Jacobians for sensors measurement models.

## Abstract

One of the well-known approaches to target tracking is the Kalman filter. The problem of applying the Kalman Filter in practice is that in the presence of unknown noise statistics, accurate results cannot be obtained. Hence the tuning of the noise covariances is of paramount importance in order to employ the filter. The difficulty involved with the tuning attracts the applicability of the concept of Constant Gain Kalman Filter (CGKF). It has been generally observed that after an initial transient the Kalman Filter gain and the State Error Covariance P settles down to steady state values. This encourages one to consider working directly with steady state or constant Kalman gain, rather than with error covariances in order to obtain efficient tracking. Since there are no covariances in CGKF, only the state equations need to be propagated and updated at a measurement, thus enormously reducing the computational load. The current work first applies the CGKF concept to heterogeneous sensor based wireless sensor network (WSN) target tracking problem. The paper considers the Standard EKF and CGKF for tracking various manoeuvring targets using nonlinear state and measurement models. Based on the numerical studies it is clearly seen that the CGKF out performs the Standard EKF. To the best of our knowledge, such a comprehensive study of the CGKF has not been carried out in its application to diverse target tracking scenarios and data fusion aspects.

### Keywords

- Constant Gain Kalman Filter
- INS
- GPS
- Wireless Sensor Network
- Tracking

## 1. Introduction

The Kalman Filter (KF) is one of the most fundamental and widely used estimation schemes in tracking application. While the KF formalism is very powerful we need to keep in mind that the solution scheme can be considered to be formal and a fundamental prerequisite for accurate results is the a prioiri knowledge of the initial state (

An alternate approach to tuning is via the direct setting of the Kalman gain as carried out in the work of Ananthasayanam et al. [9] and Ashwin et al. [10]. It is often observed that the Kalman gain converges to a steady state value which coincides with the convergence of the state error covariance

Our present work is about the application and sensitivity study of the CGKF traget tracking scheme in sensor networks scenarios and maneuvering target tracking. We look at target tracking problems in wireless sensor networks using passive infrared (PIR), acoustic and seismic sensors in stand alone (SA) and data fusion (DF) modes as given by Raol [12] for the discrete white noise acceleration (DWNA) traget motion model. We further demonstrate the capability of the CGKF to track maneuvering targets [13, 14] from acquired range and direction data for a class of coordinated turn (CT) maneuver models. The CGKF with linear measurement model was validated in [10, 15]. The present study applies the CGKF to a non linear measurement models and further demonstrates its robustness through sensitivity studies. The results obtained with respect to homogeneous and heterogeneous data fusion further demonstrate the range of applicability of the CGKF. These extensive tracking and sensitivity studies for a wide range of state and measurement models are to the best of our knowledge, unique to this paper and provide the reader with a comprehensive reference. These results also provide a firm base for application of the CGKF concept to other areas. In the sequel, Section 2 describes the CGKF concept. Section 3 introduces the various tracking scenarios based on PIR, acoustic and seismic measuerement models in SA and DF modes. In addition maneuvering targets based on CT models are discussed, since these have the potential to demonstrate the flexibility of the CGKF. Section 4 details the tracking and sensitivity studies on the above mentioned models, and Section 5 gives the conclusion of the present work.

## 2. Constant gain Kalman filtering

The KF algorithm [16] is based on the least squares principle with recursive time updates. It is a fact that optimal filter performance needs apriori knowledge of the filter statistics in terms of the state-error, system and measurement noise covariances (

The observation that the gain (reflecting

where

The following is the estimation scheme based on a predict and update mode.

### 2.1 The estimation scheme

The generic KF updates are

where the the innovations sequence is

Thus it is evident that once the optimal gain

We observe that the typically expensive covariance time update step is not needed in the constant gain approach.The CGKF is found to work quite well even with state models moderately different from that for which the gains are computed [25], suggesting a robustness of the gains calculated (Refer Tables 6 and 7). It is to be noted that the present problem is a non linear problem, in so far as the measurement model is concerned so that the filter used is the CGKF. This is one unique advantage of the CGKF over the standard KF/EKF wherein the EKF requires linearization of the measurement model via use of the Jacobian

## 3. Sensor models and modes

The focus of our study is the application of the CGKF to a variety of 2D sensor models such as those in unattended ground sensor (UGS) and Intelligence, Surveillance and Reconnaissance (ISR) systems. Sensors such as passive infrared (PIR) [21], acoustic, seismic [22] and radar have been studied. The sensor system might consist of single or multiple data inputs as required in different scenarios. They may consist of single type of sensor or multiple type of sensor nodes, as required in situations. Homogeneous and heterogeneous DF aspects of certain combination of sensors will be analyzed. We outline the regular and CGKF schemes and their application to the above mentioned systems.

### 3.1 State variable models in stand alone mode

The state model for 2D target is comprised of

The state equation for the DWNA is

where

with

where

** Sensor Measurement model for PIR sensor**[21]

** Sensor Measurement model for Acoustic sensor**[22]

** Sensor Measurement model for Seismic sensor**[22]

where

We now outline the estimation scheme by an EKF as well as a CGKF. The EKF has the following steps. For t = 0,1,2......

where

** Update/Correction:**The update of the states and covariances as per the EKF scheme are

where

where

Table 1 gives measurement Jacobians for all three sensors. In the table

Sensors Type | Measurement Equation | Jacobian H |
---|---|---|

PIR | ||

ACO-USTIC | ||

SEISMIC |

The CGKF on the other hand has the following two steps.

Once the optimized Kalman gain

In our work the optimized value of K is calculated via the application of the genetic algorithm to the innovation cost function Eqs. (1) and (2).

### 3.2 Homogeneous data fusion

In homogeneous fusion the fusion is based on the data from multiple sensors of similar type, at every time instant. Here in this section we have used mainly the centralized approach to DF in respect of the KF. The data obtained from various nodes (similar type of sensors) is combined together then applied to EKF and CGKF for tracking the target. This approach has been used as measurement fusion [12] approach in WSN of UGSs.

Measurement fusion techniques combine the raw measurements of the target obtained from the Individual Sensor Node (ISN) at the Cluster Head Node (CHN) Level. The ISN is a tier 1 node while the CHN is a tier 2 node which is capable of running a complex fusion algorithm based on KF framework. So ISNs are considered to have minimal computation capability compared to the CHNs. The two approaches which have been implemented in our work with respect to the CGKF under the homogeneous DF are Maximal Kalman filter (MKF) [12] and Weighted fusion (WF) [12] approaches. The state model is the DWNA model of the previous sub section.

#### 3.2.1 Maximal Kalman filter (MKF) method

This method is based on fusing all measurements of the ISN by incorporating them in a fused measurement vector and the corresponding measurement noise covariance and measurement matrices as described below

where

#### 3.2.2 WF method

This method is based on combining the

where

where

### 3.3 Heterogeneous DF

Heterogeneous DF differs from the homogeneous variety in that we fuse data from different types of sensors in combinations: such as, PIR and acoustic or PIR and seismic or PIR, acoustic and seismic together [12]. We have tried the architectures of centralized (measurement fusion) as well as decentralized (state fusion) data fusion. There are several methods in practice for DF but for nonlinear measurement models, it has been found that only a few models have been able to maintain the accuracy against catastrophic fusion [23]. The state model applied is the DWNA (Eq. (6)) of the subsection A.

#### 3.3.1 Centralized DF

This architecture mainly follows the measurement fusion. The measurements (data) are obtained from all ISNs and then fused at cluster head node CHNs. In our case the data obtained is nonlinear from all three sensors with different size of measurement models. The only possible approach to collate data effectively is the MKF since weighted fusion applies only to sensors based on similar measurement model. The method has been applied to both EKF and CGKF.

The MKF is an effective way to combine data from dissimilar type of sensor measurement models. At the cost of computational complexity owing to matrix size this is overall an effective method considering WF can combine data from only similar group of sensors.

#### 3.3.2 Decentralized data fusion

The method has been explicitly used to bring out the fact the CGKF did perform better as against any of these methods of combining state parameters and covariances. This method has been cited as state fusion concept [12] or hierarchical data fusion. This is based on a two tier system wherein state estimation of the target is carried out at ISNs which forms tier 1 and these states are then fused at tier 2 in the CHNs. The global state estimate and global state error covariance calculated at CHNs and these are then fed to ISNs. The KF algorithm runs in the ISN to obtain fresh state and error covariance estimates, which are again fed at the CHN and the cycle continues. There are mainly two approaches of track to track fusion as given by Raol [12] in Eq. (27) and (28) and Durrant whyte [24] in Eqs. (29) and (30). Most of the methods surveyed in this category feature a scheme where in we have to combine error covariances and state vectors to produce new covariance and state vectors. The only difference between the two methods below is the way state estimates and error covariances are used to compute fused global values of state estimate

### 3.4 Maneuvering target

The class of maneuvering targets yield particularly challenging tracking problems. The challenges include choosing a system model close to the actual target maneuvers in addition to often having to give real time solutions. In our work we now aim to demonstarate the efficacy of the CGKF framework to RADAR- measurement based coordinated turn (CT) models. We reiterate that the non necessity of prior knowledge of the system and measurement noise characteristics (often representing the nature of maneuver) make the CGKF particularly attractive. The present work builds on [15] where the CGKF algorithm has been applied to a variety of maneuvering targets based on a linear measurement model. Currently a non linear measurement model (RADAR based) has been employed in order to move a step closer to a more realistic scenario. We have applied the CGKF to the highly maneuvering class of CT models with known as well as unknown turn rates [13, 14]. In the simulation studies the turn rate is represented by

The present part is divided into the following parts.

#### 3.4.1 CT state variable model

A two dimensional model for the target tracking problem (maneuver in horizontal 2D plane) is described as follows.

where state vector is

where

A two dimensional model for the target tracking problem (maneuver in horizontal 2D plane) is described as follows.

where state vector is

where

## 4. Results and sensitivity studies

### 4.1 Stand alone mode

The tabulated result of all sensors for EKF and CGKF are given below with their respective PFE (Percentage Fit Error). The error metric

Sensor Type | No of Sensors | EKF (PFE) % | CGKF (PFE) % |
---|---|---|---|

PIR | 1 | 3.77076 | 1.03723 |

Acoustic | 1 | 2.497723 | 1.9393 |

Seismic | 1 | 4.622587 | 2.614668 |

### 4.2 Homogeneous fusion mode of sensors

The results from both, MKF and Weighted fusion have been tabulated seperately as shown in the first six enteries of Table 3. Settings of the simulations including the number of Monte Carlo runs is same as that for the Stand Alone method described above. An example of the method is illustrated in Figures 5 and 6. The PFE, RMSPE metrics (as defined in Section 4.1) metric in the plots correspond to that of the CGKF for the coresponding run.

Sensor Type | EKF (PFE) % | CGKF (PFE)% | Fusion Type |
---|---|---|---|

PIR | 6.32816 | 2.81109 | Maximal |

Acoustic | 5.76208 | 5.61651 | Maximal |

Seismic | 5.54418 | 2.38094 | Maximal |

PIR | 0.580869 | 0.579842 | Homogeneous Weighted |

Acoustic | 1.47602 | 1.33768 | Homogeneous Weighted |

Seismic | 1.7170850 | 1.0956019 | Homogeneous Weighted |

PIR & Acoustic | 18.5708 | 9.3724 | Maximal |

PIR, Acoustic & Seismic | 19.9958 | 1.34023 | Maximal |

PIR & Seismic | 10.9576 | 3.42436 | Maximal |

### 4.3 Heterogeneous fusion mode of sensors

The results in last three entries of Table 3, are those corresponding to the measurement fusion based method of heterogeneous fusion. Settings of the simulations including the number of Monte Carlo runs is same as that for the Stand Alone and homogeneous fusion method described above. One example each of the two sensor (PIR and seismic case) and three sensor (all three combined) is illustrated in Figures 7 and 8 repectively. The PFE, RMSPE metrics (as defined in Section 4.1) in the plots correspond to that of the CGKF for the coresponding run (refer Table 4).

### 4.4 Maneuvering target

The 2-D frame work study has been carried out on a set of seventy data points in order to generate a smooth trajectory. The following system and measurement covariances matrices are used to generate the simulated track

MODEL | EKF % | CGKF % | K matrix |
---|---|---|---|

CT(known | 56.75 | 10.11 | |

CT(unknown | 24 | 14.78 |

### 4.5 Sensitivity studies on constant gain in case of maneuvering targets (CT (known ω ))

Under this heading we demonstrate the robustness of the constant gain in so far as the application of gain variations to the maneuvering target tracking scenario for the CT (known

Additive variations (K) | PFE% | RMSPE% |
---|---|---|

K(1 + .1 randn) | 10.9506 | 13.3848 |

K(1 + .2 randn) | 9.02807 | 9.8478 |

K(1 + .3 randn) | 11.4343 | 13.3211 |

K(1 + .4 randn) | 11.4565 | 13.77 |

K(1 + .5randn) | 8.68005 | 9.92213 |

K(1 + .6 randn) | 10.3575 | 10.5219 |

K(1 + .7 randn) | 13.1419 | 18.0384 |

K(1 + .8 randn) | 10.4037 | 12.4931 |

Fractional Variations (K) | K/8 | K/4 | K/2 | 2 K | 4 K | 8 K |
---|---|---|---|---|---|---|

PFE% | 11.1569 | 9.34031 | 12.7851 | 13.0054 | 11.6087 | 9.11502 |

RMSPE% | 14.2031 | 11.0111 | 15.2803 | 15.1227 | 13.0654 | 10.1699 |

## 5. Conclusion

We believe that these are the only studies of a CGKF applied to tracking targets in WSN environments and maneuvering target models based on non linear measurement models. As seen the EKF is unable to effectively track the targets in WSN and for the maneuvering target case compared to the CGKF. This is a significant finding and supports the fact that CGKF effectively circumvents, or in other words trades the gains with the filter statistics which are more difficult to obtain and therein gives optimal tracking results by working directly with the Kalman Gain. The present results prove that the CGKF is successful in target tracking applications wherein the constant gain approach overcomes uncertainty regarding noise statistics that exist in the framework of the problem. The CGKF has been employed for tracking maneuvering targets and those in a WSN. The present work firmly establishes the CGKF framework thereby enabling its applicability to a wider variety of problems as deemed fit by the reader.

### 5.1 Analysis of results and future work

#### 5.1.1 Stand alone mode

Following are the deductions based on the simulation studies as summarized in Table 2.

The results and plots bring out clearly the novelty of CGKF, the overall performance of which is better than the EKF as per the PFE values.

In the case of the EKF the Acoustic sensor performs the best.

In case of the CGKF the PIR performs the best.

#### 5.1.2 Homogeneous fusion mode

Following are deductions based on the simulation studies as summarized in Table 3 [16].

The overall performance of the CGKF is better than the EKF for both the MKF and Weighted methods.

Considering the CGKF case the PIR and seismic sensors peroform better than the acoustic sensor, with the PIR performing the best overall.

Amongst the various fusion methods the overall performance of the weighted fusion is better compared to the MKF, for all types of sensors.

#### 5.1.3 Heterogeneous Fusion mode

Following are the deductions based on the simulation studies as summarized in Table 4.

Overall the CGKF performance is better than the EKF for heterogeneous fusion method.

With reference to heterogeneous fusion of PIR, acoustic and seismic sensors the Durrand Whyte method [24] gives better results compared to Global fusion method [12]. Here we note that a comparison with the CGKF is not possible since the CGKF works with purely measurements and not by propagation of state error covariances which is fundamental to these techniques. The Global fusion method [16] does not provide convergence in tracking when using PIR, acoustic and seismic sensors together.

The CGKF heterogeneous fusion model of PIR, Acoustic and seismic sensors gives optimum performance better than its EKF counterpart.

PIR based weighted fusion gives better results than the heterogeneous fusion. However we must keep in mind the fact that the simulations for heterogeneous fusion are based only one sensor of each type unlike the homogeneous fusion case where four sensor of each type are considered. The plots and result have been mentioned under. All the result has been obtained through Montecarlo simulation with runs of average of 500.

#### 5.1.4 Maneuvering target

Following are the deductions of the simulations.

Figures 9 and 10 and Table 5, clearly show that the performance of the CGKF is very much better than that of the EKF.

The results obtained show the CGKF performing better than the EKF in three models (ie. DWPA and both CT models).

The results obtained for the CT models including those of sensitivity analysis (Refer Tables 6 and 7) demonstrates the viability of applying the CGKF to this category of problems.

### 5.2 Conclusions and Suggestions for Further Studies

The efficacy of the CGKF has been demonstrated wherein a single approach yields optimal results for a varierty of linear [10] as well as non linear models in WSN and maneuvering target scenarios [15]. The extensive numerical studies establish the fact that the CGKF performs better that the conventional EKF.

Actual implementation of a target tracking application in the WSN environment shall require optimal routing, deployment, design, communication protocols and other such associated integral characteristics mentioned in the introduction. Though not directly within the purview of the scope of the work, these aspects are very important.

It would be very useful to apply this CGKF to variants of the Kalman Filter such as particle filter, ensemble filter and other formulations.

Finally CGKF could be tried out for massive data based problems like numerical weather prediction. The constant gains can be pre computed using earlier data and since the gains are robust they can be expected to handle newer data quiet efficiently similar to space debris as in [1].