5.1. Range-Only Tracking Radar Simulations
The results presented in this subsection refer to the EKF described in
Section 4.1, which was designed for the 2D radar with range-only measurements available. The following conditions were assumed for these simulations.
Simulation time of 100 seconds,
Period of measurements and filter date ,
Radar coordinates , ,
Object moving in the positive direction of the OX axis, with the initial position, velocity, and acceleration equal: , , , respectively,
Power spectral density of the white noise of the motion disturbances ,
Standard deviation of range measurements .
In the conducted tests of the algorithm, the threshold
was changed in the range from 0 to 0.1 with a step 0.01, and its influence on the number of the measurement matrix
updates and on the root-mean-squared (RMS) estimation error of the
coordinate of the object’s position. Since the results for various runs of simulations are variable due to various realizations of process disturbances and measurement errors, their values were averaged for 100 thousand simulations for each considered threshold value. The obtained values are gathered in
Table 1. Due to the mentioned averaging, the number of
matrix updates in this table is not an integer number.
In order to verify what is the influence of the intensity of the measurement errors on the optimal threshold value, the following experiment was conducted. For two different standard deviations of range errors, i.e.,
and
the threshold
was changed in the range from 0 to 0.8 with a step 0.01, and its influence on the number of the measurement matrix
updates and on the root-mean-squared estimation error of the
coordinate of the object’s position were registered. The respective results are gathered in
Table 2 and
Table 3. Even for significantly (an order of magnitude) stronger or weaker measurement noises, the same threshold
gives the best results. Thus, there is no need for adaptive threshold changes in the modified EKF even if one expects that the noise level may be variable.
Computation loads and times for arithmetic operations are largely dependent on the processor used and the implementation of functions in the applied library. However, it is generally assumed that comparisons and additions are the least computationally expensive, and more demanding, in increasing order, are multiplications, divisions, and calculations of functions, such as the square-root or trigonometric functions [
17]. Assume for illustrative purposes the times of elementary operations given in Reference [
17] for a specific computer, namely the IBM 360 Model 67-1. They are
for addition,
for multiplication,
for division, and
for the square root, respectively. Assume that the time for comparison equals that of adding two variables. The times of the calculations of unique fragments of the measurement matrix updates in the standard EKF and in the modified EKF, calculated according to Equations (16) and (17) for the considered example, are gathered in
Table 4.
The results presented in
Table 4 show significant savings in terms of processing time for the adopted assumptions, even for small thresholds
where the
matrix updates are still frequently performed.
In a further step of algorithm testing, for four selected thresholds
, values of the
element of the
matrix, values of the monitoring variable
, and the number of
H matrix updates since the beginning of a single simulation were calculated and are presented in
Figure 7,
Figure 8,
Figure 9 and
Figure 10.
In the presented figures, one can see that the number of matrix updates decreases with the increasing threshold value.
The case
is equivalent to a standard EKF with the measurement matrix update taking place in every step
, whereas the case from
Figure 10, obtained for
, represents a situation when the position estimation accuracy is still almost the same as in a standard EKF, but the filter’s computational demands are significantly reduced.
With increasing threshold value, the number of matrix updates decreases, but their uneven distribution in time is worth noting. The largest density of actualizations takes place in a portion of the object’s trajectory, where the cosine of the ϕ angle, under which the object is seen by the radar, changes most rapidly. Its quickest changes are in the central part of the assumed flight trajectory, and they result from the fact that
changes most rapidly near
.
This notion leads to establishing a practical method for choosing the threshold. The threshold
should be chosen for a specific radioelectronic system, based on simulations or on analytical considerations. One can considered “the worst case” scenario where the
matrix elements change the quickest. In case of a radar system, with known minimal detection range
the quickest
matrix updates are required for an object passing the radar in such a small distance, as presented in
Figure 11, and the threshold
can be established for this specific case only, looking for its maximal value, but still ensuring acceptable estimation accuracy.
For better visualization of the filter’s behavior, the presented example was purposefully chosen in such a way that the
matrix changes significantly in the short time of simulation. In most radar and radio-navigation systems, these changes would be much slower. For example, in the Kalman filter of a Global Positioning System (GPS) receiver, the
matrix contains direction cosines from the user to the visible GPS satellites [
2]. Since the angles under which the satellites are seen from a user’s location remain almost unchanged in short periods of time due to the very large distances between the user and the satellites, the
matrix elements would be almost constant for relatively long periods. In such a filter, the
matrix updates would be performed very rarely in comparison to the radar system example that was considered in this paper. Therefore, in real radioelectronic systems, the proposed EKF modification could lead to an even more significant reduction of its computational demands.
5.2. Angle-Only Tracking Radar Simulations
The results presented in this subsection of the paper refer to the EKF described in
Section 4.2 designed for the 2D radar with angle-only measurements available. The conditions adopted for these simulations were the same as for the range-only tracking radar, with a single difference, consisting of assuming a standard deviation of angle measurements
instead of a standard deviation of range measurements.
In the first test of the algorithm, the threshold
was changed in the range from 0 to 0.1 with a step 0.02 and from 0.1 to 0.2 with a step 0.05. Its influence on the number of the measurement matrix
updates and on the root-mean-squared estimation error of the
coordinate of the object’s position was analyzed and gathered in
Table 5. The decreasing number of matrix updates does not visibly affect the estimation error.
The times of the calculations of unique fragments of the measurement matrix updates in the standard EKF and in the modified EKF, calculated according to Equations (23) and (24) for the considered example, are gathered in
Table 6.
Similar to the range-only tracking, in this case, computational savings can be obtained. However, they are not as large as previously seen. This is due to a fact that, in the range-only tracking problem, the element of the matrix contained a time-consuming square root operation and its omitting in some steps immediately resulted in a large decrease of the processing time. In the angle-only tracking, the element is a simpler function that can be decomposed into a few additions, multiplications, and divisions. Thus, it is not computationally expensive. The savings are still possible only due to limiting the number of the mentioned elementary operations.
Based on the already presented examples, one can conclude that the proposed modified filter would be most effective in problems where the matrix elements are computationally expensive so that the obtained savings due to less frequent matrix updates are much larger than the additional computational burden due to calculations of monitoring variables and their comparisons with a threshold. As the EKFs used in GPS receivers process pseudoranges closely related to ranges, and typical radars process both ranges and angles, the matrix elements in these filters contain time-consuming operations, like square roots. Therefore, the proposed filter may be useful in these important groups of applications.
For two extreme thresholds
, values of the
element of the
matrix, values of the monitoring variable
, and the number of
H matrix updates since the beginning of a single simulation were calculated and are presented in
Figure 12,
Figure 13 and
Figure 14.
As can be seen in
Figure 12, the
element of the
matrix reaches its minimum at step 41 and then very rapidly changes its direction. This results from a very small distance between the radar and the object in the considered case. It should be mentioned that this is not a typical situation in real systems where EKFs are used, as the EKF itself is dedicated for systems with relatively benign non-linearities [
10]. In strongly non-linear systems, Unscented Kalman Filters or Particle Filters are typically applied [
9,
11].
However, the effects observed in the considered scenario give some insight into the algorithm’s behavior. If the value of
derivative is calculated around step
, it will be close to zero and, consequently, the variable
will also be close to zero. In such a case, the proposed algorithm may be too sluggish, and it will wait long with the next update of the
matrix. This effect can be observed in
Figure 13 obtained for a large
.
If rapid changes of the
matrix elements occur in a system for which the proposed algorithm is designed, one can consider forcing the update of the
matrix not only based on the observation of the monitoring variables, but also a predefined number of steps, whichever happens first. The results obtained for the above described filter but with a forced
matrix update every 10 steps are shown in
Figure 14. It is visible that such a filter can effectively work in high-response conditions and deal with its sluggish behavior when the
matrix elements reach their extreme values and their derivatives are close to zeroes.