In figure 7.39 position and velocity for one single run are shown. The blue line is the true position of the falling body, red and green line are EKF and SPF respectively. In the first seconds the velocity is almost constant, but as the falling body enters the atmosphere the velocity is decreasing with dense atmosphere. Due to this thicker air density at lower altitude, the gravity force are cancelled by a drag force, letting the falling body decrease its velocity. Looking at position in this figure it is not possible to see if one filter performed better than the other. Both filter plots lays completely on top of each other wich indicates equal performances. In figure 7.40 the system has been run with 100 Monte Carlo runs. Position errors, velocity errors and Ballistic constant errors are shown respectively. In the case of position errors, it is not large differences in performance. It may seem that SPF is slightly better from 20 seconds and out but thats negligible. In the case of velocity errors, EKF is slightly better from 0 to 15 seconds. But from here SPF performes better. But for the ballistic constant, SPF estimates the true value faster than EKF.
Similar falling body scenarios has been presented earlier in several sources, for example in [1, 4, 2]. In [1] there is no comparison of EKF and SPF, but it can be read that similar results are found by EKF. Especially the estimetion of the Ballistic constant, Arthur Gelb [1] states that EKF tracks it poor early in the trajectory because of the thin air at high altitude creates a small drag force on the body, and that the measurements includes little information of the ballistic constant. This emphasise the results found in this thesis.
In [4] the EKF and SPF are compared at almost exact the same system. Here it is shown that both filters have pikes of errors in velocity around 10 - 15 seconds, which also
Figure 7.39: Falling body position and velocity seen from radar.
is the fact in figure 7.40, and that SPF consistently gives better estimates than EKF. In [2] the same conclusions as above are made.
Conclusion
8.1
Discussion
This thesis shows a comparison of two filters, namely Extended Kalman filter (EKF) and Sigma Point filter (SPF). It is performed partly theoretical and by simulations. The sce- narios that have been simulated are a falling body and two different airplane motions.
The airplane motions are divided in two cases. One having perturbations in the hor- izontal plane as a sine wave. The other has perturbations in the horizontal and vertical plane as a sine and cosine wave respectively. Each case are simulated and estimated by use of EKF and SPF receiving measurements from a radar in spherical coordinates. The filters are compared both by unknown and known airplane motions. In the case where the motions are unknown, a linear process filter model is implemented. This gave unsat- isfactory results with high peaks of prediction. This would have been better using a more rapid measurement update. Since it is out of the scope implementing a more realistic radar model, a measurement update on 1 Hz where chosen by the author.
The tuning of the filters are a time-consuming job, some tuning where done on the process covariance matrix Q and ˆP0. Here it could have been done a more structured
analysis of both. SPF have even more tuning factors that could have been looked more into. Some tuning of these factors where done, but the author ended up using them as described in the theory of this thesis. Partly because it gave good results, and in theory should be the best choice used on stochastic variables of additive white noise.
In the case where the airplane motions are known by the filters, a non-linear process filter model is implemented. A goal for the author was to implement a model that worked well on incomming airplanes from all directions within north and east. Several models where implemented and tried. It ended up with having a more complex measurement matrix in the filters. This was an error prone process transforming to spherical coordi- nates in the measurement equation and then performe linearization for use in EKF. For the non-linear process model case, the measurements are therefore converted to cartesian coordinates before they are implemented in the filters. As the author of this thesis have
experienced by reading, this is not the usual way to do it. The transformations are gen- erally done in the measurement matrix.
The MUSIC estimate is used directly in the measurement update exchanging the Kalman filter estimate with the MUSIC estimate. This estimate could have been received as a measurement in the filters, which may have given better covariance update, since there may be crosscorrelations that could have improved other estimates.
The falling body is modelled as an object falling towards the ground from high alti- tude, straight against a radar on the ground. The radar gives measurements in cartesian coordinates, giving a non-linear process filter model and a linear measurement equation. This where implemented without use of process covariance matrix Q in the filters. The implementation of Q in the filters may have improved the filters performance as it usually do.