Open access peer-reviewed chapter

Kalman Filter for Moving Object Tracking: Performance Analysis and Filter Design

By Kenshi Saho

Submitted: April 18th 2017Reviewed: October 17th 2017Published: December 20th 2017

DOI: 10.5772/intechopen.71731

Downloaded: 2639


This chapter presents Kalman filters for tracking moving objects and their efficient design strategy based on steady-state performance analysis. First, a dynamic/measurement model is defined for the tracking systems, assuming both position-only and position-velocity measurements. Then, problems with the Kalman filter design in tracking systems are summarized, and an efficient steady-state performance index proposed by the author [termed the root-mean-squared error index (the RMS index)] is introduced to resolve these concerns. The analytical relationship between the proposed RMS index and the covariance matrix of the process noise is shown, leading to a proposed design strategy that is based on this relationship. Theoretical performance analysis is conducted using the performance indices to show the optimality of the design strategy. Numerical simulations show the validity of the theoretical analyses and effectiveness of the proposed strategy in realistic situations. In addition, the optimal performance of the position-only-measured and position-velocity-measured systems is analyzed and compared. This comparison shows that the position-velocity-measured Kalman filter tracking is accurate when compared with the position-only-measured filter.


  • Kalman tracking filter
  • moving object tracking
  • steady-state analysis
  • performance index
  • filter design
  • process noise

1. Introduction

Remote monitoring systems for cars and robots require accurate tracking of moving objects. Representative tracking algorithms include the Kalman filter [1, 2, 3, 4, 5] and its variants, such as the extended/unscented Kalman [6, 7, 8, 9] and particle filters [10, 11, 12]. These can accurately track movement based on adaptive filtering by using a state-space model.

To use the Kalman filter for the tracking of moving objects, it is necessary to design a dynamic model of target motion. The most common dynamic model is a constant velocity (CV) model [1, 10], which assumes that the velocity is constant during a sampling interval. This model has been used in many applications because of its versatility, effectiveness, and simplicity. However, in almost conventional tracking systems, the selection of process noise (zero-mean white noise in the dynamic model) is conducted empirically [4, 6, 8]. This is because conventional studies tend to assume that process noise takes one of a limited number of forms, which is known as appropriate selections. Thus, despite the large number of investigations into Kalman filter trackers, the optimal selection of a process noise model has not been discussed. The general problems of model selection for Kalman filter trackers were discussed by Ekstrand in 2012 [1]. In the years since, further research on these issues has been conducted, but no satisfactory solutions to the abovementioned problems have been presented. Crouse [13] described a general solution for optimal trackers in a steady state. However, this method also requires an empirical selection of the dynamic models. A detailed analysis of the Kalman filter has been provided for various applications, including global navigation satellite systems [14] and video trackers [15]. However, only limited systems have yet been considered, and no definitive parameter-setting procedure for the Kalman tracking filter has been provided. Although various criteria have been proposed and investigated for the design of Kalman filters and its variants to achieve better tracking accuracy, robustness, and real-time capability, relationship between these performance indices and the model parameters such as the process noise variance is not discussed even in recent studies [16].

Another significant problem in a measurement model of the conventional Kalman tracking filter is that most studies consider only position measurements and therefore cannot make full use of modern sensors that are able to measure velocity, such as ultrawideband Doppler radar [17, 18]. Moreover, sensor fusion based on Internet of Things technology also enables the simultaneous measurement of position and velocity (e.g., sensor data fusion based on the communication between radars/lasers/sonars and speedometers embedded in targets). Consequently, Kalman filters for such systems have become an important area of research [19, 20, 21, 22, 23, 24]. In Ref. [24], the extended Kalman filter for radar measurements is modified for range (position) and range-rate (velocity) measurements, and its effectiveness in realistic radar applications is verified. However, concrete design criterion is not shown. The number of conventional studies on position-velocity-measured (PVM) Kalman filters is smaller than those on the more common position-only-measured (POM) Kalman filters, and the performance and design of PVM Kalman filters are not sufficiently considered.

To resolve the two problems described above concerning the process noise selection and PVM systems, our previous work clarified the fundamental properties of PVM tracking filters [25, 26] and generated an efficient performance index to design an optimal process noise matrix [3, 5]. In the studies of PVM tracking filters [25, 26], fixed-gain PVM filter properties were analytically clarified, but there was no optimization of the PVM Kalman filters. In our work on the process noise matrix [3], an optimal POM Kalman filter, with respect to position prediction, was presented. In this chapter, an appropriate process noise design strategy, based on our proposed efficient steady-state performance index (introduced in Section 3), and its applicability are verified. Our previous work highlighted the following issues, which we address in this chapter:

  1. Analysis of the performance of a PVM Kalman filter with a CV model, based on the proposed index.

  2. Application of the proposed process noise design strategy to a PVM Kalman filter.

  3. Comparison of the performance of optimal POM and PVM Kalman filters.

This chapter presents the theoretical analyses and simulations required to tackle these issues. The remainder of this chapter is organized as follows: Section 2 defines the tracking filtering problem dealt in this chapter and explains the existing concerns and models for POM and PVM Kalman filter design. Section 3 introduces our proposed efficient performance indices with their mathematical formulations. Section 4 presents the proposed process noise design strategy based on the performance index. Section 5 shows the theoretical analysis of the optimal POM and PVM Kalman filter performance in a steady state. The effectiveness of the PVM Kalman filter is proven by the comparison with the POM filter. Section 6 shows realistic maneuvering-target-tracking application examples. Section 7 concludes this chapter and proposes future tasks.


2. Problem statement

This section introduces the Kalman filter for moving object tracking and defines the model assumed in this chapter.

2.1. Dynamic model

The Kalman filter for tracking moving objects estimates a state vector comprising the parameters of the target, such as position and velocity, based on a dynamic/measurement model. For simplicity, this chapter deals with a typical second-order one-dimensional Kalman filter tracker whose true state vector is defined as


where xtand vtare the true position and velocity of the target moving object, respectively, and T denotes the transpose. The assumed dynamic model is a CV model, which is a simple and popular model for tracking moving objects. The CV model assumes that the velocity is constant during the sampling interval, which is expressed as


where xtkdenotes the true state at time kT, Tis the sampling interval, wkis the process noise with covariance matrix Q, and Φis the transition matrix from kTto k+1T, which is expressed as


The Kalman filter predicts the target state based on this dynamic model.

2.2. Measurement model

The measurements are simply modeled as


where zkdenotes the measurement vector, Hdenotes the measurement matrix, and vkis the measurement noise with covariance matrix R. This chapter considers two types of measurement systems, which are discussed as follows.

2.2.1. Position-only-measured system

The POM system assumes that the sensors (such as radar, laser, and sonar) can measure only the position of the target. This is a general assumption in the moving object tracking. Hand Rof this model are expressed as


where Bxis the variance of the position measurement errors.

2.2.2. Position-velocity-measured system

The PVM system assumes that the sensor system can measure position and velocity simultaneously. One example of the PVM model system is a pulse Doppler radar. Sensor fusion systems using communications of position/velocity sensors can also be expressed by the PVM model. Hof this model is expressed as


We now assume that the noises of position and velocity measurements are uncorrelated, and Rof PVM systems under this assumption is defined as


where Bvis the variance of the velocity measurement errors.

2.3. Kalman filter tracking

The Kalman filter tracker based on the abovementioned models sequentially estimates state vectors via the Kalman filter equations. The prediction and estimation are calculated as


where predicts and estimates are denoted by ~ and ^, respectively, and Kkdenotes the Kalman gain that minimizes the errors in the estimated position and velocity. Kkis calculated as


where Pkis the covariance matrix of errors determined from


2.4. Aspects of tracking filter design

Moving object tracking obtains accurate and sequential estimation of the target position and velocity by using Eqs. (9)(13). As indicated in Eqs. (1)(13), the design parameters of the Kalman filter tracker are elements of the covariance matrix of the process noise Q. We must set Qto achieve tracking errors that are as small as possible. Thus, we must know how to design an appropriate Q. Moreover, we must be able to define the evaluation index of the filter performance. However, these issues have not been sufficiently deliberated because the selection of Qhas not been sufficiently addressed in previous studies. Here, the design of Qis empirically carried out.

In the conventional tracking systems, the most commonly used random acceleration (RA) process noise is often selected because it has a better performance. Its Qis


The appropriate selection of σqis important because σq(and sensor noise variance R) directly determines the performance of the tracking filter with the CV model. However, in conventional studies, process noises and their parameters are empirically selected, and the validity of the selection is discussed only casually [1, 16]. Many conventional tracking systems select the RA process noises (Qra), with variance σqset based on the assumed target motion. However, no definitive method of determining σqhas been established. Although tracking index defined by Kalata [27] is known as an effective design parameter, its empirical selection is still required. Moreover, the validity in selecting the RA process noise is also questionable. Various other forms of Qare known and have been used for different target motions [12]. For example, random velocity model [2] and the diagonal Q, which do not include correlations in process noise [7], are also frequently used. However, for the reasons discussed earlier, the differences in performance between the various process noise models are not known.

3. The efficient steady-state performance index (RMS index)

The process noise selection problems discussed in Section 2.3 must be solved to effectively design Kalman tracking filters. Thus, we must properly evaluate the performance of the filter. The effective steady-state performance index was derived [3] and is termed root-mean-squared error index (an RMS index). This section introduces the RMS index for POM and PVM systems and shows the analytical relationships between the RMS index and Q.

3.1. Definition of RMS index

In tracking filtering, the following two functions are required:

  • Function 1. Reduces random errors caused by measurement noises.

  • Function 2. Tracks targets with complicated motions (e.g., accurate tracking of an accelerating target is required for the CV model).

The RMS index is proposed for the comprehensive evaluation of the performance of these two functions and is defined as


where x˜kis the predicted target position (second element of x˜k), E[] indicates the mean with respect to k, and xtakis the true position of a constant acceleration target which is


where acis constant acceleration of the target. In the Kalman filter tracker using the CV model, it is assumed that the target velocity is constant during the sampling interval. Thus, for the constant acceleration target, a steady-state bias error occurs because of the difference between the target motion and the assumed dynamic model. Moreover, x˜kincludes random errors due to measurement noise. Thus, the RMS index εpexpresses both bias errors and random errors. With the steady-state bias error due to the model/motion difference of eacand the steady-state standard deviation of the random errors in x˜kof σp, εpis expressed as


σpexpresses the performance corresponding to Function 1 and eacexpresses the performance corresponding to Function 2. The smaller these errors are, the better is the tracking filter. Thus, the minimum εpachieves the best tracking filter in a steady state.

3.2. RMS index of a POM system

One important advantage of the RMS index is that it can be expressed in closed form. The closed form of εpfor the POM system was derived in Ref. [3]. This subsection introduces the RMS index and its relationship to the design parameter Qin the POM system.

First, the arbitrary Qis defined as


where a > 0, b > 0, and c > 0, and the dimensions of a, b, and care [m2], [m2/s], and [m2/s2], respectively. For example, substituting abc=σq2T4/4σq2T3/2σq2T2into Eq. (18) gives the Qraof Eq. (14) and b = 0 leads to the diagonal Q. The analytical relationship between Qgenand εpis expressed by the following closed form.


where αand βare components of the steady-state Kalman gain K=αβ/TTcalculated from (a, b, c) using the following equations:


The derivation process of these equations is shown in Ref. [3]. As shown in Eqs. (19)(22), the optimal (a, b, c) is designed minimizing εp.

3.3. RMS index of a PVM system

In a similar manner to the treatment of the POM system, this subsection introduces the RMS index of a PVM system and its relationship to Qgen. The RMS index of the PVM system is


where α,β,η,andθare components of the steady-state Kalman gain:




Eq. (22) is obtained from σp2and eacof the steady-state PVM Kalman filter (αβηθfilter) by using Eq. (17). The derivation processes for these are shown in Ref. [25]. The relationship between the steady-state Kalman gains and Qgenis derived as follows:




The derivation of these is given in the Appendix. Note that the dimensionless parameter Rxvcorresponds to the ratio of the measurement accuracies in position and velocity and directly affects the tracking accuracy in PVM tracking systems. From these results, we also obtain the closed form of the RMS index for PVM systems and can design optimal Qusing Eqs. (22)(32).

4. Filter design strategy based on the RMS index

Using the RMS index introduced in the previous section, we can design the Kalman filter parameters (i.e., Q) to achieve optimal tracking. This section defines the optimization problems for POM and PVM systems with a Qthat minimizes the RMS index εp.

4.1. RMS-index minimization problem

4.1.1. POM system optimization

The evaluating function to determine optimal Qis εp,pomnormalized by Bx, which is defined as




is the preset parameter for the proposed strategy. Substituting Eqs. (20)(22) into (33), we obtain μpomabcaD. Using this, the optimal (a, b, c) for the POM system is determined by solving


4.1.2. PVM system optimization

Like the POM system, a normalized RMS index can be used for the design of the PVM system. Normalizing Eq. (22) by Bxand substituting Eqs. (31) and (32) into this, the evaluating function for the PVM system is given by


To design optimal (a, b, c) for the PVM system, the optimal steady-state Kalman gains are calculated by solving the following minimization problem.


where the stability conditions with respect to Kalman gains are easily derived by the well-known Jury’s test as


Substituting the optimal (α,β,θ) calculated by Eq. (37) into Eqs. (28)(30), we obtain an optimal (a, b, c) for the PVM Kalman filter.

4.2. Procedure and notes of the proposed design strategy

The procedure of the proposed strategy for each system is summarized in this section.

4.2.1. Design procedure for a POM system

  1. Set Bxfrom the sensor performance.

  2. Preset aDbased on the approximate target acceleration.

  3. Determine (a, b, c) by solving Eq. (35).

The methodology of presetting aDis discussed in the simulation section.

4.2.2. Design procedure for a PVM system

  1. Set Bxand Bvfrom the sensor performance.

  2. Preset aDbased on the approximate target acceleration.

  3. Determine αβθby solving Eq. (37).

  4. Determine (a, b, c) from αβθusing Eqs. (28)(30).

4.2.3. Notes on computation in the proposed strategy

With respect to the proposed strategy, note that:

  • Eqs. (35) and (37) can be solved by gradient descent with several initial values. This is because that the parameter searching range is narrow due to the stability conditions.

  • The proposed design process is only carried out once before using the Kalman filter. Although the computational cost of the above optimization process is not small, it does not affect the Kalman filtering process.

4.3. Discussion on preset parameter aD

Here, the appropriate presetting for aDin practical use is discussed. The covariance matrix of process noise Qdetermined by the proposed strategy is only optimal when aDis matched to the target acceleration and the target is moving with constant acceleration corresponding to aD. However, using the proposed strategy, the tracking accuracy is always better than when using conventional models as verified in Ref. [3]. Consequently, the proposed method achieves sufficient accuracy, even if aDis not matched to the true target acceleration. This means that the relatively small difference between the true and preset acceleration is acceptable. Thus, in practical use, we estimate an approximate or a typical value for the acceleration (e.g., mean and maximum) in advance based on the assumed motion of the target and then set aDby using this estimated value. The example application presented in Section 6 assumes the approximate maximum acceleration of the target is known and is used for the Kalman filter design.

Thus, target acceleration information is required for accurate Kalman filter tracking by using the proposed strategy. As a method to obtain an approximated acceleration, communications between the tracking systems and the accelerometers embedded in targets can be considered. Many sensing targets have acceleration sensors; for example, robots and vehicles have inertial sensors, and humans have accelerometers embedded in smartphones. Soon, Internet of Things technology will make data communications between robots, smartphones, and radar possible. Thus, we can obtain approximated acceleration based on this novel technology.

5. Theoretical steady-state performance analysis

This section presents theoretical performance analyses of the Kalman tracking filters by using the proposed design strategy. With respect to POM systems, our previous study [3] verified the effectiveness of the proposed strategy by comparison with a conventional random acceleration model based filter design. Thus, the RMS indices for the following filters are compared:

  • Optimal POM filter: the Kalman filter for the POM system designed using the strategy mentioned in Section 4.2.1.

  • Optimal PVM filter: the Kalman filter for the PVM system designed using the strategy mentioned in Section 4.2.2.

  • RA filter: the Kalman filter for the PVM system with the RA process noises by using optimal σqwith respect to the RMS index.

The comparison of the optimal PVM filter with the RA filter indicates the effectiveness of the proposed strategy (i.e., considering the arbitrary covariance matrix of the process noise Qgen) and the comparison of the optimal POM and PVM filters illustrates the enhancement of tracking accuracy by using the velocity measurements in the proposed strategy. This section assumes that Bxand Tare normalized to 1.

Figure 1 shows the relationship between the design parameter aDand the minimum RMS index εp,optfor Rxv= 1 ( Figure 1 left) and Rxv= 10 ( Figure 1 right). It can be seen that the optimal PVM filter achieves the best performance. This result verifies that the proposed strategy determines steady-state gains corresponding to a better covariance matrix of process noise than the RA model. The optimal PVM filter also achieves better performance compared with the optimal POM filter even for Rxv=1, which means that the measurement accuracy of the position and velocity is the same. The addition of the velocity measurements effectively enhances the tracking accuracy. Furthermore, when the velocity measurement accuracy is high, the optimal PVM filter achieves greater accuracy than the POM filter.

Figure 1.

Analytical relationship betweenaDandεp,opt(Rxv=1(left),Rxv=10(right)).

Figure 2 shows the relationship between Rxvand εp,optfor aD2=0.01(left) and 0.1 (right). Both cases exhibit the same trend. For both optimal PVM and RA filters, better performance is achieved with better velocity measurement accuracy. The performance of the optimal PVM filter is better than that of the optimal POM filter including relatively small Rxv(the velocity measurement accuracy is low). In contrast, the performance of the RA filter is worse than that of the optimal POM filter for small Rxvbecause the covariance matrix is limited to Eq. (14). Moreover, by comparing the two insets of Figure 2 , we see the greater effectiveness of the proposed strategy for relatively large aD.

Figure 2.

Analytical relationship betweenRxvandεp,opt(aD=0.01(left),aD=0.1(right)).


6. Application to radar tracking simulation

Finally, this section provides an example of the Kalman filter tracker designed with the proposed strategy in a realistic application, namely, pulse Doppler radar tracking.

6.1. Simulation setup

We simulated the pulse Doppler radar tracking of a maneuvering target and compared the tracking errors of the filters assumed in the previous section. Figure 3 shows the simulation scenario and the true target acceleration. The true target position is xtkytk=kT220+kT1.5cosπkT/5. Two-dimensional tracking in the x-yplane of the point target is assumed. We consider two pulse Doppler radars located at (x, y) = (0.5 m, 0) and (1.0 m, 0). The sampling interval Tis 100 ms, and the observation time is 4 s. The transmitted signal is a pulse with central frequency of 60 GHz and bandwidth of 500 MHz. The received radar signals are calculated using ray tracing with the addition of the Gaussian white noise. The radar measurement parameter depends on the system under consideration: the POM system assumes the measurement of the position by using ranging results, and the PVM system assumes the position and velocity measurements where the position measurement is the same as the POM system, and the velocity measurement is based on the Doppler shift with the method presented in Ref. [18]. We determine a variance for this noise to set Bx=9×104m2 and Bv=0.09m2/s2. In these settings, Rxv=1. These values are the averages along the two axes. Using the RMS prediction error calculated from 1000 Monte Carlo simulations, the performance is defined as


where xpmkand ypmkare the predicted positions in the mth Monte Carlo simulation.

Figure 3.

Simulation setting (simulation scenario (left), true target acceleration (right)).

6.2. Implementation of Kalman filter

First, the implementation of the Kalman filters for two-dimensional system is presented. The implementation of a two-dimensional optimal POM filter is as follows:


where vytis the true velocity in the y-axis and aoptboptcoptis optimized (a, b, c), calculated using the procedure in Section 4.2.1. xtand Φof a two-dimensional PVM filter are the same as for a POM filter. Hand Rare


In addition, the formulation of Qis the same as in Eq. (43) and aoptboptcoptis calculated using the procedure in Section 4.2.2. A two-dimensional RA filter is the same as the optimal PVM filter, with the exception of Q. Qof the RA filter is


Next, the design for an appropriate aDis presented. We presume an approximate prediction of accelerations. For instance, when the maximum acceleration of the target in Figure 3 is predicted to be approximately ac=3 m/s2, aDis then 1.0, from Eq. (34). Using this aDand the radar settings described in the previous section, we have aoptboptcoptfor each filter.

6.3. Results and discussion

Figure 4 shows the simulation results. Clearly, the filters using velocity measurements achieve greater accuracy than the optimal POM filter. The mean steady-state prediction RMS errors (Eεkin 2 s < kT) of the optimal POM, RA, and optimal PVM filters are 0.59, 0.46, and 0.19 m, respectively. These results indicate that the proposed strategy achieves greater accuracy than the conventional RA filter even in realistic situations. The mean RMS error of the optimal PVM filter is 41% of that of the RA filter. This is because the RA model cannot track the abrupt motion of the high-maneuvering target because of limitations in expressing the process noise. In contrast, the optimal PVM filter can set gains corresponding to the appropriate process noise to accurately track high-maneuvering target. Moreover, the mean RMS error of the optimal PVM filter is 32% of the error in optimal POM filter, and this clearly indicates the effectiveness of the velocity measurement, even when Rxv=1(when the measurement reliability of the position and velocity are the same). These simulation results are consistent with the theoretical analyses presented in Figure 1 .

Figure 4.

Simulation results.

7. Final remarks

7.1. Conclusions

In this chapter, the efficient steady-state performance index, known as the RMS index, was introduced for both POM and PVM Kalman filters for systems that track moving objects. Automatic design (preset) of the covariance matrix of the process noise Q, to realize optimal position prediction, was achieved using the analytical relationship between Qand the RMS index. The validity of the proposed design strategy was shown via analyses and simulations. These results verified that the proposed index attained accurate tracking when compared with the conventional RA-model-based Kalman filter design. A simulation of a realistic situation indicated that the optimal performance given by the proposed strategy is 41% better than that given by the conventional design procedure for a PVM system. Moreover, the optimal performance of the optimal POM and PVM Kalman filters was compared showing that the optimal PVM Kalman filter is accurate when compared with the POM filter in a steady state.

7.2. Future works

The most important future objective is the extension of the RMS index-based design strategy to the third-order (and higher order) Kalman filters that are widely used for real applications. In third-order tracking, an acceleration is added to the state vector, becoming one of the input parameters of the Kalman filter. Performance analysis and the establishment of a design strategy for such systems (i.e., position/acceleration and position/velocity/acceleration measured Kalman filters) are important considerations for advanced sensor fusion systems under development. Moreover, considerations of other dynamic models (e.g., the constant turn model) should also be probed for use in many applications including pedestrian tracking.

A. Appendix

A.1. Derivation of Eqs. (28)(30)

Because we assume a steady state, the index kof all parameters and matrices is omitted in the following calculations. The ith row and jth column of a matrix Pare denoted as Pi,j.

Eq. (11) is also written using P̂as


As indicated in Eq. (7), Hof the PVM Kalman filter is the identity matrix. Thus, from Eq. (48), the relationship between the Kalman gains and the error covariance matrix in the estimated state P̂is calculated using Eqs. (8) and (24) as


With P1,2=P2,1and Eq. (31), we have the following relationship:


Eq. (50) is equal to Eq. (32), showing that this relationship is satisfied in the assumed PVM Kalman filter without depending on the process noise. P̂is also calculated using Eq. (13) by substituting Eqs. (7) and (24) as


Elements of P˜are required to calculate Eq. (51) and are calculated using Eqs. (3), (12), and (18) as


Substituting Eq. (52) into Eq. (51), and comparing elements of Eq. (49), we have the following linear system:


Solving this linear system with respect to (a, b, c) and substituting Eq. (50) into the solutions, we arrive at Eqs. (28)(30).

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kenshi Saho (December 20th 2017). Kalman Filter for Moving Object Tracking: Performance Analysis and Filter Design, Kalman Filters - Theory for Advanced Applications, Ginalber Luiz de Oliveira Serra, IntechOpen, DOI: 10.5772/intechopen.71731. Available from:

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