An asynchronous sensor bias estimation algorithm utilizing targets’ positions only
Introduction
Data fusion is an important technology in multi-sensor tracking systems that can improve tracking accuracy; however, without bias estimation, the integrated performance of data fusion will be poor [1]. Therefore, it is necessary to estimate sensors’ offset biases before data fusion. Thus, bias estimation is a critical problem in multi-sensor tracking systems.
To estimate the offset bias vector, two classical approaches are typically used: one is offline [2], and the other is online, which has better real-time performance. The online method was proposed by Nabaa and Bishop [3], who augmented the system state to include the bias vector as part of the state and then implemented an augmented state Kalman filter (ASKF) by stacking the states of all targets; the sensor biases were then recorded in a signal vector. The disadvantage of this method is that the dimension of the system vector may be very large, which can increase computational demand. Hsieh [4] improved this approach using a two-stage Kalman filter (KF), which could mitigate the complexity of the filtering process; however, this method could only be applied under specific conditions [5]. An alternative approach that is less computationally demanding is based on pseudo measurements under earth center earth fixed (ECEF) coordinates [6], [7], [8].
However, in practical applications, the target states detected by different sensors are not necessarily concurrently measured because of different sampling rates and the initial scanning times of the sensors. However, all of the above studies and other well-known works [9], [10], [11] have investigated only synchronous data; the bias estimation problem for asynchronous sensors [7], [12], [13], [14] has been studied considerably less.
In [7], the method transformed the measurements from different times of the sensors into pseudo measurements in “proper time slots,” creating pseudo measurements based on the measurements in the slot. Reference [13] extended the work of [7]: a pseudo measurement was obtained based on a current measurement and recent previous measurements from multiple sensors. In [14], the pseudo measurement equation of sensor biases was obtained according to all measurements in a fusion period. Reference [15] extended the work of [7] into 3D scenarios and concluded that the variance of bias estimates cannot be ignored when there are more uncertainties in bias estimation. However, the derivations of the pseudo measurements in [7], [12], [13], [14] were related to the transition matrix for CV motion, which is based on the assumption of a target with uniform motion; thus, those pseudo measurements were related to the target’s state including its velocity, and thus depended on the target state. As a result, that algorithm is sensitive to target motion.
The purpose of this paper is to propose a pseudo measurement that is only dependent on the target’s position. As a result, the merit of the proposed algorithm is insensitive to the target’s manoeuvres.
The remainder of this paper is organized as follows. The bias estimation method of synchronous sensors based on ECEF coordinates is reviewed in Section 2. The primary section of this paper discusses bias estimation of asynchronous sensors in Section 3. In Section 4, the effectiveness of the proposed method is demonstrated. Lastly, conclusions are presented in Section 5.
Section snippets
Observation model
Assuming that in the sensor’s local Cartesian coordinates, the true position of target is at time k, the corresponding measurements in the polar coordinate and local Cartesian coordinate are then and , respectively. The offset bias vector, which includes biases of range, azimuth and elevation, is . Transforming measurements into the local Cartesian coordinate, we can obtain the observation model:
Bias estimation of asynchronous sensors
As shown in Section 2, the bias estimation of synchronous sensors can be efficiently solved by the method above. However, asynchronous sensors must still be studied. This section examines the derivation of the pseudo measurement based on the interpolation time registration algorithm. It is assumed that two sensors detect a common target, and only the offset detecting biases are considered.
Numerical examples
In this section, several numeral examples are used to evaluate the performance of the bias estimator with manoeuvring and non-manoeuvring targets.
Considering an example with two sensors and one target, the offset bias of sensor A is . The stochastic noise has a zero mean and is white with a variance . The sampling interval of sensor A is 0.09 s, and the offset bias of sensor B is . The stochastic noise has a zero
Conclusions
In this paper, we proposed a novel solution to the bias estimation problem for asynchronous sensors. Using interpolation time registration, measurements from different sensors are modified to become coincident. Then, the pseudo measurement equation of the bias vector is obtained based on synchronous data. The primary advantage of the proposed algorithm is that the derivation of the pseudo measurement only depends on the target position; as a result, the proposed algorithm is not sensitive to
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