This paper presents the derivation and efectiveness of the best-acceleration (BA) filter, which minimizes the steady-state acceleration-error variance in an −− tracking filter based on a constant-acceleration motion model. The purpose is to enhance the tracking performance for a target object. Currently, the minimum-variance (MV) filter has already been proposed to minimize the steady-state position-error variance. The performance of the proposed BA filter was evaluated by comparing it with the MV filter. The results demonstrate the efectiveness of the BA filter, supporting the optimality of the BA filter in minimizing the steady-state acceleration-error variance.
Monitoring systems for robots and intelligent vehicles that use remote sensors such as cameras,
LiDAR, depth sensors, and radar require accurate tracking of moving objects. For such
applications, Kalman-filter-based trackers are commonly used to estimate position, velocity, and
acceleration [
The signal processing methodology for a tracking system is known as a tracking filter. For
moving object tracking, tracking filters estimate the future state of parameters such as position,
velocity, acceleration, and other states of a target based on observed data from sensors such as
LiDAR and radar. A tracking filter also aims to smooth the estimation results to achieve higher
prediction accuracy. Representative tracking filters include the Kalman, – , and – – filters
[
This study focuses on the optimal design of the – – filter, which is one of the simplest
tracking filters. The – – filter is a one-dimensional tracking filter that uses a
constantacceleration motion model and is known as the steady-state Kalman filter [
For the optimal gain design of the – – filter, the MV filter criterion has been proposed [
In this study, we theoretically analyze the performance of the – – filter for the best acceleration prediction. The optimal gains that minimize the variance of the predicted acceleration are derived in closed form. We term the proposed acceleration-optimized – – filter the BA iflter. The performance of the conventional MV filter and the proposed BA filter, both with optimal gains, is compared through theoretical analyses and numerical simulations. 2. − − tracking filter
The – – filter is defined based on a constant-acceleration motion model, with a sampling
interval denoted as . The filter gains are denoted by , , and , while the measured position
at time step is denoted by ,. The smoothed position, velocity, and acceleration are defined
as [
, = , + ( , − ,) (1) 2 =
4 2 (, , ) + 4
this performance is evaluated based on errors that are caused by random measurement noises, it can be evaluated in terms of the variance of the prediction errors. The true values of the target object’s position, velocity, and acceleration are denoted by , , and . We define the steady-state errors as follows. Because the steady-state error does not depend on time because of the steady-state assumption, the time index is omitted. The variance of the steady-state position error is expressed as
Similarly, the variance of the steady-state acceleration error is expressed by in terms of the filter gains , , , and the measurement error variance as
By eliminating , , and using equations (1)–(6), the variances 2 and 2 can be expressed
The predicted position, velocity, and acceleration are defined as [
− 2 − ) − (8 − 8 − 2 + + 2 = (2 − (2 − ))(4 − 2 − ) 2 ) (3) (4) (5) (6) (7) (8) (9) (10) (11)
setting them to zero, we obtain the two following equations [
From equations (12) and (13), the gains , and are determined.
Whereas the conventional MV filter minimizes the position prediction error, this study derives an – – filter that minimizes the steady-state acceleration prediction error. In this paper, the proposed filter is referred to as the BA filter.
minimum points. constant. We take partial derivatives of equation (10) with respect to and , and find their and that minimize 2 subject to being By taking the partial derivative with respect to , we have Since 2 = 0 is assumed as a condition, 2 = − [(, , )] Since 2 = 0 is assumed as a condition, is satisfied. By taking the partial derivative of equation (11) with respect to ,we can get the following equation as in terms of . From equation (22), we obtain
From the above, we obtained equations (17) and (22). Using these equations, we express 2 2 − (2 − )
2 = 0 =
By using the solution formula, is expressed in terms of , and we obtain = −(3 − 2)( − 2) ± = −(3 − 2)( − 2) ± ( − 2)
Assuming that equation (27) holds, is
= 2(2 − ) =
4( − 1)(2 − ) √︀(3 − 2) 2( − 2) 2 − 8( − 2)
2( − 1) 2 (24) (27) (21) (22) (23) (25) (26) (28) By applying the arithmetic–geometric mean inequality, Assuming that Equation (31) holds, and referring to Equation (36), we obtain Equation (37) indicates that Equation (31) is the inappropriate solution because the gain becomes negative. Therefore, equation (35) is appropriate to determine . Thus, is expressed in terms of as = 4( + √︀( + 64))(64 − − ( + 32 + √︀( + 64)) 2 √︀( + 64))
By substituting equations (35) and (38) into Equation (10), we arrive at the minimum variance of the acceleration prediction error as (29) (30) (36) (37) (31) (33) (34) (35) (38) (39)
By substituting equation (27) and (29) into equation (11), solution. Assuming that equation (28) holds, is
Equation (30) implies that the denominator of 2 becomes zero and is not appropriate as the
0 = 16( 2 − 2 + 1) − (32) By using the solution formula, is expressed in terms of , and we obtain From equation (33), we can express as or √︀( + 64)}
The equations (35), (38), and (39) are the main results of this paper that clarify the gains of the – – filter with best acceleration prediction. Furthermore, the steady-state minimum variance of the prediction errors is expressed solely in terms of .
We present numerical analyses to validate the derived results. For a model undergoing constant acceleration motion, we evaluate and compare the MV filter and the BA filter in terms of 2 and 2 using equations against the fixed . We calculate these error variances by substituting the optimal gains (MV filter: equations (12) and (13), BA filter: equations (35) and (38)). In the analyses, and are normalized to 1.
Figures 2 and 3 show the numerical analysis results of 2 and 2, respectively. Figure 2 shows the numerical analysis results of 2, and Figure 3 shows the numerical analysis results of 2. Figure 2 shows that the BA filter performs better than the MV filter in minimizing 2. Figure 3 shows that, within the range from 0 to 1, the MV filter performs better than the BA filter in minimizing 2. These results indicate the theoretical performance diferences between the two types of filters.
This subsection presents a numerical simulation to validate the derived results and to show the efectiveness under realistic sensing conditions with random noise and to corroborate the analysis. A target moved with a constant acceleration of 1 m/s2 and a sampling interval of 0.1 s and the observation period was 60 s. Observation noise was randomly applied within the range 0.01 to 0.3. Figures 4 and 5 show the simulation results for absolute prediction errors in acceleration and position, respectively. Figure 4 shows that the acceleration errors of the BA filter become smaller than those of the MV filter. As the steady-state error reflects the error in the limit as time approaches infinity, this indicates not only that the BA filter is more suitable for minimizing 2 but also that the results are consistent with Figure 2. Similarly, Figure 5 shows that the position errors of the MV filter consistently become smaller than those of the BA filter. This indicates not only that the MV filter is more suitable for minimizing 2, but also matches the results with Figure 3. Thus, the numerical simulation results prove the efectiveness of the proposed BA filter and its diference from the conventional MV filter.
In this paper, we derived the BA – – tracking filter that minimizes the steady-state variance of the predicted acceleration. The closed forms of the BA filter gains , and are given in equations (35) and (38). In addition, we conducted a performance comparison with the conventional MV filter that is designed to minimize position prediction errors. As a result, both numerical analysis and simulation demonstrate that the BA filter performs better than the MV filter for acceleration prediction.
The author(s) have not employed any Generative Al tools.