Method and Device for Estimating a Velocity of an Object

Abstract

A method is provided for estimating a velocity of an object located in the environment of a vehicle. Detections of a range, an azimuth angle and a range rate of the object are acquired for at least two different points in time via a sensor. A cost function is generated which depends on a first source and a second source. The first source is based on a range rate velocity profile which depends on the range rate and the azimuth angle, and the first source depends on an estimated accuracy for the first source. The second source is based on a position difference which depends on the range and the azimuth angle for the at least two different points in time, and the second source depends on an estimated accuracy for the second source. By minimizing the cost function, a velocity estimate is determined for the object.

Claims

1. A method comprising: acquiring, via a sensor, detections of a range, an azimuth angle, and a range rate of an object in an environment of a vehicle for at least two different points in time; via a processing unit: generating a cost function that depends on a first source and a second source, the first source based on a range rate velocity profile that depends on the rate range and the azimuth angle and the first source depends on an estimated accuracy for the first source, the second source based on a position difference that depends on the range and the azimuth angle for the at least two different points in time and the second source depends on an estimated accuracy for the second source; and determining a velocity estimate for the object by minimizing the cost function.

2. The method of claim 1, wherein: a plurality of detections of the range, the azimuth angle, and the range rate are acquired for the object for each of the at least two points in time; a respective standard deviation is estimated for the range, the azimuth angle, and the range rate based on the plurality of detections; the range rate velocity profile depends on the plurality of detections of the rate range and the azimuth angle; the position difference depends on the plurality of detections of the range and the azimuth angle for the at least two different points in time; the estimated accuracy of the first source depends on the standard deviation of the range rate and the standard deviation of the azimuth angle; and the estimated accuracy of the second source depends on the standard deviation of the range and the standard deviation of the azimuth angle.

3. The method of claim 2, wherein: different standard deviations of the range, the azimuth angle, and the range rate are estimated for at least one of: each of the at least two points in time; or each detection of the range, the azimuth angle, and the range rate.

4. The method of claim 2, wherein: the first source and the second source are based on a normalized estimation error squared (NEES) that includes the respective standard deviations of the range, the azimuth angle, and the range rate.

5. The method of claim 2, wherein: the cost function comprises a first contribution based on a normalized estimation error squared (NEES) related to the first source and a second contribution based on a normalized estimation error squared (NEES) related to the second source.

6. The method of claim 5, wherein: the first contribution and the second contribution each comprise a sum of elements over the plurality of detections and each element is estimated as a normalized estimation error squared (NEES) for the respective detection.

7. The method of claim 6, wherein: the elements of the first contribution are based on a range rate equation and on the standard deviation of the range rate.

8. The method of claim 6, wherein: the elements of the second contribution are based on: the position difference for the respective detection, a velocity covariance matrix estimated based on the standard deviations of the range and the azimuth angle, and a time interval between the at least two different points in time for which the range and the azimuth angle are acquired by the sensor.

9. The method of claim 6, wherein: the cost function is generated as an average of the first contribution and the second contribution.

10. The method of claim 1, wherein: a component of the velocity is estimated by setting a derivative of the cost function with respect to a velocity component to zero.

11. The method of claim 1, wherein the cost function and the velocity estimate are determined by assuming a constant velocity of the object (23) in order to initialize a Kalman filter state estimation of the velocity.

12. A device comprising: a sensor configured to provide data for acquiring detections of a range, an azimuth angle, and a range rate of an object in a field of view of the sensor for at least two different points in time; and a processing unit configured to: generate a cost function that depends on a first source and a second source, the first source based on a range rate velocity profile that depends on the range rate and the azimuth angle and the first source depends on an estimated accuracy for the first source, the second source based on a position difference which depends on the range and the azimuth angle for the at least two different points in time and the second source depends on an estimated accuracy for the second source; and determine a velocity estimate for the object by minimizing the cost function.

13. The device of claim 12, wherein the sensor includes at least one of: a radar sensor or a Lidar sensor.

14. The device of claim 12, wherein the device comprises a vehicle.

15. The device of claim 12, wherein: a plurality of detections of the range, the azimuth angle, and the range rate are acquired for the object for each of the at least two points in time; a respective standard deviation is estimated for the range, the azimuth angle, and the range rate based on the plurality of detections; the range rate velocity profile depends on the plurality of detections of the rate range and the azimuth angle; the position difference depends on the plurality of detections of the range and the azimuth angle for the at least two different points in time; the estimated accuracy of the first source depends on the standard deviation of the range rate and the standard deviation of the azimuth angle; and the estimated accuracy of the second source depends on the standard deviation of the range and the standard deviation of the azimuth angle.

16. The device of claim 15, wherein: the first source and the second source are based on a normalized estimation error squared (NEES) that includes the respective standard deviations of the range, the azimuth angle, and the range rate.

17. The device of claim 16, wherein: the first contribution and the second contribution each comprise a sum of elements over the plurality of detections, wherein each element is estimated as a normalized estimation error squared (NEES) for the respective detection.

18. The device of claim 17, wherein: the elements of the first contribution are based on a range rate equation and on the standard deviation of the range rate.

19. The device of claim 17, wherein: the elements of the second contribution are based on: the position difference for the respective detection, a velocity covariance matrix estimated based on the standard deviations of the range and the azimuth angle, and a time interval between the at least two different points in time for which the range and the azimuth angle are acquired by the sensor.

20. A computer-readable storage media storing computer-readable instructions that, when executed by one or more processors, cause the one or more processors to: receive, via a sensor, data for acquiring detections of a range, an azimuth angle, and a range rate of an object in a field of view of the sensor for at least two different points in time; generate a cost function that depends on a first source and a second source, the first source based on a range rate velocity profile that depends on a range rate and the azimuth angle and the first source depends on an estimated accuracy for the first source, the second source based on a position difference which depends on the range and the azimuth angle for the at least two different points in time and the second source depends on an estimated accuracy for the second source; and determine a velocity estimate for the object by minimizing the cost function.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Exemplary embodiments and functions of the present disclosure are described herein in conjunction with the following drawings, showing schematically:

[0036] FIG. 1 depicts an overview of a device according to the disclosure, which is installed in a vehicle,

[0037] FIG. 2 depicts a first scenario for verifying a method according to the disclosure,

[0038] FIGS. 3A to 5D depict verification results for the first scenario,

[0039] FIG. 6 depicts a second scenario for verifying the method according to the disclosure, and

[0040] FIGS. 7A through 9D depict verification results for the second scenario.

DETAILED DESCRIPTION

[0041] FIG. 1 schematically depicts vehicle 10 which is equipped with the device 11 according to the disclosure. The device 11 includes a radar sensor 13 for monitoring the environment of the vehicle 10 and a processing unit 15 which receives data from the sensor 13.

[0042] FIG. 1 also depicts a vehicle coordinate system 17 having an origin which is located at a center of a front bumper of the vehicle 10. The vehicle coordinate system 17 includes an x-axis 19 which is parallel to the longitudinal axis of the vehicle 10, and a y-axis 21 which is perpendicular to the x-axis 19 and points to the right in FIG. 1. Since the vehicle coordinate system 17 is a right-handed coordinate system, a z-axis (not shown) is pointing into the page.

[0043] The device 11 is provided for detecting objects in the environment of the vehicle 10 such as a target object 23 shown in FIG. 1. That is, the radar sensor 13 detects radar reflections from the target object 23 which may also be regarded as raw detections. Based on the raw detections, the device 11 is configured to determine a position for the respective detection, e.g. the position of a reflection point at the target object 23 via the processing unit 15. The position is given by a range r and an azimuth angle θ with respect to the vehicle 10, e.g. defined in the vehicle coordinate system 17. In addition, the device 11 is also configured to determine a range rate or radial velocity for each raw detection.

[0044] In addition, the raw detections are captured for a plurality of points in time (e.g. at least two). The detections for a certain point in time are also regarded as a radar scan, whereas the detections belonging to the same object are regarded as a cloud.

[0045] For each radar scan (or measurement instance), the radar sensor 13 captures m raw detections from the target object 23. The number m is typically 3 to 5. Each raw detection is indexed by i and described by the following parameters expressed in the vehicle or sensor coordinate system 17:

[0046] r.sub.i: range (or radial distance)

[0047] θ.sub.i: azimuth angle

[0048] {dot over (r)}.sub.i: raw range rate (or radial velocity), wherein i=1, . . . , m.

[0049] In addition, the accuracy of each of these detection attributes is assumed to be known (from the accuracy of the radar sensor 13) and is represented by the respective standard deviations σ.sub.θ, σ.sub.r, σ.sub.{dot over (r)}, or σ.sub.{dot over (r)}.sub.comp, wherein σ.sub.{dot over (r)}.sub.comp is the standard deviation for the compensated range rate which is explained below. Each of the m detections has a different accuracy, e.g. a different standard deviation.

[0050] The range rate equation for a single raw detection i is given as follows:

{dot over (r)}.sub.i+V.sub.s.sup.x cos θ.sub.i+V.sub.s.sup.y sin θ.sub.i=V.sub.t,i.sup.x cos θ.sub.i+V.sub.t,i.sup.y sin θ.sub.i  (1)

[0051] wherein V.sub.s denotes the current velocity of the host vehicle 10 and V.sub.t denotes the velocity to be determined for the target object 23. The components of the velocity vector of the target object 23, e.g. V.sub.x and V.sub.y, are defined along the x-axis 19 and the y-axis 21, respectively, of the vehicle coordinate system 17 (see FIG. 1). To simplify the notation, a compensated range rate is introduced and defined as:

{dot over (r)}.sub.i,cmp={dot over (r)}.sub.i+V.sub.s.sup.x cos θ.sub.i+V.sub.s.sup.y sin θ.sub.i  (2)

[0052] wherein {dot over (r)}.sub.i,cmp is the compensated range rate of the i-th raw detection.

[0053] Then the above equation can be reduced to:

{dot over (r)}.sub.i,cmp=V.sub.t,i.sup.x cos θ.sub.i+V.sub.t,i.sup.y sin θ.sub.i  (3)

[0054] The range rate equation is represented in vector form as:

[00001]$\begin{array}{cc}{\stackrel{.}{r}}_{i,cmp}=\left[\cos {\theta }_i\sin {\theta }_i\right]\left[\begin{array}{c}{V}_{t,i}^x\\ V_{t,i}^y\end{array}\right]& \left(4\right)\end{array}$

[0055] An estimate 45 (see FIGS. 3, 4, 7 and 8) for the velocity of the target object relies on two sources for estimating the velocity of the target object 23. These are defined as follows:

[0056] 1. A velocity profile which is determined based on the rate range equation:

{dot over (r)}.sub.comp.sub.i=V.sub.x cos(θ.sub.i)+V.sub.y sin(θ.sub.i)  (5)

[0057] 2. A position difference which is defined as follows:

[00002]$\begin{array}{cc}{V}_x=\frac{x_t-x_{t-\mathrm{dt}}}{dt}& \left(6\right)\end{array}$$V_y=\frac{y_t-y_{t-dt}}{\mathrm{dt}}$

[0058] For the velocity profile (which may also be referred to as range rate velocity profile), it is assumed that all m detections belonging to the “distributed” target object 23 have the same absolute value for their velocity, but different azimuth angles. Therefore, the velocity vectors belonging to detections of the target object form the velocity profile. For the position difference, dt denotes the time difference between two points in time for which two detections are acquired.

[0059] Both sources of velocity have to be combined for determining the estimate 45 for the velocity, e.g. both sources need to be weighted utilizing estimated accuracies or standard deviations of the detections. For weighting the two sources according to equations (5) and (6), a Normalized Estimation Error Squared (NEES) can be used which is generally defined as follows:

e.sub.k=(X.sub.k−{circumflex over (x)}).sup.T{circumflex over (P)}.sub.k.sup.−1(X.sub.k−{circumflex over (x)})  (7)

[0060] {circumflex over (P)}.sub.k.sup.−1 denotes the inverse of the covariance matrix, and {circumflex over (x)} denotes an estimate based on measurements X.sub.k. The Normalized Estimation Error Squared (NEES) is usually employed for a consistency check of a variance estimation. For the present disclosure, the NEES is used to define a cost function Q for the two sources as defined above for the velocity estimation:

Q=1/2(NEES.sub.VP+NEES.sub.PD)  (8)

[0061] The first term or contribution is regarded as Velocity Profile NEES and is calculated as follows:

[00003]$\begin{array}{cc}NEE{S}_{VP}={.Math.}_i\frac{{\left(V_x\cos \left({\Theta }_i\right)+V_y\sin \left({\Theta }_i\right)-{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}\right)}^2}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}& \left(9\right)\end{array}$

[0062] The second term or contribution is regarded as Position Difference NEES and can be calculated as follows:

NEES.sub.PD=Σ.sub.jV.sub.diff,j(σ.sub.V.sub.j).sup.−1V.sub.diff,j.sup.T  (10)

wherein:

V.sub.diff,j=V−V.sub.j  (11)

[0063] and V=[V.sub.x V.sub.y] is the “true” velocity of the target object 23 which is to be estimated. Here and in the following, j denotes the index of the detection within the Position Difference NEES, e.g. in the same manner as described above for the index i.

[0064] The respective vector V.sub.i includes both components of the velocity based on the position difference and given by equation (6):

V.sub.j=[V.sub.x,jV.sub.y,j]  (12)

[0065] The velocity covariance matrix of equation (10) is defined as:

[00004]$\begin{array}{cc}{\sigma }_{V_j}^2=\left[\begin{array}{cc}{\sigma }_{V_x{V}_x,j}& {\sigma }_{V_x{V}_y,j}\\ {\sigma }_{V_y{V}_x,j}& {\sigma }_{V_y{V}_y,j}\end{array}\right]& \left(13\right)\end{array}$

[0066] which may also be written as:

[00005]$\begin{array}{cc}{\sigma }_{V_j}^2=\frac{{\sigma }_A^2+{\sigma }_B^2}{{\mathrm{dt}}^2}& \left({13}^{\prime }\right)\end{array}$

[0067] wherein A and B denote two different points in time (separated by a time interval dt) for which the respective covariance matrix is determined, and wherein:

[00006]$\begin{array}{cc}{\sigma }_A^2=\left[\begin{array}{cc}{\sigma }_{A_{\mathrm{xx}}}& {\sigma }_{A_{xy}}\\ {\sigma }_{A_{yx}}& {\sigma }_{A_{yy}}\end{array}\right]=J_{\mathrm{pol}2\mathrm{cart}}{\sigma }_{A_{pol}}^2{J}_{\mathrm{pol}2\mathrm{cart}}^T& \left(14\right)\end{array}$$J_{\mathrm{pol}2\mathrm{cart}}=\left[\begin{array}{cc}\cos \left({\Theta }_{j,A}\right)& -r_i\sin \left({\Theta }_{j,A}\right)\\ \sin \left({\Theta }_{j,A}\right)& r_i\cos \left({\Theta }_{j,A}\right)\end{array}\right]$${\sigma }_{A_{pol}}^2=\left[\begin{array}{cc}{\sigma }_{r,A}^2& 0\\ 0& {\sigma }_{\Theta ,A}^2\end{array}\right]$

[0068] Hence, the velocity variance finally depends on θ and on the standard deviations of r and θ. It is noted that the above definitions are also valid accordingly for the second point in time denoted by B.

[0069] The term of Position Difference NEES may be simplified:

[00007]$\begin{array}{cc}NEE{S}_{PD}=\underset{j}{.Math.}{V_{\mathrm{diff},j}\left({\sigma }_{V_j}\right)}^{-1}{V}_{\mathrm{diff},j}^T={.Math.}_j\frac{\begin{array}{c}{\left(V_x-V_{x,j}\right)}^2{\sigma }_{V_{\mathrm{yy},j}}-\\ 2\left(V_y-V_{y,j}\right)\left(V_x-V_{x,j}\right){\sigma }_{V_{\mathrm{xy},j}}+{\left(V_y-V_{y,j}\right)}^2{\sigma }_{V_{\mathrm{xx},j}}\end{array}}{.Math.{\sigma }_{V_j}.Math.}& \left(15\right)\end{array}$

[0070] wherein:

|σ.sub.V.sub.j|=σ.sub.V.sub.xx,jσ.sub.V.sub.yy,j−σ.sub.V.sub.xy,jσ.sub.V.sub.xy,j  (16)

[0071] Finally, the entire or total NEES cost function can be written as:

[00008]$\begin{array}{cc}Q=\frac{1}{2}\left({.Math.}_i\frac{{\left(V_x\cos \left({\Theta }_i\right)+V_y\sin \left({\Theta }_i\right)-{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}\right)}^2}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}+{.Math.}_j\frac{\begin{array}{c}{\left(V_x-V_{x,j}\right)}^2{\sigma }_{V_{\mathrm{yy},j}}-\\ 2\left(V_y-V_{y,j}\right)\left(V_x-V_{x,j}\right){\sigma }_{V_{\mathrm{xy},j}}+{\left(V_y-V_{y,j}\right)}^2{\sigma }_{V_{\mathrm{xx},j}}\end{array}}{.Math.{\sigma }_{V_j}.Math.}\right)& \left(17\right)\end{array}$

[0072] To find V.sub.x and V.sub.y, the entire NEES cost function Q is minimized analytically by calculating the first derivative to find global minimum:

[00009]$\begin{array}{cc}\frac{dQ}{d{V}_x}={.Math.}_i\frac{V_x{\cos }^2\left({\Theta }_i\right)+V_y\cos \left({\Theta }_i\right)\sin \left({\Theta }_i\right)-\cos \left({\Theta }_i\right){\stackrel{.}{r}}_{{\mathrm{comp}}_i}}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}+{.Math.}_j\frac{V_x{\sigma }_{V_{\mathrm{yy},j}}-V_{x,j}{\sigma }_{V_{\mathrm{yy},j}}-V_y{\sigma }_{V_{\mathrm{xy},j}}+V_{y,j}{\sigma }_{V_{\mathrm{xy},j}}}{.Math.{\sigma }_{V_j}.Math.}& \left(18\right)\end{array}$$\frac{dQ}{d{V}_y}={.Math.}_i\frac{V_x\cos \left({\Theta }_i\right)\sin \left({\Theta }_i\right)+V_y{\sin }^2\left({\Theta }_i\right)-\sin \left({\Theta }_i\right){\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}+{.Math.}_j\frac{-V_x{\sigma }_{V_{\mathrm{xy},j}}+V_{x,j}{\sigma }_{V_{\mathrm{xy},j}}+V_y{\sigma }_{V_{\mathrm{xx},j}}-V_{y,j}{\sigma }_{V_{\mathrm{xx},j}}}{.Math.{\sigma }_{V_j}.Math.}$

[0073] For determining the minimum of the NEES cost function, these derivatives are set equal to zero:

[00010]$\begin{array}{cc}\frac{dQ}{d{V}_x}=0,\frac{dQ}{d{V}_y}=0& \left(19\right)\end{array}$

[0074] Some reorganization of the above equations can be performed as follows:

[00011]$\begin{array}{cc}\underset{i}{.Math.}\frac{\cos \left({\Theta }_i\right){\dot{r}}_{{\mathrm{comp}}_i}}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}+\underset{j}{.Math.}\frac{V_{x,j}{\sigma }_{V_{\mathrm{yy},j}}-V_{y,j}{\sigma }_{V_{\mathrm{xy},j}}}{.Math.{\sigma }_{V_j}.Math.}=V_x\left(\underset{i}{.Math.}\frac{{\cos }^2\left({\Theta }_i\right)}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}+\underset{j}{.Math.}\frac{{\sigma }_{V_{\mathrm{yy},j}}}{.Math.{\sigma }_{V_j}.Math.}\right)+V_y\left(\underset{i}{.Math.}\frac{\cos \left({\Theta }_i\right)\sin \left({\Theta }_i\right)}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}+\underset{j}{.Math.}\frac{-{\sigma }_{V_{\mathrm{xy},j}}}{.Math.{\sigma }_{V_j}.Math.}\right)& \left(20\right)\end{array}$$\underset{i}{.Math.}\frac{\sin \left({\Theta }_i\right){\dot{r}}_{{\mathrm{comp}}_i}}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}+\underset{j}{.Math.}\frac{V_{y,j}{\sigma }_{V_{\mathrm{xx},j}}-V_{x,j}{\sigma }_{V_{\mathrm{xy},j}}}{.Math.{\sigma }_{V_j}.Math.}=V_x\left(\underset{i}{.Math.}\frac{\cos \left({\Theta }_i\right)\sin \left({\Theta }_i\right)}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}+\underset{j}{.Math.}\frac{-{\sigma }_{V_{\mathrm{xy},j}}}{.Math.{\sigma }_{V_j}.Math.}\right)+V_y\left(\underset{i}{.Math.}\frac{{\sin }^2\left({\Theta }_i\right)}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}+\underset{j}{.Math.}\frac{{\sigma }_{V_{\mathrm{xx},j}}}{.Math.{\sigma }_{V_j}.Math.}\right)$$S_{\mathrm{xx}}=\underset{i}{.Math.}\frac{{\cos }^2\left({\Theta }_i\right)}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}+\underset{j}{.Math.}\frac{{\sigma }_{V_{\mathrm{yy},j}}}{.Math.{\sigma }_{V_j}.Math.};$$S_{xy}=\underset{i}{.Math.}\frac{\cos \left({\Theta }_i\right)\sin \left({\Theta }_i\right)}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}+\underset{j}{.Math.}\frac{-{\sigma }_{V_{\mathrm{xy},j}}}{.Math.{\sigma }_{V_j}.Math.}$$S_x=\underset{i}{.Math.}\frac{\cos \left({\Theta }_i\right){\dot{r}}_{{\mathrm{comp}}_i}}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}+\underset{j}{.Math.}\frac{V_{x,j}{\sigma }_{V_{\mathrm{yy},j}}-V_{y,j}{\sigma }_{V_{\mathrm{xy},j}}}{.Math.{\sigma }_{V_j}.Math.}$$S_{xy}=\underset{i}{.Math.}\frac{\cos \left({\Theta }_i\right)\sin \left({\Theta }_i\right)}{{\sigma }_{{{\stackrel{.}{r}}_{comp}}_i}^2}+\underset{j}{.Math.}\frac{-{\sigma }_{V_{\mathrm{xy},j}}}{.Math.{\sigma }_{V_j}.Math.};$$S_{yy}=\underset{i}{.Math.}\frac{{\sin }^2\left({\Theta }_i\right)}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}+\underset{j}{.Math.}\frac{{\sigma }_{V_{\mathrm{xx},j}}}{.Math.{\sigma }_{V_j}.Math.}$$S_y=\underset{i}{.Math.}\frac{\sin \left({\Theta }_i\right){\dot{r}}_{{\mathrm{comp}}_i}}{{\sigma }_{{\stackrel{.}{r}}_{c{\mathrm{omp}}_i}}^2}+\underset{j}{.Math.}\frac{V_{y,j}{\sigma }_{V_{\mathrm{xx},j}}-V_{x,j}{\sigma }_{V_{\mathrm{xy},j}}}{.Math.{\sigma }_{V_j}.Math.}$

[0075] Finally, this leads to two equations depending from the two variables V.sub.x and V.sub.y:

[00012]$\begin{array}{cc}\{\begin{array}{c}{S}_x=V_x{S}_{\mathrm{xx}}+V_y{S}_{\mathrm{xy}}\\ S_y=V_x{S}_{\mathrm{yx}}+V_y{S}_{\mathrm{yy}}\end{array}& \left(21\right)\end{array}$

wherein S.sub.xy=S.sub.yx

[0076] By defining the following abbreviations:

W=S.sub.xxS.sub.yy−S.sub.xyS.sub.yx

W.sub.x=S.sub.xS.sub.yy+S.sub.xyS.sub.y

W.sub.y=S.sub.xxS.sub.y+S.sub.xS.sub.yx  (22)

the velocity of the target object 23 is estimated as follows:

[00013]$\begin{array}{cc}\mathrm{velocity}\mathrm{estimate}:\{\begin{array}{c}{V}_x=\frac{W_x}{W}\\ V_y=\frac{W_y}{W}\end{array}& \left(23\right)\end{array}$

[0077] In summary, the two components V.sub.x and V.sub.y of the velocity vector of the target object 23 (see FIG. 1) are determined in dependence of the factors S.sub.x, S.sub.y, S.sub.xy (=S.sub.yx), S.sub.xx and S.sub.yy which dependent on the input data in the form of the range r.sub.i, the angle θ.sub.i and the rate range {dot over (r)}.sub.i and their corresponding standard deviations a for all raw detections belonging to the target object 23.

[0078] The velocity vector of the target object 23 is therefore determined analytically based on the input data and based on the NEES cost function. The NEES includes a respective first and second contribution for each of the two sources for estimating the velocity, e.g. for the velocity profile which is based on the rate range equation and for the position difference which depends on the range and the azimuth angle for each detection. When combining these two contributions, the velocity profile NEES and the position difference NEES are weighted by the respective standard deviations which reflect the accuracy of the sensor measurements.

[0079] Since two sources for estimating the velocity are considered and the cost function is minimized, the accuracy of a velocity estimate is improved in comparison to methods which rely on one of these sources for estimating the velocity only. Due to the analytical expression as shown explicitly above, performing the method requires a low computational effort and a minimum time. Hence, the method is easy to be embedded in automotive systems.

[0080] For verifying the method according to the disclosure, two different scenarios are considered in FIGS. 2 and 6, respectively. The results for the two scenarios are shown in FIGS. 3 to 5 and FIGS. 7 to 9, respectively.

[0081] The first scenario is shown in FIG. 2 and includes a non-moving vehicle 10 (e.g. host vehicle velocity=(0,0) km/h) and the moving object 23 having a velocity of (0, −50) km/h. That is, the target object 23 crosses the vehicle 10 laterally along the y-axis 21 (see FIG. 1).

[0082] The velocity estimation is performed with time steps of dt=100 ms, and single point detections are generated from the front left corner of the target object 23 for each time step or point in time. Furthermore, constant detection accuracies (standard deviations) are assumed to be: σ.sub.θ=0.3 deg, σ.sub.r=0.15 m and

[00014]${\sigma }_{{\stackrel{.}{r}}_{\mathrm{comp}}}=0.05\frac{m}{s}.$

[0083] In FIGS. 3, 4, 7 and 8, the respective results are depicted in a velocity coordinate system having a longitudinal axis 31 for V.sub.x and a lateral axis 33 for V.sub.y, wherein the respective velocity component is depicted in m/s. That is, the velocity of the moving object 23 is represented by a reference 35 at (0, −13,89) m/s for the first scenario in FIGS. 3 and 4.

[0084] In FIG. 3, estimation results are depicted based on three radar scans, e.g. three detections for the range rate and two detections for the position difference have been available for estimating the velocity. In FIG. 3A, single estimates are shown with respect to the reference 35 representing the “true” velocity. Two position difference-based velocity estimates 37A, 37B are depicted together with their respective standard deviations 39A, 39B which are represented by dashed lines. Furthermore, velocity estimates 41 based on the range rate is represented as lines. This is due to the fact that the range rate, e.g. the radial velocity with respect to the vehicle, is used only for the velocity estimate 41. The three lines which are denoted by “rr based velocity” in FIG. 3A correspond to the three detections of the range rate and are almost identical. As can be seen in FIG. 3A, the velocity estimates 37A, 37B based on the position difference deviate strongly from the reference 35.

[0085] In FIGS. 3B, 3C and 3D, respective cost functions are depicted which are calculated based on the formulas provided above. Each cost function is represented by contour lines 43. In addition, a respective estimate 45 for the velocity is shown which is determined as an absolute minimum of the respective cost function. In FIG. 3B, the total NEES cost function is shown including a first contribution for the velocity profile based on the rate range and a second contribution based on the position difference. In FIGS. 3C and 3D, the two contributions are depicted separately, e.g. a velocity profile cost function or rate range cost function in FIG. 3C and a position difference cost function in FIG. 3D.

[0086] As can be seen in FIGS. 3B and 3D, the estimates 45 for the velocity of the target object 23 are quite close to the reference 35, whereas the estimate 45 based on the rate range cost function strongly deviates from the reference 35. On the other hand, the total NEES cost function and the rate range cost function have a very small extension in the direction of V.sub.y in comparison to the position difference cost function. Therefore, the velocity estimate 45 based on the total NEES cost function has a smaller error bar than the estimate 45 based on the position difference cost function. Hence, the velocity estimate 45 based on the total NEES cost function is more reliable than the velocity estimate 45 based on the position difference cost function although the component V.sub.x is slightly closer to the reference 35 for the position difference cost function.

[0087] For FIG. 4, thousands of Monte Carlo simulations have been performed for the first scenario assuming the same conditions as in FIG. 3, e.g. respective three radar scans for each Monte Carlo simulation. In FIGS. 4B, 4C and 4D, the respective results for the estimations 45 based on the Monte Carlo simulation are shown for the same conditions as in FIGS. 3B, 3C and 3D, e.g. for the total NEES cost function (FIG. 4B), for the rate range cost function (FIG. 4C) and for the position difference cost function (FIG. 4D). The estimates 45 are depicted as point clouds with respect to the reference 35. As can be seen in FIG. 4B, the estimates 45 are generally quite close to the reference 35 for the total NEES cost function, e.g. very close to the reference in the direction of V.sub.x and quite close in an acceptable manner in the direction of V.sub.y. In contrast, the estimates 45 strongly spread in the direction of V.sub.x for the rate range cost function as shown in FIG. 4C, whereas the estimate 45 show quite a large spread in the direction of V.sub.y for the position difference cost function as shown in FIG. 4D. In summary, the estimates 45 are most “concentrated” around the reference 35 for the total NEES cost function in FIG. 4B.

[0088] This is also reflected in the statistics as shown in FIG. 4A, in which the root mean square of the error or deviation is depicted for the three types of velocity estimation, e.g. based on the total NEES cost function (depicted as bars 51), based on the range rate cost function (depicted by the bars 53), and based on the position difference cost function (depicted as bars 55). Please note that the term “cloud” is used as synonym for the range rate cost function in the legend of FIG. 4A. The respective root mean square is shown for both velocity components V.sub.x and V.sub.y, for the heading or direction of the velocity and for the magnitude or absolute value of the velocity. The velocity estimation based on the total NEES cost function shows the lowest root mean square of the deviations in all cases.

[0089] This is also confirmed by the results as shown in FIG. 5. The respective root mean square of the deviations (on the y-axis) is shown depending from the number of radar scans (on the x-axis) for the heading (FIG. 5A), for the magnitude (FIG. 5B), for the velocity component V.sub.x (FIG. 5C) and for the velocity component V.sub.y (FIG. 5D). The lines 61 represent the respective root mean square of the deviations for the velocity estimates based on the total NEES cost function, wherein the lines 63 represent the root mean square of the deviations of velocity estimates based on the range rate cost function, and the lines 65 represent the root mean square of the deviation for the velocity estimates based on the position difference cost function. As can be seen, the root mean square of the deviations has the lowest value in almost all cases for the velocity estimate based on the NEES cost function. Please note the logarithmic scale on the y-axis.

[0090] In FIG. 6, the second scenario is depicted in which the vehicle 10 and the target object 23 move perpendicular to each other with the same constant velocity, e.g. having an absolute value or magnitude of 50 km/h. Therefore, a constant azimuth angle is assumed for the target object 23 with respect to the vehicle 10. In detail, the velocity of vehicle 10 is (50,0) km/h, whereas the velocity of the target object 23 is (0, −50) km/h. For the time steps, dt=100 ms is used again, and the same detection accuracies or standard deviations are assumed as for the first scenario, e.g. σ.sub.θ=0.3 deg, σ.sub.r=0.15 m and

[00015]${\sigma }_{{\stackrel{.}{r}}_{\mathrm{comp}}}=0.05\frac{m}{s}.$

Single point detections are generated again from the left corner of the target object 23 for each time step.

[0091] The results as shown in FIG. 7 correspond to the results as shown in FIG. 3 for the first scenario. That is, all general explanations regarding the diagrams and the reference numbers as provided for FIGS. 3 and 4 are also valid for FIGS. 7 and 8. However, the velocity of the target object 23 is represented by a reference 35 at

(−13,89, −13,89) m/s which is the relative velocity with respect to the vehicle 10 for the second scenario. In other words, the velocity of the vehicle 10 is compensated for the reference 35 in FIGS. 7 and 8.

[0092] As can be seen in FIG. 7A, the single estimates 37A, 37B based on the position difference strongly deviate from the reference 35, and the range rate based estimate of the velocity is only able to cover the radial component with respect to the vehicle 10, but not the lateral component of the velocity. Hence, the range rate-based estimates 41 are again represented by lines. The three lines which are denoted by “rr based velocity” in FIG. 7A correspond again to the three detections of the range rate and are again almost identical (see also FIG. 3A).

[0093] As shown in FIGS. 7B and 7D, the velocity estimates 45 based on the total NEES cost function and based on the position difference cost function are very close to the reference 35, whereas the estimate 45 based on the range rate cost function (see FIG. 7C) strongly deviates from the reference 35. Furthermore, the spread of the cost function is much larger for the position difference cost function than for the total NEES cost function, as can be derived from the contour lines 43 representing the cost functions. As a result, the velocity estimate 45 based on the total NEES cost function has to be regarded as more reliable than the estimate based on the position difference cost function and based on the range rate cost function respectively.

[0094] FIG. 8 depicts the results of thousand Monte Carlo simulations assuming the same conditions as for FIG. 7, e.g. respective estimations based on three radar scans. As can be seen in FIGS. 8B, 8C and 8D, the total NEES cost function again shows the lowest spread with respect to the reference 35 for the velocity estimates 45 which are related to the second scenario as shown in FIG. 6. This confirms that the velocity estimate 45 based on the total NEES cost function shows the highest accuracy and reliability in comparison to the further estimates 45 since two “sources” are included for calculating the total NEES cost function.

[0095] In FIG. 8A, the root mean square of the error for deviation is shown again for the three types of velocity estimation, wherein the bars 41 represent the root mean square of the deviations for the NEES cost function, whereas the bars 53 represent the root mean square of the deviations for the estimates based on the range rate cost function, and the bars 55 represent the root mean square of the deviations for the estimates based on the position difference cost function. Please note that the term “cloud” is used as synonym for the range rate cost function in the legend of FIG. 8A. For the second scenario (see FIG. 6), the estimates 45 based on the total NEES cost function show again the lowest root mean square for the deviations, e.g. for both components V.sub.x and V.sub.y of the velocity, for the heading and for the magnitude.

[0096] This is also confirmed by the lines 61, 63 and 65 for the root mean square for the deviations of the heading (FIG. 9A), of the magnitude (FIG. 9B), of the component V.sub.x (FIG. 9C) and of the component V.sub.y (FIG. 9D) which are depicted as depending from the number of radar scans. The root mean square for the deviations of the estimates based on the total NEES cost function (line 61) is lower in almost all cases than the root mean square of the deviation for the estimates based on the position difference cost function (line 63) and based on the range rate cost function (line 65). Please note again the logarithmic scale on the y-axis.

[0097] In summary, the results for both scenarios (see FIG. 3 and FIG. 6) verify that the method according to the disclosure improves the accuracy and the reliability of the velocity estimates. This is at least partly due to the fact that the estimates rely on the total NEES cost function which includes two contributions or sources for the velocity estimates.