## 1 Introduction  
Monitoring systems for robots and intelligent vehicles that employ remote sensors, such as cameras and radar, require the tracking of moving objects. Adaptive tracking techniques such as Kalman and extended Kalman filters [1; 2; 3; 4; 5] and particle filters [6; 7] are commonly used for this purpose because of their accuracy. An alternative option is fixed-gain tracking filters, which have also been studied and used extensively for two reasons [8; 9; 10; 11; 12]:  
* Simple implementation and low computational overhead: Optimal gain calculation is not required in the fixed-gain filters. Thus, the number of matrix operations is small compared with the Kalman filter and its variants [11]. * Applicability to the analytical evaluation of the Kalman filter: Fixed-gain filters are also useful for analytical evaluations of the Kalman filter because they can be characterized as steady state Kalman filters [12]. For these reasons, fixed gain filters are still being widely used in applications that strongly require real-time capability and simple implementation, such as tracking in ultrasonography in medicine [13],motor position control [14], human fall detection [15] and vehicular radar [16]. Additionally, the analysis of the Kalman filters assuming the steady state (fixed gain) is conducted to predict their tracking performance in the filter design process. The simplest second-order fixed-gain tracking filter is known as an \(\alpha\)-\(\beta\) filter, which have been deployed in various tracking systems [17; 18; 19; 20; 21; 22; 12]. The design of the \(\alpha\)-\(\beta\) filter has been discussed based on an efficient design parameter known as the tracking index [10; 12; 22]. However, \(\alpha\)-\(\beta\) filters only consider position measurements and hence cannot make full use of modern sensors that can also measure velocity, such as ultra-wideband (UWB) Doppler radars, which have recently come into use [23; 24; 25]. In the near-field, these radars can achieve accurate sensing of moving objects, such as humans and cars. In [23; 24], position and velocity estimates for pedestrians were achieved with centimeter and cm/s accuracy, respectively (see [25] for hardware implementation). Moreover, sensor fusion based on the Internet of Things technology also enables the simultaneous measurement of position and velocity possible (e.g., sensor data fusion based on the communication between radars and speedometers embedded in targets). Consequently, tracking filters for such systems have become an important area of research [26; 27; 28; 29]. However, almost all conventional position-velocity-measured (PVM) trackers have been based on Kalman or particle filtering, whereas fixed-gain PVM filters have not seen wide use. This is because the computational performance is sufficient to drive PVM Kalman/particle filters in many applications. Additionally, the empirical design of these filters without steady state analyses can realize tolerable tracking performance. However, reiterating, fixed gain filter techniques are still important for various applications and analytical performance evaluation of tracking systems to find better parameter settings. To address the above problem, we have proposed third-order fixed-gain (\(\alpha\)-\(\beta\)-\(\gamma\)) PVM filters and have verified their performance [11]. However, a simpler second-order tracker such as an \(\alpha\)-\(\beta\) filter is often required when the number of components and/or the size of hardware is quite limited and the complexity of target motion is predicted to be relatively small (i.e., a constant velocity model assuming a second-order tracker is sufficient). To this end, we also have investigated the fundamental properties of a PVM \(\alpha\)-\(\beta\) filter [30]. However, this filter assumes an unrealistic assumption; specifically, correlated errors of position and velocity do not exist in the filtering process.