are synchronous. However,
in reality, the sensor
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