The fusion of electronics and mechanics is an ongoing process, which benefits from by downscaling and reduced production costs for various kinds of sensor technologies. For monitoring of machine states, all sorts of physical quantities are captured and analyzed. An even more sophisticated task is the prediction of future machine states and of temporary and instantaneous production steps, such as processes in milling machines. This leads to the notion of collision avoidance in automated machine tools in order to prevent damages and downtimes, which cause high maintenance and material costs .

For these kinds of industrial machine tools, different approaches of systems for damage reduction are already under investigation, e.g.,. However, for the time being, none of them is able to reliably predict the imminent risk and efficiently avoid collisions by proactive shutdown and deactivation.

In this contribution, the approach of a predictive 24 GHz Doppler surveillance and collision avoidance radar is described. Extensions and continuations of the fundamental investigations in , and are merged. A new signal processing stage for nonlinear target tracking based on Doppler shift estimates is introduced. It is essentially based on an Extended Kalman Filter (EKF). The algorithm is designed for short-range, single-target tracking. Appropriate process and measurement noise models are derived, test statistics are applied and trajectory estimation is shown. Recently, investigations on solvability, uniqueness of the solution, and the required minimum number of sensors were published by . In the application discussed herein, the solution space for target position and velocity is restricted. Symmetries and ambiguities due to unfavorable sensor placement are reduced. Another approach, based on a two-stage filter, was proposed by in order to reduce the computational costs of a stand-alone particle filter.

The next sections are structured as follows: in Sect. , we introduce the model for mono-static Doppler radar on a moving observer platform and give the resultant equations. Section  contains the involved tracking algorithm, the underlying derived process and noise models, as well as an extensive study on appropriate filter initialization. Additionally, statistical tests for filter consistency are introduced. The methods are applied to simulation examples and experimental data described in Sect. . The final conclusion is given in Sect. .

Within the scope of this work, we consider the following three-dimensional problem: The position of a single target in Cartesian coordinates shall be represented by p=[x,y,z]T and may be understood as the mass center of a point scatterer, which is assumed to be the dominant target compared to all other returns received from a cluttered environment. The observer platform comprises N sensors with states s(n)=[sx(n),sy(n),sz(n),vx,vy,vz]T, where the sensor positions are known, fixed and related to the origin of the global coordinate system, which lies in the center of the observer platform, see Fig. . The velocity vector of the platform is v=vx,vy,vzT and unknown. To get a state vector consisting of full three-dimensional position and velocity, we perform a transformation and assume the observer platform is fixed and the target is moving relative to the platform with just the reversed velocity. Hence, the full target state is x=[pT,-vT]T=[x,y,z,-vx,-vy,-vz]T. A typical 2-D scenario (z= const) is illustrated in Fig. , with two sensors at the positions s(1) and s(2), and a single moving target (scatterer) at position p with velocity v. In the following investigation, the radar sensors are assumed to have an isotropic radiation pattern, and they detect a target at distance ‖s(n)-p‖ with Doppler shift fd(n).

The system equation describes the evolution of the target state with time. The target state at time tk is given by position and velocity x=[pT,vT]T. The kinematic state of the target can be deduced from the previous step tk-1 using the process matrix F of a constant velocity (CV) model. Small deviations from the true trajectory are modeled by zero-mean, white Gaussian noise wk, with corresponding process noise covariance matrix Qk. Then, the following linear discrete-time dynamic model is obtained: xk=Fxk-1+wk-1,wk∼N(0,Qk). The measurement equation relates the measurements zk to the state xk, where the dimension of z equals the number of sensors being involved. In mono-static radar systems the observed Doppler shift, corrupted by additive zero-mean, white Gaussian measurement noise nk, can be represented by zk=hk(xk)+nk,nk∼N(0,Rk), where Rk is the measurement noise covariance matrix. The measurement model is given by the nonlinear equation of the Doppler shift observed at the nth sensor hk(n)(xk)=2λ(s(n)-pk)⋅vk‖s(n)-pk‖, where in contrast to Eq. (), s(n)=sx(n),sy(n),sz(n)T is the sensor position in Cartesian coordinates, for all n=1,…,N. ‖⋅‖ denotes the Euclidean norm. The radar wavelength λ=c/fc depends on the carrier frequency and the speed of light in the considered medium. Thus, λ has influence on the Doppler sensitivity, which is 160 Hz/(m s-1) in 24 GHz radar systems.

If only Doppler is estimated, Eq. () represents the measurement model, which is the only source of information in the localization algorithm. For state estimation, recursive Kalman filtering concepts are established as efficient algorithms. In this case, nonlinear approaches are necessary, like the EKF or the Unscented Kalman Filter (UKF), see e.g. , , . The application requires fast response on a millisecond scale (rapid estimation). Thus, it also necessitates short time for reaching a decision about shutdown. Additionally, due to the closely spaced sensor placement, the target observability is poor. Hence, a general maximum likelihood (ML) approach seems to be inappropriate here, as the number of required measurements might be very large in this nonlinear problem, to find an efficient, unbiased estimate. As a consequence, the integration time may be too long with respect to the change rate of the state. The ML approach is used only for the initialization procedure to obtain reliable initial states close to the true target state.

For the considered application the EKF is employed. Despite of the known flaws of the EKF, see e.g. , first order linearization of the nonlinear Doppler function is assumed to be sufficient, since the scenarios considered here assume either constant velocity (CV, non-maneuvering) or constant acceleration within a limited range of values. If turning maneuvers and directional changes of motion have to be tracked, UKF is preferred e.g.. However, both EKF and UKF, require appropriate initialization of expected values and covariance matrices. Otherwise, tracking with these kinds of deterministic filters may lead to poor tracking accuracy or – in case of fatal ambiguities – to completely misdirected estimates.

In this section we introduce the adapted EKF algorithm, process and noise models, and give the initialization procedure based on maximum likelihood estimation. Furthermore, statistical methods for filter consistency tests are described.

In this section, an illustrative simulation example, performed in , is described. The scenario is depicted in Fig.  (top), with the sensor platform inside the machine tool and six circularly arranged radars on the side front of it. The sensor radius is 20 cm. The global origin of the Cartesian coordinate system lies in the center of the platform. A single-target is given, moving towards the platform with constant velocity. This represents a typical situation of imminent danger, which has to be handled by a monitoring collision avoidance radar.

The initial target state x0 is given in Table , together with the initial estimate x^ML obtained from MLE. The initial state covariance matrix is set to P0=0.1I6. These values can be derived from the results in Fig. . Additionally, the minor diagonal elements are overlaid by low-power additive noise (matrix still symmetric). This leads to improved convergence. Furthermore, the process and measurement noise matrices are defined as R=σf2I6,andQ=Gσp2GT, where G=T2/2,T2/2,T2/2,T,T,TT. The variances are set to σf2= (0.4 Hz)2 resp. σp2= (0.4 m s-2)2. The sampling interval is T= 1 ms. The initial NEES=0.25≪nx has a quite low value. This comes along with low estimation error and the relative high initial covariance matrix. Better overall performance was observed during the MC simulations with larger elements in the initial covariance P0.

Figure  (middle) gives an insight into the high tracking accuracy, with steadily decreasing estimation error. After 1 s, the final true position is p=0.083,-0.014, 0.073T m, which corresponds excellently with the estimated position of p^=0.083,-0.021, 0.077T m. As can be identified from Fig.  (middle), the parameter z^ is the most erroneous one. However, this depends on the sensor configuration. If the sensors are rotated around the x axis, the estimation error z-z^ becomes lower, whereas y-y^ is slightly increased. The distance x is a crucial parameter in collision avoidance. It is estimated with sub-centimeter accuracy.

Estimated velocities are not depicted. The estimation properties of the velocities are similar to the coordinates: the estimation performance for v^y is better compared to v^z. This is again due to the sensor arrangement and could be improved by rotating the sensor arrangement. The parameter v^x is estimated with very high accuracy.

In Fig.  (bottom) the NIS is depicted. This quantity is used for outlier tests, for the gating procedure, and for noise level increasing for the initialization period. Additionally, in Fig.  lower and upper bounds of the 95 % confidence intervals (χ62-distribution) are given as dashed lines. In this simulation, 95.4 % of all estimates (NIS) are within this acceptance interval. The time-average NISavg has a value of 5.82, which matches excellently to the expectation of nz=6.

A further statistical test for correct filter design is the mean-value of the innovation νk, which has to be zero, with associated covariance matrix Sk. In this simulation the obtained mean values were in the range -0.04 to 0.08 Hz.

This paper presented investigations and results on important issues of Doppler target tracking in short-range scenarios. The utilized Extended Kalman Filter (EKF) was designed considering the requirements for predictive 24 GHz Doppler radar processing aiming at collision warning and collision avoidance in industrial machine tools. The proposed procedure for filter initialization is based on Maximum Likelihood Estimation from several sequential measurements. It delivered reliable initial guesses close to the true target state. In combination with the known restrictions of the considered application, the utilization of a deterministic EKF was investigated. Simulation results confirmed the filter models and showed very high tracking accuracy.

As a next step, we plan to extend the algorithm to handle extended targets with more irregular shapes. In this case we expect that increased Doppler spreading due to multiple scattering centers will be observed and a data association problem will arise.

Datasheet Version 2.0 of K-LC5 Radar Transceiver is available at http://www.rfbeam.ch/downloads. This work was supported by the German Research Foundation (DFG) and the Technische Universität München within the funding programme Open Access Publishing.